LETTER Communicated by Bard Ermentrout

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron

Nicolas Brunelbrunelbiomedicaleuniv-paris5frCNRS NPSM Universite Paris Rene Descartes 75270 Paris Cedex 06 France

Peter E LathampeluclaeduDepartment of Neurobiology University of California at Los AngelesLos Angeles CA 90095 USA

We calculate the ring rate of the quadratic integrate-and-re neuron inresponse to a colored noise input current Such an input current is a goodapproximation to the noise due to the random bombardment of spikeswith the correlation time of the noise corresponding to the decay timeof the synapses The key parameter that determines the ring rate is theratio of the correlation time of the colored noise iquests to the neuronal timeconstant iquestm We calculate the ring rate exactly in two limits when theratio iquests=iquestm goes to zero (white noise) and when it goes to innity Thecorrection to the short correlation time limit is Oiquests=iquestm which is qualita-tively different from that of the leaky integrate-and-re neuron where thecorrection is O

piquests=iquestm The difference is due to the different boundary

conditions of the probability density function of the membrane potentialof the neuron at ring threshold The correction to the long correlationtime limit is Oiquestm=iquests By combining the short and long correlation timelimits we derive an expression that provides a good approximation to thering rate over the whole range of iquests=iquestm in the suprathreshold regimemdashthat is in a regime in which the average current is sufcient to make thecell re In the subthreshold regime the expression breaks down some-what when iquests becomes large compared to iquestm

1 Introduction

A major challenge in computational neuroscience is to understand the be-havior of large recurrent networks of spiking neurons A rst step in thisendeavor is to be able to compute the location and stability of a networkrsquosequilibria given connectivity and single neuron properties For networksoperating in the asynchronous regime the equilibria can be determined bysolving a set of algebraic equations in which the ring rate of each neuronin the network is a function of the ring rates of all the other neurons Tosolve these equations which can be done using mean-eld techniques it is

Neural Computation 15 2281ndash2306 (2003) cdeg 2003 Massachusetts Institute of Technology

2282 N Brunel and P Latham

necessary to be able to compute the mapping from presynaptic ring ratesto postsynaptic rates For leaky integrate-and-re neurons the mapping isknown (Ricciardi 1977 Tuckwell 1988) and equilibria have been computedfor networks of irregularly ring leaky integrate-and-re neurons (Amit ampBrunel 1997a 1997b Brunel 2000) Here we compute this mapping for amore realistic reduced neuron model the quadratic integrate-and-re neu-ron (Ermentrout amp Kopell 1986)

Cortical neurons in vivo re in an irregular fashion (Burns amp Webb 1976Softky amp Koch 1993) and intracellular recordings reveal large uctuationsin the membrane potential (Destexhe amp Pare 1999 Anderson Lampl Gille-spie amp Ferster 2000) These experimental observations suggest that theinput to a neuron can be divided into two components a mean current anductuations around that mean both of which depend on the ring rates ofthe presynaptic neurons It is often reasonable to approximate the uctu-ations as colored noise with a correlation time proportional to the synap-tic time constants (Amit amp Tsodyks 1991 Brunel amp Sergi 1998 DestexheRudolph Fellous amp Sejnowski 2001) and this is the approach we take hereIn addition we assume that there is a unique synaptic decay time iquests so thatthe correlation time of the noise is equal to iquests With these assumptions theinput of the neuron is fully described by three parameters the mean currentsup1 the variance of the uctuations frac34 2 and the correlation time constant iquests

The ring rate of the leaky integrate-and-re neuron versus sup1 and frac34 wascalculated by Brunel and Sergi (1998 see also Fourcaud amp Brunel 2002)in the limit iquests iquest iquestm where iquestm is the membrane time constant Althoughthe leaky integrate-and-re neuron is a popular model it suffers from twodrawbacks it cannot be derived from conductance-based models using arigorous reduction procedure and its f -I curve in the absence of noise hasa pathological behavior close to threshold For these reasons it is usefulto consider another simplied model neuron the quadratic integrate-and-re neuron This neuron which is related by a change of variables to themicro-neuron (Ermentrout 1996 Gutkin amp Ermentrout 1998) represents thenormal form of a neuron with a type I bifurcation leading to spike generation(Ermentrout amp Kopell 1986 Ermentrout 1996) The quadratic integrate-and-re neuron is thus expected to describe the dynamics of any type Ineuron close to bifurcation where ring rates are low In particular its ringrate in the absence of noise scales as

pI iexcl Ithr where Ithr is the threshold

current This is the behavior exhibited by all type I neurons near thresholdHere we compute the ring rate of the quadratic integrate-and-re neu-

ron in two limits small and large iquests=iquestm Interpolation between these limitsprovides a good approximation to the ring rate over the whole range ofiquests=iquestm in the suprathreshold regime and a reasonable zeroth-order approx-imation in the subthreshold regime The ring rate must be computed nu-merically but for applications such as mean-eld analysis where speed isimportant two lookup tables are sufcient to characterize the ring rate forall values of sup1 frac34 iquestm and iquests

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2283

2 Neuron Model and Analytical Methods

The quadratic integrate-and-re neuron with synaptic noise obeys the dif-ferential equations

iquestmdvdt

D f v C w (21a)

iquestsdwdt

D iexclw C iquest1=2m frac34acutet (21b)

where iquestm is the neuronal time constant (hereafter called the membrane timeconstant) vt is a ldquovoltagerdquo variable (hereafter called the membrane poten-tial)

f v D v2 C sup1 (22)

with sup1 the mean input current owing into the cell w represents the stochas-tic component in the synaptic currents iquests is the synaptic time constant acute iswhite noise with hacutetacutet0i D plusmnt iexcl t0 and frac34 is the overall amplitude of thenoise

In the noiseless version of the model when the membrane potentialbecomes positive the quadratic term on the right-hand side of equation 21aensures that the membrane potential diverges to innity in nite time Thetime at which the membrane potential reaches innity denes the timea spike is emitted Another way of dening the spike time would be toset a threshold for ring and then formally send this threshold to innityImmediately after ring the membrane potential is reset to iexcl1 Againthe quadratic term in the right-hand side of equation 21a ensures that themembrane potential comes back to a nite value in nite time To determinethe ringrate from equation 21 we reformulate the problem using a Fokker-Planck equation (Risken 1984)

iquestmtPv w t C v[ f v C wP] C iquestm

iquestsw[iexclwP] D frac34 2

2iquest 2

m

iquest 2s

2wP

where Pv w t is the joint probability density function (pdf) of v w attime t Once Pv w t is known the ring rate is given by the total proba-bility ux at v D C1 (see equation 25 below) For other studies of the pdfapproach to neural modeling mostly in the context of integrate-and-reneurons see Treves (1993) Abbott and van Vreeswijk (1993) Brunel andHakim (1999) Brunel (2000) Knight Omurtag and Sirovich (2000) andNykamp and Tranchina (2000)

The Fokker-Planck equation can be rewritten as a continuity equation

tPv w t C vJvv w t C wJwv w t D 0 (23)

2284 N Brunel and P Latham

where Jv and Jw are probability uxes

Jvv w t D1

iquestm f v C wPv w t (24a)

Jwv w t D1iquests

sup3iexcl iquestm

iquests

frac34 2

2wPv w t iexcl wPv w t

acute (24b)

The ring rate ordmt is the total probability ux at v D C1 it is given by

ordmt DZ C1

iexcl1dw lim

v1Jvv w t (25)

We are interested in computing the ring rate in the limit t 1 Inthis limit the probability distribution becomes time independent We thusset tP to zero in equation 23 resulting in a continuity equation involvingonly v and w That equation cannot be solved exactly but it can be solvedas an expansion in the ratio of the synaptic to the membrane time constant(or its inverse) To facilitate this expansion we introduced a new variablez acute kw=frac34 where k acute iquests=iquestm1=2 The advantage of using z rather than wis that its variance remains O1 in both the large and small k limits aproperty not shared by w With this change of variable and with tP set tozero equation 23 may be written as

LPv z D kfrac34zvPv z C k2v[ f vPv z] (26)

where

Lcent acute12

2z cent Czzcent

In the next two sections we solve this equation in the short and long corre-lation time limits

3 Short Correlation Time Limit

To solve equation 26 in the short correlation time limit (k iquest 1) we expandboth the probability distribution and the ring rate in powers of k

Pv z D1X

iD0

kiPiv z (31a)

ordm D1X

iD0

kiordmiS (31b)

where the subscript S on ordmiS is to remind us that we are working in the shortcorrelation time limit (Note that Pi should also have a subscript S we do

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2285

not include it for clarity It should be clear from the context whether we arereferring to the short or long correlation time limit the latter is discussedin section 4) Since Pv z is a probability distribution it must integrate toone independent of k Consequently we must have

Zdv dz P0v z D 1 (32a)

Zdv dz Piv z D 0 for i cedil 1 (32b)

To derive equations for the Pi we insert equation 31a into equation 26and match powers of k This results in a coupled set of equations

LP0v z D 0 (33a)

LP1v z D frac34 zvP0v z (33b)

LPiv z D frac34 zvPiiexcl1v z C v[ f vPiiexcl2v z] for i cedil 2 (33c)

Using equations 24a 25 and 31b the ring rate at order i is

ordmiS D1

iquestm

Z C1

iexcl1dz lim

v1[ f vPiv z C frac34zPiC1v z]

D1

iquestm

Z C1

iexcl1dz lim

v1f vPiv z (34)

where the last equality follows because Pv z must vanish as v 1otherwise it will not be normalizable

31 Probability Distribution in the Short Correlation Time Limit To com-pute the probability distribution as an expansion in powers of k we simplysolve equations 33a through 33c order by order The zeroth-order solu-tion P0 corresponds to white noise the next orders give corrections asso-ciated with the nite synaptic time constant It turns out as we will seethat we have to go to the fourth order to determine the lowest nonvan-ishing correction to the ring rate In this section however we sketch thesolutions to equations 33a through 33c only through second order thesolutions through fourth order are derived in appendix A We begin withequation 33a

Examining equation 33a we see that the zeroth-order contribution to theprobability distribution P0 can be written as a linear combination of twofunctions one even in z and the other odd Those two functions denotedAacute0 and Aacute1 are given by

Aacute0z D frac14iexcl1=2 expiexclz2

Aacute1z D expiexclz2

Z z

0expu2 du

2286 N Brunel and P Latham

The second of these Aacute1z scales as 1=z for large z Consequently it willbe negative when z approaches either C1 or iexcl1 and large compared toAacute0 This violates the condition Pv z cedil 0 (recall that Pv z is a probabilitydistribution) Thus P0 must be proportional to Aacute0 only The constant ofproportionality can be v-dependent so we have

P0v z D Q0vAacute0z (35)

The function Q0v is determined at higher order as we will see shortlyThe rst-order correctionto the probability distribution P1v z is found

by combining equations 33b and 35

P1v z D Q1vAacute0z C frac34vQ0vLiexcl1zAacute0z (36)

Because we explicitly included the homogeneous term in the above expres-sion Liexcl1 must pick out only inhomogeneous terms This leads to

Liexcl1Fz D 2 expiexclz2

Z z

0dz0 expz02

Z z0

iexcl1dz00 Fz00 (37)

which is valid for any L1 function Fz that vanishes at sect1 The lower limitof iexcl1 in the second integral ensures that Liexcl1Fz does not pick up a pieceproportional to Aacute1z as we will see below

The second-order correction P2v z which comes from combiningequations 33c 35 and 36 may be written after a small amount of algebraas

P2v z D Q2vAacute0z C frac34vQ1vLiexcl1zAacute0z

C frac34 22v Q0vLiexcl1zLiexcl1zAacute0z C v[ f vQ0v]Liexcl1Aacute0z (38)

Using equation 37 it is straightforward to show that

Liexcl1zAacute0z D iexclzAacute0z and

Liexcl1z2Aacute0z D z2=2Aacute0z iexcl 1=2Liexcl1Aacute0z

With these relations equation 38 becomes

P2v z D Q2vAacute0z iexcl frac34vQ1vzAacute0z C frac34 2

22

v Q0vz2Aacute0z

iexclmicro

frac34 2

22

vQ0v iexcl v[ f vQ0v]para

Liexcl1Aacute0z (39)

Again using equation 37 we see that for large positive z Liexcl1Aacute0z raquo 1=jzjThis makes Liexcl1Aacute0z nonnormalizable which means that it cannot make

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2287

any contribution to P2v z Thus the solvability condition on equation 39is

frac34 2

22

vQ0v iexcl v[ f vQ0v] D 0 (310)

This solvability condition can also be derived by applying the Fredholmalternative theorem (Reed amp Simon 1980)

To solve equation 310 we rst integrate once with respect to v and mul-tiply both sides by iexcl1 yielding

f vQ0v iexcl frac34 2

2vQ0v D iquestmordm0S (311)

In this equation we chose a specic form for the constant of integration(the term on the right-hand side) To see that this is the correct choice notethat in the large v limit vQ0v must approach zero to ensure that Q0 isnormalizable Thus in this limit the only term that survives on the left-hand side is f vQ0v Multiplying both sides by iquest iexcl1

m Aacute0z integrating overz and taking the limit v 1 equation 311 becomes equation 34 withi D 0 indicating that iquestmordm0S is the correct constant of integration

Equation 311 has the solution

Q0v D2ordm0Siquestm

frac34 2

Z 1

vdu exp[Atildev iexcl Atildeu] (312)

where

Atildev acuteZ v

0

2 f u

frac34 2 du (313)

The value of ordm0S is determined from the condition that Q0v must be nor-malized to 1 (see equation 32a and note that

Rdz Aacute0z D 1) which implies

that

ordm0S D frac34 2

2iquestm

microZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu]

paraiexcl1

(314)

We will compute this integral explicitly in the next sectionTo determine the next order correction to the ring rate we need to nd

Q1v This is done by deriving a solvability condition at third order Simi-larly Q2v which is needed for the second-order correction is determinedby deriving a solvability condition at fourth order Deriving these condi-tions requires that we compute P3 and P4 This is conceptually no moredifcult than computing P1 and P2 the only real difference is that the al-gebra is slightly more involved We thus relegate those computations to

