Paul C. Bressloff and S. Coombes- Dynamics of Strongly Coupled Spiking Neurons

8/3/2019 Paul C. Bressloff and S. Coombes- Dynamics of Strongly Coupled Spiking Neurons

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ARTICLE Communicated by Carl van Vreeswijk

Dynamics of Strongly Coupled Spiking Neurons

Paul C. BressloffS. CoombesNonlinear and Complex Systems Group, Department of Mathematical Sciences,

Loughborough University, Loughborough, Leicestershire LE11 3TU, U.K.

We present a dynamical theory of integrate-and-fire neurons with strongsynaptic coupling. We showhow phase-locked statesthat arestable in the

weak coupling regime can destabilize as the coupling is increased, lead-ing to states characterized by spatiotemporal variations in the interspikeintervals (ISIs). The dynamics is compared with that of a correspondingnetwork of analog neurons in which the outputs of the neuronsare takento be mean firing rates. A fundamental result is that for slow interac-tions, there is good agreement between the two models (on an appropri-ately defined timescale). Various examples of desynchronization in thestrong coupling regime are presented. First, a globally coupled networkof identical neurons with strong inhibitory coupling is shown to exhibitoscillator death in which some of the neurons suppress the activity ofothers. However, the stability of the synchronous state persists for verylarge networks and fast synapses. Second, an asymmetric network with amixture of excitationand inhibition is shown to exhibit periodic burstingpatterns. Finally, a one-dimensional network of neurons with long-rangeinteractions is shown to desynchronize to a state with a spatially periodicpattern of mean firing rates across the network. This is modulated by de-terministic fluctuations of the instantaneous firing rate whose size is anincreasing function of the speed of synaptic response.

1 Introduction

There are two basicclasses of neurod ynam ical mod el that are distinguished

by their representation of neuronal output activity (see Abbott, 1994, and

references therein). The first considers the output of a neuron to be a mean

firing rate that either specifies the number of spikes emitted in some fixed

time window (Hopfield, 1984; Amit & Tsodyks, 1991; Ermentrout, 1994) or

corresponds to the probability of firing within some population (Wilson &

Cowan, 1972; Gerstner, 1995). Recently, however, a number of experiments

on sensory neurons have shown that the precise timing of spikes may besignificant in neu ronal information processing, and this has led to renew ed

interest in the second class of neurodynamical models based on spiking

neurons. For example, spike train recordings from H1, a motion-sensitive

Neural Computation 12, 91129 (2000) c 2000 Massachusetts Institute of Technology


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92 Paul C. Bressloff and S. Coombes

neuron in the fly visual system, exhibit variability of response to constant

stimuli but a high degree of reproducibility for more natural dynamic stim-

uli (Strong, Koberle, va n Steveninck, & Bialek, 1998). This rep rodu cibility

provid es an enh anced cap acity for carrying informat ion (Rieke, Warland , &

van Steveninck, 1996). Similar findings have been obtained in retinal gan-

glion cells of the tiger salamander and rabbit (Berry, Warland, & Meister,

1997). It is also well kn own that precise spike timing is essential for soun d

localization in the auditory system of animals such as the barn owl (Carr

& Konishi, 1990). Recent ad vances in comp utationa l neu roscience sup port

the notion that synapses are capable of supporting computations based

on h ighly stru ctured temp oral codes (Mainen & Sejnowski, 1995; Gerstner,

Kreiter, Markram, & H erz, 1997). Moreover, this m ode of compu tation sug-

gests how only one typ e of neu roarchitecture can sup port th e processing of

several very different sensory modalities (Hopfield, 1995). Such codes areund oubtedly imp ortant in sensory and cognitive processing. The simplest

and most popular example of a spiking neuron is the so-called integrate-

and-fire (IF) model (Keener, Hoppensteadt, & Rinzel, 1981; Tuckwell, 1988)

and its generalizations (Gerstner, 1995). The state of an IF neuron changes

discontinuously (resets) whenever it crosses some threshold and fires, so a

complete description in term s of smooth differential equations is no longer

possible. This type of model can be derived systematically from more de-

tailed Hod gkin-Huxley equations d escribing the process of action potential

generation (Abbott & Kepler, 1990; Kistler, Gerstner, & van Hemmen , 1997).

There are major differences in the an alytical treatments of firing-rate and

spiking neural network models. A common starting point for the former

is to consider conditions under which destabilization of a homogeneous

low-activity state occurs, leading to the formation of a state with inhomo-

geneous and /or time-depend ent firing rates (see, for example, Wilson &Cowan , 1973; Ermen trout & Cowan , 1979; Atiya & Baldi, 1989; Li & H op-

field, 1989; Ermentrout, 1998a). On the other hand, most of the work on IF

network d ynamics has been concerned w ith the existence and stability of

phase-locked solutions, in which the neurons fire at a fixed common fre-

quency. Analysis of globally coupled IF oscillators in term s of return map s

has shown that synchronization almost always occurs in the presence of in-

stantaneou s excitatory interactions (Mirollo & Strogatz, 1990).Subsequen tly

this result has been extended to take into account the effects of inhomo-

geneities and various synaptic and axonal delays (Treves, 1993; Tsodyks,

Mitkov, & Somp olinsky, 1993; Abbott & van Vreeswijk,1993; van Vreeswijk,

Abbott, & Ermentrou t, 1994; Ernst, Pawelzik, & Giesel, 1995; Han sel, Mato,

& Meunier, 1995; Coombes & Lord, 1997; Bressloff, Coombes, & De Souza,

1997; Bressloff & Coombes, 1998, 1999). One finds that for small transmis-

sion delays, inhibitory (excitatory) synapses tend to synchronize if the risetime of a synapse is longer (shorter) than th e du ration of an action poten-

tial. Increasing the tran smission delay leads to a lternating ban ds of stability

and instability of the synchronous state. Analogous results have been ob-


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Dynam ics of Strongly Cou pled Spiking N eu rons 93

tained in the spike response model (Gerstner, Ritz, & van Hemmen, 1993;

Gerstner, 1995; Gerstner, van Hem men, & Cowan , 1996). Traveling wav es

of synchronized activity have been investigated in finite chains of IF oscilla-

tors mod eling locomotion in simple vertebrates (Bressloff &Coom bes,1998),

and in two-dimensional networks where spirals and target patterns are also

observed (Chu,Milton,&Cowan,1994;Kistler,Seitz,&van Hemmen,1998).

In this article we p resent a theory of spike train d ynam ics in IF netw orks

that bridges the gap between firing-rate and spiking models. In particular,

we show that synaptic interactions that are synchronizing in the weak cou-

pling regime can become desynchronizing for sufficiently strong coupling.

The resulting dynamics is compared with the behavior of a corresponding

analog model in which the outputs of the neurons are taken to be mean

firing rates. A basic result of ou r work is that for slow interactions, there

is good agreement between the two models (on an appropriately definedtimescale). On the other hand, discrepancies can arise for fast synapses

where IF neurons may remain phase locked.

We take as our starting p oint the nonlinear mapping of the neuronal fir-

ing times and show how a bifurcation analysis of this map can serve as a

basis for understanding the extremely rich dynamical structure seen in net-

works of spiking n eurons. Explicit criteria for the stability of p hase-locked

solutions are derived in both the weak and strong coupling regimes by con-

sidering the propagation of perturbations of the firing times throughout a

network. In the strong coupling case, the analysis p redicts regions in p a-

rameter space where instabilities in the firing times may cause transitions

to nonp hase-locked states. Num erical simulations are used to establish that

in these regions, the full nonlinear firing map can supp ort several distinct

types of behavior. For small networks, these includ e mod e-locked bur sting

states, where packets of spikes may be generated, separated by periods ofinactivity, and inhomogeneous states in which some of the oscillators be-

come inactive. For large global inhibitory n etworks w ith vanishing mean

perturbations of the firing times, we show that such bifurcations can be

suppressed, in agreement with the mode locking theorem of Gerstner et al.

(1996). As a fina l example of th e imp ortance of strong cou pling instabilities,

we consider a ring of IF neurons with a Mexican hat interaction function.

A discrete Turing-Hopf bifurcation of the firing times from a synchronous

state to a state with periodic or qu asi-periodic variations of th e interspike

intervals (ISIs) on closed orbits is shown to occur in the strong coupling

regime. Further, it is shown how the separation of these orbits in phase-

space results in a spatially periodic pattern of mean firing rate across the

network.

2 Integrate-and-Fire Model of a Spiking Neuron

The standard dynamical system for describing a neuron with spatially con-

stant membrane p otential V is based on conservation of electric charge, so


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that

Cd V

d t= F + Is + I, (2.1)

where C is the cell capacitance, F is the membrane current, Is the sum ofsynaptic currents en tering the cell, and Idescribes any external cur rents. Inthe Hodgkin-Huxley model the membrane current arises mainly through

the conduction of sodium and potassium ions through voltage-depend ent

channels in the m embrane. The contribution from other ionic currents is

assumed to obey Ohms law. In fact F is considered to be a function of Vand of three time-depend ent and voltage-depend ent conductance variables

m, n, and h,

F(V, m, n, h) = gL(V VL) + gKn4(V VK) + gNahm

3(V VNa), (2.2)

where gL, gK, and gNa are constants and VL, VK, and VNa represent the con-stant membrane reversal potentials associated with the leakage, potassium,

and sodium channels, respectively. The conductance variables m, n, and htake values between 0 and 1 and approach the asymptotic values m(V),n(V) an d h(V) with time constants m(V), n(V) an d h(V), respectiv ely.

