Neurocomputing 26}27 (1999) 87}91

Is there a problem matching real and model CV(ISI)?

David Brown*, Jianfeng FengLaboratory of Computational Neuroscience, The Babraham Institute, Cambridge, CB2 4AT, UK

Accepted 18 December 1998

Abstract

Previous studies have claimed that (a) an exact balance between numbers of randomly timedEPSPs and IPSPs and (b) high numbers of such inputs are necessary in single neuron models toobtain a coe$cient of variation of interspike intervals in the physiological range (i.e. between0.5 and 1). We show here that these are not necessary. For the perfect integrate-and-"re model,0.5(CV(ISI)(1 for 0.82(r(0.96. For the leaky integrator with reversal potentials, CV(ISI)'0.5 whenever the ratio between the number of inhibitory and excitatory inputs, r, exceeds 0.27 fora total Poisson EPSP input rate of 10,000 Hz. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Random synaptic input; Variability of interspike interval; Integrate and "re neuron

1. Introduction

KoK nig et al. [7] recently reviewed some of the strands of argument in the debateabout whether neurones operate as integrators or coincidence detectors. In particular,they dispute the biological relevance of Shadlen and Newsome's [9] results using theintegrate-and-"re model. The latter authors show that if the input to model corticalneurones is balanced in the sense that it comprises equal numbers of IPSPs andEPSPs the coe$cient of variation, CV, of interspike interval (ISI) is broadly consistentwith an approximately random, i.e. Poissonian, temporal pattern, indicating consider-able noise in the cellular output. In their words

`in essence the random walk model with balanced excitatory and inhibitory inputsallows the neurone to behave as an integrate-and-"re device and maintain a reason-able response rate. The cost, however, is an irregular ISI. If this conjecture is correct,then the timing of output spikes is stochastic and can convey little, if any, information.a

*Corresponding author. Tel.: #44-1223-496224; fax: #44-1223-496031; e-mail: db10@cus.cam.ac.uk.

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 1 0 - 7

KoK nig et al. have two major objections to the arguments supporting this statement.(a) For real biological systems, the a!erent inputs used are unrealistically high

involving a total rate of 300*100 i.e. 30,000 Hz.(b) In a real neuronal system, excitatory and inhibitory inputs are unlikely to be

exactly balanced.However, recent systematic and quantitative analyses con"rmed by simulations of

ours [2}5] show that CV(ISI) is in the physiological range for a smaller total numberof inputs, and under substantial departures from exact balance. In fact, solving oneproblem } by examining the e!ects of departures from exact balance } also solves theother: when the level of inhibitory input is lower than the excitatory input, fewer inputneurones are needed to achieve "ring rates observed in cortical neurones. In oursimulations and analytical results, the size of EPSPs and IPSPs were the same (exceptfor the leaky integrator with reversal potentials in which case they were the same whenthe membrane potential was at the resting level), but this could easily be adjusted todeal with any di!erences in size. The ratio of the inhibitory and excitatory input ratesis denoted by r.

2. The perfect integrate-and-5re model

For the perfect integrate-and-"re (I&F) model, i.e. without leakage, and for EPSPsand IPSPs arriving according to independent Poisson processes, it can be shown[1,11] that the relationships between expected ISI, standard deviation (ISI) } alsoknown as output jitter, p

065} and hence CV(ISI) on the one hand, and r and N

5), the

number of EPSPs required to reach threshold on the other, are given by

E[ISI]"N

5)N

EjE(1!r)

,

p065

"S1#r

(1!r)3

JN5)

NEjE

,

CV(ISI)"1

JN5)S

1#r

1!r.

Two important features of the expression for CV(ISI) are as follows.

1. It is independent of the total input rate, only depending on the ratio of inhibitory toexcitatory input rates, r, and N

5). Asssume for the sake of de"niteness that N

5)"40.

Therefore, for any total input rate when r lies between 0.82 and 0.95, CV(ISI) lies inthe physiological range, i.e. between 0.5 and 1. This answers the "rst objection (a)above. We do not need a high input rate to ensure that CV(ISI) lies in thephysiological range.

2. An exact balance is not needed to ensure 0.5(CV(ISI)(1. In fact, CV(ISI) isa continuous function lying within the range 1/JN

5)to in"nity. Then for any given

value of CV(ISI), C0, there is a value of r such that CV(ISI)"C

0. As the relative

88 D. Brown, J. Feng / Neurocomputing 26}27 (1999) 87}91

numbers of inhibitory and excitatory inputs gets nearer to equality, i.e. exactbalance occurs (rP1), then CV(ISI)PR. So, for the case of exact balance,CV(ISI) is then well above the physiological range, at least for the perfectintegrator.

3. The leaky integrator

We can incorporate more biological realism into the model without altering thequalitative nature of the results. In the next two paragraphs, we extend the results tothe leaky integrator, and to the case where reversal potentials are also included.

For the leaky integrator,

dZ5"!