2288 N Brunel and P Latham

appendix A There we nd that Q1v D 0 which means that the rst-ordercorrection to the ring rate ordm1S vanishes and that Q2v is given by

Q2v D ordm2S

ordm0S

Q0v iexcl iquestmordm0S

frac34 2

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (315)

where

h0v acute iexcl3 f 00v iexcl 4 f 0vAtilde 0v (316)

To calculate ordm2S we use equation 32b which tells us thatR

dvdz P2v z D 0and equation 39 along with the facts that the coefcient in front of Liexcl1Aacute0vanishes

R 1iexcl1 dv 2

vQ0v D 0 andR 1

iexcl1 dv vQ0v D 0 which tell us thatthe only way for P2v z to integrate to zero is to have

R 1iexcl1 dv Q2v D 0

Then integrating both sides of equation 315 and usingR 1

iexcl1 dv Q0v D 1we arrive at

ordm2S D ordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (317)

32 Firing Rate in the Short Correlation Time Limit The zeroth-order con-tribution to the ring rate is given by equation 314 and the lowest nonva-nishing correction which enters at Ok2 is given by equation 317 Theseexpressions involve double integrals and thus are not so easy to computeHowever we show in appendix B that both can be reduced to a single inte-gral resulting in

ordm0S D1

frac14iquestm

microZ 1

iexcl1

draquo

frac14 1=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]paraiexcl1

(318a)

ordm2S D iexclordm0Sfrac14frac34 2 iquestmordm0S

2

Z 1

iexcl1

draquo

frac141=2 raquo 2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48] (318b)

The ring rate through second order which we denote ordmS to indicate thatthe expression is valid in the short correlation time limit is then given by

ordmS D ordm0S C k2ordm2S (319)

In the absence of noise (frac34 D 0) thep

sup1 behavior is recovered for thezeroth-order contribution to the ring rate

ordm0S D sup11=2

frac14iquestm

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2289

-05 0 05 1Input current

0

10

20

30

40F

iring

rat

e (H

z)

Figure 1 Firing rate as a function of mean input current sup1 for white noise withfrac34 D 0 (solid line) 01 (dot-dashed line) 02 (dashed line) and 05 (dotted line)iquestm D 10 ms

and in the absence of a mean current (sup1 D 0) ordm0S increases with noise asfrac34 2=3

ordm0S D35=6frac34 2=3

22=3frac14 1=201=6iquestm

Plots of ordm0S versus sup1 are given in Figure 1 for several values of frac34 The second-order contribution to the ring rate ordm2S (the lowest nonvan-

ishing contribution) is negative Thus the ring rate decreases linearly withk2D iquests=iquestm as the correlation time constant increases This is inconvenientas the total ring rate through second order equation 319 can become neg-ative at large k It is possible to derive alternative expressions that coincideup to second order in k with equation 319 but stay positive for any valueof k One such alternative expression is

ordmS D ordm0S

1 iexcl k2ordm2S=ordm0S

(320)

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2282 N Brunel and P Latham

necessary to be able to compute the mapping from presynaptic ring ratesto postsynaptic rates For leaky integrate-and-re neurons the mapping isknown (Ricciardi 1977 Tuckwell 1988) and equilibria have been computedfor networks of irregularly ring leaky integrate-and-re neurons (Amit ampBrunel 1997a 1997b Brunel 2000) Here we compute this mapping for amore realistic reduced neuron model the quadratic integrate-and-re neu-ron (Ermentrout amp Kopell 1986)

Cortical neurons in vivo re in an irregular fashion (Burns amp Webb 1976Softky amp Koch 1993) and intracellular recordings reveal large uctuationsin the membrane potential (Destexhe amp Pare 1999 Anderson Lampl Gille-spie amp Ferster 2000) These experimental observations suggest that theinput to a neuron can be divided into two components a mean current anductuations around that mean both of which depend on the ring rates ofthe presynaptic neurons It is often reasonable to approximate the uctu-ations as colored noise with a correlation time proportional to the synap-tic time constants (Amit amp Tsodyks 1991 Brunel amp Sergi 1998 DestexheRudolph Fellous amp Sejnowski 2001) and this is the approach we take hereIn addition we assume that there is a unique synaptic decay time iquests so thatthe correlation time of the noise is equal to iquests With these assumptions theinput of the neuron is fully described by three parameters the mean currentsup1 the variance of the uctuations frac34 2 and the correlation time constant iquests

The ring rate of the leaky integrate-and-re neuron versus sup1 and frac34 wascalculated by Brunel and Sergi (1998 see also Fourcaud amp Brunel 2002)in the limit iquests iquest iquestm where iquestm is the membrane time constant Althoughthe leaky integrate-and-re neuron is a popular model it suffers from twodrawbacks it cannot be derived from conductance-based models using arigorous reduction procedure and its f -I curve in the absence of noise hasa pathological behavior close to threshold For these reasons it is usefulto consider another simplied model neuron the quadratic integrate-and-re neuron This neuron which is related by a change of variables to themicro-neuron (Ermentrout 1996 Gutkin amp Ermentrout 1998) represents thenormal form of a neuron with a type I bifurcation leading to spike generation(Ermentrout amp Kopell 1986 Ermentrout 1996) The quadratic integrate-and-re neuron is thus expected to describe the dynamics of any type Ineuron close to bifurcation where ring rates are low In particular its ringrate in the absence of noise scales as

pI iexcl Ithr where Ithr is the threshold

current This is the behavior exhibited by all type I neurons near thresholdHere we compute the ring rate of the quadratic integrate-and-re neu-

ron in two limits small and large iquests=iquestm Interpolation between these limitsprovides a good approximation to the ring rate over the whole range ofiquests=iquestm in the suprathreshold regime and a reasonable zeroth-order approx-imation in the subthreshold regime The ring rate must be computed nu-merically but for applications such as mean-eld analysis where speed isimportant two lookup tables are sufcient to characterize the ring rate forall values of sup1 frac34 iquestm and iquests

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2283

2 Neuron Model and Analytical Methods

The quadratic integrate-and-re neuron with synaptic noise obeys the dif-ferential equations

iquestmdvdt

D f v C w (21a)

iquestsdwdt

D iexclw C iquest1=2m frac34acutet (21b)

where iquestm is the neuronal time constant (hereafter called the membrane timeconstant) vt is a ldquovoltagerdquo variable (hereafter called the membrane poten-tial)

f v D v2 C sup1 (22)

with sup1 the mean input current owing into the cell w represents the stochas-tic component in the synaptic currents iquests is the synaptic time constant acute iswhite noise with hacutetacutet0i D plusmnt iexcl t0 and frac34 is the overall amplitude of thenoise

In the noiseless version of the model when the membrane potentialbecomes positive the quadratic term on the right-hand side of equation 21aensures that the membrane potential diverges to innity in nite time Thetime at which the membrane potential reaches innity denes the timea spike is emitted Another way of dening the spike time would be toset a threshold for ring and then formally send this threshold to innityImmediately after ring the membrane potential is reset to iexcl1 Againthe quadratic term in the right-hand side of equation 21a ensures that themembrane potential comes back to a nite value in nite time To determinethe ringrate from equation 21 we reformulate the problem using a Fokker-Planck equation (Risken 1984)

iquestmtPv w t C v[ f v C wP] C iquestm

iquestsw[iexclwP] D frac34 2

2iquest 2

m

iquest 2s

2wP

where Pv w t is the joint probability density function (pdf) of v w attime t Once Pv w t is known the ring rate is given by the total proba-bility ux at v D C1 (see equation 25 below) For other studies of the pdfapproach to neural modeling mostly in the context of integrate-and-reneurons see Treves (1993) Abbott and van Vreeswijk (1993) Brunel andHakim (1999) Brunel (2000) Knight Omurtag and Sirovich (2000) andNykamp and Tranchina (2000)

The Fokker-Planck equation can be rewritten as a continuity equation

tPv w t C vJvv w t C wJwv w t D 0 (23)

2284 N Brunel and P Latham

where Jv and Jw are probability uxes

Jvv w t D1

iquestm f v C wPv w t (24a)

Jwv w t D1iquests

sup3iexcl iquestm

iquests

frac34 2

2wPv w t iexcl wPv w t

acute (24b)

The ring rate ordmt is the total probability ux at v D C1 it is given by

ordmt DZ C1

iexcl1dw lim

v1Jvv w t (25)

We are interested in computing the ring rate in the limit t 1 Inthis limit the probability distribution becomes time independent We thusset tP to zero in equation 23 resulting in a continuity equation involvingonly v and w That equation cannot be solved exactly but it can be solvedas an expansion in the ratio of the synaptic to the membrane time constant(or its inverse) To facilitate this expansion we introduced a new variablez acute kw=frac34 where k acute iquests=iquestm1=2 The advantage of using z rather than wis that its variance remains O1 in both the large and small k limits aproperty not shared by w With this change of variable and with tP set tozero equation 23 may be written as

LPv z D kfrac34zvPv z C k2v[ f vPv z] (26)

where

Lcent acute12

2z cent Czzcent

In the next two sections we solve this equation in the short and long corre-lation time limits

3 Short Correlation Time Limit

To solve equation 26 in the short correlation time limit (k iquest 1) we expandboth the probability distribution and the ring rate in powers of k

Pv z D1X

iD0

kiPiv z (31a)

ordm D1X

iD0

kiordmiS (31b)

where the subscript S on ordmiS is to remind us that we are working in the shortcorrelation time limit (Note that Pi should also have a subscript S we do

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2285

not include it for clarity It should be clear from the context whether we arereferring to the short or long correlation time limit the latter is discussedin section 4) Since Pv z is a probability distribution it must integrate toone independent of k Consequently we must have

Zdv dz P0v z D 1 (32a)

Zdv dz Piv z D 0 for i cedil 1 (32b)

To derive equations for the Pi we insert equation 31a into equation 26and match powers of k This results in a coupled set of equations

LP0v z D 0 (33a)

LP1v z D frac34 zvP0v z (33b)

LPiv z D frac34 zvPiiexcl1v z C v[ f vPiiexcl2v z] for i cedil 2 (33c)

Using equations 24a 25 and 31b the ring rate at order i is

ordmiS D1

iquestm

Z C1

iexcl1dz lim

v1[ f vPiv z C frac34zPiC1v z]

D1

iquestm

Z C1

iexcl1dz lim

v1f vPiv z (34)

where the last equality follows because Pv z must vanish as v 1otherwise it will not be normalizable

31 Probability Distribution in the Short Correlation Time Limit To com-pute the probability distribution as an expansion in powers of k we simplysolve equations 33a through 33c order by order The zeroth-order solu-tion P0 corresponds to white noise the next orders give corrections asso-ciated with the nite synaptic time constant It turns out as we will seethat we have to go to the fourth order to determine the lowest nonvan-ishing correction to the ring rate In this section however we sketch thesolutions to equations 33a through 33c only through second order thesolutions through fourth order are derived in appendix A We begin withequation 33a

Examining equation 33a we see that the zeroth-order contribution to theprobability distribution P0 can be written as a linear combination of twofunctions one even in z and the other odd Those two functions denotedAacute0 and Aacute1 are given by

Aacute0z D frac14iexcl1=2 expiexclz2

Aacute1z D expiexclz2

Z z

0expu2 du

2286 N Brunel and P Latham

The second of these Aacute1z scales as 1=z for large z Consequently it willbe negative when z approaches either C1 or iexcl1 and large compared toAacute0 This violates the condition Pv z cedil 0 (recall that Pv z is a probabilitydistribution) Thus P0 must be proportional to Aacute0 only The constant ofproportionality can be v-dependent so we have

P0v z D Q0vAacute0z (35)

The function Q0v is determined at higher order as we will see shortlyThe rst-order correctionto the probability distribution P1v z is found

by combining equations 33b and 35

P1v z D Q1vAacute0z C frac34vQ0vLiexcl1zAacute0z (36)

Because we explicitly included the homogeneous term in the above expres-sion Liexcl1 must pick out only inhomogeneous terms This leads to

Liexcl1Fz D 2 expiexclz2

Z z

0dz0 expz02

Z z0

iexcl1dz00 Fz00 (37)

which is valid for any L1 function Fz that vanishes at sect1 The lower limitof iexcl1 in the second integral ensures that Liexcl1Fz does not pick up a pieceproportional to Aacute1z as we will see below

The second-order correction P2v z which comes from combiningequations 33c 35 and 36 may be written after a small amount of algebraas

P2v z D Q2vAacute0z C frac34vQ1vLiexcl1zAacute0z

C frac34 22v Q0vLiexcl1zLiexcl1zAacute0z C v[ f vQ0v]Liexcl1Aacute0z (38)

Using equation 37 it is straightforward to show that

Liexcl1zAacute0z D iexclzAacute0z and

Liexcl1z2Aacute0z D z2=2Aacute0z iexcl 1=2Liexcl1Aacute0z

With these relations equation 38 becomes

P2v z D Q2vAacute0z iexcl frac34vQ1vzAacute0z C frac34 2

22

v Q0vz2Aacute0z

iexclmicro

frac34 2

22

vQ0v iexcl v[ f vQ0v]para

Liexcl1Aacute0z (39)

Again using equation 37 we see that for large positive z Liexcl1Aacute0z raquo 1=jzjThis makes Liexcl1Aacute0z nonnormalizable which means that it cannot make

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2287

any contribution to P2v z Thus the solvability condition on equation 39is

frac34 2

22

vQ0v iexcl v[ f vQ0v] D 0 (310)

This solvability condition can also be derived by applying the Fredholmalternative theorem (Reed amp Simon 1980)

To solve equation 310 we rst integrate once with respect to v and mul-tiply both sides by iexcl1 yielding

f vQ0v iexcl frac34 2

2vQ0v D iquestmordm0S (311)