The conductance-based Hod gkin-Huxley equations depend on four d y-

namical variables. A reduction of this number is often desirable in order

to facilitate any mathematical analysis and to ease the computational bur-

den for simu lations oflarge netw orks. A systematic app roach for doing this

involves the use ofequ ivalent potentials (Abbott & Kepler, 1990; Kepler, Ab-

bott, & Marder, 1992). Following th is app roach, one can d erive an IF model,

which provides a caricature of the capacitative nature of cell membrane atthe expense of a d etailed mod el of the refractory p rocess. The IF model sat-

isfies equation 2.1w ith F = F(V), together with the condition that wheneverthe neuron reaches a threshold h, it fires, and V is immediately reset to theresting potential V0. The dynamics of the membrane conductances m, n, his eliminated completely. Using a curve-fitting procedure, it is possible to

approximate F(V) by a cubic, F(V) = a(V V0)(V V1)(V h) where theconstants a, V0,1, and h can be determined explicitly from the reduction ofthe u nderlying Hod gkin-Huxley equations.

At a synapse,p resynaptic firing results in the releaseof neurotransmitters

that causes a change in the membrane conductance of the postsynaptic

neuron. This postsynaptic current may be written

Is = gss(Vs V), (2.3)

where V is the voltage of the postsynaptic neuron, Vs is the membranereversal potential, and gs is a constant. The variable s corresponds to theprobability that a synaptic receptor channel is in an open conducting state.


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This probability depends on the presence and concentration of neurotrans-

mitter released by the p resynaptic neuron. Und er certain assumptions, it

may be shown th at a second-order Markov scheme for the synaptic chan-

nel kinetics describes the so-called alph a function response comm only used

in syn aptic m odeling (Destexhe, Mainen, & Sejnowski, 1994): the syna ptic

response to an incoming spike at time t0 is

s(t) = s0(t t0)e(tt0), t > t0 (2.4)

for constants s0, . Here determines the inverse rise time for synapticresponse. Ifw e now combine equations 2.1and 2.3 und er the approximation

F = F(V), we obtain an equation of the form (after setting C = 1)

d Vd t

= F(V) + I+ gs m

s(t tm)[Vs V]. (2.5)

We are assuming that the neuron receives a sequence of action potential

spikes at times {tm} and each spike generates a synaptic response accordingto equation 2.4. The neuron itself fires a spike whenever V(t) reaches thethreshold h, and Vis immediately reset to the resting potential, V0.Denotingthe firing times of the neuron by {Tm}, we can view the neuron as a devicethat maps {tm} {Tm}. We introduce two additional simplifications. First,we neglect shunting effects by setting Vs V Vs; the possible nonlineareffects of shunting are d iscussed by Abbott (1991). Second, we redu ce F(V)further by considering the linear approximation F(V) = b(V V0) (Abbott& Kepler, 1990). This linear IF model of a spiking neuron will be used in our

subsequent analysis of network dynamics (see sections 3 and 4). However,

before proceeding further, we briefly mention an alternative formulation ofspiking neurons based on the so-called spike response model.

The IF model assumes that it is the capacitative nature of the cell that in

conjunction with a simple thresholding process d ominates the p roduction

of spikes. The spike resp onse (SR) mod el (Gerstner & van Hem men, 1994;

Gerstner, 1995; Gerstner et al., 1996) is a m ore genera l framew ork that can

accommodate the apparent reduced excitability (or increased threshold) of

a neu ron after the em ission of a spike. Spike reception and spike generation

are combined with the use of two separate response functions. The first,

hs(t), describes the postsynaptic response to an incoming spike in a similarfashion to the IF model, whereas the second, hr(t), mimics the effect ofrefractoriness. The refractory function hr(t) can in principle be related to thedetailed d ynam ics un derlying the d escription of ionic channels. In pr actice,

an idealized functional form is often used, although numerical fits to the

Hodgkin-Huxley equations during the spiking process are also possible(Kistler et al.,1997).In more detail,a sequence ofincomingsp ikes {tm} evokesa postsynaptic potential in the neuron via Vs(t) =

m hs(t tm), where hs(t)

incorporates d etails of axonal, synaptic, and dend ritic processing. The total


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membrane potential of the neuron is taken to be V(t) = Vr(t) + Vs(t), whereVr(t) =

m hr(t Tm) an d {Tm} is the sequence of output firing times.

Whenever V(t) reaches some threshold, the neuron fires a spike, and at thesame time a negative contribution hr is added to V so as to approximatethe reduced excitability seen after firing. Since the reset condition of the

IF model is equivalent to a sequence of current pulses, h

m (Tm), thelinear IF model is a special case of the SR model. That is, ifI= 0, F(V) = Vand we n eglect shunting, then we can integrate equation 2.5 to obtain the

equivalent formulation

V(t) =

m

hr(t Tm) +

m

hs(t tm), (2.6)

where

hr(t) = het, hs(t) =

t0

e(tt)s(t)d t, t > 0, (2.7)

and there is no reset.

Most work to d ate on the an alysis of the SR model has been based on the

stud y oflarge networks using dynam ical mean field theory (Gerstner &v an

Hemm en, 1994; Gerstner, 1995), wh ich can be viewed as a gen eralization of

the pop ulation averaging ap proach of Wilson and Cowa n (1972). This is dif-

ferent from the ap proach w e take here, wh ich is principally concerned w ith

the dynamics of finite networks of spiking neurons. However, we will link

the two in section 4.3, where we discuss large, globally coupled networks

and the m ode-locking th eorem of Gerstner et al. (1996).

3 Averaging Methods for Analyzing IF Network Dynamics

We now consider a network of IF neurons that interact via synapses by

transmitting spike trains to one another. Let Vi(t) denote the state of the ithneuron at time t, i = 1, . . . , N, where Nis the total number of neurons in thenetwork. Suppose that the variables Vi(t) evolve according to th e equations(cf. equ ation 2.5)

d Vi(t)

d t=

Vi(t)

d+ Ii + Xi(t), (3.1)

where Ii is a constant external bias, d is the membrane time constant, and

Xi(t) is the totalsynaptic current into the cell. We shallfix the units of time bysetting d = 1; typical values for d are in the r ange 520 msec. Equation 3.1is supplemented by the reset condition that whenever Vi = h, neuron i firesand is reset to Vi = 0. We shall set the threshold h = 1. The synap tic curren t


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Xi(t) is generated by the arrivalof spikes from neuron j and takes the explicitform

Xi(t) = N

j=1

Wij

m=

J(t Tmj ), (3.2)

where Wij is the effective synaptic weight of th e connection from the jth toth e ith neuron, J(t) determines the time course of the postsynaptic responseto a single spike with J(t) = 0 for t < 0, and Tmj denotes the sequence

of firing times of the jth neuron with m running through the integers. Wehave introduced a coupling constant characterizing the overall strength

of synaptic interactions. From the discussion in section 2, a biologically

motivated choice for J(t) is

J(t) = s(t a)(t a), s(t) = 2tet, (3.3)

where s(t) is an alpha function (with unit normalization), a is a discreteaxonal transmission delay, and is the unit step function, (t) = 1 ift > 0 and zero otherwise. The maximum synaptic response then occurs ata nonzero delay t = a +

1. In a compan ion pa per, the effects of dend ritic

interactions (both passive and active) will be analyzed by taking J(t) to bethe Greens function of some cable equation (Bressloff, 1999).

Although equations 3.1 and 3.2 are considerably simpler than those de-

scribing a network of Hodgkin-Huxley neurons,the analysis of the resulting

dyn amics is still a nontr ivial problem since one has to ha nd le both the pres-

ence of delays and discontinuities arising from reset. Most work to date

has therefore been based on some form of ap proximation scheme. We shall

describe two such schemes: phase redu ction in the w eak coupling regimeand time averaging with slow synapses. In section 4 we shall then carry

out a d irect analysis of the IF network d ynamics without recourse to such

approximations.

