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Kv7 channels regulate pairwise spiking covariability in health and disease

Gabriel Koch Ocker1,3 and Brent Doiron2,3

1Department of Neuroscience, University of Pittsburgh, Pittsburgh, Pennsylvania; 2Department of Mathematics,University of Pittsburgh, Pittsburgh, Pennsylvania; and 3Center for the Neural Basis of Cognition, Pittsburgh,Pennsylvania

Submitted 27 January 2014; accepted in final form 24 April 2014

Ocker GK, Doiron B. Kv7 channels regulate pairwise spikingcovariability in health and disease. J Neurophysiol 112: 340–352,2014. First published April 30, 2014; doi:10.1152/jn.00084.2014.—Low-threshold M currents are mediated by the Kv7 family of potas-sium channels. Kv7 channels are important regulators of spikingactivity, having a direct influence on the firing rate, spike timevariability, and filter properties of neurons. How Kv7 channels affectthe joint spiking activity of populations of neurons is an important andopen area of study. Using a combination of computational simulationsand analytic calculations, we show that the activation of Kv7 conduc-tances reduces the covariability between spike trains of pairs ofneurons driven by common inputs. This reduction is beyond thatexplained by the lowering of firing rates and involves an activecancellation of common fluctuations in the membrane potentials of thecell pair. Our theory shows that the excess covariance reduction isdue to a Kv7-induced shift from low-pass to band-pass filtering of thesingle neuron spike train response. Dysfunction of Kv7 conductancesis related to a number of neurological diseases characterized by bothelevated firing rates and increased network-wide correlations. Weshow how changes in the activation or strength of Kv7 conductancesgive rise to excess correlations that cannot be compensated for bysynaptic scaling or homeostatic modulation of passive membraneproperties. In contrast, modulation of Kv7 activation parametersconsistent with pharmacological treatments for certain hyperactivitydisorders can restore normal firing rates and spiking correlations. Ourresults provide key insights into how regulation of a ubiquitouspotassium channel class can control the coordination of populationspiking activity.

model; noise correlations; potassium channels; model neurons; syn-chrony

POTASSIUM-MEDIATED M CURRENTS are a common intrinsic prop-erty of neurons in cortical, subcortical, and hippocampal re-gions (Brown and Passmore 2009; Delmas and Brown 2005;Jentsch 2000). M currents have slow activation kinetics, lackinactivation dynamics, and decrease overall cellular excit-ability (Aiken et al. 1995; Gu et al. 2005; Higgs et al. 2007;Lawrence et al. 2006; Peters et al. 2005). M currents are dueto heteromeric channels composed of Kv7.2/3 and Kv7.3/5subunits that are encoded by the KCNQ gene family (Wanget al. 1998; Selyanko et al. 2001; Peters et al. 2005).Dysfunction of Kv7 channels is related to a number ofdisease: shifts in Kv7 activation have been associated withtinnitus (Li et al. 2013), mutations involved in peripheralnerve hyperexcitability decrease Kv7 surface expression(Wuttke et al. 2007), and changes in both voltage-sensingand surface expression have been related to neonatal epi-

lepsy (Soldovieri et al. 2006). Kv7 mutations are alsoassociated with the most common childhood epilepsy con-dition, rolandic epilepsy (Coppola et al. 2003; Neubauer etal. 2008). Despite the evidence of reduced Kv7 activation intinnitus, chronic pain, and epilepsy, there lacks a coherentmechanistic theory for how Kv7 channels contribute tophysiological signatures of such disease states.

Two common neural correlates of tinnitus and epilepsy areelevated firing rates and excess spike train synchronization(Norea and Eggermont 2003; McCormick and Contreras2001). While a decrease in Kv7 activation is expected toincrease firing rates, it remains unclear how Kv7 channelsshape the coordinated activity of populations of neurons. Thetemporal correlation between the spike trains of pairs of neu-rons is an important signature of network function. Spikingcorrelations are modulated by attention, stimulus tuning andpresentation, learning, behavioral context, and sensory adapta-tion (Adibi et al. 2013; Gutnisky and Dragoi 2008; Wang et al.2011; Cohen and Kohn 2011). While specific arrangements ofcorrelated activity across a population of neurons can benefitcortical representation (Abbott and Dayan 1999; Averbeck etal. 2006), in many cases widespread and unstructured correla-tions are deleterious to coding (Josic et al. 2009; Sompolinskyet al. 2001). Previous studies of slow voltage-activated con-ductances like Kv7 have focused on single neuron statistics(Benda et al. 2010; Muller et al. 2007; Naud and Gerstner2012; Schwalger et al. 2010; Fisch et al. 2012), yet little isknown about their impact on the correlation of spiking activitybetween neurons. The dysfunction of Kv7 conductances indiseases characterized by increased correlations makes this acritical subject to study.

We consider a pair of model spiking neurons with volt-age-gated Kv7 conductances that receive fluctuating, par-tially correlated synaptic inputs. We show that Kv7 conductancesreduce firing rates and spike train correlations. Using a pertur-bative theoretical framework (de la Rocha et al. 2007; Shea-Brown et al. 2008), we relate the reduction in pairwise corre-lation to how Kv7 conductances shape single-neuron filterproperties. The theory shows that the reduction in correlationsis beyond that expected from the reduction in firing rates (de laRocha et al. 2007) and involves an active cancellation ofcorrelated inputs. This decoupling of firing rates and correla-tions by Kv7 conductances prevents synaptic scaling or ho-meostatic plasticity of passive membrane properties from si-multaneously correcting for Kv7 pathology-induced increasesin firing rates and correlations. However, we show that treat-ments that directly affect the Kv7 conductances can restorenormal spiking activity. Our work provides important predic-tions about how regulation of a common membrane potassium

Address for reprint requests and other correspondence: B. Doiron, Dept. ofMathematics, Univ. of Pittsburgh, Thackeray 528, Pittsburgh, PA 15213(e-mail: [email protected]).

J Neurophysiol 112: 340–352, 2014.First published April 30, 2014; doi:10.1152/jn.00084.2014.

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channel controls the temporally coordinated activity acrosspopulations of neurons in both health and disease states.

METHODS

Modeling

We describe pyramidal neurons with an exponential integrate-and-fire model (Fourcaud-Trocme et al. 2003) including a voltage-acti-vated potassium (Kv7) conductance. The membrane potential Vobeys:

CdV

dt� gL(VL � V) � gxx(Vx � V) � gL�exp

V � VT

�� I(t), (1)

�x

dx

dt� x�(V) � x , (2)

x�(V) � �1 � e�V�Vxh

Dx ��1

. (3)

The first three terms on the right-hand side of Eq. 1 describe theleak, Kv7, and spiking currents, respectively. The passive membraneproperties are determined by the leak conductance gL and reversalpotential VL. The Kv7-mediated M current has a gating variable x,with equilibrium x�(V). The voltage-dependence of x� is given by itshalf-activation voltage Vxh and activation slope Dx. As the voltagechanges, x relaxes to x� with time constant �x. Finally, the Kv7 currentis determined by the maximum conductance gx and K� reversalpotential Vx. The exponential term describes a phenomenologicalaction potential with an initiation threshold VT and steepness �. Whenthe voltage reaches VT, it depolarizes exponentially until it hits athreshold Vth at time tj and then decays linearly to the reset potential

Vre: V(t) � Vth �t�tj

�ref(Vth � Vre). When the voltage reaches Vre, its

dynamics again become governed by Eqs. 1–3. This explicit spikeshape, in contrast to the usual reset rule for integrate-and-fire models,allows action potentials to directly activate the Kv7 conductance.Parameter values are given in Table 1.