In this equation we chose a specic form for the constant of integration(the term on the right-hand side) To see that this is the correct choice notethat in the large v limit vQ0v must approach zero to ensure that Q0 isnormalizable Thus in this limit the only term that survives on the left-hand side is f vQ0v Multiplying both sides by iquest iexcl1

m Aacute0z integrating overz and taking the limit v 1 equation 311 becomes equation 34 withi D 0 indicating that iquestmordm0S is the correct constant of integration

Equation 311 has the solution

Q0v D2ordm0Siquestm

frac34 2

Z 1

vdu exp[Atildev iexcl Atildeu] (312)

where

Atildev acuteZ v

0

2 f u

frac34 2 du (313)

The value of ordm0S is determined from the condition that Q0v must be nor-malized to 1 (see equation 32a and note that

Rdz Aacute0z D 1) which implies

that

ordm0S D frac34 2

2iquestm

microZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu]

paraiexcl1

(314)

We will compute this integral explicitly in the next sectionTo determine the next order correction to the ring rate we need to nd

Q1v This is done by deriving a solvability condition at third order Simi-larly Q2v which is needed for the second-order correction is determinedby deriving a solvability condition at fourth order Deriving these condi-tions requires that we compute P3 and P4 This is conceptually no moredifcult than computing P1 and P2 the only real difference is that the al-gebra is slightly more involved We thus relegate those computations to

2288 N Brunel and P Latham

appendix A There we nd that Q1v D 0 which means that the rst-ordercorrection to the ring rate ordm1S vanishes and that Q2v is given by

Q2v D ordm2S

ordm0S

Q0v iexcl iquestmordm0S

frac34 2

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (315)

where

h0v acute iexcl3 f 00v iexcl 4 f 0vAtilde 0v (316)

To calculate ordm2S we use equation 32b which tells us thatR

dvdz P2v z D 0and equation 39 along with the facts that the coefcient in front of Liexcl1Aacute0vanishes

R 1iexcl1 dv 2

vQ0v D 0 andR 1

iexcl1 dv vQ0v D 0 which tell us thatthe only way for P2v z to integrate to zero is to have

R 1iexcl1 dv Q2v D 0

Then integrating both sides of equation 315 and usingR 1

iexcl1 dv Q0v D 1we arrive at

ordm2S D ordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (317)

32 Firing Rate in the Short Correlation Time Limit The zeroth-order con-tribution to the ring rate is given by equation 314 and the lowest nonva-nishing correction which enters at Ok2 is given by equation 317 Theseexpressions involve double integrals and thus are not so easy to computeHowever we show in appendix B that both can be reduced to a single inte-gral resulting in

ordm0S D1

frac14iquestm

microZ 1

iexcl1

draquo

frac14 1=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]paraiexcl1

(318a)

ordm2S D iexclordm0Sfrac14frac34 2 iquestmordm0S

2

Z 1

iexcl1

draquo

frac141=2 raquo 2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48] (318b)

The ring rate through second order which we denote ordmS to indicate thatthe expression is valid in the short correlation time limit is then given by

ordmS D ordm0S C k2ordm2S (319)

In the absence of noise (frac34 D 0) thep

sup1 behavior is recovered for thezeroth-order contribution to the ring rate

ordm0S D sup11=2

frac14iquestm

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2289

-05 0 05 1Input current

0

10

20

30

40F

iring

rat

e (H

z)

Figure 1 Firing rate as a function of mean input current sup1 for white noise withfrac34 D 0 (solid line) 01 (dot-dashed line) 02 (dashed line) and 05 (dotted line)iquestm D 10 ms

and in the absence of a mean current (sup1 D 0) ordm0S increases with noise asfrac34 2=3

ordm0S D35=6frac34 2=3

22=3frac14 1=201=6iquestm

Plots of ordm0S versus sup1 are given in Figure 1 for several values of frac34 The second-order contribution to the ring rate ordm2S (the lowest nonvan-

ishing contribution) is negative Thus the ring rate decreases linearly withk2D iquests=iquestm as the correlation time constant increases This is inconvenientas the total ring rate through second order equation 319 can become neg-ative at large k It is possible to derive alternative expressions that coincideup to second order in k with equation 319 but stay positive for any valueof k One such alternative expression is

ordmS D ordm0S

1 iexcl k2ordm2S=ordm0S

(320)

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2283

2 Neuron Model and Analytical Methods

The quadratic integrate-and-re neuron with synaptic noise obeys the dif-ferential equations

iquestmdvdt

D f v C w (21a)

iquestsdwdt

D iexclw C iquest1=2m frac34acutet (21b)

where iquestm is the neuronal time constant (hereafter called the membrane timeconstant) vt is a ldquovoltagerdquo variable (hereafter called the membrane poten-tial)

f v D v2 C sup1 (22)

with sup1 the mean input current owing into the cell w represents the stochas-tic component in the synaptic currents iquests is the synaptic time constant acute iswhite noise with hacutetacutet0i D plusmnt iexcl t0 and frac34 is the overall amplitude of thenoise

In the noiseless version of the model when the membrane potentialbecomes positive the quadratic term on the right-hand side of equation 21aensures that the membrane potential diverges to innity in nite time Thetime at which the membrane potential reaches innity denes the timea spike is emitted Another way of dening the spike time would be toset a threshold for ring and then formally send this threshold to innityImmediately after ring the membrane potential is reset to iexcl1 Againthe quadratic term in the right-hand side of equation 21a ensures that themembrane potential comes back to a nite value in nite time To determinethe ringrate from equation 21 we reformulate the problem using a Fokker-Planck equation (Risken 1984)

iquestmtPv w t C v[ f v C wP] C iquestm

iquestsw[iexclwP] D frac34 2

2iquest 2

m

iquest 2s

2wP

where Pv w t is the joint probability density function (pdf) of v w attime t Once Pv w t is known the ring rate is given by the total proba-bility ux at v D C1 (see equation 25 below) For other studies of the pdfapproach to neural modeling mostly in the context of integrate-and-reneurons see Treves (1993) Abbott and van Vreeswijk (1993) Brunel andHakim (1999) Brunel (2000) Knight Omurtag and Sirovich (2000) andNykamp and Tranchina (2000)

The Fokker-Planck equation can be rewritten as a continuity equation

tPv w t C vJvv w t C wJwv w t D 0 (23)

2284 N Brunel and P Latham

where Jv and Jw are probability uxes

Jvv w t D1

iquestm f v C wPv w t (24a)

Jwv w t D1iquests

sup3iexcl iquestm

iquests

frac34 2

2wPv w t iexcl wPv w t

acute (24b)

The ring rate ordmt is the total probability ux at v D C1 it is given by

ordmt DZ C1

iexcl1dw lim

v1Jvv w t (25)

We are interested in computing the ring rate in the limit t 1 Inthis limit the probability distribution becomes time independent We thusset tP to zero in equation 23 resulting in a continuity equation involvingonly v and w That equation cannot be solved exactly but it can be solvedas an expansion in the ratio of the synaptic to the membrane time constant(or its inverse) To facilitate this expansion we introduced a new variablez acute kw=frac34 where k acute iquests=iquestm1=2 The advantage of using z rather than wis that its variance remains O1 in both the large and small k limits aproperty not shared by w With this change of variable and with tP set tozero equation 23 may be written as

LPv z D kfrac34zvPv z C k2v[ f vPv z] (26)

where

Lcent acute12

2z cent Czzcent

In the next two sections we solve this equation in the short and long corre-lation time limits

3 Short Correlation Time Limit

To solve equation 26 in the short correlation time limit (k iquest 1) we expandboth the probability distribution and the ring rate in powers of k

Pv z D1X

iD0

kiPiv z (31a)

ordm D1X

iD0

kiordmiS (31b)

where the subscript S on ordmiS is to remind us that we are working in the shortcorrelation time limit (Note that Pi should also have a subscript S we do

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2285

not include it for clarity It should be clear from the context whether we arereferring to the short or long correlation time limit the latter is discussedin section 4) Since Pv z is a probability distribution it must integrate toone independent of k Consequently we must have

Zdv dz P0v z D 1 (32a)

Zdv dz Piv z D 0 for i cedil 1 (32b)

To derive equations for the Pi we insert equation 31a into equation 26and match powers of k This results in a coupled set of equations

LP0v z D 0 (33a)

LP1v z D frac34 zvP0v z (33b)

LPiv z D frac34 zvPiiexcl1v z C v[ f vPiiexcl2v z] for i cedil 2 (33c)

Using equations 24a 25 and 31b the ring rate at order i is

ordmiS D1

iquestm

Z C1

iexcl1dz lim

v1[ f vPiv z C frac34zPiC1v z]

D1

iquestm

Z C1

iexcl1dz lim

v1f vPiv z (34)

where the last equality follows because Pv z must vanish as v 1otherwise it will not be normalizable

31 Probability Distribution in the Short Correlation Time Limit To com-pute the probability distribution as an expansion in powers of k we simplysolve equations 33a through 33c order by order The zeroth-order solu-tion P0 corresponds to white noise the next orders give corrections asso-ciated with the nite synaptic time constant It turns out as we will seethat we have to go to the fourth order to determine the lowest nonvan-ishing correction to the ring rate In this section however we sketch thesolutions to equations 33a through 33c only through second order thesolutions through fourth order are derived in appendix A We begin withequation 33a

Examining equation 33a we see that the zeroth-order contribution to theprobability distribution P0 can be written as a linear combination of twofunctions one even in z and the other odd Those two functions denotedAacute0 and Aacute1 are given by

Aacute0z D frac14iexcl1=2 expiexclz2

Aacute1z D expiexclz2

Z z

0expu2 du

2286 N Brunel and P Latham

The second of these Aacute1z scales as 1=z for large z Consequently it willbe negative when z approaches either C1 or iexcl1 and large compared toAacute0 This violates the condition Pv z cedil 0 (recall that Pv z is a probabilitydistribution) Thus P0 must be proportional to Aacute0 only The constant ofproportionality can be v-dependent so we have

P0v z D Q0vAacute0z (35)

The function Q0v is determined at higher order as we will see shortlyThe rst-order correctionto the probability distribution P1v z is found

by combining equations 33b and 35

P1v z D Q1vAacute0z C frac34vQ0vLiexcl1zAacute0z (36)

Because we explicitly included the homogeneous term in the above expres-sion Liexcl1 must pick out only inhomogeneous terms This leads to

Liexcl1Fz D 2 expiexclz2

Z z

0dz0 expz02

Z z0

iexcl1dz00 Fz00 (37)

which is valid for any L1 function Fz that vanishes at sect1 The lower limitof iexcl1 in the second integral ensures that Liexcl1Fz does not pick up a pieceproportional to Aacute1z as we will see below

The second-order correction P2v z which comes from combiningequations 33c 35 and 36 may be written after a small amount of algebraas

P2v z D Q2vAacute0z C frac34vQ1vLiexcl1zAacute0z

C frac34 22v Q0vLiexcl1zLiexcl1zAacute0z C v[ f vQ0v]Liexcl1Aacute0z (38)

Using equation 37 it is straightforward to show that

Liexcl1zAacute0z D iexclzAacute0z and

Liexcl1z2Aacute0z D z2=2Aacute0z iexcl 1=2Liexcl1Aacute0z

With these relations equation 38 becomes

P2v z D Q2vAacute0z iexcl frac34vQ1vzAacute0z C frac34 2

22

v Q0vz2Aacute0z

iexclmicro

frac34 2

22

vQ0v iexcl v[ f vQ0v]para

Liexcl1Aacute0z (39)

Again using equation 37 we see that for large positive z Liexcl1Aacute0z raquo 1=jzjThis makes Liexcl1Aacute0z nonnormalizable which means that it cannot make

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2287

any contribution to P2v z Thus the solvability condition on equation 39is

frac34 2

22

vQ0v iexcl v[ f vQ0v] D 0 (310)

This solvability condition can also be derived by applying the Fredholmalternative theorem (Reed amp Simon 1980)

To solve equation 310 we rst integrate once with respect to v and mul-tiply both sides by iexcl1 yielding

f vQ0v iexcl frac34 2

2vQ0v D iquestmordm0S (311)

In this equation we chose a specic form for the constant of integration(the term on the right-hand side) To see that this is the correct choice notethat in the large v limit vQ0v must approach zero to ensure that Q0 isnormalizable Thus in this limit the only term that survives on the left-hand side is f vQ0v Multiplying both sides by iquest iexcl1

m Aacute0z integrating overz and taking the limit v 1 equation 311 becomes equation 34 withi D 0 indicating that iquestmordm0S is the correct constant of integration

Equation 311 has the solution

Q0v D2ordm0Siquestm

frac34 2

Z 1

vdu exp[Atildev iexcl Atildeu] (312)

where

Atildev acuteZ v

0

2 f u

frac34 2 du (313)

The value of ordm0S is determined from the condition that Q0v must be nor-malized to 1 (see equation 32a and note that

Rdz Aacute0z D 1) which implies

that

ordm0S D frac34 2

2iquestm

microZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu]

paraiexcl1

(314)

We will compute this integral explicitly in the next sectionTo determine the next order correction to the ring rate we need to nd

Q1v This is done by deriving a solvability condition at third order Simi-larly Q2v which is needed for the second-order correction is determinedby deriving a solvability condition at fourth order Deriving these condi-tions requires that we compute P3 and P4 This is conceptually no moredifcult than computing P1 and P2 the only real difference is that the al-gebra is slightly more involved We thus relegate those computations to

2288 N Brunel and P Latham

appendix A There we nd that Q1v D 0 which means that the rst-ordercorrection to the ring rate ordm1S vanishes and that Q2v is given by

Q2v D ordm2S

ordm0S

Q0v iexcl iquestmordm0S

frac34 2

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (315)

where

h0v acute iexcl3 f 00v iexcl 4 f 0vAtilde 0v (316)

To calculate ordm2S we use equation 32b which tells us thatR

dvdz P2v z D 0and equation 39 along with the facts that the coefcient in front of Liexcl1Aacute0vanishes

R 1iexcl1 dv 2

vQ0v D 0 andR 1

iexcl1 dv vQ0v D 0 which tell us thatthe only way for P2v z to integrate to zero is to have