3.1 Weak Coupling: Phase-Oscillator Models. We first show how inthe weak coupling limit, equation 3.1 with reset can be reduced to a phase-

oscillator equation. For concreteness, assume that Ii = I > 1 for all i =1, . . . , N, so that in the absence of any coup ling ( = 0), each neuron acts asa regular oscillator by firing spikes with a constant period T = ln[I/(I 1)].Following van Vreeswijk et al. (1994), we introdu ce the pha se variable i(t)according to

(m o d 1) i(t) +t

T

= (Vi(t)) 1

TVi(t)

0

dx

F(x), (3.4)

where F(x) = Ix for the linear IFm odel. (One can also apply this transformto the nonlinear IF model discussed in section 2 with F(x) now given by a


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cubic.) Under such a t ransformation, equa tion 3.1 becomes

i(t) = RT(i(t) + t/T)Xi(t), (3.5)

with

RT( ) 1

T

1

F[1( )]=

1 eT

eT

T(3.6)

for 0 < 1 and RT( + k) = RT( ) for all integers k. In the absence ofany coupling, the phase variable i(t) is constant in time, and all oscillatorsfire with their natural periods T. For weak coupling, each oscillator stillapproximately fires at the unperturbed rate, but now the phases slowly

drift according to equation 3.5 since Xi(t) = O(). To first order in , w ecan take the firing times to be Tnj = (n j(t))T. Under this approximation,

equations 3.2 and 3.5 lead to the following equation for the shifted phases

i(t) = i(t) + t/T:

d i

d t=

1

T+

Nj=1

WijRT(i)PT(j) +O(2), (3.7)

where PT( + 1) = PT( ) for all an d

PT( ) =

m=

J(( + m)T), 0 < 1. (3.8)

The summation over m in equation 3.8 is easily performed for J( ),satisfyingequation 3.3, and gives PT( ) = JT( a/T), where JT( ) is a periodicfunction of with

JT( ) = 2e T1 eT

T+

TeT

1 eT

, 0 < 1. (3.9)

The function RT may be interpreted as the phase-response curve (PRC)of an individual IF oscillator, and PT is the corresponding pulselike func-tion that contains all details concerning the synaptic interactions. Note that

phase equations of the form 3.7 can also be derived for a system of weakly

coupled limit cycle oscillators based on more detailed biophysical models

such as Hodgkin-Huxley neurons (Ermentrout & Kopell, 1984; Kuramoto,

1984; Han sel et al., 1995).As discussed in some detail by Hansel et al.(1995),

linear IF oscillators have a type I PRC, which means that an instantaneousexcitatory stimulus always ad vances its phase (RT( ) is positive for all ).On the other hand, Hodgkin-Huxley neurons are of type II since a stimulus

can either advan ce or retard th e phase d epending on the point on th e cycle


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at which the stimulus is applied (RT( ) takes on both positive and negativevalues over the domain [0, 1]). Equation 3.7 can be simplified further by

averaging over the natural p eriod T. This leads to an equation of the form

d i

d t= +

Nj=1

WijHT(j i) +O(2) (3.10)

with = 1/Tan d

HT() =

10

RT( )PT( + )d. (3.11)

For positive kernels J( ), the interaction fun ction HT() is positive for all since IF oscillators are of typ e I. How ever, it is possible to m imic an effec-

tive type II interaction function by introd ucing a combination of excitatoryand inhibitory interactions between the IF neurons in the definition of J( )(Bressloff & Coombes, 1998a). To illustrate this idea, suppose that we de-

compose J( ) as

J( ) = J+( ) J(), J( ) = s( )( ),

s( ) = 2e

, (3.12)

where J( ) represent functions for excitatory (+) and inhibitory ()synapses. Assuming that the inhibitory pathways are delayed with respect

to the excitatory ones ( > +) then th e effective interaction fun ction of an

IF oscillator can be ap proximately sinusoid al. This is illustrated in Figure 1.

It is not clear that such a combination of synapses is directly realized in

cortical microcircuits. However, it is known that recurrent excitatory con-

nections also stimulate inhibitory interneurons, and this might lead to an

effective delay kern el of the form 3.14 at the p opu lation level.

We define a phase-locked solution of equation 3.10 to be of the form

i(t) = i + t, where i is a constant phase and = 1/T + O() is thecollective frequency of th e coup led oscillators. Substitution of this solution

into equation 3.10 and working to O() leads to the fixed-point equations,

=1

T+

Nj=1

WijHT(j i). (3.13)

After choosing some reference oscillator, the N equations (3.13) determ inethe collectivep eriod an d N1 relative phases,w ith the latter independentof . In order to analyze the local stability of a phase-locked solution =

(1, . . . , N), w e linearize equation 3.10 by setting i(t) = i + t +

i(t)

and expanding to first order in i:did t

= N

j=1

Hij()j, (3.14)


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Figure 1: (Top) Delay kernel J(t) and (bottom) associated interaction functionHT() for a combination of excitatory and delayed inhibitory synaptic interac-tions given by equation 3.12. Here = 10, + = 0, = 0.6, and T = ln 2.

where

Hij() = Hij() i,jN

k=1Hik(), Hij() = WijHT(j i) (3.15)

an d HT() = dHT()/d . One ofth e eigenvalues ofth e JacobianH is always

zero, and the corresponding eigenvector points in the direction of the flow,

that is (1, 1, . . . , 1). The phase-locked solution will be stable provided that

all other eigenvalues have a negative real part (Ermentrout, 1985).

The existence and stability ofp hase-locked solutions in the case ofa sym -

metric pair of excitatory or inhibitory IF neurons w ith synap tic interactions

have been studied in some detail by van Vreeswijk et al. (1994). In this case,

N = 2 and W11 = W22 = 0, W12 = W21 = 1. Equation 3.13 then showsthat the allowed solutions for the relative phase = 2 1 are given

by the zeroes of the odd interaction function HT() = HT() HT().The und erlying symm etry of the p air of neurons guarantees the existence

of the in-phase or synchronous solution = 0 and the antiphase or an-tisynchronous solution = 1/2. Suppose that a = 0. For small , one

finds that the in-phase and antiphase solutions are the only phase-locked

solutions. However, as is increased, a critical value c is reached, where


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0.0

0.2

0.4

0.6

0.8

1.0

2 6 10 14 18

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6

Figure 2: (a) Relative phase = 2 1 for a pair of IFoscillators with symm etric

inhibitory coupling as a function of with I= 2. In each case the antiphase stateundergoes a bifurcation at a critical value of = c, where it becomes stable,

and two additional unstable solutions , 1 are created. The synchronous

state remains stable for all . (b) Relative p hase versus discrete delay a for a

pair of p ulse-coup led IF oscillators with I = 2 and = 2.

there is a bifurcation of the antiphase solution, leading to the creation of

two partially synchronized states a n d 1 , w ith 0 < < 1/2 and

0 as (see Figure 2a an d van Vreeswijk et al., 1994). As

shown by Coombes and Lord (1997), variation of a for fixed produces a

checkerboard pattern of alternating stable and unstable solution branches

(see Figure 2b) that can overlap to produce multistable solutions. Using

equation 3.15, one obtains th e following necessary and sufficient condition

for local asymptotic stability of a phase-locked state in the weak coupling

regime:

dHT()

d > 0. (3.16)

One finds that for a = 0 and inhibitory coupling ( < 0), the synchronousstate is stable for all 0 < < . Moreover, the antiphase solution = 1/2

is unstable for < c, but it gains stability when > c with the creation

of two unstable partially synchronized states. The stability properties of


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0

0.4

0.8

1.2

1.6

2

0 0.2 0.4 0.6 0.8 1

1

Figure 3: Stability of the synchronou s solution = 0 as a function of1 an d afor weak excitatory coup ling with I= 2.Stable and un stable regions are denotedby s an d u, respectively. The stability diagrams are periodic in a with periodln[I/(I 1)]. (This periodicity would be distorted in th e strong coupling regimesince the collective period of oscillation d epend s on th e strength of coupling).

all solutions are reversed in the case of excitatory coupling ( > 0) so that

the synchronous state is now unstable for all . If the discrete delay a is

increased from zero, then alternating bands of stability and instability are

created. An example of such regions is d isplayed in Figure 3. The corre-

sponding stability diagram for the antisynchronous state may be obtained

by shifting the delay according to a a + T/2.

3.2 Slow Synapses: Analog Models. A second approximation schemefor an alyzing IF network dynam ics is to assume that the synaptic interac-

tions are sufficiently slow so that the outp ut of a neuron can be characterized

reasonably well by a mean (time-averaged) firing rate (see, for example,

Amit & Tsodyks, 1991; Ermentrout, 1994). Therefore, let us consider the

case in which J( ) is given by the alpha function (see equation 3.3) with asynaptic rise time 1 significantly longer than all other timescales in the

system. Supp ose that the total synapticcurrent Xi(t) to neuron i is describedby a slowly varying function of time t. If the n euronal d ynam ics is fast rela-tive to 1, then the actual firing rate Ei(t) of a neuron will quickly relax toapproximately its steady-state value, that is,

Ei(t) = f(Xi(t) + Ii), (3.17)

where, from equ ation 3.1, the firing-rate fun ction f is of the form

f(X) =

ln

X

X 1

1(X 1). (3.18)


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(Noteth at we have ignored the effectsof absolute refractory period, which is

reasonable when th e system is operating w ell below its optimal firing rate).

Equation 3.17 relates the d ynam ics of the firing rate d irectly to the stimu lus

dynamics Xi(t) throu gh the steady-state response function. In effect, the useof a slowly varying kernel J( ) allows a consistent d efinition of the firingrate so that a dynamical network model can be based on the steady-state

properties of an isolated neuron.

