Neurocomputing 44–46 (2002) 81–84www.elsevier.com/locate/neucom

On the �ring variability of the integrate-and-�reneurons with partial reset in the presence of

inhibitionChris Christodoulou∗;1

School of Computer Science and Information Systems, Birkbeck College, University of London,Malet Street, London WC1E 7HX, UK

Abstract

It has been reported (Phys. Rev. Lett. 82 (1999) 4731) that the �ring variability of integrate-and-�re (I&F) neurons is strongly dependent on the level of inhibitory input unlike the Hodgkin–Huxley and FitzHugh–Nagumo neurons. In this paper we demonstrate that when an I&F neuronwith random synaptic input is subjected to partial somatic reset, its �ring is only very weaklydependent on the level of inhibitory input. Therefore, in contrast to what has been suggestedabove, the I&F neuron equipped with the partial somatic reset mechanism can replace biophysicalmodels in stochastic network modelling. c© 2002 Published by Elsevier Science B.V.

Keywords: Integrate-and-�re neuron; Firing variability; Partial somatic reset; Inhibition

1. Introduction

It has recently been suggested by Brown et al. [1] that the use of integrate-and-�re(I&F) neurons in stochastic network modelling as opposed to biophysical models maycompromise the network behaviour. These authors based their suggestion on their re-sults which demonstrated that I&F neurons require strong inhibitory input in order toachieve high �ring variability unlike the Hodgkin–Huxley (HH) [6] and FitzHugh–Nagumo (FHN) [5] neurons which �re irregularly independently of the level of inhi-bition. From our previous work we note that: (i) an I&F neuron with partial somatic

∗ Tel.: +44-20-7631-6718; fax: +44-20-7631-6727.E-mail address: chris@dcs.bbk.ac.uk (C. Christodoulou).

1 Also with the Department of Electronic Engineering, King’s College, University of London, Strand,London, WC2R 2LS, UK and visitor in the Department of Computer Science, University of Cyprus, 75Kallipoleos Street, P.O. Box 20537, CY-1678 Nicosia, Cyprus.

0925-2312/02/$ - see front matter c© 2002 Published by Elsevier Science B.V.PII: S0925 -2312(02)00360 -0

82 C. Christodoulou /Neurocomputing 44–46 (2002) 81–84

reset and excitatory inputs only can produce irregular �ring [2] and that partial somaticreset is the most likely mechanism used in the brain for reproducing irregular �ring [4]and (ii) with 80% inhibition to concurrent excitation the �ring of an I&F neuron withtotal reset at high rates is nearly consistent with Poisson-type variability [3]. Inspiredby these �ndings, in this paper we are examining whether the variability and mean ofthe �ring rate of an I&F neuron equipped with the partial somatic reset mechanismis still dependent on the level of inhibitory input. Subsequently, we are readdressingthe question of whether I&F neurons can, in general, substitute biophysical models instochastic neural network modelling, despite their simpli�ed nature.

2. Models and methods

For the simulations we used a simple leaky I&F neuron with partial somatic reset[2]; the partial somatic reset is modelled as follows: when the neuron �res at timet, the potential of the capacitor resets to a value V (t + Kt) = �V (t), where � isthe reset parameter (06 �6 1) and Kt the time step. An absolute refractory periodtR =2 ms was used and the threshold voltage was set to Vth =15 mV above the restingmembrane potential. The leaky membrane capacitance is given by C = 85 pF and theresistance by R = 235 MN, giving a membrane time constant � � 20 ms (same valueas the one used by Brown et al. [1]). The neuron was subjected to random synap-tic inputs which could either be excitatory or inhibitory, producing EPSCs and IPSCs(excitatory and inhibitory postsynaptic currents) respectively. In particular, EPSCs aremodelled by triangular excitatory (+ve) current pulses of 2 ms duration and 42:5 pAamplitude while IPSCs are modelled by identical inhibitory (−ve) current pulses.The current amplitude of 42:5 pA gives a potential of 0:5 mV when it passes throughthe leaky integrator (same value as Brown et al. [1] used for the EPSPs and IP-SPs). The simulation time step Kt was set to 1ms and the system was left to run forT = 10000 ms.The number of excitatory inputs (NE) was kept constant while the number of in-

hibitory inputs (NI) was varied to give a variable range of r (where r = NI=NE). Thefrequency of the random synaptic input was altered for diOerent values of r such thatthe total mean input current (IM) had always the same value of 100 pA for all r values.In addition, for each value of r the optimal value for the somatic reset parameter �was calculated. The optimal value of � was chosen to be the one which produced acoePcient of variation (CV—measure of spike train irregularity de�ned as the stan-dard deviation divided by the mean interspike interval) closest to the theoretical onefor a random spike train with discrete time steps and a refractory period tR given byCV =

√(KtM − tR)=KtM [2] where KtM is the mean interspike interval (ISI).