R 1iexcl1 dv Q2v D 0

Then integrating both sides of equation 315 and usingR 1

iexcl1 dv Q0v D 1we arrive at

ordm2S D ordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (317)

32 Firing Rate in the Short Correlation Time Limit The zeroth-order con-tribution to the ring rate is given by equation 314 and the lowest nonva-nishing correction which enters at Ok2 is given by equation 317 Theseexpressions involve double integrals and thus are not so easy to computeHowever we show in appendix B that both can be reduced to a single inte-gral resulting in

ordm0S D1

frac14iquestm

microZ 1

iexcl1

draquo

frac14 1=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]paraiexcl1

(318a)

ordm2S D iexclordm0Sfrac14frac34 2 iquestmordm0S

2

Z 1

iexcl1

draquo

frac141=2 raquo 2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48] (318b)

The ring rate through second order which we denote ordmS to indicate thatthe expression is valid in the short correlation time limit is then given by

ordmS D ordm0S C k2ordm2S (319)

In the absence of noise (frac34 D 0) thep

sup1 behavior is recovered for thezeroth-order contribution to the ring rate

ordm0S D sup11=2

frac14iquestm

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2289

-05 0 05 1Input current

0

10

20

30

40F

iring

rat

e (H

z)

Figure 1 Firing rate as a function of mean input current sup1 for white noise withfrac34 D 0 (solid line) 01 (dot-dashed line) 02 (dashed line) and 05 (dotted line)iquestm D 10 ms

and in the absence of a mean current (sup1 D 0) ordm0S increases with noise asfrac34 2=3

ordm0S D35=6frac34 2=3

22=3frac14 1=201=6iquestm

Plots of ordm0S versus sup1 are given in Figure 1 for several values of frac34 The second-order contribution to the ring rate ordm2S (the lowest nonvan-

ishing contribution) is negative Thus the ring rate decreases linearly withk2D iquests=iquestm as the correlation time constant increases This is inconvenientas the total ring rate through second order equation 319 can become neg-ative at large k It is possible to derive alternative expressions that coincideup to second order in k with equation 319 but stay positive for any valueof k One such alternative expression is

ordmS D ordm0S

1 iexcl k2ordm2S=ordm0S

(320)

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2284 N Brunel and P Latham

where Jv and Jw are probability uxes

Jvv w t D1

iquestm f v C wPv w t (24a)

Jwv w t D1iquests

sup3iexcl iquestm

iquests

frac34 2

2wPv w t iexcl wPv w t

acute (24b)

The ring rate ordmt is the total probability ux at v D C1 it is given by

ordmt DZ C1

iexcl1dw lim

v1Jvv w t (25)

We are interested in computing the ring rate in the limit t 1 Inthis limit the probability distribution becomes time independent We thusset tP to zero in equation 23 resulting in a continuity equation involvingonly v and w That equation cannot be solved exactly but it can be solvedas an expansion in the ratio of the synaptic to the membrane time constant(or its inverse) To facilitate this expansion we introduced a new variablez acute kw=frac34 where k acute iquests=iquestm1=2 The advantage of using z rather than wis that its variance remains O1 in both the large and small k limits aproperty not shared by w With this change of variable and with tP set tozero equation 23 may be written as

LPv z D kfrac34zvPv z C k2v[ f vPv z] (26)

where

Lcent acute12

2z cent Czzcent

In the next two sections we solve this equation in the short and long corre-lation time limits

3 Short Correlation Time Limit

To solve equation 26 in the short correlation time limit (k iquest 1) we expandboth the probability distribution and the ring rate in powers of k

Pv z D1X

iD0

kiPiv z (31a)

ordm D1X

iD0

kiordmiS (31b)

where the subscript S on ordmiS is to remind us that we are working in the shortcorrelation time limit (Note that Pi should also have a subscript S we do

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2285

not include it for clarity It should be clear from the context whether we arereferring to the short or long correlation time limit the latter is discussedin section 4) Since Pv z is a probability distribution it must integrate toone independent of k Consequently we must have

Zdv dz P0v z D 1 (32a)

Zdv dz Piv z D 0 for i cedil 1 (32b)

To derive equations for the Pi we insert equation 31a into equation 26and match powers of k This results in a coupled set of equations

LP0v z D 0 (33a)

LP1v z D frac34 zvP0v z (33b)

LPiv z D frac34 zvPiiexcl1v z C v[ f vPiiexcl2v z] for i cedil 2 (33c)

Using equations 24a 25 and 31b the ring rate at order i is

ordmiS D1

iquestm

Z C1

iexcl1dz lim

v1[ f vPiv z C frac34zPiC1v z]

D1

iquestm

Z C1

iexcl1dz lim

v1f vPiv z (34)

where the last equality follows because Pv z must vanish as v 1otherwise it will not be normalizable

31 Probability Distribution in the Short Correlation Time Limit To com-pute the probability distribution as an expansion in powers of k we simplysolve equations 33a through 33c order by order The zeroth-order solu-tion P0 corresponds to white noise the next orders give corrections asso-ciated with the nite synaptic time constant It turns out as we will seethat we have to go to the fourth order to determine the lowest nonvan-ishing correction to the ring rate In this section however we sketch thesolutions to equations 33a through 33c only through second order thesolutions through fourth order are derived in appendix A We begin withequation 33a

Examining equation 33a we see that the zeroth-order contribution to theprobability distribution P0 can be written as a linear combination of twofunctions one even in z and the other odd Those two functions denotedAacute0 and Aacute1 are given by

Aacute0z D frac14iexcl1=2 expiexclz2

Aacute1z D expiexclz2

Z z

0expu2 du

2286 N Brunel and P Latham

The second of these Aacute1z scales as 1=z for large z Consequently it willbe negative when z approaches either C1 or iexcl1 and large compared toAacute0 This violates the condition Pv z cedil 0 (recall that Pv z is a probabilitydistribution) Thus P0 must be proportional to Aacute0 only The constant ofproportionality can be v-dependent so we have

P0v z D Q0vAacute0z (35)

The function Q0v is determined at higher order as we will see shortlyThe rst-order correctionto the probability distribution P1v z is found

by combining equations 33b and 35

P1v z D Q1vAacute0z C frac34vQ0vLiexcl1zAacute0z (36)

Because we explicitly included the homogeneous term in the above expres-sion Liexcl1 must pick out only inhomogeneous terms This leads to

Liexcl1Fz D 2 expiexclz2

Z z

0dz0 expz02

Z z0

iexcl1dz00 Fz00 (37)

which is valid for any L1 function Fz that vanishes at sect1 The lower limitof iexcl1 in the second integral ensures that Liexcl1Fz does not pick up a pieceproportional to Aacute1z as we will see below

The second-order correction P2v z which comes from combiningequations 33c 35 and 36 may be written after a small amount of algebraas

P2v z D Q2vAacute0z C frac34vQ1vLiexcl1zAacute0z

C frac34 22v Q0vLiexcl1zLiexcl1zAacute0z C v[ f vQ0v]Liexcl1Aacute0z (38)

Using equation 37 it is straightforward to show that

Liexcl1zAacute0z D iexclzAacute0z and

Liexcl1z2Aacute0z D z2=2Aacute0z iexcl 1=2Liexcl1Aacute0z

With these relations equation 38 becomes

P2v z D Q2vAacute0z iexcl frac34vQ1vzAacute0z C frac34 2

22

v Q0vz2Aacute0z

iexclmicro

frac34 2

22

vQ0v iexcl v[ f vQ0v]para

Liexcl1Aacute0z (39)

Again using equation 37 we see that for large positive z Liexcl1Aacute0z raquo 1=jzjThis makes Liexcl1Aacute0z nonnormalizable which means that it cannot make

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2287

any contribution to P2v z Thus the solvability condition on equation 39is

frac34 2

22

vQ0v iexcl v[ f vQ0v] D 0 (310)

This solvability condition can also be derived by applying the Fredholmalternative theorem (Reed amp Simon 1980)

To solve equation 310 we rst integrate once with respect to v and mul-tiply both sides by iexcl1 yielding

f vQ0v iexcl frac34 2

2vQ0v D iquestmordm0S (311)

In this equation we chose a specic form for the constant of integration(the term on the right-hand side) To see that this is the correct choice notethat in the large v limit vQ0v must approach zero to ensure that Q0 isnormalizable Thus in this limit the only term that survives on the left-hand side is f vQ0v Multiplying both sides by iquest iexcl1

m Aacute0z integrating overz and taking the limit v 1 equation 311 becomes equation 34 withi D 0 indicating that iquestmordm0S is the correct constant of integration

Equation 311 has the solution

Q0v D2ordm0Siquestm

frac34 2

Z 1

vdu exp[Atildev iexcl Atildeu] (312)

where

Atildev acuteZ v

0

2 f u

frac34 2 du (313)

The value of ordm0S is determined from the condition that Q0v must be nor-malized to 1 (see equation 32a and note that

Rdz Aacute0z D 1) which implies

that

ordm0S D frac34 2

2iquestm

microZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu]

paraiexcl1

(314)

We will compute this integral explicitly in the next sectionTo determine the next order correction to the ring rate we need to nd

Q1v This is done by deriving a solvability condition at third order Simi-larly Q2v which is needed for the second-order correction is determinedby deriving a solvability condition at fourth order Deriving these condi-tions requires that we compute P3 and P4 This is conceptually no moredifcult than computing P1 and P2 the only real difference is that the al-gebra is slightly more involved We thus relegate those computations to

2288 N Brunel and P Latham

appendix A There we nd that Q1v D 0 which means that the rst-ordercorrection to the ring rate ordm1S vanishes and that Q2v is given by

Q2v D ordm2S

ordm0S

Q0v iexcl iquestmordm0S

frac34 2

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (315)

where

h0v acute iexcl3 f 00v iexcl 4 f 0vAtilde 0v (316)

To calculate ordm2S we use equation 32b which tells us thatR

dvdz P2v z D 0and equation 39 along with the facts that the coefcient in front of Liexcl1Aacute0vanishes

R 1iexcl1 dv 2

vQ0v D 0 andR 1

iexcl1 dv vQ0v D 0 which tell us thatthe only way for P2v z to integrate to zero is to have

R 1iexcl1 dv Q2v D 0

Then integrating both sides of equation 315 and usingR 1

iexcl1 dv Q0v D 1we arrive at

ordm2S D ordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (317)

32 Firing Rate in the Short Correlation Time Limit The zeroth-order con-tribution to the ring rate is given by equation 314 and the lowest nonva-nishing correction which enters at Ok2 is given by equation 317 Theseexpressions involve double integrals and thus are not so easy to computeHowever we show in appendix B that both can be reduced to a single inte-gral resulting in

ordm0S D1

frac14iquestm

microZ 1

iexcl1

draquo

frac14 1=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]paraiexcl1

(318a)

ordm2S D iexclordm0Sfrac14frac34 2 iquestmordm0S

2

Z 1

iexcl1

draquo

frac141=2 raquo 2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48] (318b)

The ring rate through second order which we denote ordmS to indicate thatthe expression is valid in the short correlation time limit is then given by

ordmS D ordm0S C k2ordm2S (319)

In the absence of noise (frac34 D 0) thep

sup1 behavior is recovered for thezeroth-order contribution to the ring rate

ordm0S D sup11=2

frac14iquestm

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2289

-05 0 05 1Input current

0

10

20

30

40F

iring

rat

e (H

z)

Figure 1 Firing rate as a function of mean input current sup1 for white noise withfrac34 D 0 (solid line) 01 (dot-dashed line) 02 (dashed line) and 05 (dotted line)iquestm D 10 ms

and in the absence of a mean current (sup1 D 0) ordm0S increases with noise asfrac34 2=3

ordm0S D35=6frac34 2=3

22=3frac14 1=201=6iquestm

Plots of ordm0S versus sup1 are given in Figure 1 for several values of frac34 The second-order contribution to the ring rate ordm2S (the lowest nonvan-

ishing contribution) is negative Thus the ring rate decreases linearly withk2D iquests=iquestm as the correlation time constant increases This is inconvenientas the total ring rate through second order equation 319 can become neg-ative at large k It is possible to derive alternative expressions that coincideup to second order in k with equation 319 but stay positive for any valueof k One such alternative expression is

ordmS D ordm0S

1 iexcl k2ordm2S=ordm0S

(320)

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2285

not include it for clarity It should be clear from the context whether we arereferring to the short or long correlation time limit the latter is discussedin section 4) Since Pv z is a probability distribution it must integrate toone independent of k Consequently we must have

Zdv dz P0v z D 1 (32a)

Zdv dz Piv z D 0 for i cedil 1 (32b)

To derive equations for the Pi we insert equation 31a into equation 26and match powers of k This results in a coupled set of equations

LP0v z D 0 (33a)

LP1v z D frac34 zvP0v z (33b)

LPiv z D frac34 zvPiiexcl1v z C v[ f vPiiexcl2v z] for i cedil 2 (33c)

Using equations 24a 25 and 31b the ring rate at order i is

ordmiS D1

iquestm

Z C1

iexcl1dz lim

v1[ f vPiv z C frac34zPiC1v z]

D1

iquestm

Z C1

iexcl1dz lim

v1f vPiv z (34)

where the last equality follows because Pv z must vanish as v 1otherwise it will not be normalizable

31 Probability Distribution in the Short Correlation Time Limit To com-pute the probability distribution as an expansion in powers of k we simplysolve equations 33a through 33c order by order The zeroth-order solu-tion P0 corresponds to white noise the next orders give corrections asso-ciated with the nite synaptic time constant It turns out as we will seethat we have to go to the fourth order to determine the lowest nonvan-ishing correction to the ring rate In this section however we sketch thesolutions to equations 33a through 33c only through second order thesolutions through fourth order are derived in appendix A We begin withequation 33a

Examining equation 33a we see that the zeroth-order contribution to theprobability distribution P0 can be written as a linear combination of twofunctions one even in z and the other odd Those two functions denotedAacute0 and Aacute1 are given by