We model the total synaptic input to a pair of pyramidal neurons as

partially correlated Gaussian white noise: Ii(t) � � � gLD[�cc(t) �

�1�ci(t)]. Here, Ii denotes the input to neuron i of the pair, withi � 1, 2. The input is composed of a baseline � and fluctuations,which are broken up into two parts. One part, c(t), is common to bothneurons while the other part, i(t), is specific to neuron i. Both c(t)and i(t) are Gaussian white noise processes with 0 mean and unitamplitude. The parameter denotes the strength of the noise and D �

�2C

gLscales the noise amplitude to be independent of the membrane

timescale. With this noise scaling, the infinitesimal variance of diffu-sion of the passive membrane voltage is (gLD)2. The parameter cdefines the fraction of shared input to the two neurons so that theinfinitesimal covariance of the passive membrane voltage diffusion isc(gLD)2. Simulations were performed using an Euler-Maryamamethod with a timestep of 0.1 ms and a 1-s initial transient periodexcluded to ensure that the system had reached statistical equilibrium.

Statistics

Spike count statistics. A spike count from neuron i, niT(t), is the

number of spikes occurring within the window (t, t � T). For a givenwindow length T and trial length L, we compute a sequence of spikecounts from neuron i on each trial using windows that overlap by T/2:ni

T(0), niT(T/2), . . . , ni

T(L � T)} We use angular brackets, �·�, to denoteaveraging over trials. The firing rate of neuron i is simply:

ri � �niL� ⁄ L .

The spike count variance of neuron i and covariance between thespike counts of neurons 1 and 2 are, respectively:

Var�niT� � ��ni

T�2� � �niT�2,

Cov(n1T, n2

T) � �n1Tn2

T� � �n1T��n2

T� .(4)

Finally, the correlation coefficient of the spike counts 1 and 2 is:

Corr�n1T, n2

T� �Cov�n1

T, n2T�

�Var�n1T�Var�n2

T�. (5)

Spike train covariance. The spike train from neuron i is the pointprocess yi � �j�1

niL

�(t � tj), where tj is the time of the jth spike, �(t)the Dirac delta function, and ni

L the number of spikes emitted byneuron i during the trial of length L. The spike train cross-covariancebetween neurons 1 and 2 describes the expectation that an actionpotential will occur in each spike train, separated by a time lag s:

q12(s) �1

L0

L�y1(t)y2(t � s)�dt � r1r2.

By subtracting the product of the firing rates r1r2, we correct for thechance coincidence of action potentials. The spike count covariance isrelated to the integral of the cross-covariance function (Cox and Isham1980):

Cov�n1T, n2

T� � �T

Tq12(s)(T � �s�)ds , (6)

where the term (T � |s|) arises from the spike counting window.

Linear Response Theory

As in past studies (de la Rocha et al. 2007; Shea-Brown et al. 2008;Vilela and Lindner 2009; Litwin-Kumar et al. 2011), we use awell-established linear approximation to relate y(t) to an input stim-ulus s(t). To transition from the temporal to frequency domain we

define the Fourier transform of the time series x(t) as X(f) � [x(t)] �

��e�2 iftx(t)dt. We consider the rate-corrected spike train Y(f) �

[y(t) � r]. Our linear theory is based on the following ansatz:

�Yi�s(f)� S(f)Ai(f), (7)

where S(f) is the Fourier transform of s(t) and Ai(f) is the neuron linear

response function (or transfer function). In brief, �Yi|s�f�� is then thetrial-averaged fluctuation of spike train i about the time-averagedfiring rate conditioned on the realization of the stimulus s(t).

Table 1. Model parameters

Parameter Description Value

C Membrane capacitance 1 �F/cm2

gL Leak conductance 0.1 mS/cm2

VL Leak reversal potential �55 mVgx Peak Kv7 conductance 0.3 mS/cm2

�x Kv7 activation time constant 200 msVxh Kv7 half-activation voltage �40 mVDx Slope of Kv7 activation 8 mVVx Kv7 reversal potential �85 mV� Action potential steepness 1.4 mVVT Action potential initiation threshold �50 mVVth Action potential threshold 30 mVVre Action potential reset �72 mV�ref Action potential width 2 ms� Input mean 0 �A/cm2

Input SD 8 mVc Fraction of common input 0.1

341Kv7 CHANNELS REGULATE SPIKING COVARIABILITY

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For numerical simulations, we compute the transfer function Ai(f) as

Ai(f) �Qis(f)

Qss(f), (8)

where Qis(f) is the cross-spectrum between the stimulus and spiking

response and Qss(f) is the stimulus power spectrum. We computedthese in MATLAB using the pwelch function with a Bartlett window.We approximate the cross-spectrum of the activity of two neurons,assuming that their spike trains are conditionally independent giventhe stimulus, as:

Qij(f) Ai(f)A j*(f)Qss(f) (9)

where A* denotes the complex conjugate of A. To relate spike countstatistics to spike train statistics, we use the Wiener-Khinchin theorem(Risken 1996) to relate the cross-covariance function to the cross-spectrum of the activity of the two neurons and rewrite Eq. 6:

Cov�n1T, n2

T� � �T

T ��

�Qij(f)e ift(T � �s�)dfds (10)

��

�Qij(f)kT(f)df . (11)

Here kT(f) �1

2f2Tsin2( fT) is the Fourier transform of the window

function (T � |s|).Inserting Eq. 9 into Eq. 11 and using the fact that the stimulus has

a white power spectrum Qss(f) � c2, we then estimate the spikecount covariance as:

Cov�n1T, n2

T� � c2��

�A1(f)A 2

*(f)kT(f)df . (12)

The theory states that the way a pair of neurons transfer correlatedinputs to covariable spiking outputs is determined by how each neurontransfers the correlated portion of its input to modulations of itsspiking output. For a general exposition of linear response for cova-riance transfer in networks of spiking neurons, see Trousdale et al.(2012).

Solving for the Firing Rates and Response Functions with theFokker-Planck Theory

In this section we review the calculation for the transfer functionfirst developed in Richardson (2007) and extended to models withvoltage-activated conductances in Richardson (2009). We refer thereader to these past studies for a full derivation of these techniques.For earlier treatments of spiking responses in stochastic neural net-work activity, see Abbott and van Vreeswijk (1993), Ginzburg andSompolinsky (1994), and for a useful exposition of Fokker-Plancktechniques in neural networks, see Fourcaud and Brunel (2002).