Within th e above app roximation, we can derive a closed set of equations

for the synap tic currents Xi(t). This is achieved by rewriting equation 3.2as a pair of differential equations that generate the alpha function J(t) ofequation 3.3, and replacing the output spike train of a neuron by the firing-

rate function (see equ ation 3.18):

1 d Xi(t)d t

+ Xi(t) = Yi(t), (3.19)

1d Yi(t)

d t+ Yi(t) =

Nj=1

WijEj(t a). (3.20)

Here Yi(t) is an auxiliary current.A common starting point for the analysis of analog models is to consider

conditions und er which destabilization of a homogeneou s low-activity state

occurs,leading to the formation ofa state with inhomogeneous and /or time-

dep end ent firing rates (see Erment rout, 1998a, for a review). To simplify our

analysis, we shall impose the following condition on the external bias Ii,

I= Ii + f(I)N

j=1 Wij (3.21)for some fixed I > 1. Then for su fficiently weak coupling, || 1, the ana-log model has a single stable fixed point given by Yi = Xi = f(I)

j Wij,

such that the firing rates are kept at the value f(I). Suppose that we lin-earize equations 3.19 and 3.20 about this fixed p oint and substitute into th e

linearized equations a solution of the form (Xk(t), Yk(t)) = et(Xk, Yk). This

leads to the eigenvalue equation

p

= 1

f(I)pe

p a/2, p = 1, . . . , N, (3.22)

where p,p = 1, . . . , N,are the eigenvaluesofthe weight matrix W.The fixedpoint w ill be asymptotically stable if and only if Re tp < 0 for all p. A s || is

increased from zero, an instability may occur in at least two distinct ways.If a single real eigenvalue crosses the origin in the comp lex -plane, then

a static bifurcation can occur, leading to the em ergence of add itional fixed-

point solutions that correspond to inhomogen eous firing rates. For example,


14/39


if > 0 and W has real eigenvalues 1 > 2 > > N with 1 > 0, thena static bifurcation will occur at the critical coupling c for which

+1 = 0,

that is, 1 =

c f(I)1. On the other hand, if a pair of complex conjugateeigenvalues = R iI crosses the imaginary axis (R = 0) from left toright in the complex plane, then a Hopf bifurcation can occur, leading to

the formation of period ic solutions, that is, time-depen den t firing rates. For

example, suppose that a = 0 and W has a pair of complex eigenvalues(,) with = rei and 0 < < . Denote the corresponding solutionsof equation 3.22 by (,

). Assuming that all other eigenvalues have

negative real part, a Hopf bifurcation will occur at the critical coupling cfor which Re+ = 0, that is, 1 =

c f(I)r cos( /2). An alternative way

of generating oscillatory solutions is to have nonzero delays a (Marcus &

Westervelt, 1989).N ote that the basic stability results are independ ent of the

inverse rise time .To illustrate, consider a symmetric pair of analog neurons with inhibitory

coupling, < 0, and W11 = W22 = 0, W12 = W21 = 1. The weight matrix Whas eigenvalues 1 and eigenmodes (1, 1). Let xi = Xi f(I) so that thefixed-point equations become x1 = f(x2 + I) f(I) an d x2 = f(x1 + I) f(I). One solution is the homog eneous fixed point xi = 0. A full bifu rcationdiagram showing the location of the fixed points x1 as a function of || isshown in Figure 4 (top). The homogeneous fixed point xi = 0 is stable forsufficiently small coupling || but destabilizes at the critical point || = cwith c = 1/f(I), where it coalesces with two unstable fixed points. For|| > c, the unstable fixed p oint at the or igin coexists with tw o stable fixed

points (arising from sadd le-nod e bifurcations). Just beyon d th e bifurcation

point, the system jumps from a homogeneous state to a state in w hich one

neuron is active with a constant firing rate f(I f(I)) and the other is

passive w ith zero firing rate. Note that a p air of excitatory analog n euronswould bifurcate into another homogeneous state. For example, since Ii ha sto decrease to keep the firing rate constant when > 0 (see equation 3.21),

for strong enough coupling Ii < 1 so that the quiescent state is also a validsolution.

A well-known result from the analysis of analog neurons is that an

excitatory-inhibitory pair can undergo a Hopf bifurcation to an oscillatory

solution (Atiya & Baldi, 1989). As a n illustration, consider the case > 0,

a = 0 and W11 = W22 = 0, W12 = 2, W21 = 1. Here neuron 2 inhibitsneuron 1, wh ereas neuron 1 excites neuron 2. The eigenvalues of the weight

matrix are i so that a Hopf bifurcation arises as is increased. (See thediscussion below equation 3.22.) Let x1 = X1 + 2f(I) an d x2 = X2 f(I) sothat there exists a fixed point at xi = 0. The bifurcation d iagram for the am -plitude x1 as a function of is shown in Figure 4 (bottom). It can be seen that

the system undergoes a so-called subcritical Hopf bifurcation in which thehomogeneous fixed point xi = 0 becomes un stable and the system jump s toa coexisting stable limit cycle,signaling a solution w ith period ic firing rates.


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active

passive

1

1

Figure 4: (Top) Bifurcation diagram for a pair of analog neurons with sym-

metric inhibitory coupling and external input I = 2. (Bottom) Subcritical Hopfbifurcation for a p air of an alog oscillators w ith self-interactions. = 0.5, I = 2,W11 = W22 = 0, W21 = 1, and W12 = 2. Open circles denote the amplitudeof the resulting limit cycle from the Hopf bifurcation point 2.0, and s (u)

stands for stable (unsta ble) dynamics.

This form of jump is often referred to as a hard excitation. The system also ex-

hibits hysteresis.It is interesting to note that ifth e firing-rate function f(X) ofequation 3.18 were taken to be the u sual smooth sigmoid function, then, forthe given weights, the Hopf bifurcation would be supercritical in the sense

that the limit cycle would grow smoothly from the unstable fixed point so

that there is no jump phenomenon or hysteresis. This is called a soft excita-tion. (See Atiya & Baldi, 1989.) Another importa nt p oint is that if the ana logmodel were d escribed by a first-order equation rather than a second-order

equation (as given by equations 3.19 and 3.20), then it would be necessary

to introduce additional self-interactions (W12, W21 = 0) in order for a Hopfbifurcation to occur. This is d ifficult to justify on neurobiological ground s.

(A first-order equation would be obtained if the delay kernel were taken

to be an exponential function rather than the alpha function 3.3.) We shall

return to this issue in section 4.6.

4 Spike Train Dynamics in the Strong-Coupling Regime

Recall from section 3 that netw orks of weakly coup led IF neurons can h ave

stable phase-locked solutions in which all the neurons have the same con-


16/39


stant ISI. On the other hand, a strongly coupled network of analog neu-

rons can bifurcate from a stable homogeneous state with identical time-

independent firing rates toa statew ith inhomogeneousand /or time-varying

firing rates. (Note that a homogeneous state of the analog model d oes not

distinguish between different phase-locked solutions since all phase infor-

mation is lost during time averaging.) This suggests that there should exist

some mechanism for destabilizing pha se-locked solutions of the IFm odel in

the strong-coupling regime. Here we identify such a mechanism based on a

discrete Hopf bifurcation of the firing times. This induces inhomogeneous

and period ic variations in the ISIs, which supp orts a variety of comp lex dy-

namics, including oscillator d eath (section 4.3), bur sting (section 4.4), and

pattern formation (section 4.5).

4.1 Phase Locking for Arbitrary Coupling. An important simplifyingaspect ofth e dynam ics of pu lse-coupled (linear) IFn eurons is that it is possi-

ble to derive phase-locking equations without the assumption of weak cou-

pling (van Vreesw ijk et al., 1994; Bressloff et a l., 1997; Bressloff & Coom bes,

1999).This can be achieved by solving equ ation 3.1 directly und er the ansatz

that the firing times are of the form Tnj = (n j)Tfor some self-consistent

period T and constant phases j. Integrating equation 3.1 over the inter-

va l t (Ti, T Ti) and incorporating the reset condition by settingUi(iT) = 0 and Ui(T iT) = 1 leads to the result

1 = (1 eT)Ii + N

j=1

WijKT(j i), (4.1)

where

KT() = eT

T0

e

m=

J( + (m + )T)d =T2eT

1 eTHT(). (4.2)

Equation 4.1 has an id entical structure to that of equa tion 3.13 and involves

the same phase interaction function (up to a multiplicative factor). Indeed,

in the weak coupling regime with Ii = I for all i, equation 4.1 redu ces toequation 3.13 with 1/T = . This means that techniques previously devel-oped for studying phase locking in weakly coupled oscillator networks can

be extended to strongly coupled IF networks. For example, in the case of

a ring of identical IF oscillators with symmetric coupling, group-theoretic

method s can be used to classify all phase-locked solutions and construct bi-

furcation diagrams showing h ow new solution branches emerge via spon-

taneous symmetry breaking (Bressloff et al., 1997; Bressloff & Coombes,1999). Furth ermore, in t he case of a finite chain of IF oscillators w ith a gra-

dient of external inputs and sinusoidal-like phase interaction functions of

the form shown in Figure 1, one can establish the existence of traveling


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wave solutions in w hich the p hase varies monotonically along th e chain

(except in some narrow bou nd ary layer);su ch systems can be used to model

locomotion in simple vertebrates (Bressloff & Coombes, 1998a).

There are, however, a number of significant differences between phase-

locking equations 3.13 and 4.1. First, equation 4.1 is exact, whereas equa-

tion 3.13 is valid only to O() since it is derived under the assumption of

weak coupling. Second, the collective period of oscillations T must be de-termined self-consistently in equation 4.1. Assume for the moment that Tis given. Supp ose that we choose 1 as a reference oscillator and subtract

from equation 4.1for i = 2, . . . , Nthe corresponding equ ation for i = 1.This

leads to N1 fixed-point equations for the N1 relative phases j = j 1,j = 2, . . . , N, which for Ii = Itake the form

0 =N

j=1

WijKT(j i) N

j=1

W1jKT(j),

where 1 0. The resu lting solutions for j, j = 2, . . . , N, are functions ofT,wh ich can th en be substituted back into equation 4.1 for i = 1 to give an im-plicit self-consistency condition for T. The analysis is considerably simplerin the weak-coupling regime since the relative phases are then functions of

the natural period T. The third difference between weak and strong cou-pling is that although the equations for phase locking in the two models

are formally the same, the underlying dynamical systems are distinct, thus

leading to differences in their stability properties. For example, in the spe-

cial case of a pair of IF neuron s, a return m ap argu ment can be used to show

that equation 3.16 with Treplaced by the collective period Tis a necessary

condition for the stability of a phase-locked state for any (van Vreeswijket al., 1994). However, as we shall establish below, it is no longer a sufficient

condition for stability in the strong coupling regime. (See also Chow, 1998.)

