The ground state of cortical feedforward networks

The Ground State of Cortical Feed-Forward

Networks

Tom Tetzla??,Theo Geisel,Markus Diesmann Department of Nonlinear Dynamics,Max-Planck-Institut f¨u r Str¨o mungsforschung,

G¨o ttingen,Germany

Abstract

The occurrence of spatio-temporal spike patterns in the cortex is explained by mod-els of divergent/convergent feed-forward subnetworks–syn?re chains.Their excited mode is characterized by spike volleys propagating from one neuron group to the next.We demonstrate the existence of an upper bound for group size:Above a crit-ical value synchronous activity develops spontaneously from random?uctuations. Stability of the ground state,in which neurons independently?re at low rates,is https://www.docsj.com/doc/d114353789.html,parison of an analytic rate model with network simulations shows that the transition from the asynchronous into the synchronous regime is driven by an instability in rate dynamics.

Key words:Spike patterns,Spike synchronization,Syn?re chain,Pulse packet, Rate model

1Introduction

Cortical neurons in vivo exhibit ongoing spiking activity at rates of a few Hz. In the presence of this background“noise”subnetworks are able to process rel-evant information[3].The syn?re model[1]was introduced to explain the task related occurrence of precise spatio-temporal spike patterns[7].A syn?re chain consists of groups of w neurons which are linked by divergent/convergent con-nections in feed-forward manner.In a completely connected chain each neuron in group i receives w inputs from the preceding group i?1and projects to all w neurons of the succeeding group i+1.We assume coupling to be purely ?Corresponding author.MPI f¨u r Str¨o mungsforschung,Postfach2853,37018 G¨o ttingen,Germany.

Email address:tom@chaos.gwdg.de(Tom Tetzla?).

Preprint submitted to Elsevier Science15October2001

A B

Fig.1.Synchronous activity in syn?re chains of group sizes w=100(A)and w=400(B)arising from asynchronous states.Each box represents the activity of a speci?c group.Each vertical position in a box is reserved for the spikes(marked by dots)of a particular neuron.(A)Synchronous activity following the injection of a slowly increasing current(curve)into the neurons of the?rst group.(B)Pulse packets developing from spontaneous discharges.

excitatory(in contrast,[6]).The chain is regarded to constitute a subgraph embedded into a large cortical network.The excited mode of a syn?re chain is characterized by propagating volleys of synchronized spikes(e.g.Fig.1A). Stable propagation of such“pulse packets”requires a minimal number of neu-rons per group[5].The functionality of syn?re chains[1]requires a clear separation between the excited state and the ground state,in which neurons independently?re at low rates:Pulse packets should not be evoked by ran-dom?uctuations of background activity.Though the synchronous state is well described[4],little is known about the transition from the asynchronous into the synchronous regime.Fig.1A shows a network simulation in which this transition is initiated by injection of a slowly increasing subthreshold current, leading to an increase of?ring rate in the stimulated?rst group.From this elevated asynchronous state,pulse packets self organize in consecutive neuron groups.In contrast,no stimulus is applied in Fig.1B.However,even though the spike rate in the?rst few groups is at ground state level,pulse packets spontaneously develop.Apart from the stimulus,the two cases(Fig.1)dif-fer in a single parameter:While neuron groups consist of w=100neurons in Fig.1A,this value is increased to w=400in Fig.1B.Obviously,there is an upper bound for group size above which synchronous activity is spon-taneously ignited.In the following,we develop a rate model that provides a common framework for the two situations in Fig.1and allows us to predict the point of transition,leading from the asynchronous into the synchronous regime,in dependence of single neuron properties and background activity.

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2Methods

Analytical and numerical work is based on a leaky-integrate-and-?re neuron model withα-function shaped synaptic currents[5].Amplitudes of excitatory and inhibitory post-synaptic potentials(PSPs)di?er only in sign.Thus,mean μm and varianceσ2m of the membrane potential can be calculated as:μm=(K EλE?K IλI)·

u(t)dt,σ2m=(K EλE+K IλI)· u2(t)d t,(1)

where u(t)denotes the PSP andλE the(Poisson)input rate arriving at K E excitatory synapses,λI,K I respectively.With reset potential V0,spike thresh-oldθ,and membrane time constantτm the output spike rateλout can be approximated by

1λout =τr+τm

√πθ?μm√m

V0?μm

√2σ

m

e x2·[1+erf(x)]d x.(2)

The r.h.s.is the sum of absolute refractory periodτr and mean?rst-passage-time(e.g.[8]).Combining(1)and(2)yields a rate transmission function

λout=Φ(K EλE,K IλI)(3)

for stationary inputs.In a stable network state characterized by ratesλ?E and λ?I each excitatory neuron reproducesλ?E:

λ?E=Φ(K Eλ?E,K Iλ?I),(4)

inhibitory unitsλ?I,respectively.It has been shown that in random networks such attractor states can indeed exist(e.g.[2]).Here,we consider a given state (λ?E,λ?I)ful?lling(4)as the ground state of the network.In an embedded syn-?re chain the total excitatory input of each neuron in group i+1is composed of w channels arriving from the preceding group i?ring at ratesλi and con-tributionsλ?E from the excitatory background.In a basic type of embedding the chain is assumed to represent a structure on top of a random network:

λi+1=Φ(wλi+K Eλ?E,K Iλ?I)(Model I).(5)

Thus,the total number of excitatory inputs of each neuron increases with w. In a more realistic model intra-chain connections are assumed to be taken from the embedding network:

λi+1=Φ(wλi+[K E?w]λ?E,K Iλ?I)(Model II).(6)

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A

Fig.2.Fixed points of stationary rate in a syn?re chain as a function of group size w and rate perturbationλ(log-scaled).(A)Constant number of synapses from background(Model I).(B)Constant total number of inputs(Model II).Background activityλ?E=2Hz,λ?I=12.6Hz,with number of inputs K=K E+K I=20000, K E/K=0.88.Leaky I&F modelθ?V0=15mV,PSP amplitude:0.14mV, PSP rise time:1.7ms,membrane time constantτm=10ms,membrane capacity C=250pF,absolute refractory timeτr=2ms(cf.[5]).

Here,the total number of inputs remains constant.(5)and(6)describe the relation between spike rates in consecutive groups in a stationary situation. 3Results

The results of a stability analysis of iterative maps(5)and(6)with respect to w are summarized in the bifurcation diagrams in Fig.2.For both embedding schemes the system exhibits a stable?xed point at low rates for small group sizes(w<130).At moderate w an additional attractor at high rates is created. While the lower attractor in Model I is?nally annihilated by collision with the unstable?xed point,bifurcation is transcritical in Model II.Thus,the requirement of a stable ground state in both cases implies an upper bound for w.This approach is only valid as long as all involved processes are stationary, i.e.as long as rates remain constant in time.The model must fail as soon as the system switches into the synchronous mode.To check the relevance of our considerations we perform a set of network simulations(Model II)in which stationary rate perturbations are applied to the neurons of the?rst group. For perturbations relaxing to the ground state in the rate model,the output rates of the10th group are in good agreement with predictions.However,at a critical rate perturbation the system enters the synchronous regime where spike packets travel through the network.Spike rate obtained by averaging over time is no longer a useful measure.The measured rate of pulse packet occurrence in the10th group is visualized in Fig.3A as a function of rate perturbationλand group size w.Fixed points of the rate model are shown superimposed.Panels(A),(D)and(E)refer to di?erent background states (λ?E,λ?I).Inhibitory background ratesλ?I were adjusted in order to ful?ll the self-consistency condition(4).The three cases are distinguished by di?erent

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A

D

00000

ρ(H z )ρ(H z )Fig.3.Pulse packet rate ρ(gray coded contours,measured in 10th group)as a func-tion of group size w and rate perturbation λ(log-scaled)applied to the ?rst group.

Background activity:(A)λ?E =2Hz ,λ?I =12.6Hz,(D)λ?E =1Hz ,λ?I =4.8Hz,

(E)λ?E =10Hz ,λ?I =72.2Hz (other parameters as in Fig.2).Stable (solid curves)

and unstable ?xed points (dashed curve)of the corresponding rate model (Model II)are superimposed.White area indicates regime in which no simulations were per-formed.(B)Vertical cross-section of (A)at w =200.(C)Horizontal cross-section of (A)at λ=3Hz.In both panels simulation results (dots)are connected by gray lines,vertical dashed line indicates position of the unstable ?xed point of the rate model.

membrane potential characteristics (1).Surprisingly,the transition from the asynchronous into the synchronous state coincides with the unstable ?xed point of the rate model (cf.Fig.3B,C).Furthermore,Fig.3reveals that the critical group size,at which the ground state looses its stability,decreases with increasing background rate.At high background rates the available range between the lower and the upper group size shrinks to zero (not shown),rendering background rate an additional bifurcation parameter.

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4Discussion

The?ndings summarized in Fig.3suggest that asynchronous stimulation of syn?re activity is caused by an instability in rate dynamics.Note that the rate model correctly predicts the basin of attraction of the ground state.However, the nature of the high-rate regime di?ers dramatically from an asynchronous state.Ignition of synchronous activity in syn?re chains does not necessarily require rate instability if correlations are large enough[4].Therefore,we sug-gest that in our network model rate instability occurs before correlations due to shared input are large enough to elicit syn?re activity.Synchronous activ-ity arises as the result of a rate induced increase of correlation:a rapid rate increase in successive groups(following rate instability)eventually leads to correlations large enough to ignite syn?re activity.For parameters ensuring rate stability of the ground state correlations seem to be negligible.Stability of the asynchronous ground state imposes an upper bound on group size.In the present analysis we have assumed stationary background activity.Future work has to address feedback of activity from the chain into the embedding network.Feedback to inhibitory populations may stabilize the systems ground state leading to a predominant role of correlations.

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