In the equilibrium state, the voltage distribution associated with Eqs. 1and 2, P0(V, x), obeys the continuity and flux equations (Risken 1996)

��J0

�V�

� P0

� t� r0��(V � Vth) � �(V � Vre)� (13)

�� P0

�V�

1

gL2�CJ0 � I0P0� (14)

where I0 � gL(V � VL) � gL�e(V � VT)/� � gxx0 (V � Vx) � �0, with�0 being the mean of the input current. We use a separation oftimescales to self-consistently solve for the steady-state Kv7 activa-tion x0 and density of nonrefractory neurons, P0. We then recover thefiring rate from the normalization condition ��

��P0dV � r0�ref � 1.The density P0 does not describe the neuron during an action poten-tial. To take into account the membrane voltage during action poten-tials, we add the distribution of the membrane voltage during spikes. For

the linear spike shape described above, the full equilibrium density isP0(V) � r0�ref�(V � Vre), where �(V) is the Heaviside function.

We investigate the response of the system to inputs that fluctuate intime by considering the time-varying responses to a periodic input asa perturbation from the equilibrium state: Ii(t) � �0 � �1e

2 ift �gLDi(t). Here �1 is the amplitude of the input modulation and istaken to be small. To first order in �1, the periodic input inducesperiodic modulations in the system at the same frequency f. Wedecompose the probability density, probability flux, firing rate, andKv7 activation into steady-state and modulated components:

P � P0 � P1e2 ift, J � J0 � J1e2 ift,

r � r0 � Ae2 ift, x � x0 � x1e2 ift.

We solve for the modulated components in the Fourier domain afterobtaining the equilibrium solution to Eqs. 13 and 14. They obey thefirst order continuity and flux equations:

� J1

�V� 2 if P1 � A��(V � Vth) � e�2 if�ref�(V � Vre)� (15)

� P1

�V�

1

gL2��0P1 � CJ1 � gxx1(V � Vx)P0 � �1P0�. (16)

We also take into account the distribution of voltage during actionpotentials so that:

P1 � P� � �xx1Px � A�refPsp, A � A� � �xx1Ax. (17)

�x � gx/gL is the nondimensionalized peak Kv7 conductance. For thelinear spike shape we have, the voltage density occupied by action

potentials is Psp �e�2 if�ref�Vth�V��Vth�Vre�

Vth�Vre�(V � Vre).

Since the first order flux given by Eq. 16 is linear in x1 and �1, wecan separate Eqs. 15 and 16 into two systems: one set of equations forthe modulations related to x1, and one set of equations for themodulations related to �1. The modulations due to �1 obey:

�J�

�V� 2 if P� � A��(V � Vth) � e�2 f�ref�(V � Vre)� (18)

�� P�

�V�

1

gL2�I0P� � CJ� � �0P0� (19)

and those due to x1 obey:

�Jx

�V� 2 if Px � Ax��(V � Vth) � e�2 f�ref�(V � Vre)� (20)

�� Px

�V�

1

gL2�I0Px � CJx� �V � Vx

2 P0. (21)

We solve Eqs. 18 and 19 and 20 and 21 numerically for

P�, Px, A�, and Ax. Again taking advantage of the slow kinetics of Kv7

activation, we then self-consistently calculate P1 and x1, which we use

in Eq. 17 to calculate the transfer function A. See Richardson (2009),where this method was developed, for a full exposition of theself-consistent solution of Fokker-Planck equations for neurons withslow voltage-activated conductances. This method can easily beadapted to the case of spike-activated conductances or currents.

RESULTS

Effect of Kv7 Conductances on Single-Neuron Activity

Kv7 channels are ubiquitous in the nervous system, and theirmodulation is a cellular correlate of many neurological disor-

342 Kv7 CHANNELS REGULATE SPIKING COVARIABILITY


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ders. The influence of these channels on the transfer of com-mon input fluctuations to correlated variability in output spiketrains is unknown. We used an exponential integrate-and-firemodel to examine how Kv7 conductances affect the statisticsof pairwise spiking activity (see METHODS). We modeled Kv7 asa dynamic, voltage-gated conductance with slow activation x(t)and no inactivation kinetics. The model neuron received inputfrom a large pool of presynaptic neurons, which we approxi-mated as Gaussian white noise with mean � and variance 2

(Fig. 1A). The stochastic input induced variability in the neuronmodel membrane potential and spike train outputs. We con-sidered how the presence of Kv7 conductances in the modelneuron (�Kv7) affects its input-output transfer when com-pared with a neuron that lacked Kv7 conductances (�Kv7).We first describe how Kv7 conductances impact the statisticsof the output of a single neuron before presenting our mainresults for correlation transfer by pairs of neurons.

When the input mean � is constant in time, the outputstatistics are also time invariant. In this equilibrium state, themembrane potential spent the majority of its time in thesubthreshold voltage range. Even though the Kv7 activationx�(V) was small for subthreshold V, the slow Kv7 activationcaused it to interact strongly with the equilibrium membranepotential distribution (Fig. 1B). The Kv7 activation x(t) fluc-tuated around a steady-state value, reducing the neuron firingrate from 22 spikes (sp)/s (�Kv7) to 8 sp/s (�Kv7) through ahyperpolarizing shift of the steady-state membrane potentialdistribution (Fig. 1B). In response to a step increase of thestimulus mean �, the neuron firing rate transiently increasedand then decreased to a new steady-state level as the Kv7conductance accumulated (Fig. 1C). The neuron steady-statefiring rate increased with � but to a lesser degree for the �Kv7neuron than the �Kv7 neuron (Fig. 1D). This shows that Kv7activation divisively scales firing rates, a well-known conse-quence of outward currents that provide spike-driven negativefeedback (Ermentrout 1998; Benda and Herz 2003).

We next asked how the Kv7 conductance affects the single-neuron responses to time-varying inputs (Kongden et al. 2008).To study the dynamic response we let I(t) � �0 � �1 sin(2 ft)with f being low frequency (f � 1 Hz) and �1 being smallrelative to the background fluctuations 2. The Kv7 currenttracked the sinusoidal input and provided negative feedback(Fig. 2B, orange curve) that cancelled off both the static (�0)and fluctuating [�1sin(2 ft)] components of the input current(Fig. 2B, compare green and blue curves). The membranevoltage was noisy and varied over repeated presentations(cycles) of the stimulus (Fig. 2, A and B, bottom). Performingsufficient cycle averaging allowed for an estimate of the effectof Kv7 conductance on the instantaneous firing rate of theneuron over the cycle. The stimulus induced a sinusoidalmodulation of the instantaneous firing rate, r(t) � r0 ��1 A sin(2 ft � �), where r0 is the firing rate without thedynamic stimulus, and A and � are the amplitude and phaseshift of the response respectively (Fig. 2C). Importantly, the�Kv7 neuron reduced the response amplitude A comparedwith the �Kv7 neuron (Fig. 2C, compare blue and blackcurves), reflecting the Kv7 induced cancellation of the dynamicinput current (Fig. 2B).