Aacute0z D frac14iexcl1=2 expiexclz2

Aacute1z D expiexclz2

Z z

0expu2 du

2286 N Brunel and P Latham

The second of these Aacute1z scales as 1=z for large z Consequently it willbe negative when z approaches either C1 or iexcl1 and large compared toAacute0 This violates the condition Pv z cedil 0 (recall that Pv z is a probabilitydistribution) Thus P0 must be proportional to Aacute0 only The constant ofproportionality can be v-dependent so we have

P0v z D Q0vAacute0z (35)

The function Q0v is determined at higher order as we will see shortlyThe rst-order correctionto the probability distribution P1v z is found

by combining equations 33b and 35

P1v z D Q1vAacute0z C frac34vQ0vLiexcl1zAacute0z (36)

Because we explicitly included the homogeneous term in the above expres-sion Liexcl1 must pick out only inhomogeneous terms This leads to

Liexcl1Fz D 2 expiexclz2

Z z

0dz0 expz02

Z z0

iexcl1dz00 Fz00 (37)

which is valid for any L1 function Fz that vanishes at sect1 The lower limitof iexcl1 in the second integral ensures that Liexcl1Fz does not pick up a pieceproportional to Aacute1z as we will see below

The second-order correction P2v z which comes from combiningequations 33c 35 and 36 may be written after a small amount of algebraas

P2v z D Q2vAacute0z C frac34vQ1vLiexcl1zAacute0z

C frac34 22v Q0vLiexcl1zLiexcl1zAacute0z C v[ f vQ0v]Liexcl1Aacute0z (38)

Using equation 37 it is straightforward to show that

Liexcl1zAacute0z D iexclzAacute0z and

Liexcl1z2Aacute0z D z2=2Aacute0z iexcl 1=2Liexcl1Aacute0z

With these relations equation 38 becomes

P2v z D Q2vAacute0z iexcl frac34vQ1vzAacute0z C frac34 2

22

v Q0vz2Aacute0z

iexclmicro

frac34 2

22

vQ0v iexcl v[ f vQ0v]para

Liexcl1Aacute0z (39)

Again using equation 37 we see that for large positive z Liexcl1Aacute0z raquo 1=jzjThis makes Liexcl1Aacute0z nonnormalizable which means that it cannot make

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2287

any contribution to P2v z Thus the solvability condition on equation 39is

frac34 2

22

vQ0v iexcl v[ f vQ0v] D 0 (310)

This solvability condition can also be derived by applying the Fredholmalternative theorem (Reed amp Simon 1980)

To solve equation 310 we rst integrate once with respect to v and mul-tiply both sides by iexcl1 yielding

f vQ0v iexcl frac34 2

2vQ0v D iquestmordm0S (311)

In this equation we chose a specic form for the constant of integration(the term on the right-hand side) To see that this is the correct choice notethat in the large v limit vQ0v must approach zero to ensure that Q0 isnormalizable Thus in this limit the only term that survives on the left-hand side is f vQ0v Multiplying both sides by iquest iexcl1

m Aacute0z integrating overz and taking the limit v 1 equation 311 becomes equation 34 withi D 0 indicating that iquestmordm0S is the correct constant of integration

Equation 311 has the solution

Q0v D2ordm0Siquestm

frac34 2

Z 1

vdu exp[Atildev iexcl Atildeu] (312)

where

Atildev acuteZ v

0

2 f u

frac34 2 du (313)

The value of ordm0S is determined from the condition that Q0v must be nor-malized to 1 (see equation 32a and note that

Rdz Aacute0z D 1) which implies

that

ordm0S D frac34 2

2iquestm

microZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu]

paraiexcl1

(314)

We will compute this integral explicitly in the next sectionTo determine the next order correction to the ring rate we need to nd

Q1v This is done by deriving a solvability condition at third order Simi-larly Q2v which is needed for the second-order correction is determinedby deriving a solvability condition at fourth order Deriving these condi-tions requires that we compute P3 and P4 This is conceptually no moredifcult than computing P1 and P2 the only real difference is that the al-gebra is slightly more involved We thus relegate those computations to

2288 N Brunel and P Latham

appendix A There we nd that Q1v D 0 which means that the rst-ordercorrection to the ring rate ordm1S vanishes and that Q2v is given by

Q2v D ordm2S

ordm0S

Q0v iexcl iquestmordm0S

frac34 2

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (315)

where

h0v acute iexcl3 f 00v iexcl 4 f 0vAtilde 0v (316)

To calculate ordm2S we use equation 32b which tells us thatR

dvdz P2v z D 0and equation 39 along with the facts that the coefcient in front of Liexcl1Aacute0vanishes

R 1iexcl1 dv 2

vQ0v D 0 andR 1

iexcl1 dv vQ0v D 0 which tell us thatthe only way for P2v z to integrate to zero is to have

R 1iexcl1 dv Q2v D 0

Then integrating both sides of equation 315 and usingR 1

iexcl1 dv Q0v D 1we arrive at

ordm2S D ordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (317)

32 Firing Rate in the Short Correlation Time Limit The zeroth-order con-tribution to the ring rate is given by equation 314 and the lowest nonva-nishing correction which enters at Ok2 is given by equation 317 Theseexpressions involve double integrals and thus are not so easy to computeHowever we show in appendix B that both can be reduced to a single inte-gral resulting in

ordm0S D1

frac14iquestm

microZ 1

iexcl1

draquo

frac14 1=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]paraiexcl1

(318a)

ordm2S D iexclordm0Sfrac14frac34 2 iquestmordm0S

2

Z 1

iexcl1

draquo

frac141=2 raquo 2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48] (318b)

The ring rate through second order which we denote ordmS to indicate thatthe expression is valid in the short correlation time limit is then given by

ordmS D ordm0S C k2ordm2S (319)

In the absence of noise (frac34 D 0) thep

sup1 behavior is recovered for thezeroth-order contribution to the ring rate

ordm0S D sup11=2

frac14iquestm

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2289

-05 0 05 1Input current

0

10

20

30

40F

iring

rat

e (H

z)

Figure 1 Firing rate as a function of mean input current sup1 for white noise withfrac34 D 0 (solid line) 01 (dot-dashed line) 02 (dashed line) and 05 (dotted line)iquestm D 10 ms

and in the absence of a mean current (sup1 D 0) ordm0S increases with noise asfrac34 2=3

ordm0S D35=6frac34 2=3

22=3frac14 1=201=6iquestm

Plots of ordm0S versus sup1 are given in Figure 1 for several values of frac34 The second-order contribution to the ring rate ordm2S (the lowest nonvan-

ishing contribution) is negative Thus the ring rate decreases linearly withk2D iquests=iquestm as the correlation time constant increases This is inconvenientas the total ring rate through second order equation 319 can become neg-ative at large k It is possible to derive alternative expressions that coincideup to second order in k with equation 319 but stay positive for any valueof k One such alternative expression is

ordmS D ordm0S

1 iexcl k2ordm2S=ordm0S

(320)

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2286 N Brunel and P Latham

The second of these Aacute1z scales as 1=z for large z Consequently it willbe negative when z approaches either C1 or iexcl1 and large compared toAacute0 This violates the condition Pv z cedil 0 (recall that Pv z is a probabilitydistribution) Thus P0 must be proportional to Aacute0 only The constant ofproportionality can be v-dependent so we have

P0v z D Q0vAacute0z (35)

The function Q0v is determined at higher order as we will see shortlyThe rst-order correctionto the probability distribution P1v z is found

by combining equations 33b and 35

P1v z D Q1vAacute0z C frac34vQ0vLiexcl1zAacute0z (36)

Because we explicitly included the homogeneous term in the above expres-sion Liexcl1 must pick out only inhomogeneous terms This leads to

Liexcl1Fz D 2 expiexclz2

Z z

0dz0 expz02

Z z0

iexcl1dz00 Fz00 (37)

which is valid for any L1 function Fz that vanishes at sect1 The lower limitof iexcl1 in the second integral ensures that Liexcl1Fz does not pick up a pieceproportional to Aacute1z as we will see below

The second-order correction P2v z which comes from combiningequations 33c 35 and 36 may be written after a small amount of algebraas

P2v z D Q2vAacute0z C frac34vQ1vLiexcl1zAacute0z

C frac34 22v Q0vLiexcl1zLiexcl1zAacute0z C v[ f vQ0v]Liexcl1Aacute0z (38)

Using equation 37 it is straightforward to show that

Liexcl1zAacute0z D iexclzAacute0z and

Liexcl1z2Aacute0z D z2=2Aacute0z iexcl 1=2Liexcl1Aacute0z

With these relations equation 38 becomes

P2v z D Q2vAacute0z iexcl frac34vQ1vzAacute0z C frac34 2

22

v Q0vz2Aacute0z

iexclmicro

frac34 2

22

vQ0v iexcl v[ f vQ0v]para

Liexcl1Aacute0z (39)

Again using equation 37 we see that for large positive z Liexcl1Aacute0z raquo 1=jzjThis makes Liexcl1Aacute0z nonnormalizable which means that it cannot make

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2287

any contribution to P2v z Thus the solvability condition on equation 39is

frac34 2

22

vQ0v iexcl v[ f vQ0v] D 0 (310)

This solvability condition can also be derived by applying the Fredholmalternative theorem (Reed amp Simon 1980)

To solve equation 310 we rst integrate once with respect to v and mul-tiply both sides by iexcl1 yielding

f vQ0v iexcl frac34 2

2vQ0v D iquestmordm0S (311)

In this equation we chose a specic form for the constant of integration(the term on the right-hand side) To see that this is the correct choice notethat in the large v limit vQ0v must approach zero to ensure that Q0 isnormalizable Thus in this limit the only term that survives on the left-hand side is f vQ0v Multiplying both sides by iquest iexcl1

m Aacute0z integrating overz and taking the limit v 1 equation 311 becomes equation 34 withi D 0 indicating that iquestmordm0S is the correct constant of integration

Equation 311 has the solution

Q0v D2ordm0Siquestm

frac34 2

Z 1

vdu exp[Atildev iexcl Atildeu] (312)

where

Atildev acuteZ v

0

2 f u

frac34 2 du (313)

The value of ordm0S is determined from the condition that Q0v must be nor-malized to 1 (see equation 32a and note that

Rdz Aacute0z D 1) which implies

that

ordm0S D frac34 2

2iquestm

microZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu]

paraiexcl1

(314)

We will compute this integral explicitly in the next sectionTo determine the next order correction to the ring rate we need to nd

Q1v This is done by deriving a solvability condition at third order Simi-larly Q2v which is needed for the second-order correction is determinedby deriving a solvability condition at fourth order Deriving these condi-tions requires that we compute P3 and P4 This is conceptually no moredifcult than computing P1 and P2 the only real difference is that the al-gebra is slightly more involved We thus relegate those computations to

2288 N Brunel and P Latham

appendix A There we nd that Q1v D 0 which means that the rst-ordercorrection to the ring rate ordm1S vanishes and that Q2v is given by

Q2v D ordm2S

ordm0S

Q0v iexcl iquestmordm0S

frac34 2

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (315)

where

h0v acute iexcl3 f 00v iexcl 4 f 0vAtilde 0v (316)

To calculate ordm2S we use equation 32b which tells us thatR

dvdz P2v z D 0and equation 39 along with the facts that the coefcient in front of Liexcl1Aacute0vanishes

R 1iexcl1 dv 2

vQ0v D 0 andR 1

iexcl1 dv vQ0v D 0 which tell us thatthe only way for P2v z to integrate to zero is to have

R 1iexcl1 dv Q2v D 0

Then integrating both sides of equation 315 and usingR 1

iexcl1 dv Q0v D 1we arrive at

ordm2S D ordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (317)

32 Firing Rate in the Short Correlation Time Limit The zeroth-order con-tribution to the ring rate is given by equation 314 and the lowest nonva-nishing correction which enters at Ok2 is given by equation 317 Theseexpressions involve double integrals and thus are not so easy to computeHowever we show in appendix B that both can be reduced to a single inte-gral resulting in

ordm0S D1

frac14iquestm

microZ 1

iexcl1

draquo

frac14 1=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]paraiexcl1

(318a)

ordm2S D iexclordm0Sfrac14frac34 2 iquestmordm0S

2

Z 1

iexcl1

draquo

frac141=2 raquo 2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48] (318b)

The ring rate through second order which we denote ordmS to indicate thatthe expression is valid in the short correlation time limit is then given by

ordmS D ordm0S C k2ordm2S (319)

In the absence of noise (frac34 D 0) thep

sup1 behavior is recovered for thezeroth-order contribution to the ring rate

ordm0S D sup11=2

frac14iquestm

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2289

-05 0 05 1Input current

0

10

20

30

40F

iring

rat

e (H

z)

Figure 1 Firing rate as a function of mean input current sup1 for white noise withfrac34 D 0 (solid line) 01 (dot-dashed line) 02 (dashed line) and 05 (dotted line)iquestm D 10 ms

and in the absence of a mean current (sup1 D 0) ordm0S increases with noise asfrac34 2=3

ordm0S D35=6frac34 2=3

22=3frac14 1=201=6iquestm

Plots of ordm0S versus sup1 are given in Figure 1 for several values of frac34 The second-order contribution to the ring rate ordm2S (the lowest nonvan-

ishing contribution) is negative Thus the ring rate decreases linearly withk2D iquests=iquestm as the correlation time constant increases This is inconvenientas the total ring rate through second order equation 319 can become neg-ative at large k It is possible to derive alternative expressions that coincideup to second order in k with equation 319 but stay positive for any valueof k One such alternative expression is

ordmS D ordm0S

1 iexcl k2ordm2S=ordm0S

(320)

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2287

any contribution to P2v z Thus the solvability condition on equation 39is

frac34 2

22

vQ0v iexcl v[ f vQ0v] D 0 (310)

This solvability condition can also be derived by applying the Fredholmalternative theorem (Reed amp Simon 1980)

To solve equation 310 we rst integrate once with respect to v and mul-tiply both sides by iexcl1 yielding

f vQ0v iexcl frac34 2

2vQ0v D iquestmordm0S (311)

