Elsevier

Neurocomputing

Volumes 38–40, June 2001, Pages 1141-1149
Neurocomputing

Coefficient of variation vs. mean interspike interval curves: What do they tell us about the brain?

https://doi.org/10.1016/S0925-2312(01)00480-5Get rights and content

Abstract

A number of models have been produced recently to explain the high variability of natural spike trains (Softky and Koch, J. Neurosci. 13 (1) (1993) 334). These models use a range of different biological mechanisms including partial somatic reset, concurrent inhibition and excitation, correlated inputs and network dynamics effects. In this paper we examine which model is more likely to reflect the mechanisms used in the brain and we evaluate the ability of each model to reproduce the experimental coefficient of variation (CV) vs. mean interspike interval (ISI) curves (CV=standard deviation/mean ISI). The results show that the partial somatic reset mechanism is the most likely candidate to reflect the mechanism used in the brain for reproducing irregular firing.

Introduction

The available experimental data indicating the degree of irregularity of neuronal firing is intrinsically noisy. The best analysis of such data has been performed by Softky and Koch [18], who demonstrated that real neural firing at high firing rates is consistent with a Poisson process and plotted the experimental curve of the coefficient of variation (CV) vs. mean interspike interval (ISI) (Fig. 9 in [18]; the CV is a measure of spike train irregularity defined as the standard deviation divided by the mean interspike interval). Analysis of experimental data has also been performed by Shadlen and Newsome [15] who plotted the experimental ISI histogram distribution (recorded from the area MT of an alert monkey, see Fig. 1C in [15]), which can be fitted to an exponential probability density function, pointing to an underlying generating process of Poisson type. Poisson-type firing is verified if the interspike intervals are both exponentially distributed and independent [21]. Although independence has not been verified experimentally, we will postulate that neuronal firing in cortical cells is of Poisson type. Apart from the work mentioned above [15], [18], other earlier experimental studies have shown that cortical neuron firing is highly irregular; Smith and Smith [17] investigated the spontaneous cortical activity in the biologically isolated forebrain of a cat and pointed out that trains of action potentials in this preparation sometimes represent a series of events that is almost random with respect to time. They showed that there is a large range of intervals over which the probability of occurrence of any chosen interval can often be predicted from the Poisson distribution. Similarly, Burns and Webb [5], have observed that nerve cells in the cerebral cortex of an unanesthetized mammal appear to exhibit spontaneous activity. In other words, they discharge in an irregular fashion at times which often bear no obvious relation to the events in the animal's environment. In this paper we evaluate current models of production of irregular spike trains in terms of their ability to reproduce a typical Poisson firing (limited by the refractory time). It has to be pointed out that measurements with models have been taken by selecting a long time window in a spike train produced by a simulated stationary process. In real firing spike trains, however, there is no stationary data for sufficiently long time and one needs to devise a special technique for measuring statistical properties in dynamically changing firing patterns (as done in [18]).

The aim of the work presented in this paper is primarily to identify the determinants of the highly variable firing that has been observed in neurons. Secondly, the paper aims to precisely clarify the firing mechanism of individual spikes, which might give us an insight to the controversial issue of the ‘neural code’. The neural code controversy refers to the two conflicting interpretations given to the highly irregular timing of successive action potentials. The first supports that this irregularity reflects noise and does not convey information suggesting that the neuron carries information by a mean firing rate obtained by temporarily integrating input signals [14], [15]. The second one, on the other hand, supports that this irregular timing does convey information suggesting that the neuron operates by precise processing of coincident input signals [2], [11], [18].

Section snippets

Partial somatic reset

In an attempt to model high irregularity, we have demonstrated, using a simple leaky integrator model with partial reset on the somatic membrane potential, that irregular firing can be produced at high firing rates resulting from current fluctuation detection [4], despite long somatic temporal integration times. We have also showed that partial somatic reset is a powerful parameter to control the gain of the neuron. A partial reset model (Vreset being 6mV below threshold) was also used by

Conclusions

It was pointed out in Section 2.2 that the mere statement that CV values ∈[0.5,1] neither implies that the CV vs. mean ISI curve will follow the experimental one, nor indicates that firing is Poissonian. For proving that ISIs are Poissonian, it is necessary to show that they are both exponentially distributed and independent. Poissonian spike trains will produce CV vs. mean ISI curves with the characteristic shape as in the theoretical curves in Fig. 1, Fig. 4, but that shape alone is not a

Chris Christodoulou received a B.Eng. degree in Electronic Engineering from Queen Mary and Westfield College, University of London and a Ph.D. in Neural Networks from King's College, University of London. From 1991 to 1997 he worked as a researcher in the Centre for Neural Networks, King's College. He joined Birkbeck College, University of London, as a Lecturer in October 1997 and he is also a Visiting Research Fellow at King's College. His research interests cover many aspects of neural

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      For the purposes of our study we used β = 0.91; this value of the reset parameter was chosen as it was found to produce the observed high firing irregularity at high rates by cortical neurons (Bugmann et al., 1997; Christodoulou and Bugmann, 2001). More specifically, in Christodoulou and Bugmann (2001), it was shown that with the somatic reset value set at β = 0.91, the firing ISIs at high rates are: (i) exponentially distributed and (ii) independent; in addition, in Bugmann et al. (1997), it was demonstrated that the coefficient of variation (CV) vs mean firing ISI curve with β = 0.91 shows a close similarity, firstly with the experimental one (Softky and Koch, 1992, 1993) and secondly with the theoretical curve for a random spike train with discrete time steps and a refractory time. In the respective simulations in this paper the CV was approximately equal to 0.85.

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    Chris Christodoulou received a B.Eng. degree in Electronic Engineering from Queen Mary and Westfield College, University of London and a Ph.D. in Neural Networks from King's College, University of London. From 1991 to 1997 he worked as a researcher in the Centre for Neural Networks, King's College. He joined Birkbeck College, University of London, as a Lecturer in October 1997 and he is also a Visiting Research Fellow at King's College. His research interests cover many aspects of neural networks including models, applications and computational neuroscience.

    Guido Bugmann was born in 1953 and has two children. He studied Physics at the University of Geneva in Switzerland. In 1986 he completed a Ph.D. on ‘Fabrication of photovoltaic solar cells with a-Si : H produced by anodic deposition in a DC plasma’. He then worked at the Swiss Federal Institute of Technology in Lausanne on the development of a measurement system using an ultra-sound beam and neural networks to measure the size of air bubbles in bacterial cultures. In 1989 he joined the Fundamental Research Laboratories of NEC in Japan and modelled the function of biological neurons in the visual system. In 1992 he joined Prof. John G. Taylor at King's College London to develop applications of the pRAM neuron model and develop a theory of visual latencies. In 1993 he joined the group of Prof. Mike Denham at the University of Plymouth (UK) where he is developing vision-based navigation systems for robots and investigates biological planning and working memory. Dr. Bugmann has three patents and over 90 publications. He is member of the Swiss Physical Society, The Neuroscience Society and The British Machine Vision Association.

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