To examine whether the Kv7-mediated reduction in re-sponse gain was frequency specific, we carried out a similar

analysis for inputs across frequency f and calculated the trans-fer function A(f) of the �Kv7 and �Kv7 model neurons. Thetransfer function measures how strongly a neuron modulates itsinstantaneous firing rate in response to an input of a givenfrequency. We saw that the �Kv7 model had a reduced A(f)at all frequencies compared with the �Kv7 neuron. This wasespecially true at low f, imparting a band-pass shape to the

V (

mV

)K

v7 A

ctiv

.x

0 200 4000.02

.04

.06

.08

Time (ms)

Kv7

Con

d.

(mS

/cm

2)

−2 −1 0 10

10

20

30

40

Current (μA/cm2)

Mea

n R

ate

(sp/

s)

B

C D

x∞

P0

−500

−500

100 ms.05

.1

−100

−50

0

V (

mV

)

-Kv7

+Kv7

. . .

. . . + μ }σ

Presynaptic Spike Trains

Net Input Current ModelNeuron

μ

A

Rat

e (s

p/s)

Fig. 1. Kv7 conductances control a neuron equilibrium state and overall firingrate. A: schematic of our model. A large pool of presynaptic neurons projectsto the model neuron. We approximate the total synaptic input as Gaussianwhite noise with mean � and variance 2. B, top: the membrane potential ofthe model neuron without (black) and with (blue) a Kv7 conductance, bothwith the same realization of input noise. B, bottom: the Kv7 activationfluctuates over time around a steady-state mean value. Dashed line: mean Kv7activation x0, computed from a Fokker-Planck theory of the model neuron witha Kv7 conductance (see METHODS). The neuron dynamics are governed by Eqs.1–3, and the parameters of the neuron and the noisy input are given in Table1. B, right: activation of Kv7 conductances decreases the neuron firing rate bya slight hyperpolarization and tightening of the equilibrium membrane poten-tial distribution. Solid lines: membrane potential distributions computed fromthe Fokker-Planck theory. Shaded lines: simulations. C: in response to a stepinput (top) the Kv7 conductance accumulates until it reaches a new steady state(bottom). This implements spike- frequency adaptation, as reflected by theinstantaneous firing rate computed by averaging the spike trains from 20,000realizations (middle); sp/s, spikes/s. D: Kv7 conductances reduce the gain ofthe firing rate curve. Circles mark the firing rates at � � 0, used in B.Steady-state firing rates computed using the Fokker-Planck theory (r0 ofEq. 13).



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�Kv7 transfer function (Fig. 2D). For a mathematical exposi-tion of how adaptation currents give rise to band-pass transferfunctions, see Benda and Herz (2003).

Throughout our study we made use of a theoretical frame-work to predict the transfer functions of the �Kv7 and �Kv7neuron models. The theory requires that �1 is sufficiently smallto ensure a linear input-output relationship [r(t) inherits the

sinusoidal time dependence of I(t)] and that the activationdynamics of the Kv7 conductance are much slower than thepassive membrane time constant. These assumptions allowedus to calculate the transfer function A(f) and the time-averagedfiring rate r0 using a well-documented perturbation technique(Richardson 2007, 2009; see METHODS). The theory gave excel-lent estimates of both the steady-state and instantaneous firingrates (Figs. 1B and 2, C and D; compare theory to simulationscurves).

Effect of Kv7 Conductances on Covariability of PairwiseActivity

The central aim of our study was to explore how Kv7conductances affect the covariability of the spike train outputsfrom pairs of neurons. We modeled two pairs of neurons: a pairwith Kv7 conductances (�Kv7 pair) and a pair without (�Kv7pair). As before, each neuron received a fluctuating input withmean � and variance 2. The neurons were not synapticallycoupled, but they shared a fraction of their input c 0(Fig. 3A). This shared input drove coherent membrane poten-tial fluctuations and sometimes caused the neurons to spikesynchronously (Fig. 3A, top, arrows). To isolate the effect ofthe Kv7 conductance on how the neuron pair transferred inputcorrelations to their outputs, we kept the statistics of the input(�, 2, and c) and all other model parameters identical betweenthe neuron pairs. The magnitude of the peak of the spike traincross-covariance function q12(s) (see METHODS) of the �Kv7pair was 34% that of the �Kv7 pair, and the �Kv7 covariancefunction was lower for all time lags (Fig. 3B). Thus activationof Kv7 conductances decorrelated the spike train responses ofneuron pairs receiving common inputs.

To quantify this reduction of output covariation acrossdifferent time scales, we counted the number of spikes thateach neuron of a pair emitted in windows of T milliseconds, n1

T

and n2T, and computed the covariance of n1

T and n2T, Cov(n1

T,n2

T)Kv7, as a function of window size (see METHODS). For shortT, Cov(n1

T, n2T) corresponds to synchrony of individual action

potentials. For long T, it corresponds to covariation of the firingrates. The �Kv7 neuron pair had a much lower spike countcovariance than the �Kv7 pair across all window sizes (Fig.3C, top). To correct for the trend that Cov(n1

T, n2T) increases

with T, we considered the ratio of the �Kv7 and �Kv7 pairs’spike count covariances. This showed that the �Kv7 pair had,depending on window size, 20–30% as much spiking covari-ability as the �Kv7 pair (Fig. 3C, bottom).

To relate pairwise covariance to single neuron filter proper-ties, we used a perturbation theory that has been previouslyused for �Kv7 integrate-and-fire models (de la Rocha et al.2007; see METHODS). For weak input correlations ( c �� 1),our theory relates the output spike count covariance Cov(n1

T,n2

T) to the input covariability, c2, and the single-neurontransfer function, A(f), via:

Cov�n1T, n2

T� � c2��

� �A(f)�2kT(f)df . (22)

Here the kernel kT(f) relates spike count statistics over awindow of length T to the spike train statistics. It is useful toconsider this theory in two regimes, the long time windowregime T ¡ � and the short time window regime T ¡ 0. Inthese opposing cases Eq. 22 reduces to:

Vol

tage

0 500 1000Time (ms)

InputIKv7

0 500 1000Time (ms)

0 500 1000 1500 20000

20

40

Time (ms)

Firi

ng R

ate

(sp/

s)

10−1 100 101 1020

1

2

3

4

5

6

Input Frequency, (Hz)

Res

pons

e A

mpl

itude

, |A

(ω)|

(sp/

s)

A B

C

D

−2

0

2

Cur

rent

(nA

)

Trial 1

Trial 2

Input+IKv7

ω

2|A(ω=1)|

}-Kv7+Kv7

Fig. 2. Kv7 conductances shape responses to dynamic inputs. A, top: themembrane currents of a �Kv7 model in response to a sinusoidal currentinjection. We added a static bias of gLD�c so that the minimum of the sinewave was at 0 for visualization purposes in A and B. A, bottom: the membranepotential responses to the sine input on 2 different trials. B: Kv7 currentprovides negative feedback of the input, canceling its mean and reducing itsamplitude. IKv7 and Input � IKv7 currents are from a single trial, and low-passfiltered for ease of visualization. C: firing rate response to the sine input isobtained by averaging the spike train responses across trials. The sine inputinduces a sinusoidal modulation of the firing rate in response. Kv7 conduc-tances reduce the equilibrium firing rate (dashed line) and the amplitude of theresponse modulation. The peak-to-trough response modulation is given bytwice the transfer function A(f). Solid lines: theory. Shaded lines: simulations.D: response functions showing the amplitude of the modulation induced in thefiring rate by a sine input of fixed strength. Kv7 conductances reduce theresponse amplitude for all input frequencies and shape the response function tobe band-pass rather than low-pass. Solid lines: theory. Shaded lines:simulations.