In this equation we chose a specic form for the constant of integration(the term on the right-hand side) To see that this is the correct choice notethat in the large v limit vQ0v must approach zero to ensure that Q0 isnormalizable Thus in this limit the only term that survives on the left-hand side is f vQ0v Multiplying both sides by iquest iexcl1

m Aacute0z integrating overz and taking the limit v 1 equation 311 becomes equation 34 withi D 0 indicating that iquestmordm0S is the correct constant of integration

Equation 311 has the solution

Q0v D2ordm0Siquestm

frac34 2

Z 1

vdu exp[Atildev iexcl Atildeu] (312)

where

Atildev acuteZ v

0

2 f u

frac34 2 du (313)

The value of ordm0S is determined from the condition that Q0v must be nor-malized to 1 (see equation 32a and note that

Rdz Aacute0z D 1) which implies

that

ordm0S D frac34 2

2iquestm

microZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu]

paraiexcl1

(314)

We will compute this integral explicitly in the next sectionTo determine the next order correction to the ring rate we need to nd

Q1v This is done by deriving a solvability condition at third order Simi-larly Q2v which is needed for the second-order correction is determinedby deriving a solvability condition at fourth order Deriving these condi-tions requires that we compute P3 and P4 This is conceptually no moredifcult than computing P1 and P2 the only real difference is that the al-gebra is slightly more involved We thus relegate those computations to

2288 N Brunel and P Latham

appendix A There we nd that Q1v D 0 which means that the rst-ordercorrection to the ring rate ordm1S vanishes and that Q2v is given by

Q2v D ordm2S

ordm0S

Q0v iexcl iquestmordm0S

frac34 2

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (315)

where

h0v acute iexcl3 f 00v iexcl 4 f 0vAtilde 0v (316)

To calculate ordm2S we use equation 32b which tells us thatR

dvdz P2v z D 0and equation 39 along with the facts that the coefcient in front of Liexcl1Aacute0vanishes

R 1iexcl1 dv 2

vQ0v D 0 andR 1

iexcl1 dv vQ0v D 0 which tell us thatthe only way for P2v z to integrate to zero is to have

R 1iexcl1 dv Q2v D 0

Then integrating both sides of equation 315 and usingR 1

iexcl1 dv Q0v D 1we arrive at

ordm2S D ordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (317)

32 Firing Rate in the Short Correlation Time Limit The zeroth-order con-tribution to the ring rate is given by equation 314 and the lowest nonva-nishing correction which enters at Ok2 is given by equation 317 Theseexpressions involve double integrals and thus are not so easy to computeHowever we show in appendix B that both can be reduced to a single inte-gral resulting in

ordm0S D1

frac14iquestm

microZ 1

iexcl1

draquo

frac14 1=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]paraiexcl1

(318a)

ordm2S D iexclordm0Sfrac14frac34 2 iquestmordm0S

2

Z 1

iexcl1

draquo

frac141=2 raquo 2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48] (318b)

The ring rate through second order which we denote ordmS to indicate thatthe expression is valid in the short correlation time limit is then given by

ordmS D ordm0S C k2ordm2S (319)

In the absence of noise (frac34 D 0) thep

sup1 behavior is recovered for thezeroth-order contribution to the ring rate

ordm0S D sup11=2

frac14iquestm

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2289

-05 0 05 1Input current

0

10

20

30

40F

iring

rat

e (H

z)

Figure 1 Firing rate as a function of mean input current sup1 for white noise withfrac34 D 0 (solid line) 01 (dot-dashed line) 02 (dashed line) and 05 (dotted line)iquestm D 10 ms

and in the absence of a mean current (sup1 D 0) ordm0S increases with noise asfrac34 2=3

ordm0S D35=6frac34 2=3

22=3frac14 1=201=6iquestm

Plots of ordm0S versus sup1 are given in Figure 1 for several values of frac34 The second-order contribution to the ring rate ordm2S (the lowest nonvan-

ishing contribution) is negative Thus the ring rate decreases linearly withk2D iquests=iquestm as the correlation time constant increases This is inconvenientas the total ring rate through second order equation 319 can become neg-ative at large k It is possible to derive alternative expressions that coincideup to second order in k with equation 319 but stay positive for any valueof k One such alternative expression is

ordmS D ordm0S

1 iexcl k2ordm2S=ordm0S

(320)

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2288 N Brunel and P Latham

appendix A There we nd that Q1v D 0 which means that the rst-ordercorrection to the ring rate ordm1S vanishes and that Q2v is given by

Q2v D ordm2S

ordm0S

Q0v iexcl iquestmordm0S

frac34 2

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (315)

where

h0v acute iexcl3 f 00v iexcl 4 f 0vAtilde 0v (316)

To calculate ordm2S we use equation 32b which tells us thatR

dvdz P2v z D 0and equation 39 along with the facts that the coefcient in front of Liexcl1Aacute0vanishes

R 1iexcl1 dv 2

vQ0v D 0 andR 1

iexcl1 dv vQ0v D 0 which tell us thatthe only way for P2v z to integrate to zero is to have

R 1iexcl1 dv Q2v D 0

Then integrating both sides of equation 315 and usingR 1

iexcl1 dv Q0v D 1we arrive at

ordm2S D ordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu

pound exp[Atildev iexcl Atildeu][hu iexcl hv C 4 f 0u] (317)

32 Firing Rate in the Short Correlation Time Limit The zeroth-order con-tribution to the ring rate is given by equation 314 and the lowest nonva-nishing correction which enters at Ok2 is given by equation 317 Theseexpressions involve double integrals and thus are not so easy to computeHowever we show in appendix B that both can be reduced to a single inte-gral resulting in

ordm0S D1

frac14iquestm

microZ 1

iexcl1

draquo

frac14 1=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]paraiexcl1

(318a)

ordm2S D iexclordm0Sfrac14frac34 2 iquestmordm0S

2

Z 1

iexcl1

draquo

frac141=2 raquo 2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48] (318b)

The ring rate through second order which we denote ordmS to indicate thatthe expression is valid in the short correlation time limit is then given by

ordmS D ordm0S C k2ordm2S (319)

In the absence of noise (frac34 D 0) thep

sup1 behavior is recovered for thezeroth-order contribution to the ring rate

ordm0S D sup11=2

frac14iquestm

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2289

-05 0 05 1Input current

0

10

20

30

40F

iring

rat

e (H

z)

Figure 1 Firing rate as a function of mean input current sup1 for white noise withfrac34 D 0 (solid line) 01 (dot-dashed line) 02 (dashed line) and 05 (dotted line)iquestm D 10 ms

and in the absence of a mean current (sup1 D 0) ordm0S increases with noise asfrac34 2=3

ordm0S D35=6frac34 2=3

22=3frac14 1=201=6iquestm

Plots of ordm0S versus sup1 are given in Figure 1 for several values of frac34 The second-order contribution to the ring rate ordm2S (the lowest nonvan-

ishing contribution) is negative Thus the ring rate decreases linearly withk2D iquests=iquestm as the correlation time constant increases This is inconvenientas the total ring rate through second order equation 319 can become neg-ative at large k It is possible to derive alternative expressions that coincideup to second order in k with equation 319 but stay positive for any valueof k One such alternative expression is

ordmS D ordm0S

1 iexcl k2ordm2S=ordm0S

(320)

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2289

-05 0 05 1Input current

0

10

20

30

40F

iring

rat

e (H

z)

Figure 1 Firing rate as a function of mean input current sup1 for white noise withfrac34 D 0 (solid line) 01 (dot-dashed line) 02 (dashed line) and 05 (dotted line)iquestm D 10 ms

and in the absence of a mean current (sup1 D 0) ordm0S increases with noise asfrac34 2=3

ordm0S D35=6frac34 2=3

22=3frac14 1=201=6iquestm

Plots of ordm0S versus sup1 are given in Figure 1 for several values of frac34 The second-order contribution to the ring rate ordm2S (the lowest nonvan-

ishing contribution) is negative Thus the ring rate decreases linearly withk2D iquests=iquestm as the correlation time constant increases This is inconvenientas the total ring rate through second order equation 319 can become neg-ative at large k It is possible to derive alternative expressions that coincideup to second order in k with equation 319 but stay positive for any valueof k One such alternative expression is

ordmS D ordm0S

1 iexcl k2ordm2S=ordm0S

(320)

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2290 N Brunel and P Latham

which after a small amount of algebra reduces to

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1raquo 2 iexcl frac34 4raquo 6=48]1 C k2frac34 2raquo 2=2

paraiexcl1

(321)

A second alternative expression can be derived by replacing 1 C k2frac34 2raquo 2=2with expk2frac34 2raquo 2=2 in equation 321 which leads to the expression

ordmS D1

frac14iquestm

microZ 1

iexcl1

draquo

frac141=2 exp[iexclsup1 iexcl k2frac34 2=2raquo 2 iexcl frac34 4z6=48]paraiexcl1

(322)

Both equations 320 and 322 have the same behavior at small k as equa-tion 319 However they have the advantage over equation 319 that thering rate stays positive for any value of iquests

Equations 320 and 322 are plotted in Figure 2 along with the results ofnumerical simulations of equation 21 Numerical simulations were doneusing the theta model (Gutkin amp Ermentrout 1998) which is related to thequadratic neuron by the change of variables v D tanmicro=2 In the theta modelspikes are identied by passage of micro through frac14 The equations were inte-grated using a Euler scheme with dt D 0001iquestm Note that the fact that noiseis colored avoids the complications associated with numerical integrationof the multiplicative white noise where Ito or Stratonovich prescriptionsyield different results (Lindner Longtin amp Bulsara 2003)

Figure 2 shows the ring rate as a function of the ratio iquests=iquestm for three dif-ferent regimes suprathreshold (sup1 gt 0) threshold (sup1 D 0) and subthreshold(sup1 lt 0) In the suprathreshold regime the second-order expansion holdsonly for small iquests (roughly speaking iquests lt 05iquestm) and when iquests becomes oforder iquestm or larger both equations 320 and 322 underestimate the ringrate When sup1 D 0 equation 320 gives a very good approximation to thering rate in a large range of iquests at least up to iquests D 2iquestm Equation 322on the other hand strongly underestimates the ring rate unless iquests=iquestm issmall Finally when sup1 is sufciently subthreshold equation 322 gives agood approximation to the ring rate while equation 320 overestimates it

Although both equations 320 and 322 have advantages in the remain-der of this article we focus on the former This is because it more easilymerges with the long correlation time ring rate (which we derive in thenext section) and can thus be used to provide an expression for the ringrate that is approximately valid for all synaptic time constants

4 The Long Correlation Time Limit

In the suprathreshold regime the short synaptic time expansion describesthe ring rate only when iquests is small compared to iquestm that is only when kis small This does not mean we cannot compute the ring rate when k is

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2291

0 05 1 15 2t

st

m

0

01

02

03

04

05

Firi

ng r

ate

(H

z)

0 05 1 15 2t

st

m

6

7

8

9

10

11

Firi

ng r

ate

(Hz)

0 05 1 15 2t

st

m

21

22

23

Firi

ng r

ate

(Hz)

m = 05 s = 05

m = 0 s = 05

m = -05 s = 05

Figure 2 Firing rate as a function of synaptic time constant for three differentinput statistics (indicated at the top of each graph) In each graph solid curveequation 320 dotted curve equation 322 full circles simulations (error barsare smaller than symbol size in all cases) dashed horizontal curve in the upperpanel ring rate for frac34 D 0 or iquests D 1 iquestm D 10 ms

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2292 N Brunel and P Latham

large it just means that we need to perform an expansion in powers of 1=krather than k Analogous to equations 31a and 31b we write

Pv z D1X

iD0

kiexcliPiv z (41a)

ordm D1X

iD0

kiexcliordmiL (41b)

where the subscript L is to remind us that we are working in the long corre-lation limit As mentioned in section 3 the Pi in equation 41a are differentfrom the ones in equation 31a but it should be clear what we mean fromthe context To ensure that Pv z has the proper normalization we demandthat the Pi obey the conditions given in equation 32

To derive the equations for the Pi we insert equation 41a into equation 26and match powers of k We nd that the Pi obey the relations

v f vP0v z D 0 (42a)

v f vP1v z D iexclfrac34 zvP0 (42b)

v f vPiv z D iexclfrac34 zvPiiexcl1 C LPiiexcl2 for i cedil 2 (42c)

In the next section we solve these equations through second order andin the section after that we compute the ring rate again through secondorder (the lowest nonvanishing order)

41 Probability Distribution in the Long Correlation Time Limit Thestrategy for solving equations 42a through 42c is essentially identical to thatused for equations 33a through 33c compute Pi at each order by invertinga differential operator express Pi in terms of a set of arbitrary functions anddetermine those functions by deriving solvability conditions at higher or-der The analysis is however considerably easier in the long than the shortcorrelation time limit for two reasons First the operator v[ f vcent] whichappears on the left-hand side of equations 42a through 42c is much easierto invert than L the corresponding operator for equations 33a through 33cSecond we need solve for Pi only through third order rather than fourth

We begin by writing down the solutions of equations 42a through 42cthrough second order which are easily found by inspection

P0v z D p0z

f v(43a)

P1v z D p1z

f viexcl frac34zp0z

f 2v(43b)

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3vC Lp0z

f v

Z v duf u

(43c)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2293

The piz are constants of integration that arise from inverting the oper-ator v[ f vcent] they are analogous to the Qiv that appeared in the shortcorrelation time analysis

Note that equations 43a through 43c are valid only for sup1 gt 0 For sup1 lt 0the distribution at zero order is a delta function at v D iexcl

piexclsup1 and the ring

rate is ordm0L D 0 Thus in the following we consider the case sup1 gt 0 we willcome back to the subthreshold regime in section 5

From equations 43a and 43b we see that the probabilityux f vPiv zis continuous at z D sect1 when i D 0 or 1 However this is not the case forequation 43c for which there is a jump discontinuity in the second-orderux

f vP2v zjvDC1 iexcl f vP2v zjvDiexcl1 D Lp0z

Z 1

iexcl1

duf u

To eliminate the discontinuity we must have Lp0z D 0 Using the samereasoning that led to equation 35 this implies that

p0z D sup11=2

frac14Aacute0z (44)