4.2 Desynchronization in the Strong Coupling Regime. In order toinvestigate the linear stability of phase-locked solutions of equation 4.1, we

consider perturbations ni of the firing times (van Vreeswijk, 1996; Gerstner

et al., 1996; Bressloff & Coombes, 1999). That is, we set Tni = (n i)T+ ni in

equation 3.2 and th en integrate equ ation 3.1 from Tni to Tn+1i using the reset

condition. This leads to a mapping of the firing times that can be expanded

to first order in th e p ertur bations (Bressloff & Coombes, 1999):

Ii 1 + N

j=1

WijPT(j i)

n+1i ni

= N

j=1

Wij

m=

Gm(j i, T)

nmj ni

, (4.3)


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where PT satisfies equation 3.8 and

Gm(, T) =eT

T

T0

etJ(t + (m + )T) dt. (4.4)

The linear d elay-difference equation 4.3 has solutions of th e form nj = enj

with 0 Im () < 2 . The eigenvalues and eigenvectors (1, . . . , N)

satisfy the equation

Ii 1 + N

j=1

WijPT(j i)

(e 1)i

= N

j=1

WijGT(j i, )j N

j=1

WijGT(j i, 0)i, (4.5)

with

GT(,) =

m=

emGm( , T). (4.6)

One solution to equation 4.5 is = 0 with i = for all i = 1, . . . , N. Thisreflects the invarian ce of the dy nam ics with resp ect to uniform ph ase shifts

in the firing times, Tni Tni + . Thus the condition for linear stability of

a phase-locked state is that all remaining solutions of equation 4.5 have

negative real part. This ensures that nj 0 a s n and, hence, that the

pha se-locked solution is asymptotically stable.By performing an exp ansionin powers of the coupling , it can be established tha t for sufficiently small

coupling, equation 4.5 yields a stability condition that is equivalent to the

one based on the Jacobian of the phase-averaged model, equations 3.14

and 3.15. (See Bressloff & Coombes, 1999.) We wish to determine whether

this stability condition breaks down as || is increased (with the sign of

fixed).

For concreteness, we shallfocus on the stability of synchronous solutions.

In order to ensure that such solutions exist, we impose the condition

Ii = I

1 KT(0) N

j=1

Wij

, i = 1, . . . , N (4.7)

for some fixed I > 1 and T = ln[I/(I 1)]. The condition on Ii for the IFmodel plays an analogous role to equation 3.21 for the analog model insection 3.2. The synchronou s state i = for all i and arbitrary is then asolution of equ ation 4.1 with collective p eriod T. By fixing T we can make


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a more d irect comparison with the analog model whose fixed p oint at the

origin will have a firing rate f(I) = 1/T in equa tion 3.18. Define the matrixW according toWij = Wij i,j N

k=1

Wik. (4.8)

For sufficiently w eak coup ling, equations 3.14, 3.15, and 4.2 imply that the

synchronous state is linearly stable if and only if

KT(0)Rep < 0, p = 1, . . . , N 1, (4.9)

wherep, p = 1, . . . , Nare the eigenvalues of the matrix W withN = 0, andKT( ) = T1dKT()/d . In the particular case ofa symmetricpair ofneurons,equation 4.9 reduces to the condition KT(0) > 0, which is equivalent toequation 3.16 for = 0. (The case = 0 is obtained in exactly the same

fashion). It follows that the stability diagram of Figure 3 displays the sign of

KT(0) as a function of an d a. For example,in the case of zero axonal delays(a = 0),Figure3impliesthat K

T(0) < 0for allfinite an d T,a nd equa tion 4.9

reduces to the stability condition Rep > 0 for all p = 1, . . . , N 1.The stability of the synchronous state for weak coupling implies that

the zero eigenvalue is nondegenerate and all other eigenvalues satisfy-

ing equation 4.5 have negative real parts. As the coupling strength || is

increased from zero, destabilization of the synchronous state will be sig-

naled by one or more complex conjugate pairs of eigenvalues crossing the

imaginary axisin the complex -plane from left to right.(It is simple to estab-

lish that the synchronous state cannot destabilize due to a real eigenvalues

crossing the origin by setting = 0 in equation 4.5.) We proceed, there-fore, by substituting i = , i = 1, . . . , N and imposing the condition 4.7.Equation 4.5 redu ces to the form

e 1

I 1 + iIK

T(0)

+ iK

T(0)

i = GT(0, )

Nj=1

Wijj, (4.10)

where i =N

j=1 Wij. We have used the identities PT(0) IKT(0) = IKT(0)

an d GT(0, 0) = KT(0). We then substitute = i into equation 4.10 and

look for the smallest value of the coupling strength, || = c, for which a

real solution exists. This determines the Hopf bifurcation point at which

the synchronous state becomes unstable. (In the special case = this

redu ces to a period dou bling bifurcation.)N ote that when i is independ ent

of i, equation 4.10 can be simplified by choosing i to lie along one of theeigenvectors of the weight matrix W.We shall explore the nature of the Hopf instability through a num ber of

specific examp les. We shall also establish tha t for slow synap ses, the strong


20/39


coupling behavior of the IF mod el is consistent w ith that of the mean firing -

rate model of section 3.2 (on an appropriately defined timescale). In order

to compare the two types of model, it will be useful to introduce a few

definitions. For a network ofNIF neurons labeled i = 1, . . . , N, let us d efine

the long-term firing rate of a neuron to be ai = 1

i , w here i is the mea n ISI,

i = limM

1

2M + 1

Mm=M

mi , (4.11)

with mi = Tm+1i T

mi .A necessaryrequirement for good agreement between

theanalog and IFmodels isthat them ean firing rates ai oftheIFmodelmatchthe correspond ing firing rates of the analog model. In general, one wou ld

also expect to observe fluctuations in the ISIs of the IF network th at are notresolved by the analog model. A Hopf bifurcation for maps (also known as

a Neimark-Sacker bifurcation) is usually associated with the formation of

periodic (or quasiperiodic) orbits, wh ich in the current context corresponds

to periodic variations in the ISIs.In cases where the analog mod el bifurcates

to a state with time-independ ent firing rates, we expect the periodic orbits

of the ISIs to be small relative to the firing p eriod, at least for slow synap ses;

that is, |mi i|/i 1 for all i, m. (See the example of pattern formationin section 4.5, in particular, Figure 12.) On the other hand, in cases where

an analog network destabilizes to a state with time-varying firing rates, we

expect the periodic orbits of the ISIs in the IF model to be relatively large.

Under such circumstances, we can introduce a sliding window of width

2P + 1 for the IF model in order to d efine a short-term avera ge firing rate ofthe form ai(m) = i(m)

1 where

i(m) =1

2P + 1

Pp=P

m+pi .

One can then see if there is a good match between the time-depen den t firing

rates of the two models for an appropriately chosen value ofP. Actu ally, inpractice it is simpler to compare variations in the firing rate of an analog

networ k directly with variations in the ISIs of the corresponding IF netw ork

(see Figure 8).

4.3 Oscillator Death in a Globally Coupled Inhibitory Network. Asour first example illustrating the Hopf instability identified in section 4.2,

we consider a network ofNidentical IF oscillators with all-to-all inhibitory

coupling and no self-interactions. This type of architecture has been used,for example,to model the reticular thalamicnu cleus (RTN),which is thought

to act as a pacemaker for synchronous spind le oscillations observed d uring

sleep or anesthesia (Wang & Rinzel, 1992; Golomb & Rinzel, 1994). In the


21/39


biophysical model of RTN develop ed by Wang and Rinzel (1992), neur al os-

cillations are sustained by postinhibitory rebound, a phenomenon whereby

a neuron can fire after being hyp erpolarized over an extended period and

then released. This shou ld be contrasted w ith our simp le IF model in which

oscillations are sustained by an external bias. We shall show that for slow

synapses, desynchronization via a Hopf bifurcation in the firing times oc-

curs, leading to oscillator death in the strong coup ling regime, that is,certain

cells suppress the activity of others. (See also the recent study of mutually

inhibitory H odgkin-H uxley neu rons by White, Chow, Ritt, Soto-Trevino, &

Kopell, 1998.)