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Cov�n1T→�, n2

T→�� A(0)�2, (23)

Cov�n1T→0, n2

T→0� � ��

� �A(f)�2df . (24)

Thus for long T only the low frequency transfer A(0) isimportant, while for short T the entire frequency transfer A(f)impacts Cov(n1

T, n2T).

The spike count covariance predicted with linear responsetheory in Eq. 22 was in excellent agreement with spike countcovariance computed from simulations for both the �Kv7 and�Kv7 neuron pairs (Fig. 3, B and C, solid lines). Further, therelative reduction in covariance was larger for long T than forshort T (Fig. 3C, bottom). This is expected from Eqs. 23 and 24and the fact that the slow negative feedback Kv7 conductanceshaped A(f) more prominently at low f (Fig. 2D). In total, ourtheory shows that the shaping of single-neuron filtering by Kv7conductances (Fig. 2) can account for the large decrease in thecovariability of pairwise activity (Fig. 3).

A widely used, normalized measure of the similarity be-tween two spike trains is the spike count correlation coeffi-

cient, Cov(n1T, n2

T)/�Var�n1T�Var�n2

T� (Averbeck et al. 2006;Cohen and Kohn 2011). The weak input correlation c provideda theoretical estimate for Cov(n1

T, n2T) in Eq. 22; however, a

similar theory for Var(nT) remains elusive (Farkhooi et al.2011; Muller et al. 2007; Naud and Gerstner 2012; Toyoizumiet al. 2008). The difficulty stems from the history dependenceof the slow adaptation current that makes the spike trains intononrenewal processes, precluding a simple relationship be-tween spike train and spike count variance (Cox and Isham1980).

A number of studies have investigated the role of M currentsand similar mechanisms in reducing the variability of single-neuron spike trains (Benda et al. 2010; Muller et al. 2007;Schwalger et al. 2010; Fisch et al. 2012). This raised the

question of how the effect of Kv7 conductances on pairwisecovariability related to their effect on single-neuron variability.The reduction in Cov(n1

T, n2T) due to Kv7 conductances could

be accompanied by an equivalent reduction in Var(nT), so thatthe spike count correlation would remain unaffected. Analysisof the spiking simulations showed that this was not the case(Fig. 3D) and rather the reduction in the covariance was thedominant effect over changes in single-neuron variability.Thus, while our remaining analysis will be restricted toCov(n1

T, n2T), allowing the use of the linear response theory, the

influence of Kv7 on spike count correlation is expected to bequalitatively similar. We next used our theory to build amechanistic understanding of how Kv7 conductances controlthe transfer of covariability.

Kv7 Conductance Decouples Firing Rates and PairwiseSpiking Covariability

Including Kv7 conductances in the model dynamics drasti-cally reduced both firing rates and pairwise spiking correla-tions. Previous work has linked changes in firing rates tochanges in spike count correlations at long time intervals T (dela Rocha et al. 2007). Thus it is important to determine if thereduction of covariance with Kv7 (Fig. 3) can be explainedsolely by a reduction in firing rate (Fig. 1D).

To this end, we ranged over the input mean (�) and variance(2) to effectively explore a range of firing rates and covari-ance values accessed by �Kv7 and �Kv7 pairs. For higherfiring rates, there was a broad range of (�, 2) pairs thatproduced the same output firing rate but a distribution of outputcovariance values (Fig. 4A, shaded regions). Nevertheless, forany fixed firing rate the presence of Kv7 conductances shiftedthe region of spike count covariance to lower values. This wasespecially true for the low firing rates that characterize spon-taneous activity in many cortical areas (Hromadka et al. 2008),

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Fig. 3. Kv7 conductances reduce pairwise covariability.A: pairs of neurons (shaded and solid) with (blue) and without(black) Kv7 conductances were stimulated by partially corre-lated Gaussian white noise. Fluctuations of the common inputgive rise to synchronous spikes (top, arrows) and to covariablemodulations of the spike counts of the two neurons (bottom).B: cross-covariance functions of pairs with (blue) and without(black) Kv7 conductances. C, top: covariance of spike countsas a function of the counting window length T. C, bottom: ratioof the spike count covariances, with Kv7/no Kv7. The Kv7conductance reduces spike count covariances by 70–80%. D,top: spike count correlation for pairs with and without Kv7conductances. D, bottom: ratio of the correlations. Kv7 con-ductance reduces the correlation by 40–50%.



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where there was little overlap between the rate-covariancespaces of pairs with Kv7 conductance and pairs without (Fig.4A). Thus the reduction of spiking covariability by Kv7 con-ductances cannot be explained by a simple lowering of firingrate.

To gain intuition into the mechanisms behind the reductionin spike train covariability with Kv7, we focused on a specificexample within the (�, 2) plane. Specifically, we consideredthe (�, 2) studied earlier (Figs. 1–3; indicated with black andblue markers in Fig. 4, A and B). To correct for firing rates westudied a second �Kv7 neuron pair with a higher input mean� so that they matched the firing rate (22 sp/s) of the �Kv7pair (indicated with orange markers in Fig. 4, A and B). Thisrate correction produced an approximate multiplicative scalingof the transfer function A(f), so that the rate-corrected �Kv7pair responded to high-frequency inputs with the same re-sponse gain as the �Kv7 pair (Fig. 4C, black and orangecurves for f 10 Hz). However, the Kv7 conductance stillproduced an overall band-pass A(f) so that the low-frequencytransfer was below that of the �Kv7 transfer for low frequen-cies (Fig. 4C, black and orange curves for f � 10 Hz). Thus,while the Kv7-induced change in single-neuron filtering ofhigh-frequency inputs A(f) could be accounted for by thechange in firing rates, the change in low-frequency filteringcould not.

Our theory shows that the low-frequency transfer A(f) affectsspiking covariability at both short and long windows T (Eqs. 23and 24). Thus the shift from low-pass to band-pass transferwith Kv7 conductances accounted for much of the reduction inspiking covariability, especially at long timescales (Fig. 4D).Since matching firing rates between the �Kv7 and �Kv7neuron pairs did not correct for the low-frequency attenuationof the transfer function by Kv7 conductances, then we con-clude that the Kv7-induced reduction in spiking covariabilitywas not solely due to a lowering of firing rates.