The factor sup11=2=frac14 was chosen to ensure thatR

dvdzP0v z D 1 which fol-lows from

RdzAacute0z D 1 and

Rdv=f v D frac14sup1iexcl1=2 With Lp0z D 0 P2v z

is given by

P2v z D p2z

f viexcl frac34zp1z

f 2vC frac34 2z2p0z

f 3v (45)

With the last term in equation 43c eliminated we nd by combiningequations 42c 43b and 45 that P3v z is given by

P3v z D3X

jD0

iexclfrac34z jp3iexcljz

f jC1v

C Lp1z

f v

Z v duf u

iexcl frac34 Lzp0z

f v

Z v duf 2u

(46)

Again there is a jump discontinuity in the probability ux

f vP3v zjvDC1 iexcl f vP3v zjvDiexcl1 D Lp1z

Z 1

iexcl1

duf u

iexcl frac34 Lzp0z

Z 1

iexcl1

duf 2u

(47)

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2294 N Brunel and P Latham

To determine how to choose p1z to eliminate the discontinuity we use

Z 1

iexcl1

dvf nv

D iexclsup1niexcl1

n iexcl 1

Z 1

iexcl1

dvf v

D frac14

n iexcl 1iexclsup1niexcl1 1

psup1

D frac14

sup11=2

2n iexcl 32sup1niexcl1n iexcl 1

(48)

where the rst and second equalities follow from the fact that f v D v2 C sup1

(see equation 22) and n is dened implicitly via n acute n pound n iexcl 2 andiexcl1 acute 1 Then using equation 48 to compute the integrals over u inequation 47 we nd that the discontinuity in the third-order ux vanisheswhen p1z obeys the equation Lp1z D frac34=2sup1Lzp0z Consequently

p1z D frac34

2sup1[zp0z C c1Aacute0z] (49)

where c1 is an arbitrary constant To determine the constant we use the factthat

Zdv dzP1v z D 0

which tells us that c1 D 0Knowledge of p1z is it turns out sufcient to calculate the lowest non-

vanishing correction to the ring rate We turn now to that calculation

42 Firing Rate in the Long Correlation Time Limit In the long correlationtime limit the probability distribution at each order can be written as anexpansion in powers of 1=f v In fact it is not hard to show that

Piv z DiX

jD0

iexclfrac34 z jpiiexcljz

f jC1v

which is a straightforward generalization of equation 46 with all termsthat produce jump discontinuities at sect1 removed Thus since f v raquo v2

for large v (see equation 22) the ith order probability ux at v D C1 isproportional to piz This in turn means via equation 34 that the ith orderring rate is given by

ordmiL D1

iquestm

Z 1

iexcl1dz piz (410)

Thus to compute the ring rate at order i we need to know only the integralof the pi which simplies the analysis somewhat

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2295

To compute ordm0L we use equation 410 with i D 0 and equation 44 forp0z and we nd that

ordm0L D sup11=2

frac14iquestm

As stated above ordm0L D 0 when sup1 lt 0 which means that the long correlationtime limit affects only the suprathreshold regime To compute ordm1L we useequation 410 with i D 1 and equation 49 for p1z Performing the integralin equation 410 we nd that ordm1L D 0

With ordm1L D 0 we need to go to second order to nd a correction to ordm0L Atthis order the situation changes slightly since we do not have an explicitexpression for p2z However we can nd the integral of p2z which isall we need for ordm2L by integrating both sides of equation 45 Applyingequation 32b which tells us that this integral is zero and using equation 48for the integrals over v we arrive at

Zdz p2z D frac34

2sup1

Zdz zp1z iexcl

3frac34 2

8sup12

Zdz z2p0z

With equations 44 and 49 for p0z and p1z the integrals are elementaryand we nd via equation 410 that

ordm2L D iexcl sup11=2

frac14iquestm

frac34 2

16sup12

We will adopt the convention that ordm2L D 1 if sup1 middot 0Thus the ring rate through second order in 1=k is given by

ordmL D sup11=2

frac14iquestm

sup31 iexcl frac34 2

16sup12k2

acute (411)

Interestingly in the long correlation time limit noise decreases the ringrate This is the opposite of what we saw in the short correlation timelimit where even at zeroth order noise increases the ring rate (see equa-tion 318a)

Equation 411 suffers from the same inconvenience as equation 319 itbecomes negative for small k An alternative expression that is identical toit through second order in 1=k is

ordmL D sup11=2

frac14iquestm

11 C frac34 2=16sup12k2

(412)

Figure 3 shows how equation 412 compares with simulations

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2296 N Brunel and P Latham

0 1 2 3 4 5t

st

m

31

315

32F

iring

rat

e (H

z)

0 1 2 3 4 5t

st

m

21

215

22

225

23

Firi

ng r

ate

(Hz)

m = 1 s = 05

m = 05 s = 05

Figure 3 Firing rateasa functionof synaptic time constant in the suprathresholdregime Filled circles simulations (error bars are smaller than symbol size in allcases) Thin solid lines short synaptic time expansion (equation 320) Dottedline long synaptic time expansion (equation 412) Long-dashed line ring ratefor frac34 D 0 or iquests D 1 Thick solid line simple rational function that interpolatesbetween short and long synaptic time constants equation 51

5 Combining the Short and Long Correlation Time Limits

One might wonder if a simple function could interpolate between the shortand the long synaptic time expansions The simplest function that repro-duces the calculated asymptotics is a rational function of the form

ordmk D ordm0S C ak2 C bordm0Lk4

1 C ck2 C bk4 (51)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2297

This function was chosen so that it asymptotes to ordm0S when k 0 and ordm0L

when k 1 The three parameters a b and c are chosen by requiring that

ordmk raquok0 ordm0S C ordm2Sk2

ordmk raquok1 ordm0L C ordm2Lkiexcl2

Since there are only two conditions and three parameters we have anadditional degree of freedom when choosing a b and c We can break thedegeneracy associated with this degree of freedom by demanding that inthe subthreshold regime where ordm0L D 0 ordmk reduce exactly to equation 320This forces a to be zero and we nd that

ordmk D ordm0S C ordm2S=ordm0Sordm0L=ordm2Lordm0Lk4

1 iexcl ordm2S=ordm0Sk2 C ordm2S=ordm0Sordm0L=ordm2Lk4 (52)

Because both ordm2S and ordm2L are negative and ordm0S and ordm0L are positive ordmk isnever negative

Figure 3 shows that equation 52 provides a good approximation to thesimulated data in the suprathreshold regime even in the intermediate range(around iquests D iquestm) where both short and long time expansions fail In the sub-threshold regime ordm0L=ordm2L is zero and equation 52 reduces to equation 320Thus we can evaluate how well ordmk approximates the true ring rate in thesubthreshold regime by examining the solid line in Figure 2 Although ordmk

slightly overestimates the ring rate the overestimation is small in absoluteterms mainly because the ring rate is small

Interestingly the synaptic time constant has different effects on the ringrate in the subthreshold and suprathreshold regimes In the subthresholdregime (see Figure 2) the ring rate decreases monotonically with synap-tic time constant In the suprathreshold range (see Figure 3) however thedependence is nonmonotonic ring rate decreases with time constant forsmall iquests but increases with large iquests and in between there is a minimum

The overall quality of the interpolation can be seen from the f -I curvewhich is plotted in Figure 4 for iquests=iquestm D 05 and iquests=iquestm D 2 This plot indicatesthat equation 52 does a good approximation of the true ring rate at allvalues of sup1 both subthreshold and suprathreshold

As can be seen from Figure 4 colored and white noise produce aboutthe same ring rate in the suprathreshold regime This is not surprisingsince noise has very little effect in the suprathreshold regime whatever itscharacteristics (all curves approach the dotted line when the input currentis large) In the subthreshold regime however the ring rate with colorednoise is small compared to that with white noise For example when iquests D 20ms and iquestm D 10 ms (both reasonable values for real neurons) the ratio ofthe colored noise ring rate to the white noise ring rate is 037 when theinput current is iexcl025

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2298 N Brunel and P Latham

-05 0 05 1Input current

0

10

20

30

40

Firi

ng r

ate

(Hz)

Figure 4 Firing rate as a function of mean input current sup1 when frac34 D 05iquestm D 10 ms and at several values of the synaptic time constant Curves arefrom equation 51 symbols are simulations Dashed line iquests D 0 (white noise)Circlesthick solid curve iquests D 5 ms (colored noise) Squaresthin solid curveiquests D 20 ms (colored noise) Dotted line iquests D 1 (no noise) Error bars for thesimulations are smaller than the symbol size in all cases

A second difference between colored and white noise is the dependenceof the ring rate on the amplitude of the noise frac34 For short correlationtimes iquests iquest iquestm increasing the noise amplitude increases the ring rate(see equation 318a from which it is easy to see that ordm0S is an increasingfunction of frac34 ) For long correlation times on the other hand noise decreasesring rate (see equation 412) This difference is important for models thatpropose noise as a mechanism of gain control (Chance Abbott amp Reyes2002 Prescott amp De Koninck 2003)

To compute ring rate via equation 52 there are two quantities that mustbe determined numerically ordm0S and ordm2S the zeroth- and second-order ringrates in the short correlation time limit Fortunately both of these quantitiesdepend primarily on a single variable deg D 481=3sup1=frac34 4=3 Using equations

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2299

318a and 318b it is straightforward to show that

ordm0S D 1frac14iquestm

sup3frac34 4

48

acute1=6 1Atilde0deg

ordm2S D iexclfrac14p

12iquestmordm20SAtilde2deg

where

Atildepdeg acuteZ 1

iexcl1

draquo

frac141=2 raquo p exp[iexcldeg raquo 2 iexcl raquo 6]

Thus the ring rate of the quadratic integrate-and-re neuron in the pres-ence of colored noise depends on two functions Atilde0deg and Atilde2deg This isextremely convenient for applications in which rapid calculation of ringrates is important (such as mean-eld calculations) Atilde0deg and Atilde2deg canbe computed for a range of values of deg and put into lookup tables and thering rates for all values of iquests iquestm sup1 and frac34 can then be rapidly computedusing those tables

6 Discussion

The ring rate of the noisy quadratic integrate-and-re (QIF) neuron wascomputed using expansions in the ratio of the noise correlation time to theneuronal time constant iquests=iquestm Our approach was to rst write down theFokker-Planck equation for the time evolution of the voltage and synapticcurrent the two variables describing the neuron We then expanded thisequation in either iquests=iquestm or iquestm=iquests In both limits small iquests=iquestm and small iquestm=iqueststhe Fokker-Planck equation reduced to an innite set of ordinary differentialequations (ODEs) These ODEs could be solved at successive orders byrecursion

The calculation for short correlation times turned out to be much simplerfor the quadratic than for the more standard linear integrate-and-re (LIF)neuron (Brunel amp Sergi 1998 Fourcaud amp Brunel 2002) This is because themain difculty with the LIF is that the voltage is reset from threshold to zeroin a rather unnatural way which leads to complicated boundary conditionsfor the Fokker-Planck equation With the QIF the boundary conditions aremuch more natural (C1 is identied with iexcl1) which dramatically sim-plies the calculation

Because of the different boundary conditions the ring rate of the QIFneuron is much less sensitive to temporal correlations in the noise than theLIF neuron Indeed at small iquests the ring rate for the QIF neuron decreasesas iquests=iquestm rather than

piquests=iquestm as it does for the LIF neuron At large correlation

times and large input currents on the other hand the behavior of the QIFand LIF neurons is qualitatively similar (Moreno de la Rocha Renart ampParga 2002 Moreno amp Parga 2003)

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2300 N Brunel and P Latham

The problem we studied is a particular case of the calculation of themean rst passage time of a dynamical system passing through a saddle-point bifurcation This problem has been studied extensively over the pastseveral decades Sigeti and Horsthemke (1989) Colet San Miguel Casade-munt and Sancho (1989) and Lindner Longtin and Bulsara (2003) studiedthe problem for white noise and derived an expression analogous to ourequation 314 Ram otilde rez-Piscina amp Sancho (1991) using other methods con-sidered the weakly colored noise case and derived our equations 318a and318b These results have been extended here in three ways we have pre-sented an alternative computation of the mean rst passage time in the shortcorrelation time limit we have obtained the mean rst passage time in thelong correlation time limit and we showed how a simple analytical formulacan interpolate between the two limits and provide a good approximationof the ring rates for any correlation time

These results extend our knowledge of the basic properties of the QIFmodel This model has seen increasing interest in recent years among theo-reticians primarily because the QIF neuron has the same properties as anytype I neuron whenever the ring rate is low (Ermentrout amp Kopell 1986 Er-mentrout 1996) Among other studies Roper Bressloff and Longtin (2000)studied the critical slowing down and stochastic resonancetrapping ef-fects that occur close to the saddle-node bifurcation QIF neurons have alsobeen used in network studies Latham Richmond Nelson and Nirenberg(2000) studied the equilibrium properties of the background state of largenetworks and Hansel and Mato (2001 2003) studied the synchronizationproperties of networks of QIF neurons in the absence of noise In this con-text our results can be useful in at least two ways First they can be usedin the mean-eld analysis of large networks of QIF neurons in the presenceof noise as has been done with LIF neurons (Amit amp Brunel 1997a 1997bBrunel 2000) Second they can be used to understand the trade-off betweennoise and synaptic and membrane time constants in type I neurons This isimportant as synaptic noise may be a mechanism for changing the gain ofneurons and may even act as a form of gain control (Chance et al 2002Prescott amp De Koninck 2003)

Appendix A Correction to the Probability Distribution Through FourthOrder in k in the Short Correlation Time Limit

In this appendix we compute P3v z and P4v z which allows us to com-pute the correction to the ring rate through second order (the lowest non-vanishing correction) To simplify the calculation we dene the two linearoperators