The weight matrix W of a globally coupled inhibitory network with < 0 is given by Wii = 0 and Wij = 1/(N 1) so that i = 1 for alli = 1, . . . , N. It follows that W has a nondegenerate eigenvalue + = 1

with corresponding eigenvector (1, 1, . . . , 1) and an (N 1)-fold degenerateeigenvalue = 1/(N 1). The eigenvalues ofthe matrix W, equation 4.8,are + = 0 and = N/(N 1), so that the synchronous state is stablein the weak coupling regime provided that KT(0) > 0 (see equation 4.9).This is certainly true for zero axonal delays (see Figure 3). Note that the

und erlying permu tation symmetry of the system means that there are addi-

tionalp hase-locked states in which the neurons are divided into two or more

cluster s of synchron ized cells (Golomb & Rinzel, 1994). We shall inv estigate

the stability of the synchronous state as || is increased. Take (1, . . . , N) in

equation 4.10 to be an eigenvector corresponding to either = + or = and set = i. Equating real and imaginary p arts then leads to the p air ofequations,

KT(0) + [cos() 1][I 1 + IKT(0)] = C()

sin ()[I 1 + IKT(0)] = S(), (4.12)

where C() = ReGT(0, i, ), S() = Im GT(0, i).In Figure 5 w e plot the solutions of equation 4.12 as a function of the

inverse rise time for a pair of inhibitory neu rons (N = 2) with T = ln 2an d a = 0. The solid (dashed) solution branch corresponds to the eigen-

value = 1 ( = 1). The lower branch determines the critical coupling

|| = c() for a Hopf instability. An important result that emerges from

this figure is that there exists a critical value 0 of the inverse rise time be-

yond which a Hopf bifurcation cannot occur. That is, if > 0, then the

synchronous state remains stable for arbitrarily large inhibitory coupling.

On the other hand, for < 0, d estabilization of the synchronous state

occurs as the coupling strength crosses the solid curve of Figure 5 from

below.This signals the activation of the eigenmode (1, 2) = (1, 1), which

suggests that an inhomogeneous state will be generated. Indeed, direct nu-merical simulation of the IF model shows that just after destabilization of

the synchronous state (|| > c), the system jumps to an inhomogeneous

state consisting of one active neuron and one passive neuron. We conclude


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Figure 5: Region of stability for a synchronized pair of identical IF neurons

with inhibitory coupling and collective period T = ln 2. The solid curve || =c() denotes the bou nd ary of the stability region, which is obtained by solving

equation 4.12 for = 1 as a function of with a = 0. Crossing the boundaryfrom below signals excitation of the linear m ode (1, 1), leading to a stable

state in w hich one n euron becomes quiescent (oscillator death ). For > 0, the

synchronous state is stable for all . The dashed curve denotes the solution of

equation 4.12 for = 1.

that for sufficiently slow synapses, a pair of IF neurons displays similar

behavior to a correspond ing pair of analog neu rons in the strong coupling

regime (see section 3.2 and Figure 4 (top)). A rough order estimate for the

critical inverse rise time 0 is 10

T, where T is the collective periodof oscillations before destabilization. Such a result is not surprising since

one would expect that for a reasonable match between the IF and (time-

averaged) analog models to occur, the IF neurons should sample incoming

spike trains over a sliding window that is not too small. The width of sucha sliding wind ow is determined by the rise time 1.

The above result appears to be quite general. For example, suppose that

we have a nonzero d iscrete delay a such that the synchronous state is un-

stable and the antiphase state is stable for a given . The latter state also

destabilizes via a Hopf bifurcation in the firing times for small , lead-

ing to oscillator death. The result extends to larger values ofN, for which = 1/(N1) in equation 4.12. Here desynchron ization ofa p hase-lockedstate leads to clusters of active and passive neurons. Interestingly, the crit-

ical value 0 decreases with N, indicating the greater tendency for phaselocking to occur in large, globally coupled networks. We plot c as a func-

tion of for a range of values of network size Nin Figure 6, where it can beseen that 0(N) 0 as N . This implies that for large networks, theneurons remain synchronized for arbitrarily large coupling, even for slow

synapses. Indeed, since real neurons typically have an inverse rise time of the order 2 or larger, it follows that for realistic synaptic time constants,

the network can never experience oscillator death for Nlarger than 5 or so.(There is one subtlety to be n oted here. The dashed solution curve shown


23/39


c

Figure 6: Plot of critical coup ling c as a function of for various network sizes

N. The critical inverse r ise time 0(N) is seen to be a decreasing function ofN,with 0(N) 0 a s N .

in Figure 5 corresponds to excitation of the uniform mode (1, 1, . . . , 1) an d

is independ ent ofN. A s Nincreases, it is crossed by th e curve c() so thatfor a certain range of values of , it is possible for the synchronous state to

destabilize due to excitation of the uniform mode. The neurons in the new

state will still be synchronized, but the spike trains w ill no longer have a

simple periodic structure.)

The persistence of synchrony in large networks is consistent with the

mode-locking theorem obtained by Gerstner et al. (1996) in their analysis

of the spike response model. We shall briefly discuss their result within

the context of the IF mod el. For the given globally coupled network, the

linearized m ap of the firing times, equat ion 4.3, becomesI 1 + IKT(0)

n+1i

ni

=

m=

Gm(0, T)

1

N 1

j=i

nmj ni

. (4.13)

The major assumption underlying the analysis of Gerstner et al. (1996) is

that for large N, the mean p erturbation m =

j=i m

j /(N 1) 0 for all

m 0 say. Equation 4.13 then simplifies to the one-dimensional, first-ordermapping,

n+1i =I 1 + (I 1)KT(0)

I 1 + IK

T(0)

ni kTni . (4.14)

The synchronous (coherent) state will be stable if an d only if |kT| < 1. Itis instructive to establish explicitly that equation 4.14 is equivalent to the


24/39


corresponding result d erived for the spike-response mod el (see equation 4.8

of Gerstner et al., 1996), which in our notation can be written as

n+1i =

l1 h

r(lT)

n+1li

l1 hr(lT) +

l1 h

s(lT)

. (4.15)

Here hr(t) = dhr(t)/dt. First, using equations 2.7,3.3,and 4.2,it can be shownthat

l1 h

r(lT) = e

T/(1 eT) = I 1 and

l1 hs(lT) = IK

T(0). Setting

A(T) = I 1 + IKT(0), we can then rewrite equation 4.15 as A(T)n+1i =

l1 elTn+1li . It follows that A(T)

n+1i e

TA(T)ni = eTni , which is

identical to equation 4.14since eT = (I1)/I. Equation 4.15 or equation 4.14implies that the synchronous state is stable in the large N limit providedthat

l1hs(lT) > 0, that is, K

T(0) > 0 (cf. equation 3.16). This is the

essence of the m ode-locking theorem of Gerstner et al. (1996). Our analysisalso h ighlights one of the simplifying features of the IF model with alpha

function response kernels: that the summations over l in equ ation 4.15 canbe performed explicitly. It would be of interest, however, to investigate to

wha t extent the results of this article carry over to more general choices ofhran d hs in the construction of the spike-response model. We expect that theinclusion of details concerning refractoriness will not alter the basicp icture,

for oscillator death is also found in a pair of mutually inhibitory Hodgkin-

Hu xley neuron s, as shown by White et al.(1998) in the case ofsyn apses with

first-order channel kinetics, and also confirmed num erically by ourselves

in the case of second -order kinetics.

Finally, in other related w ork, van Vreeswijk (1996)h as shown h ow a net-

work of IF neurons with global excitatory coupling can d estabilize from an

asynchronou s state via a Hopf bifurcation in the firing times. However, this

leads to small-amplitude quasiperiodic variations in the ISIs of the oscilla-tors (the ISIs lie on relatively small, closed orbits). This can be u nd erstood

by looking at a corresponding n etwork of excitatory ana log neurons, which

can bifurcate only to another homogeneous time-independent state.

4.4 Bursting in an Excitatory-Inhibitory Pair of IF Neurons. Now letus consider an excitatory-inhibitory pair of IF neurons with > 0, W11 =W22 = 0 and W12 = 2, W21 = 1 and zero axonal delays a = 0. Theanalog version of this networ k, studied in section 3.2, was shown to exhibit

periodic variations in the mean firing rate in the strong coupling regime.

It can be established from equation 4.9 that the synchronous state is stable

for sufficiently weak coupling. In order to investigate Hopf instabilities in

the strong coupling regime, we set = i in equation 4.10 and solve for(,) as a function of. For each , the sma llest solution = c determines

the H opf bifurcation p oint, which leads to a stability d iagram of the formshown in Figure 7. In contrast to the previous example, a critical coupling

for a Hopf bifurcation in the firing times exists for all . Direct numerical

solution of the system shows that beyond the H opf bifurcation p oint, the


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Figure 7: Region of stability for an excitatory-inhibitory pair of IF neuron s with

inhibitory self-interactions, > 0 and W11 = W22 = 0, W21 = 1, W12 = 2. Thecollective period of synchronized oscillations is taken to be T = ln 2. Solid anddashed curves denot e solutions of equation 4.10 for with = i, real . Cross-ing the solid bou nd ary of the stability region from below signals d estabilization

of the synchronous state, leading to the formation of a periodic bursting state.

system jumps to a state in which the two neurons exhibit periodic bursting

patterns (see Figure 8). This can be understood in terms of mode-locking

associated w ith period ic variations of the ISIs on closed attracting orbits (see

Figure 9). Sup pose that the kth oscillator has a periodic solution oflength Mk

so that n+pMkk

= nk for all integers p. If

1k

nk for all n = 2, . . . ,Mk, say,

then th e resulting spike train exhibits bursting w ith the interburst interval

equal to 1

kand the num ber of spikes per burst equal to

Mk. Although both

oscillators have different interburst intervals (11 = 12) and numbers of

spikes per burst (M1 = M2), their spike trains have the same total period,

that is,M1

n=1 n1 =

M2n=1

n2 . Due t o the periodicity of the activity, the ISIs

fall on only a number of discrete points on the orbit, which is radically

different from the case found in van Vreeswijk (1996), where the whole curve

is visited due to the quasiperiod ical natu re of the firing. (Quasiperiodicity is

also observed in the pattern formation example presented in section 4.5.)The

variation of the ISIs ni with n is compared directly with the correspondingvariation of the firing rates ofthe analog model in Figure 8.Good agreement

can be seen between the two m odels in the case of small (see Figure 8a),

but discrepancies between the tw o arise as increases (see Figure 8b). As

in the case of oscillator d eath, the occurrence of bursting through a H opf

bifurcation in the firing times appears to be quite general. For example,

suppose that we have a nonzero discrete delay a such that the synchronousstate is unstable and the antiphase state is stable for a given . We find that

the latter state also d estabilizes via a H opf bifurcation in the firing tim es for

small , leading to a periodic bursting state.