Sub- and Suprathreshold Kv7 Activation can Contribute toCovariance Cancellation

Kv7 conductances impact both single neuron (Figs. 1 and 2)and pairwise spiking statistics (Fig. 3) through attenuation of

the transfer of low frequency inputs to spike train outputs. Thiswas because Kv7 currents actively cancelled input fluctuations(Fig. 5A, compare black and blue curves), even when wecompensated for the mean Kv7 current (Fig. 5A, compare blackand orange curves). To understand how this input cancellationshaped A(f) we next investigated the impact of Kv7 conduc-tances on the membrane potential response to fluctuating in-puts.

We considered the linear approximation V(t) � �V� � �1V1(f)sin(2 ft � �), where �1 again defines the amplitude of theinput modulation. V1(f) is the transfer function for the mem-brane potential response, defined as V1(f) � ��

��VP1(V, f)dV,with P1(V, f) being the first order perturbation of the densityP1(V, f) induced by I(t) (see METHODS). In contrast to the effectof Kv7 conductance on firing rate responses, Kv7 conduc-tances increased the amplitude of membrane potential re-sponses to intermediate frequency inputs (Fig. 5B, blue vs.black curves). In the case of high firing rates (�Kv7), themembrane potential did not respond strongly to inputs becauseit was comparatively dominated by spiking and repolarization.In the �Kv7 case, the membrane potential was better able torespond to inputs without being influenced by spiking dynam-ics because of the lower firing rate (Rosenbaum and Josic2011). This effect disappeared at low frequencies, however, asthe Kv7-mediated current canceled off the input-driven mem-brane potential fluctuations (Fig. 5B, black vs. orange curves).When the mean effective input was the same (in the rate-corrected model of Fig. 4, B–D), the Kv7 conductance clearlyreduced the membrane potential response to slow inputs, pre-venting them from driving spiking activity (Fig. 5, A and B).

Previous theoretical investigations of spike-frequency adap-tation have often focused on spike-driven recruitment of neg-ative feedback (Ermentrout 1998; Benda and Herz 2003; Fark-hooi et al. 2011; Schwalger et al. 2010; Naud and Gerstner2012). To isolate the relative contributions of spike-driven andsubthreshold activation of Kv7 conductances to stimulus trans-fer, we defined an activation threshold of �47 mV for the Kv7conductance. We chose it 3 mV above the initiation thresholdVT � �50 mV to ensure that only successfully generatedspikes would contribute to spike-driven activation. We then

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Fig. 4. Kv7 conductance shapes the relationship be-tween firing rate and covariance. A: range of firing ratesand covariances obtained by sweeping over the inputmean � and variance 2 for pairs with Kv7 (blue shadedregion) and without (grey shaded region). Inset: rangeof � and 2. B: increasing the input mean can correct forthe difference in firing rates between pairs with andwithout Kv7. C: firing rate linear response functions forthe �Kv7 (black), �Kv7 (blue), and �Kv7 with high �(orange) models. D: spike count covariance as a functionof window size. The Kv7 conductances reduce spikingcovariability even when firing rates are corrected.



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examined two models of Kv7: one with only the portion of theKv7 activation curve x�(V) below �47 mV contributing to x(t)dynamics (subthreshold activation, Fig. 5C) and one with onlythe portion of the activation curve above �47 mV contributingto x(t) (suprathreshold activation, Fig. 5D).

Kv7 conductances are strongly activated by depolarizedmembrane potentials. However, in our model the subthresholdactivation accounted for most of the effect of Kv7 on cancelingthe input mean �, as reflected by the firing rates and membranepotential transfer functions (Fig. 5, E and G). This was due tothe fact that the timescale of Kv7 activation was long, so thatit interacted strongly with the equilibrium distribution of themembrane potential, which was mainly subthreshold (Fig. 1B).Subthreshold activation of Kv7 conductance accounted formost of its reduction of spiking covariability on both short andlong timescales (Fig. 5H, black vs. solid green vs. blue curves).Here, subthreshold activation of x(t) led to a cancelation ofinput fluctuations on long timescales (Figs. 2B and 5A), whichnecessarily reduced the transfer of fluctuations to spikingresponses. In contrast, suprathreshold Kv7 activation had asmaller impact on correlation transfer (Fig. 5H, black vs.dashed green vs. blue curves). We remark that while the sub-and suprathreshold activation curves sum to the total x�, theeffects of sub- and suprathreshold activation on the membranevoltage and spiking responses do not sum to give the responsewith the full Kv7 activation, since the voltage and Kv7 acti-

vation interact nonlinearly (Eqs. 1–3). In total, the subthresholdactivation of Kv7 cancelled slow input fluctuations and re-duced pairwise spike train covariance beyond what was ex-pected by a simple lowering of firing rates (de la Rocha et al.2007).

Kv7 Conductances, Homeostatic Plasticity, and PathologicalActivity

A number of neurological diseases characterized by hyper-activity and hypersynchrony have been related to Kv7 channeldysfunction (Jentsch 2000; Cooper and Jan 2003). We modeledKv7-related hyperactivity disorders with a reduced Kv7 con-ductance strength, where we decreased the mean Kv7 conduc-tance gxx0 from 0.025 to 0.01 mS/cm2. A decrease in gx isconsistent with a decrease in Kv7 surface expression, as seen inneonatal and rolandic epilepsies (Coppola et al. 2003; Neu-bauer et al. 2008; Soldovieri et al. 2006). This modulationincreased the spontaneous firing rate from 8.1 to 15.7 sp/s andalso led to a 2.3–3.2 times increase in the covariance of spikecounts (depending on the window size). Modeling Kv7-relatedhyperactivity instead with a shift in the V1/2 had a similar effecton rates and spike count covariances (Fig. 7).

We modeled a homeostatic modulation in response to thereduced Kv7 conductance with a shift of model parameters.The parameters VL and gL control the passive membrane

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Fig. 5. Sub- and suprathreshold activation of Kv7 contributes to covariance reduction. A: I(t) � IKv7 for the 3 cases of Fig. 4B–D. Currents were low-pass filteredfor visualization. B: filter response of the membrane potential as a function of input frequency f. Kv7 conductance lowers the firing rate, allowing the membraneto better track inputs without being interrupted by spiking (black vs. blue). Kv7 also cancels low-frequency membrane potential responses (blue and compareblack vs. orange). Solid lines: theory. Shaded lines: simulation. C and D: separate sub- and suprathreshold activation functions of the Kv7 conductance. Mostof the activation curve lies in the suprathreshold regime. E: subthreshold activation accounts for most of the steady-state Kv7 activation, since the membranepotential spends the most time in the subthreshold regime. F: suprathreshold activation accounts for the spike-activated portion of the Kv7 conductance.G: membrane potential response functions. The subthreshold activation of Kv7 cancels off the input mean and slow inputs, and the suprathreshold activationshapes the membrane potential responses to slow and intermediate inputs. H, top: spike count covariance as a function of window size T for the models withonly sub- or suprathreshold Kv7 activation and the �Kv7 and �Kv7 models. H, bottom: ratio of spike count covariance in the various conditions compared withthe control case.