3. Results

Very few or no spikes at all were produced with the lowest number of excitatoryinputs: NE = 20; 40 for the higher values of r considered of 0.8 and 0.95. The meanISI for 206NE6 100 varies between 4.07 and 12:94 ms (see Fig. 1b) and shows

C. Christodoulou /Neurocomputing 44–46 (2002) 81–84 83

0 0.2 0.4 0.6 0.8 10.5

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r ( Ratio o f I nhibitorytoExcitatoryinputs)

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Ne=100 Ne=80 Ne=40 Ne=20

0 0.2 0.4 0.6 0.8 10

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r (Ratio of Inhibitory to Excitatory inputs)

mea

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Ne=100 Ne=80 Ne=40 Ne=20

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(b)

Fig. 1. Response of a leaky I&F neuron with partial somatic reset subject to NE excitatory random inputsand a variable number of inhibitory random inputs NI, where NI = rNE; the frequency of the inputs wasaltered for diOerent values of r such that the total mean input current (IM) had always the same value of100 pA for all r values; an optimal value of the somatic reset parameter � corresponds to each value of r(see text and Fig. 2). (a) Firing variability response, CV vs. r for each value of NE. (b) Mean ISI vs. r foreach value of NE.

a very weak relation with r. Similarly for the same range of NE values, the CV isapproximately independent of r, taking values of about 0.72–0.94 (see Fig. 1a).Fig. 2 shows the optimal value for the somatic reset parameter � for each value

of r for the results presented in Fig. 1 (206NE6 100). As it can be observed fromFig. 2, the optimal somatic reset level appears to be inversely proportional to the levelof inhibitory input.

4. Conclusions

Our results show that a leaky I&F neuron equipped with partial somatic reset mech-anism, can produce high �ring variability when subjected to random synaptic input,

84 C. Christodoulou /Neurocomputing 44–46 (2002) 81–84

0 0.2 0.4 0.6 0.8 10.4

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r (Ratio of Inhibitory to Excitatory inputs)

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a(S

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Ne=100reset value per `r`

Ne=80reset value per `r`

Ne=40reset value per `r`

Ne=20reset value per `r`

Fig. 2. Optimal value for the somatic reset parameter � for each value of r for the results presented inFig. 1 (206NE6 100). This optimal value was chosen to be the one which produced a CV closest to thetheoretical one for a random spike train with discrete time steps given by CV =

√(KtM − tR)=KtM [2].

which is only weakly dependent on the level of inhibitory input. This was observedfor a range of excitatory input level (NE varying from 20 to 100). These �ndings arein contrast to the results of Brown et al. [1]. Similar weak dependence on the level ofinhibitory input is reported by Brown et al. [1] for the HH and FHN biophysical neu-ron models. Therefore, the I&F neurons when equipped with the partial somatic resetmechanism can in general substitute biophysical models when building stochastic net-work models, since their response to random synaptic input is similar. In addition wehave observed that the optimal somatic reset level appears to be inversely proportionalto the level of inhibitory input.

References

[1] D. Brown, J. Feng, S. Feerick, Variability of �ring of Hodgkin–Huxley and FitzHugh–Nagumo neuronswith stochastic synaptic input, Phys. Rev. Lett. 82 (1999) 4731–4734.

[2] G. Bugmann, C. Christodoulou, J.G. Taylor, Role of temporal integration and Tuctuation detection in thehighly irregular �ring of a leaky integrator neuron with partial reset, Neural Comput. 9 (1997) 985–1000.

[3] C. Christodoulou, G. Bugmann, Near-poisson-type �ring produced by concurrent excitation and inhibition,Biosystems 58 (2000) 41–48.

[4] C. Christodoulou, G. Bugmann, CoePcient of variation (CV) vs. mean interspike interval (ISI) curves:what do they tell us about the brain? Neurocomputing 38–40 (2001) 1141–1149.

[5] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membranes, Biophys. J.65 (1961) 81–90.

[6] A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application toconduction and excitation in a nerve, J. Physiol. (London) 117 (1952) 500–544.