Neurocomputing 26}27 (1999) 117}122

Origin of "ring varibility of the integrate-and-"remodel

Jianfeng FengComputational Neuroscience Laboratory, The Babraham Institute, Cambridge CB2 4AT, UK

Abstract

It has been reported that neurones in the visual cortex "re with a high CV (standarddeviation/mean) of interspike interval, greater than 0.5. In terms of the integrate-and-"re modelwith and without reversal potentials, we elaborate the underlying mechanism of producingspike trains with CV greater than 0.5. When the attractor of deterministic part of models isabove the threshold of the membrane potential, the "ring is mainly due to deterministic forcesand its output CV is usually lower than 0.5; whereas if the attractor is below the threshold, thegeneration of spikes results from random oscillations and the CV is usually greater than 0.5.The critical value of the number of active inhibitory synapses at which CV is greater or smallerthan 0.5 is determined, which gives a clear picture of how neurones adjust their synaptic inputsto elicit irregular spike trains. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: The integrate-and-"re model; Coe$cient of variation; Interspike intervals

1. Introduction

In vivo cell recordings show that most neurones "re in an irregular way, more orless like a Poisson process. For example the coe$cient of variation of interspikeintervals of e!erent spike trains for neurones in the visual cortex of the monkey isgreater than 0.5 [7]. A comparison between in vitro and in vivo experiments stronglysupports the assertion that the irregularity results from the intrinsic inputs of otherneurones, both inhibitory and excitatory. How do neurones manipulate their synapticinputs so that the elicited spike trains behave like a stochastic process? This isa fundamental question and is one of the central themes in computational neuro-science [8]. A better understanding of the origin of the irregularity of spike trains willhelp us to elaborate the principle of the neuronal circuitry, and to test whether the ratecoding or timing coding assumption is the fundamental building block of neuralcomputations.

0925-2312/99/$ } see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 2 3 1 2 ( 9 9 ) 0 0 0 0 6 - 5

In this paper, based upon the simple leaky integrate-and-"re model (Stein's model),we explore the origin of the appearance of CV greater than 0.5 in physiologicalparameter region. Essentially there are two kinds of mechanism to ensure that themodel "res: one is driven by a deterministic force, together with a random oscillation,and in this case CV is usually less than 0.5; the other corresponds to the case thata random oscillation causes the model to "re and in this scenario CV is normallygreater than 0.5. This suggests that CV greater than 0.5 is a result of the randomoscillation and answers the &how' question. In keeping with the theory, we estimatehow many inhibitory synapses need to be active during each epoch of neuronalactivity for the CV of e!erent spike trains to be greater than 0.5. This gives an answeron the &when' question.

For con"rmation of our theory we further consider a model with reversalpotentials. Numerical results show that in this more complex model CV greater than0.5 is also due to a random oscillation. The number of inhibitory input synapses atwhich the CV is greater than 0.5 almost coincides with that at which the randomoscillation causes the model to "re. We refer the reader to our full paper [6] for moredetails.

2. Models

The basic idea of Stein's model [9,3}5] is that neurones are integrate-and-"redevices charged with incoming EPSPs and IPSPs. The inter-arrival times of singleEPSPs and IPSPs are exponentially distributed with rate N

EjE

and NIjIrespectively,

where NE(N

I) is the number of a!erent, excitatory (inhibitory) synapses and j

E(j

I) is

the rate of EPSPs(IPSPs) propagating along each synapse.The membrane potential <

tat time t is thus governed by

d<t"!

1

c<

tdt#adNE

t!bdNI

t(1)

with <0"<

3%45, where a'0 and b'0 are the magnitude of single EPSP and IPSP,

<3%45

is the resting potential, 1/c the decay rate and NEt, NI

tare Poisson processes with

rate NEjE

and NIjI. Once the membrane potential crosses the threshold potential

<5)

a spike is elicited and<tis reset to<

3%45. We take<

3%45"0 mV,<

5)is 20 mV above

the resting potential and a"b"0.5 mV [8] for Stein's model. Usually a discreteprocess like Stein's model (a birth-and-death process [2]) is hard to deal withtheoretically and so various approximations have been sought. It is known that thefollowing di!usion process [1] serves well for such a purpose [9]:

dyt"!

1

cytdt#kdt#pdB

t(2)

with y0"<

3%45, where B

tis the standard Brownian motion and

k"aNEjE!bN

IjI, p2"a2N

EjE#b2N

IjI. (3)

118 J. Feng / Neurocomputing 26}27 (1999) 117}122

For the convenience of discussion we have "xed a few parameters NE"100 (see [8]

for a discussion of this choice), jE"j

I"100 Hz (we are going to return to this

assumption later on) and c"20.2$14.6 ms.

3. Results

We are only able to present results of Stein's model without reversal potentials heredue to the space limitation. However, similar results are true for Stein's model withreversal potentials [6].

As we have pointed out in the previous subsection it is hard to "nd an analyticalformula for the "rst exiting time of <

tor y

tfrom (!R, <

5)] that will give informa-

tive analytical results. However, a direct check on the dynamics de"ned by Eq. (1)or (2) shows that there are two essentially di!erent cases:

f <5)'kc, i.e. the threshold is above the position of the attractor y of the determinis-

tic part given by

!y/c#k"0. (4)

The process starting from <3%45

will initially be driven by the deterministic force!y

t/cdt#kdt to approach kc, oscillating with a random motion p dB

t. Once it

arrives at kc the deterministic force to depolarize or hyperpolarize the membranepotential vanishes. The random force causes the membrane potential to oscillatearound kc: if the membrane potential is above kc the deterministic force will pull itdown, while if the membrane potential is below kc it pushes the membranepotential up. The possibility of hitting the threshold is due to the purely randommotion.

f <5)(kc. This case is quite di!erent from the case above. The process will cross the

threshold driven by the deterministic force !yt/cdt#k dt, in conjunction with the

random oscillation.