1

c(Z

5!<

3%45) dt#c dNE

5!c dNI

5,

where Z5is the membrane potential at time t, <

3%45is the resting membrane potential,

c is the EPSP/IPSP size, and 1/c is the membrane potential decay rate, the range ofr which ensures CV(ISI) is between 0.5 and 1 depends on c, as might be expected. Thetotal excitatory input rate is 10,000 Hz in our simulations [5]; the frequency of EPSPsis therefore one third of that used by Shadlen and Newsome. We use a range of valuesof c similar to those used by Troyer and Miller [10]: 20.2$14.6 ms. We simulated themodel for various values of c viz 5.6, 10.1, 20.2 ms and for N

5)"40. CV(ISI) is in the

physiological range for di!erent ranges of r, depending on the value of c: 0.5(CV(ISI)(1 for 0.20(r(0.31 approximately when c"5.6 ms; for 0.45( r (0.56when c"10.1 ms; and for 0.57(r(0.69 when c"20.2 ms.

When reversal potentials, <E

and <I, are included, the model becomes

dZ5"!

1

c(Z

5!<

3%45) dt#a(<

E!Z

5) dNE

5#b(<

I!Z

5) dNI

5,

where a(<E!Z

5) and b(<

I!Z

5) are the magnitudes of EPSPs and IPSPs arriving

according to Poisson processes NE5

and NI5, with rates j

EN

Eand j

IN

I, respectively. We

set the "ring threshold <5)3%

"!30 mV, the resting potential <3%45

"!50 mV,<

I"!60 mV, <

E"50 mV, and a, b are adjusted so that a(<

E!<

3%45)"

b(<3%45

!<I) "1 mV. Fig. 1 illustrates the ranges of r within which CV(ISI) is in the

physiological range for each value of c. For all values of c, CV(ISI) is approximatelybetween 0.5 and 1 when r'0.27, indicating that very substantial departures fromexact balance with this model are consistent with 0.5(CV(ISI)(1, a result alsocon"rmed in other studies [8].

4. Discussion

Results of this type have also been con"rmed in other simulations of ours for quitelow levels of input for the leaky integrator: down to total Poisson EPSP input rates of

D. Brown, J. Feng / Neurocomputing 26}27 (1999) 87}91 89

Fig. 1. Variability of output interspike interval (CV(ISI)) of the leaky integrator with reversal potentials,<

E, <

I, with membrane potential decay rate of 1/c, for various values of c spanning the range used also by

Troyer and Miller [10] i.e. 20.2$14.6 ms. Key: (]*]) c"5.6 ms; (O*O) c"10.1 ms; (#- - -#)c"20.2 ms; (***)c"34.8 ms. The total excitatory input rate is 10,000 Hz. <

5)3%"!30 mV, the resting

potential <3%45

"!50 mV, <I"!60 mV, <

E"50 mV. EPSP and IPSP sizes are a(<

E!<

3%45) and

b(<3%45

!<I) respectively, where a and b are adjusted so that a(<

E!<

3%45)"b(<

3%45!<

I) "1 mV. It is

readily seen that CV(ISI)'0.5 for r taking values above quite low thresholds.

4,000 Hz; and for some more biophysically based models, such as the classicalHodgkin}Huxley model of squid giant axon, in which case CV(ISI) is in the physiolo-gical range for a much wider range of values of r for moderate levels of synaptic input.Another study involving more detailed biophysical models of cortical pyramidalneurons has also shown a high CV(ISI) in the case of purely excitatory input [12]. Wehave also shown for the integrate and "re model [3,4] that CV(ISI) depends on the taillength of the EPSP and IPSP inter-arrival time distribution: for tails longer than theexponential (e.g. the Pareto distribution, "rst suggested as an ISI distribution byGerstein and Mandelbrot [6]) CV(ISI) is greater than for the exponential; and forshorter tail distributions (e.g. positive-normal), CV(ISI) is lower, for "xed values ofother parameters.

Thus in a wide range of model neurons, a high CV(ISI) appears to be fairly easy toachieve with physiologically plausible levels of random synaptic input.

Acknowledgements

This work was "nancially supported by the Biotechnology and Biological SciencesResearch Council and by the Royal Society.

90 D. Brown, J. Feng / Neurocomputing 26}27 (1999) 87}91

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submitted.[6] G.L. Gerstein, B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys.

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4 (1994) 569}579.[10] T.W. Troyer, K.D. Miller, Physiological gain leads to high ISI variability in a simple model of

a cortical regular spiking cell, Neural Comput. 9 (1997) 733}745.[11] H.C. Tuckwell, Stochastic Processes in the Neurosciences, Society for Industrial and Applied

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David Brown is Head of the Laboratory of Computational Neuroscience at theBabraham Institue, Cambridge, UK. Trained initially in mathematics and statis-tics, his research interests are modelling and analysis of neuroendocrine andsensory systems, statistical analysis of electrophysiological data, and more gen-erallly in stochastic models of neuronal systems.

Jianfeng Feng is a postdoctoral research scientist in the Laboratory of Computa-tional Neuroscience at the Babraham Institute, Cambridge, UK. His training andearly research at Peking University was in mathematics and probability. Hiscurrent reasearch interests are in stochastic models of neuronal systems, fromsingle cells to networks.

D. Brown, J. Feng / Neurocomputing 26}27 (1999) 87}91 91