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Activity Patterns in Purely Excitatory Networks

Jonathan Rubin

Department of Mathematics, University of Pittsburgh

rubin@math.pitt.edu

SIAM Life Sciences Meeting - July 11, 2004

MAIN POINTS:

• spatially localized, temporally sustained activity (bumps) canoccur in purely excitatory networks

• the go curve provides a useful construct for understanding theunderlying mechanism and properties

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Bumps: what are they and why do we care?

cell population/stimulus feature

firing rate/firing probability/averaged voltage

• spatially localized, temporally sustained activity

•which cells are active depends on some external feature

• activity persists without persistent stimulus

• seen in visual system, head direction system, and pre-frontal cortex (working memory):

stimulus shown, subject must later recall position;localized group of cells fire until recall

stimulus memory recall

3

Recipes for bumps

1. E

I

e.g. Wilson/Cowan/Amari: ut(x, t) = h− σu(x, t) +∫ ∞−∞w(x− y)f(u(y, t))dy

−4 −2 0 2 4

−0.2

0

0.2

0.4

0.6

x

w

w(x) > 0 on (−x, x)

w(−x) = w(x) = 0

w(x) < 0 on (−∞,−x)∪ (x,∞)

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Bumps in neuroscience models with Mexican hat

• Guo and Chow, 2003

• Coombes, Lord, and Owen, 2003

• Renart, Song, and Wang, 2003

• Laing and Troy, 2002 & 2003

• Laing, Troy, Gutkin, and Ermentrout, 2002

• Gutkin, Laing, Colby, Chow, and Ermentrout, 2001

• Laing and Chow, 2001

• Pinto and Ermentrout, 2001

•Werner and Richter, 2001

• Compte, Brunel, Goldman-Rakic, and Wang, 2000

• Taylor, 1999

• Camperi and Wang, 1998

• Hansel and Sompolinsky, 1998

• Amit and Brunel, 1997

• Kishimoto and Amari, 1979

• Amari, 1977

•Wilson and Cowan, 1972 & 1973

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2. Rubin, Terman, and Chow, JCNS, 2001: no E-E; PIR

E

I

TC cells

time

7500

9000

10 40

-80 10

mV

RE cells

time

6000

9000

10 40

-80 10

mV

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3. Rubin and Troy, SIAP, to appear:

E

I

⇓

w(x)

x

off-center coupling

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4. One-layer network with purely excitatory coupling[Drover/Ermentrout, SIAP, 2003; Rubin/Bose, 2004]:

Equations

v′i = f(vi,wi)− gsyn[vi−Esyn]cosi + Σ

j=3j=1cj[si−j + si+j]

w′i = [w∞(vi)−wi]/τw(vi)

s′i = α[1− si]H(vi− vθ)− βsi (on a ring)

−0.5 0 0.50

0.1

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v

w

. o

SNIC

(movie)

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Geometry - θ model:

θ′i = 1− cosθi + (1 + cosθi)(b+ gisyn)

gisyn = gsync0si + Σ

j=3j=1cj(si−j + si+j)

s′j = α[1− sj]e−γ(1+cos θj)− βsj, j = i− 3, . . . , i+ 3

with α,γ large

synaptic decaydynamics:

θ′ = f(θ, gisyn),

g′isyn = −βgisyn

−1.5 −1 −0.5 0 0.5 1 1.50

0.1

0.2

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θ

gisyn

−b

(θS,0) (θ

U,0)

P

go curve

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Geometry - θ model - (2):

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θ

gisyn

−b

(θS,0) (θ

U,0)

P

go curve

synaptic decaydynamics:

θ′ = f(θ, gisyn),

g′isyn = −βgisyn

main ingredients:

• threshold of synaptic decay dynamics determinesresult of stimulation

• delay to firing depends on proximity to threshold

• variable delays desynchronize and stop spread

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Geometry - Morris-Lecar with w′ = 0:

v′ = f(v,w)− gisyn(v−Esyn)

w′ = 0

s′i = α[1− si]H(vi− vθ)− βsi

decay dynamics: replace si-equations with g′isyn = −βgisyn

l 2

(v , w )l

w=w

w

v

g=gg=g

1g=0

g

v

g=g

g=g2

1P

(v , 0) (v , 0)l mc

g=0

l

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V

gisyn

go curve

P

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Geometry - Morris-Lecar full system:

v′ = f(v,w)− gisyn(v−Esyn)

w′ = g(v,w)

s′i = α[1− si]H(vi− vθ)− βsi

decay again: replace si-equations with g′isyn = −βgisyn

w

v

w=w

m

go curvefor w=w

(v , w , 0)

RK

RK

m

gisyn

v’=0,isyn

v’=0, g =0

g >0

isyn w’=0, g =0isyn

W s

note: can still project into (v, gisyn) phase plane!

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Analysis: who jumps?

w

v

w=w

m

go curvefor w=w

(v , w , 0)

RK

RK

m

gisyn

v’=0,isyn

v’=0, g =0

g >0

isyn w’=0, g =0isyn

W s

• suppose that cell i gets a synaptic input from cell j

• check the projected (v, gisyn) phase plane for cell i atthe moment cell j falls down through vthresh

• position of cell i relative to go curve determinesrecruitment

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Implications - identify recruitment

−0.32 −0.3 −0.28 −0.26 −0.24 −0.22 −0.20.005

0.01

0.015

0.02

0.025

0.03

0.035

V

g isyn

−0.24 −0.22

0.026

0.028

0.03

0.032

V

g isyn

first

second

−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 00.005

0.01

0.015

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0.025

0.03

0.035

V

g isyn

shut off

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Implications - recruitment depends on more than |gisyn|

−0.32 −0.3 −0.28 −0.26 −0.24 −0.22 −0.20.005

0.01

0.015

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0.025

0.03

0.035

V

g isyn

0 50 100 150 2000

0.005

0.01

0.015

0.02

0.025

0.03

time

gisyn

• also, faster synaptic decay hurts in two ways...

• ...and bump size is non-unique and nonrobust

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Bump formation:

• recruitment as above; initial synchrony helps

−− start near same rest state

−− common input overrules synaptic coupling

• localization via desynchronization after shock (delayedescape from go surface)

• sustainment via self-coupling (not sufficient!) andasynchrony

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Synaptic depression:

Bose et al., SIAP, 2001:

s′ = α[d− s]H(v− vthresh)− βsd′ = ([1− d]/τγ)H(vthresh− v)− (d/τη)H(v− vthresh)

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0

0.5

v

30 40 50 60 70 80 90 1000

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1

time

sd

Results: filtering and toggling

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Filtering: strong input moderate input

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Toggling:

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Discussion

• bumps can exist in purely excitatory networks -inhibition is not necessary

– initiated by transient input

– go curve forms recruitment threshold

– localized by desychronization (Type I dynamics)

– sustained by self-coupling (or e.g. ICaL) anddesynchronization

• bump details are sensitive to parameters

• synaptic depression ⇒ band-pass filter, toggling

•OPEN: rigorous analysis, role of short-term spikefrequency changes, bump mechanism for Type II (D/E)

• IDEA: go curve/surface is useful for predicting theimpact of transient inputs

• IDEA: intrinsic dynamics can be important, andexcitatory networks can do interesting things

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• IDEA: go curve/surface is useful for predicting theimpact of transient inputs

• IDEA: intrinsic dynamics can be important, andexcitatory networks can do interesting things

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