L1 acute frac34Liexcl1zv (A1a)

L2 acute Liexcl1v f v (A1b)

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2301

where Liexcl1 is given by equation 37 and both Liexcl1 and v operate on every-thing to their rightmdashfor example L2gv z D Liexcl1v[ f vgv z] With thisnotation the solution to equation 33c is

Piv z D QivAacute0z C L1Piiexcl1v z C L2Piiexcl2v z (A2)

Iterating equation A2 several times with i D 3 it is straightforward to showthat

P3 D Q3Aacute0 C L1Q2Aacute0 C L21 C L2Q1Aacute0 C [L1L2

1 C L2 C L2L1]Q0Aacute0

For clarity we have suppressed the arguments our convention is that Qi isa function of v and Aacute0 is a function of z

In what follows we will need to compute Liexcl1znAacute0 through n D 4 thesequantities are given via equation 37 by

Liexcl1z Aacute0z D iexclzAacute0z (A3a)

Liexcl1z2Aacute0z D iexclz2=2Aacute0z C 1=2Liexcl1Aacute0z (A3b)

Liexcl1z3Aacute0z D iexclz3=3 C zAacute0z (A3c)

Liexcl1z4Aacute0z D iexclz4=4 C 3z2=4Aacute0z C 3=4Liexcl1Aacute0z (A3d)

Using equations A1 and A3a through A3c it is straightforward to showthat

P3 Dmicro

Q3 iexcl frac34zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

v Q0v C frac34zv[ f vvQ0v]para

pound Aacute0z C frac34 2

22

v Q1z2Aacute0 iexclmicro

frac34 2

22

v Q1 iexcl v f Q1

paraLiexcl1Aacute0

Since the coefcient in front of Liexcl1Aacute0 must vanish Q1v obeys the samedifferential equation as Q0v (see equation 310) However Q1v differsfrom Q0v in that the former integrates to zero while the latter integratesto 1 (see equations 32a and 32b) This can happen only if Q1 vanishes Infact it is not hard to show in general that Qi vanishes for i odd We will notprove this here but we will assume it in the remainder of this appendixThus both Q1v and Q3v are zero and P3 is given by

P3 Dmicro

iexclfrac34 zvQ2v iexcl frac34 3

2

sup3z3

3C z

acute3

vQ0v C frac34zv[ f vvQ0v]para

Aacute0z

We now turn to P4 Iterating equation A2 with both Q1 and Q3 zero wearrive at

P4 D Q4Aacute0 C L21 C L2Q2Aacute0 C [L2

1 C L22 C L1L2L1]Q0Aacute0

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2302 N Brunel and P Latham

Using equation A1 for L1 and L2 and equation A3 to invert L we nd afterstraightforward but tedious algebra that

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

Cmicro

frac34 2

22

v Q2 iexcl v fQ2 C3frac34 4

84

v Q0 iexcl frac34 2

22

v f vQ0 iexcl frac34 2

4v f2

v Q0

para

pound z2Aacute0 iexcl Liexcl1Aacute0 (A4)

The solvability condition on Q2 is thus

frac34 2

22

v Q2 iexcl v fQ2 D iexcl 3frac34 4

84

v Q0 C frac34 2

22

v f vQ0 C frac34 2

4v f2

v Q0 (A5)

Inserting equation A5 into A4 we arrive at

P4 D Q4Aacute0 C v fQ2z2Aacute0 C frac34 4

244

v Q0z4Aacute0

To nd Q2v we rst simplify the right-hand side of equation A5 Usingequations 310 and 312 equation A5 may be rewritten

frac34 2

22

v Q2 iexcl v fQ2 D iexcl frac34 2

4v

micro3 f 00Q0 C 4 f 0Atilde 0Q0 iexcl

8ordm0iquestm

frac34 2 f 0para

where a prime denotes a derivativeTo solve this equation we use the same method as with equation 310

we integrate both sides with respect to v and multiply by iexcl1 This produces

f vQ2v iexcl frac34 2

2vQ2v

D iquestmordm2S iexclmicro

frac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para (A6)

The constant of integration was set to iquestmordm2S To see why note rst of allthat we can combine equations 34 and 39 to write

iquestmordm2S D limv1

microf vQ2v C frac34 2

4f v2

v Q0v

paraD lim

v1f vQ2v (A7)

where the rst equality follows from the facts that Q1v D 0 the coef-cient in front of Liexcl1Aacute0 vanishes and

R 1iexcl1 z2Aacute0z D 1=2 The second equality

follows because in the large v limit Q0 raquo 1=f and f raquo v2 Consequentlyequation A6 is correct if both vQ2 and the term inbrackets on the right-hand

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2303

side of that equation vanish as v 1 The rst of these vQ2 must vanishat large v to ensure normalizability To verify that the term in brackets van-ishes use limv1 Q0v D ordm0Siquestm=f v (see equation 311 and the discussionfollowing it) to write

limv1

microfrac34 2

4[iexcl3 f 00v iexcl 4 f 0vAtilde 0v]Q0v C 2ordm0Siquestm f 0

para

D ordm0Siquestm limv1

microiexcl

3frac34 2 f 00

4 fiexcl frac34 2 f 0Atilde 0

fC 2 f 0

para

Using equation 313 which tells us that Atilde 0 D 2 f=frac34 2 the last two terms on theright-hand side cancel This leaves us with f 00=f which vanishes for large vThus equation A6 in the large v limit really is equivalent to equation A7and ordm2S is the correct constant of integration

Equation A6 has the solution

Q2v D ordm2S

ordm0S

Q0v iexclZ 1

vdu exp[Atildev iexcl Atildeu]

poundmicro

12

h0uQ0u C4ordm0Siquestm

frac34 2 f 0u

para

where hv is given in equation 321 This can be simplied by integratingthe rst term in brackets by parts and using equation 312 for Q0v theresulting expression is given in equation 315

Appendix B Simplifying the Expressions for Firing Rate in the ShortCorrelation Time Limit

The expressions for the zeroth- and second-order ring rates in the shortcorrelation time limit equations 314 and 317 consist of rather complicatedtwo-dimensional integrals We can transform them into one-dimensionalintegrals by making the change of variables

v D x iexcl y (B1a)

u D x C y (B1b)

and integrating over xWith this change of variables for which the Jacobian is 2 equation 314

becomes

ordm0S D frac34 2

4iquestm

microZ 1

iexcl1dx

Z 1

0dy exp[Atildex iexcl y iexcl Atildex C y]

paraiexcl1

(B2)

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2304 N Brunel and P Latham

For the quadratic integrate-and-re neuron f v D v2 Csup1 equation 22 andwe have via equation 313

Atildev D 2frac34 2

microv3

3C sup1v

para (B3)

Inserting equation B3 into B2 and performing the integral over x we ndthat

ordm0S D frac34

2frac14iquestm

microZ 1

0

dyfrac14 1=2 yiexcl1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

paraiexcl1

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318aWe now turn to the expression for the second-order contribution to the

ring rate ordm2S equation 317 This equation is more difcult to reduce toa one-dimensional integral because of the term hu iexcl hv C f40u in theintegrand Our rst step then is to simplify this term Integrating equation316 we nd that hv D iexcl2 f vAtilde 0v iexcl 3 f 0v so that

hu iexcl hv C f 40u D iexcl2[ f uAtilde 0u iexcl f vAtilde 0v] C f 0u C 3 f 0v (B4)

The rst term on the right-hand side can be further modied by integratingequation 317 by parts

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f uAtilde 0u iexcl f vAtilde 0v]

DZ 1

iexcl1dv

microZ 1

vdu exp[Atildev iexcl Atildeu] f uAtilde 0u

iexclZ v

iexcl1du exp[Atildeu iexcl Atildev] f uAtilde 0u

para

DZ 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u C f 0v] (B5)

The intermediate step is useful because it allows explicit cancellation ofinnities that appear in the naive integration by parts Using equation B5we can replace equation B4 with iexcl f 0uC f 0v and equation 317 becomes

ordm2S D iexclordm0S

iquestmordm0S

frac34 2

Z 1

iexcl1dv

Z 1

vdu exp[Atildev iexcl Atildeu][ f 0u iexcl f 0v] (B6)

Again making the change of variables given in equation B1 using f v Dv2 C sup1 and integrating over x equation B6 becomes

ordm2S D iexcl4ordm0S

iquestmordm0S

frac34 2 frac14frac34 21=2Z 1

0dy y1=2 exp[iexcl4sup1y=frac34 2 iexcl 4y3=3frac34 2]

Finally letting y D frac34 2raquo 2=4 we arrive at equation 318b

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron 2305

Acknowledgments

We acknowledge the hospitality of the Institute for Theoretical Physics atthe University of California Santa Barbara where this work was performedNB thanks Ruben Moreno for discussing his results on the long synaptictime expansion of the leaky integrate-and-re neuron PL was supportedby NIMH grant R01 MH62447

References

Abbott L F amp van Vreeswijk C (1993) Asynchronous states in a network ofpulse-coupled oscillators Phys Rev E 48 1483ndash1490

Amit D J amp Brunel N (1997a) Dynamics of a recurrent network of spikingneurons before amp following learning Network 8 373ndash404

Amit D J amp Brunel N (1997b) Model of global spontaneous activity andlocal structured activity during delay periods in the cerebral cortex CerebralCortex 7 237ndash252

Amit D J amp Tsodyks M V (1991) Quantitative study of attractor neuralnetwork retrieving at low spike rates I Substratemdashspikes rates and neuronalgain Network 2 259ndash274

Anderson J S Lampl IGillespie D C amp Ferster D (2000)The contributionofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 1968ndash1972

Brunel N (2000) Dynamics of sparsely connected networks of excitatory ampinhibitory spiking neurons J Comput Neurosci 8 183ndash208

Brunel N amp Hakim V (1999) Fast global oscillations in networks of integrate-amp-re neurons with low ring rates Neural Computation 11 1621ndash1671

Brunel N amp Sergi S (1998) Firing frequency of integrate-amp-re neurons withnite synaptic time constants J Theor Biol 195 87ndash95

Burns B D amp Webb A C (1976) The spontaneous activity of neurones in thecatrsquos cerebral cortex Proc R Soc Lond B 194 211ndash223

Chance F Abbott L amp Reyes A (2002) Gain modulation from backgroundsynaptic input Neuron 35 773ndash782

Colet P San Miguel M Casademunt J amp Sancho J M (1989) Relaxationfrom a marginal state in optical bistability Phys Rev A 39 149ndash156

Destexhe A amp Pare D (1999) Impact of network activity on the integrativeproperties of neocortical pyramidal neurons in vivo J Neurophysiol 811531ndash1547

Destexhe A Rudolph M Fellous J-M amp Sejnowski T J (2001) Fluctuatingsynaptic conductances recreate in vivondashlike activity in neocortical neuronsNeuroscience 107 13ndash24

Ermentrout G B (1996) Type I membranes phase resetting curves amp syn-chrony Neural Computation 8 979ndash1001

Ermentrout G B amp Kopell N (1986) Parabolic bursting in an excitable systemcoupled with a slow oscillation SIAM J Appl Math 46 233ndash253

Fourcaud N amp Brunel N (2002) Dynamics of ring probability of noisyintegrate-amp-re neurons Neural Computation 14 2057ndash2110

2306 N Brunel and P Latham

Gutkin B amp Ermentrout G B (1998) Dynamics of membrane excitabilitydetermine interspike interval variability A link between spike generationmechanisms and cortical spike train statistics Neural Comp 10 1047ndash1065

Hansel D amp Mato G (2001) Existence and stability of persistent states in largeneuronal networks Phys Rev Lett 10 4175ndash4178

Hansel D amp Mato G (2003) Asynchronous states and the emergence of syn-chrony in large networks of interacting excitatory and inhibitory neuronsNeural Comp 15 1ndash56

Knight B W Omurtag A amp Sirovich L (2000) The approach of a neuronpopulation ring rate to a new equilibrium An exact theoretical result NeuralComputation 12 1045ndash1055

Latham P E Richmond B J Nelson P G amp Nirenberg S (2000) Intrinsicdynamics in neuronal networks I Theory J Neurophysiol 83 808ndash827

Lindner B Longtin A amp Bulsara A (2003) Analytic expressions for rate and CVof a type I neuron driven by white gaussian noise Neural Comp in press

Moreno R de la Rocha J Renart A amp Parga N (2002) Response of spikingneurons to correlated inputs Phys Rev Lett 89 288101

Moreno Ramp PargaN (2003)Simple model neurons with slow ltering operateas detectors of rare events Submitted

Nykamp D Q amp Tranchina D (2000) A population density approach thatfacilitates large-scale modeling of neural networks Analysis and an applica-tion to orientation tuning J Comp Neurosci 8 19ndash50

Prescott S amp De Koninck Y (2003) Gain control of ring rate by shuntinginhibition Roles of synaptic noise and dendritic saturation Proc Natl AcadSci 100 2076ndash2081

Ram otilde rez-Piscina L amp Sancho J M (1991) First-passage times for a marginalstate driven by colored noise Phys Rev A 43 663ndash668

Reed M amp Simon S (1980) Methods of modern mathematical physics I Functionalanalysis New York Academic Press

Ricciardi L M (1977) Diffusion processes and related topics on biology BerlinSpringer-Verlag

Risken H (1984) The Fokker-Planck equation Methods of solution and applicationsBerlin Springer-Verlag

Roper P Bressloff P C amp Longtin A (2000) A phase model of temperature-dependent mammalian cold receptors Neural Comp 12 1067ndash1093

Sigeti D amp Horsthemke W (1989) Pseudo-regular oscillations induced byexternal noise J Stat Phys 54 1217

Softky W R amp Koch C (1993) The highly irregular ring of cortical cells isinconsistent with temporal integration of random EPSPs J Neurosci 13334ndash350

Treves A (1993) Mean-eld analysis of neuronal spike dynamics Network 4259ndash284

Tuckwell H C (1988) Introduction to theoretical neurobiology Cambridge Cam-bridge University Press

Received December 13 2002 accepted April 4 2003