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properties, and their modulation can mimic homeostaticchanges in slow currents (Brickley et al. 2001; Hong andLnenicka 1995; Desai et al. 1999; van Welie et al. 2004; Fanet al. 2005) and single-neuron integration (Litwin-Kumar et al.2011; Burkitt et al. 2003). The scaling of synaptic strength isalso a common target for homeostatic regulation (Turrigiano etal. 1998). We modeled a change in synaptic strength as a globalscaling of all inputs to the neuron. This corresponded tomultiplying the input by a factor �: � ¡ �� and ¡ �. Werequired that the homeostatic regulation of each model param-eter (gL, VL, and �) restore the firing rates from the pathologicmodels to that of the control model (Fig. 6, A–C).

Despite firing rate matching, all three homeostatic regula-tions failed to restore the low-frequency transfer function A(f)(Fig. 6D). Since the low-frequency response affects spikecount covariances at both short and long timescales, none ofthe three homeostatic corrections restored Cov(n1

T, n2T) to con-

trol levels (Fig. 6E). This failure was robust over a wide rangeof homeostatic modulation. For instance, if the modulation wasdefined so that Cov(n1

T, n2T) for large T-matched control values,

then firing rates for the pathologic model would be much lowerthan those in the control model (Fig. 6F). In sum, homeostatic

modulation of passive membrane properties or synaptic scalingcould not correct for pathological activity due to changes in theKv7 conductance.

Neurological disorders related to Kv7 pathology are oftentreated pharmacologically. The antiepileptic and analgesicdrugs flupirtine and retigabine, for instance, act by hyperpo-larizing the voltage-activation curves of Kv7.2 and Kv7.3channels (Main et al. 2000; Tatulian et al. 2001). We nextinvestigated whether homeostatic or exogenous modulation ofthe Kv7 half-activation voltage V1/2 could correct for hyperac-tivity and hypersynchrony induced by pathological changes inthe maximum Kv7 conductance gx, and vice versa. We startedwith the same reduction in gx as explored above. DecreasingV1/2 by 12 mV restored the mean Kv7 conductance to thecontrol value 0.025 mS/cm2 and restored the control firing rate(Fig. 7A). Furthermore, shifting Kv7 activation allowed thetreated Kv7 conductance to cancel slow inputs in the same wayas in the control case, restoring the single-neuron responsefunction and pairwise covariance to control levels (Fig. 7, Band C). We also examined a disease model characterized by a10 mV increase of V1/2 (Fig. 7D), which decreased the meanKv7 conductance gxx0 to 0.013 mS/cm2. This shift of V1/2 ofKv7 channels is associated with hyperexcitability in a mousemodel of tinnitus (Li et al. 2013). An increase in the maximalKv7 conductance gx restored the mean activation as well as thecontrol levels of activity and spiking covariability (Fig. 7, Eand F). Both parameters of the Kv7 activation modulated rateand covariance similarly, allowing modulations of each param-eter to compensate for pathological changes in the other (Fig.7G). These results suggest that, while intrinsic homeostaticprocesses triggered by changes in firing rate may not be able tocorrect for pathological changes in Kv7 conductances, drugsdirectly targeting any parameter of the Kv7 activation canrestore spiking activity to normal levels.

DISCUSSION

Kv7 channels mediate voltage-activated K� conductancesthat control overall cellular excitability (Aiken et al. 1995; Guet al. 2005; Higgs et al. 2007; Lawrence et al. 2006; Peters etal. 2005). These channels are present throughout the nervoussystem (Brown and Passmore 2009; Delmas and Brown 2005;Jentsch 2000), and their dysfunction is involved in severaldisease states (Cooper and Jan 2003; Li et al. 2013; Passmoreet al. 2003; Shah and Aizenman 2014; Zheng et al. 2012).Using a well-established theoretical framework to study cor-relation transfer in spiking neurons (de la Rocha et al. 2007),we showed that the recruitment of Kv7 conductances providesan input-driven negative feedback that cancels correlatinginputs and reduces the covariability of pairwise spiking activityin model neurons. Pathological loss of Kv7 conductance due tochanges either in its strength or activation leads to bothincreased firing rates and synchrony. Using our theory weshowed that homeostatic regulation of passive membrane prop-erties or input strength, triggered by changes in firing rates, didnot maintain normal pairwise spiking activity. However, mod-ulation of the activation of the Kv7 conductance could correctfor both hyperactivity and hypersynchrony induced by patho-logical reduction of Kv7 conductances.

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Fig. 6. Homeostatic compensation for Kv7 dysfunction corrects for patholog-ical firing rates but not covariances. A: correction for firing rates through theleak reversal potential. B: correction for firing rates through the leak conduc-tance. C: correction for firing rates through synaptic scaling of the input meanand variance. We set VL � Vr and � � 1.7 �A/cm2 to give the same firingrates, response functions, and covariances as VL � �55 mV and � � 0�A/cm2. D and E: firing rate response functions and spike count covariancesfor all three homeostatic mechanisms. F: paths through rate-covariance spaceobtained by varying gx (black), VL (magenta), gL (orange), and � (green),starting from the disease state (intersection at black square). The pointscorresponding to the cases of A–E are marked by color-matched squares.



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Adaptation and Population Coding

Correlations in trial-variable neural activity (“noise correla-tions”) play a large role in shaping population-wide activity(Averbeck et al. 2006). In particular, they can facilitate orimpair population coding, depending on the spatial structure ofthe noise correlations and the relationship between them and

the encoded signal (Abbott and Dayan 1999; Josic et al. 2009;Sompolinsky et al. 2001). Adaptation to repeated stimulationreduces both single-neuron variability and pairwise noise cor-relations in sensory cortex (Adibi et al. 2013; Gutnisky andDragoi 2008). In specific cases this has promoted improvedpopulation-based stimulus estimation (Gutnisky and Dragoi2008) and discrimination (Wang et al. 2011). However, thespecific biophysical mechanisms by which adaptation proto-cols affect population responses are, in general, unknown.

The main two mechanisms of sensory adaptation, as under-stood from single-unit activity, are thought to be synapticdepression and the recruitment of intrinsic slow outward cur-rents (Kohn 2007). The reduction of firing rates by both ofthese mechanisms is well understood. More recently, synapticdepression has been predicted to reduce pairwise correlationsby causing failures in shared, correlating inputs (Rosenbaum etal. 2013). Previous theoretical investigations of the intrinsicmechanisms of adaptation and population coding have focusedon how spike-driven processes shape the variability of uncor-related neurons (Naud and Gerstner 2012; Farkhooi et al.2011). We have now shown that activity-dependent outwardcurrents often associated with adaptation can also reduce noisecorrelations by canceling correlating inputs at the membranepotential. The combination of these results suggests that Kv7channels improve population coding. This prediction is con-sistent with recent studies that document, after cochlear dam-age, an overall reduction of Kv7 conductance in auditorypathways and associated behavioral deficits in acoustic detec-tion tasks (Li et al. 2013).