When <5)(kc the generation of a spike is caused by the combination of the

deterministic force and the random oscillation. We can imagine that the e!erent spiketrains are quite regular. When <

5)'kc the emission of a spike is due to a purely

random oscillation and we would expect the e!erent spike trains to be irregular. It isnecessary to point out that this case does not correspond to the well-known theory ofrandom perturbations for a complete description) where the random force becomessmall and the trajectories of the random process concentrate around the deterministictrajectories. Only with a small probability does the random process deviate from thedeterministic process. As a consequence its CV approaches zero. In the model weconsider here the variation of the random term is p which is always greater than

aJNEjE. This large random force will always introduce &noise' into the model.

How do neurones manipulate their synaptic inputs so that <5)(kc or 'kc?

According to the de"nition of k we see that NE!N

I"<

5)/(2cj

E). Let us denote

J. Feng / Neurocomputing 26}27 (1999) 117}122 119

Fig. 1. Position of attractor y, a linear function of NI, de"ned by Eq. (4) vs. number of inhibitory synapses,

NE"100. When there are only excitatory inputs, y is above the threshold (20 mV) no matter what c is;

a better balance between excitatory and inhibitory inputs pulls it down to the resting potential 0 mV. N!Iat

which attractor is the same as the threshold is indicated by arrows for di!erent c"5.6, 10.1, 20.2 and34.8 ms (from left to right). (b) CV of interspike intervals.

N!I"N

E!<

5)/(2cj

E) as the critical point, the critical number of inhibitory synapses,

at which the dynamical attractor equals the threshold. Commencing from NI"0 (see

Fig. 1), inputs with purely excitatory synapses, we see that the position of attractor y isabove the threshold. For c"5.60 ms, N!

I"29 is the critical point (indicated by arrow

in Fig. 1): below it the position of attractor is above the threshold; above it theposition of attractor is below the threshold. For c"10.1, 20.2 and 34.8 ms,N!

I"61,80,89 are the critical points, respectively (indicated by arrows in Fig. 1). In

particular when an exact balance between excitatory and inhibitory inputs is attained,

120 J. Feng / Neurocomputing 26}27 (1999) 117}122

the position of attractor y reaches the position of resting potential: no deterministicinputs from synapses at all.

Corresponding to the pictures above, in order to see what is the e!ect of the abovetwo cases on the CV of e!erent spike trains, we carry out systematic numericalsimulations for Stein's model. We "nd that the threshold at which CV is greater than0.5 almost coincides with the requirement<

5)"kc as shown in numerical results (Fig.

1(b)). More speci"cally we have the following conclusion:

N$I!10(N!

I(N$

I#10, (5)

where N$I

is the critical point at which CV equals 0.5 as shown in Fig. 1(b)).In conclusion we see that the reason CV is greater than 0.5 is mainly due to

the random force, the increase in which is caused by the increase in inhibitorysynapses inputs. When inhibitory inputs and excitatory inputs are poorly balanced,output CV is less than 0.5, whereas a good balance ensures output CV greaterthan 0.5. CV is thus a good measurement of the underlying dynamics: CV'0.5indicates that the attractor is almost always below the threshold; CV(0.5the attractor is almost always above the threshold. Finally we want to point outthat it is impossible to assert N!

I"N$

Isince when the attractor is close to the

threshold, the e!ect we described above becomes vague because the &noise' term p dBt

is always large.

Acknowledgements

We are grateful to Martin Baxter for his valuable comments. The paper waspartially supported by BBSRC and an ESEP of the Royal Society.

References

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[2] J. Feng, The hydrodynamic limit for the reaction di!usion equation} an approach in terms of the GPVmethod, J. Theoret. Probab. 9 (1996) 285}299.

[3] J. Feng, Behaviours of spike output jitter in the integrate-and-"re model, Phys. Rev. Lett. 79 (1997)4505}4508.

[4] J. Feng, D. Brown, Spike output jitter, mean "ring time and coe$cient of variation, J. Phys. A: Math.Gen. 31 (1998) 1239}1252.

[5] J. Feng, D. Brown, Impact of temporal variation and the balance between excitation and inhibition onthe output of the perfect integrate-and-"re model, Biol. Cybernet. 78 (1998) 369}376.

[6] J. Feng, D. Brown, Coe$cient of variation greater than 0.5 how and when? Biol. Cybernet. 1999,in press.

[7] W. Softky, C. Koch, The highly irregular "ring of cortical-cells is inconsistent with temporalintegration of random EPSPs, J. Neurosci. 13 (1993) 334}350.

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[8] M.N. Shadlen, W.T. Newsome, Noise, neural codes and cortical organization, Curr. Opin. Neurobiol.4 (1994) 569}579.

[9] H.C. Tuckwell, Stochastic Processes in the Neurosciences, Society for Industrial and Applied Mathe-matics, Philadelphia, PA, 1988.

J. Feng is a senior research scientist (project leader) at the Babraham Institute,Cambridge CB2 4AT, UK. He is mainly interested in modelling neuronal systems,from single to system levels.

122 J. Feng / Neurocomputing 26}27 (1999) 117}122