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Figure 8: Spike train dy namics for a pair of IF neurons w ith both excitatory and

inhibitory coupling corresponding to points A an d B in Figure 7, respectively.The firing times of the two oscillators are represented with lines of different

heights (marked with a +). Smooth curves represent variation of firing rate in

the analog version of the model.

Figure 9: A plot of th e ISIs (n1k , nk ) of the spike trains sh own in Figure 8a, il-

lustrating how they lie on closed periodicorbits. Points on an orbit are connected

by lines.Each triangular region is associated with only one of the neurons, high-lighting the difference in interburst intervals (see also Figure 8a). The inset is a

blowup of orbit points for one of the neurons w ithin a burst.


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Figure 10: (a) Example of a Mexican h at function w eight distribution W(k) an d(b) its eigenvalue d istribution (p).

Our analysis of a pair of excitatory-inhibitory IF neurons suggests that

bursting can arise at a network level due to strong synaptic coupling w ith-out the need for additional slow ionic currents (see Wang & Rinzel, 1995).

An alternative mechanism for generating bursts in networks of interacting

neurons is based on electrical (diffusive) coupling between cells. This has

been studied in small networks of leaky-integrator neurons with postin-

hibitory reboun d (Mulloney, Perkel, & Bud elli, 1981) and in large n etworks

of Hodgkin-Huxley neurons (Han, Kurrer, & Kuramoto, 1995). Our analy-

sis also shows that bursting can occur in th e absence of the self-interaction

terms considered in ou r p revious w ork (Bressloff & Coombes, 1998b); it is

hard to justify the presence of such terms on neurophysiological grounds.

(See section 4.6.)

4.5 Pattern Formation in a One-Dimensional Network. As our finalexample of strong-coupling Hopf instabilities in IF networks, we consider

a ring ofN = 2M + 1 neurons evolving according to equations 3.1 and 3.2with > 0, a = 0 and a w eight matrix Wij = W(i j) where

W(k) = A1 exp (k2/(221 )) A2 exp (k

2/(222 )), 0 < k M. (4.16)

W(k) = 0 for |k| > M an d W(0) = 0 (no self-interactions). We chooseA1 > A2 an d 1 < 2. W(k) then represents a lattice version of a Mexicanhat interaction function in which there is competition between short-range

excitation and long-range inhibition. A continu um version ofW(k) isp lottedin Figure 10a for illustration. This type of architecture is well known to

support spatial pattern formation in analog neural systems (Ermentrout

& Cowan, 1979; Ermentrout, 1998a). We shall show that similar behavior

occurs in the IF model.

For the given weight matrix W and homogeneous external inputs Ii = I,

i = 1, . . . , N, the network is translationally invariant. This means that oneclass of solution to the phase-locking equations 4.1 consists of travelingwaves of the form k = qk/N for integers q = 0, . . . , N 1. This type ofphase-locked solution also arises in studies of the spike-response model


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region of stability

^

0 1 2 3 4 5 6

patterns patterns

0

5

10

15

20

Figure 11: Region of stability for a ring ofN = 51 IF neurons having the weightdistribution W(k) of equation 4.16 with 1 = 2.1, 2 = 3.5, A1 = 1.77, and

k W(k) = 0. The collective period of synchronized oscillations is taken to beT = ln 1.5. Solid and dashed curves denote solutions of equation 4.18 for .Crossing the solid boundary of the stability region from below signals destabi-

lization of the synchronous state, leading to the formation of spatially periodic

patterns of activity as show n in Figure 12.

associated with the dynamics on the periodic orbits of Figure 12, which is

not resolved by the analog model.Thisis illustrated in theinsets ofFigure 13,

where we plot temporal variations in the inverse ISI (or instantaneou s firing

rate) of a single oscillator for two different values of . It can be seen that

there are deterministic fluctuations in the mean firing rate. In order to char-

acterize the size of these fluctuations, we define a deterministic coefficient

of variation CV(k) for a neuron k according to

CV(k) =

k k

2k

, (4.19)

with averages d efined by equation 4.12 for some sliding wind ow of width

P. The CV for a single neuron is plotted as a function of in Figure 13. Thisshows that the relative size of the deterministic fluctuations in the mean

firing rate is an increasing function of. For slow synapses ( 0),the CV isvery small,in dicating an excellent match between the IF and an alog models.

However, the fluctuations become m ore significant when the synapses are

fast. For IF networks with a type of disordered Mexican hat interaction,

instantaneous synapses, and stochastic external input, it is also known that

a further component to the CV arises from the amplification of correlatedfluctua tions (Usher,Stemmler,Koch, & Olami, 1994).Interestingly Softy andKoch (1992) have shown that stochastic input alone cannot account for the

high ISI variability exhibited by cortical neurons in aw ake mon keys.


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0.4

0.5

0.6

0.7

0.8

0.4 0.5 0.6 0.7 0.8

k

k

n-1

n0

015 30 45

1

2

k

a

k

k

Figure 12: Separation of the ISI orbits in phase-space for a ring of N = 51IF corresponding to point A in Figure 11 ( = 2, = 0.4). The attractor ofthe embedded ISI with coordinates, (n1k ,

nk ), is shown for all N neurons.

(Inset) Regular sp atial variations in the long-term av erage firing rate ak (dashedcurve) are in good agreement with the corresponding activity pattern (solid

curve) found in the analog version of the network constructed in section 3.2,

with ak now determined by the mean firing-rate function (see equation 3.18).

One p otential app lication of the above analysis is to the study of an IF

model of orientation selectivity in simple cells of cat visual cortex. Such a

mod el has been investigated numerically by Somers, Nelson, and Sur (1996),

who consider a r ing of IF neurons with k labeling th e orientation p reference = k/N of a given neuron. The neurons interacted via synapses withspike-evoked conductance changes described by alpha functions and with

a pattern of connectivity consisting of short-range excitation and long-range

inhibition analogous to the Mexican hat function of Figure 10. Numerical

investigation show ed how the recurrent interactions led to sharpening of

orientation tuning curves. Note that the model of Somers et al. (1996) also

included shunting terms and time-depend ent thresholds; it should be pos-

sible to extend the an alysis of our article to accoun t for such features.

4.6 Bursting and Oscillator Death Revisited. In section 3.2 we pointedout that in the case of a first-order analog model, an excitatory-inhibitorypair of neurons w ithout self-interactions cannot u ndergo a Hopf bifurca-

tion. This suggests that the bursting behavior studied in section 4.4 might


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Figure 13: Plot of the coefficient of variation CV for a single IF oscillator as afunction oft he inverse rise time . All other param eter values are as in Figure 12.

For each , the oscillator is chosen to be one with the maximu m mean firing r ate.

(Insets) Time series show ing v ariation of th e inverse ISI (n)1 with n for twoparticular values of.

be sensitive to th e rise time of synaptic response even in the case of slow

synaptic decay. In order to explore the d istinct roles played by the rate of

rise an d fall of synap tic interactions, we generalize the alpha function of

equation 3.3 by considering t he d ifference of exponen tials

J(t) = 122 1

e1t e2t(t). (4.20)In the limit 2 1 , this reduces to the alpha function (see equa-

tion 3.3). If1 > 2, then the asymptotic behavior of a synapse is approx-

imately given by e2t, and the time to peak is tp = [1 2]1 ln (1/2).

The roles of1 an d 2 are reversed if2 > 1.

In Figure 14a, we p lot the critical coup ling for a discrete Hop f bifurcation

in the firing times for the excitatory-inhibitory pair previously analyzed in

section 4.4. Now, however, the delay distribution J(t) is taken to be givenby equ ation 4.20, with 2 = 0.2 and 1 a free parameter. Since the synaptic

decay time is relatively slow, we expect there to be a good match between

the IF and an alog mod els. This is indeed found to be the case, as illustrated

in Figure 14a, which shows that the critical coupling for bursting in the

analog an d IF mod els is in very close agreement. More surpr ising, it is seenthat bu rsting persists even for large values of 1, that is, for fast synaptic

rise times. (Bursting still occurs even when 1 is of the ord er 100 or more.)