Intrinsic Membrane Properties and Pairwise Activity

There have been a recent suite of studies investigating themechanics of correlation transfer by spiking neurons. Many ofthese studies have focused on the role of spike initiationdynamics in shaping the timescale over which correlations aretransferred (Galan et al. 2006; de la Rocha et al. 2007;Shea-Brown et al. 2008; Tchumatchenko et al. 2010; Abouzeidand Ermentrout 2011; Barreiro et al. 2012; Hong et al. 2012;Ratte et al. 2013). In contrast, we have shown how slowintrinsic currents that are not involved in spike initiation canalso determine the transfer of correlation by neuron pairs.

Our linear response theory explicitly linked Kv7-inducedmodulations of correlation transfer to modulations of single-neuron transfer functions, particularly at low frequencies. Forvery low frequency inputs this corresponds to a divisive mod-ulation of the firing rate-current curve (Fig. 1D). Previous invitro experimental work in somatosensory and motor cortexusing the drug XE991, a Kv7 antagonist, has shown a subtrac-tive, rather than divisive, effect of Kv7 conductances on thefiring rate curves in low-noise conditions (Guan et al. 2011). Inlow-noise conditions ( � 0), we also saw that Kv7 conduc-tance induced a subtractive shift in the firing rate curves (datanot shown). It is known that strong fluctuations can “linearize”firing rate curves so that subthreshold static inputs can elicitresponses (Burkitt et al. 2003). Subtractive modulations offiring rate-current curves in low noise conditions can becomedivisive modulations with higher noise (Doiron et al. 2001). Infact, when Guan et al. (2011) drove their neurons with largefluctuations to measure A(f) they saw that XE991 caused anoverall increase in A(f), even when firing rates were matched

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Fig. 7. Treatments targeting Kv7 channels can correct for pathological firingrates and covariances. A, top: model of pathologically weak Kv7 conductance.A, bottom: treatment that hyperpolarizes the half-activation voltage of thepathological Kv7 conductance can restore the firing rate. Blue: control. Black:disease. Green: hyperpolarized V1/2. B and C: compensating for the weakenedconductance with a hyperpolarization of the activation curve can restore theresponses and spiking covariability to control levels. D, top: model of apathological shift in the activation of Kv7 conductance. D, bottom: treatmentthat increases the strength of the conductance (orange) can restore the firingrate. E and F: compensating for a pathological depolarization of the activationcurve by strengthening the conductance can restore the responses and spikingcovariability to control levels. G: paths through rate-covariance space obtainedby varying gx and V1/2 from the control state (black) and from the disease states(green and orange curves). All paths lie in a tight region of rate-covariancespace.



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(compare their Fig. 7C to our Fig. 4C). Thus there is strongexperimental evidence that Kv7 conductances shape single-neuron transfer functions, and given our theory we expect thatKv7 conductances also determine the correlation transfer bypairs of neurons. This prediction can be verified using well-established experimental protocols (de la Rocha et al. 2007;Litwin-Kumar et al. 2011).

We modeled the Kv7 conductance as a deterministic con-ductance, separate from the membrane noise. This correspondsto the limit of an infinite population of Kv7 channels. Thestochastic opening of a finite population of M channels givesrise to distinct effects on single neuron spiking statistics(Schwalger et al. 2010; Fisch et al. 2012). The effect ofstochastic M-channel dynamics on pairwise activity remains anopen question. We expect, however, that it could limit theability of M channels to cancel input correlations by introduc-ing noise into the relationship between the input and Kv7current.

Diseases and Homeostatic Plasticity

Homeostatic processes regulate the strength of both intrinsicand synaptic currents in neocortical neurons (Desai et al. 1999;Turrigiano et al. 1998). They are mediated by calcium sensorsand other signaling pathways and maintain stable firing rates(Ibata et al. 2008; MacLean et al. 2003; Brickley et al. 2001).Failures of homeostatic mechanisms lead to pathological ac-tivity. In models of Kv7 dysfunction, we showed that changesin passive membrane properties or synaptic scaling that correctfor an increase in firing rates could not correct for hypersyn-chrony (Fig. 6F). We did not investigate homeostatic plasticityof the spike initiation threshold or the action potential slope,both of which would be controlled by fast sodium channels. Aprevious study of homeostatic plasticity of the intrinsic prop-erties of the neurons has shown that the amplitude of fastsodium currents increases in response to activity deprivation,increasing their excitability (Desai et al. 1999). In our model,this would have a similar effect to changing the leak reversalpotential: bringing the resting potential and spike thresholdcloser. It would be a less effective modulation with respect toKv7 activation since decreasing spike threshold would de-crease the subthreshold activation of the Kv7 conductance,which plays a strong role in covariance cancellation (Fig. 5H).Changes in the conductance density of fast sodium channelscould also change the steepness of action potential initiation(the parameter �), but since the effects of this are limited to thehigh-frequency region of the response function A(f) (Fourcaud-Trocme et al. 2003), it would not have a large effect on spikingcovariability (Eq. 22). Therefore, changes in the strength offast sodium conductances would be unlikely to correct forhypersynchrony induced by Kv7 dysfunction.

The interaction between different mechanisms of correlationgeneration and cancellation remains an open area of research.Homeostatic or pharmacological down-regulation of inwardcurrents with similar kinetics to Kv7, such as persistent sodiumcurrent, could perhaps compensate for pathological loss of Kv7conductance. Indeed, persistent sodium current is implicated insome forms of epilepsy and is a target of some antiepilepticdrugs (Stafstrom 2007). Likewise, homeostatic upregulation ofother subthreshold-activated potassium conductances (Pingand Tsunoda 2012; Ransdell et al. 2012) could perhaps par-

tially compensate for loss of Kv7 conductance. Changes in thecovariation of the activity of the two neurons could also affectsynaptic currents through spike timing-dependent plasticity orcould trigger changes in intrinsic excitability (Cudmore et al.2010). We have shown how one intrinsic property, Kv7 con-ductance, can contribute to decorrelating neural activity. Fur-thermore, dysfunction of Kv7 conductance gives rise to hyper-synchronous, high-firing rate activity that cannot be correctedfor by simple homeostatic mechanisms. How dysfunction ofKv7 currents interacts with other activity-dependent processesremains an open question with important implications for theunderstanding of healthy neural activity and the correction ofpathological hyperactivity.

ACKNOWLEDGMENTS

We thank Thanos Tzounopoulos and Anne-Marie Oswald for useful dis-cussions.

GRANTS

Funding was provided by National Science Foundation Grants NSF-DMS-1313225 (to B. Doiron) and NSF-IGERT-0549352 (to G. K. Ocker).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: G.K.O. and B.D. conception and design of research;G.K.O. performed experiments; G.K.O. analyzed data; G.K.O. and B.D.interpreted results of experiments; G.K.O. and B.D. prepared figures; G.K.O.and B.D. drafted manuscript; G.K.O. and B.D. edited and revised manuscript;G.K.O. and B.D. approved final version of manuscript.

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