One might still expect the frequency of the oscillations to approach zero


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1

bursting

synchrony

(a)

||

oscillator death

synchrony

(b)

1

Figure 14: Critical coupling for a Hop f bifurcation in the firing times as a func-

tion of the rise time 1 for fixed decay time 2 = 0.2. (a) Excitatory-inhibitory

pair with > 0, W11 = W22 = 0, W21 = 1, and W12 = 2. Bursting can oc-cur even for fast rise times. The corresponding critical coupling for the analog

model is shown as a dashed curve. (b) Inhibitory pair of IF neurons with < 0,

W11 = W22 = 0, W21 = W12 = 1. Critical coupling is only weakly dependenton 1.

in the limit 1 . However, this is not found to be the case, which canbe understood from the fact that the Hopf bifurcations are subcritical (cf.

Figure 4b). Finally, in Figure 14b w e p lot the critical coup ling for oscillator

death in an inhibitory pair of IF neurons. There is only a weak dependence

on 1, which is consistent with the analog model.

5 Discussion

We have presented a dynam ical theory of spiking neurons that bridges the

gap between weakly coupled phase oscillator models and strongly coupled

firing-rate (analog) models. The basic results of our article can be summa-

rized as follows:

1. A fundam ental mechanism for the desynchronization of IF neurons

with strong synaptic coupling is a discrete Hopf bifurcation in thefiring times. This typically leads to nonphase-locked behavior char-

acterized by periodic or q uasiperiod ic variations of the ISIs on closed

orbits. In th e case of slow syn apses, the coarse-grained d ynam ics can


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be represented very well by a corresponding analog mod el in which

the outpu t of a neuron is taken to be a m ean firing rate.

2. A homogeneous network ofN synchronized IF neurons with globalinhibitory coupling can d estabilize in the strong coup ling regime to a

state in w hich some of the n eurons became ina ctive (oscillator death ).

Thus the stability criterion obtained by van Vreeswijk et al. (1994) for

N= 2 is valid only in the w eak coupling regime. There exists a criticalvalue of the inverse rise time, 0(N), such that synchrony persists forarbitrarily large coupling when > 0(N) (fast synapses). Moreover,0(N) 0 as N . This establishes that the stability conditionspecified in the mod e-locking theorem of Gerstner et al. (1996) is valid

in the thermodynamic limit but no longer holds for finite networks

with slow synapses.

3. Rhythmicbursting patterns can occur in asymmetricIF networks with

a mixture of inhibitory and excitatory synaptic coupling. The ISIs

evolve on period ic orbits that are large relative to the m ean ISI within

a bu rst. There is good agreement between th e temporal variation of

the ISIs in the IF model and the variation of the firing rate in the

corresponding analog mod el.

4. A network of IF neuron s with long-range interactions can desynchro-

nize to a state with a regular spatial variation in the m ean firing rate

that is in good agreement with the corresponding analog model. The

ISIs evolve on closed orbits that are w ell separated in p hase-space.

The fine temporal structure of the IF model leads to deterministic

fluctuations of the activity patterns whose coefficient of variation is

an increasing function of the inverse rise time . In other words, fast

synapses lead to higher levels of deterministic noise.

An important qu estion that w e hope to ad dress elsewhere concerns the

extent to which the comp lex behavior exhibited by the IF networks stud ied

in this article is mirrored by more bioph ysically detailed m odels of spiking

neurons. Preliminary nu merical studies of H odgkin-Huxley n eurons sug-

gest a strong connection between the tw o classes of mod el. (See also White

et al., 1998.) The advantage of working with the IF model is that it allows

precise analytical statements to be made. Moreover, there are a number of

ways of extending ou r an alysis to take into account add itional aspects of

neuronal processing.

Theeffects ofd iffusion alongth e dendritictree ofa neuron can be incorpo-

rated into the model by taking J( ) in equa tion 3.2 to be the Greens functionof the corresponding cable equation. It is also possible to take into account

active membrane properties under the assumption that variations of thedendritic membrane p otential are small so that the channel kinetics can be

linearized along the lines developed by Koch (1984). The resulting mem-

brane imped ance ofthe den drites displays resonant-like behavior due to the


34/39


additional presence of inductances. This leads to a corresponding form of

resonant-like synchronization. In other word s, there is a strongly enhanced

(reduced) tend ency to synchronize for excitatory (inhibitory)coup ling wh en

the resonant frequency of the dend ritic membrane approximately m atches

the frequency of the oscillators.Active dend rites can also desynchronize ho-

mogeneous networks of IF oscillators with strong excitatory or inhibitory

coupling, leading to p eriodic bu rsting p atterns (Bressloff, 1999).

The response of a network to a sinusoidal external stimulus may be

studied by taking Ii = A+B sin (t) in equation 3.1,w here A, B are constants.Since the resulting dynam ical system is now non auton omou s, it is no longer

possible to specify phase locking in terms of N 1 relative phases and aself-consistent collective period (see equa tion 4.1). How ever, one can d erive

conditions und er wh ich the network is entrained to the external stimulus.

For example,in the case of1:1m ode locking,the firingtimes ofthe IFneuronsare of the form Tmj = (m j)T, with T = 2/. The absolute phases j are

determined from the set of equations

1 = (1 eT)A + (1 eT)F(i) + N

j=1

WijKT(j i), (5.1)

with KTgiven by equation 4.2an d F() = B[sin(2 ) + cos(2 )]/(1+2).

It is also possible to investigate th e stability of such solutions by a relatively

straightforward generalization of the ana lysis presented in section 4.2. An-

other interesting extension is to p: q mode locking. For example, supposethat the neurons have T-periodic firing patterns in which the jth neuronfires qj times over an interval T. In order to generate such solutions, it isnecessary to take the firing times to have the general form (Coombes &

Bressloff, 1999):

Tnj = T

n

qj

j,n m od qj

, (5.2)

where [.] denotes the integer part. This generates a set ofN

j=1 qj closed

equations for th e absolute phases j,n mo d qj. One can u se these equations

to determine the borders in parameter space where the ansatz 5.2 for mode

locking fails. The set of such bord ers will define an Ar nold ton gue d iagram,

with overlapping tongues indicating possible regions of multistability and

chaos. Similar techniques can be used to stu dy th e bursting solutions found

in section 4.4. The existence of chaotic solutions can be studied using an

extension of the notion of a Lyapunov exponent that takes into account

the presence of discontinuities in the dynamics due to reset (Coombes &

Bressloff, 1999). Incidentally, the construction of a Lyapunov exponent alsoallows one to understand the spike-time reliability of an IF neuron in re-

sponse to small, aperiodic signals as recently observed by Hunter, Milton,

Thomas, & Cowan (1998).


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Another important issue concerns the effects of noise. We expect that

sufficiently small levels of noise will not alter the basic results of this ar-

ticle. On the other hand, the presence of noise can lead to new and in-

teresting phenomena as illustrated by the nu merical study of pattern for-

mation by Usher et a l. (1994) and Usher, Stemmler, & Olami (1995). They

considered a two-dimensional network of IF neurons with short-range ex-

citation and long-range inhibition, fast synapses, and a stochastic exter-

nal input. They observed patterns of activity in the mean firing rate that

are two-dimensional versions of the patterns shown in the inset of Fig-

ure 12. It was found that in a certain parameter regime, these patterns

were metastable in the presence of noise and tended to diffuse through

the netw ork. This macroscopic dynamics resulted in 1/f power spectra andpow er law distributions of the ISIs on the m icroscopic scale of a single neu-

ron, and led to a large coefficient of variation, CV > 1. We are currentlycarrying out a detailed study of pattern formation in two-dimensional IF

networks using the methods p resented in ou r article and hop e to gain fur-

ther insight into the observations of Usher et al. (1994, 1995), in par ticular,

the interplay between noise and the underlying deterministic dynamics.

One way of including noise in the IF model or the spike-response model

is by introducing a p robability P of firing du ring an infinitesimal time t:P(U; t) = 1(U)t. To mimicm ore realisticm odels,the responsetime (U)is chosen to be large ifU < h and to vanish ifU h. This is achieved bywriting (U) = 0e(Uh), where 0 is the response time at threshold and determ ines the level of noise. The determ inisticcase is recovered in the limit

. In the case of large, homogeneous networks, one can use a mean-

field approach to reduce the dynamics to an analytically tractable form

(see Gerstner, 1995). Note that another consequence of noise is to smooth

out the firing-rate function (see equation 3.18) in the analog version of themodel.

A final aspect we would like to highlight here is the distinction between

excitable and oscillatory networks. In this article we have focused on the

latter case by taking Ii > 1 in equation 3.1 so that the network may be viewedas a coupled system of oscillators with natural periods Ti = ln [Ii/(Ii 1)].However, it is also possible for spontaneous network oscillations to occur

in the excitable regime, whereby Ii < 1 for all i = 1, . . . , N. All of the basicanalytical results presented in section 4 can be carried over to th e excitable

regime, in particular, the phase-locking equation 4.1 and the eigenvalue

equation 4.5. On the other hand, the weak coupling theory of section 3.1 is

no longer app licable since one requires a finite level of coupling to maintain

oscillatory behaviour. There have also been a number of studies of waves

in excitable spiking netw orks in both on e dimen sion (Ermentrou t & Rinzel,

1981; Ermentrout, 1998b) and two dimensions (Chu et al., 1994; Kistler etal., 1998). We are curre

Documents

Paul C. Bressloff and S. Coombes- Dynamics of Strongly Coupled Spiking Neurons