Irregularity in neocortical spike trains: Influence of measurement factors and another method of estimation

Abstract

Irregularity in the interspike interval is a common phenomenon especially in the neocortex. A measure of this random variation in the spacing between neuronal spikes is usually obtained with the coefficient of variation CV (standard deviation/mean interspike interval). In excitable cells, the standard deviation in the interspike interval can be large and the mean firing rate often fluctuates. As a result, there can be substantial variability in the value of the CV computed for the same spike train using only slightly different samples as we show. Moreover, these CV values can be comparatively meaningless unless certain conditions are met. In doing so some researchers have selectively sampled data over a stable mean while others have used a wide range of trial times or subsets thereof (capture window) to compute the CV. This has made interpretation of the raw CV cumbersome. We demonstrate that the CV has a triple sensitivity, namely, for the size of the capture window, the spike count and the refractory period. We assuage these difficulties by introducing a modified term, the coefficient of variation proportion of maximum (CVpm) that offers transportability across different experimental conditions by compensating for the triplet.

Introduction

Reliable acquisition and processing of sensory information is essential to an organism for appropriate sensory motor behaviour. The rapid flow of information in the nervous system, particularly over relatively long distances such as between the eye and the brain, is believed to propagate through a series of action potentials. As the shape of the action potential (spike) waveform is generally regarded as irrelevant, it permits reduction of the waveform to a discrete event in time. We assume that only the duration of the spike waveform (refractory period) sets the theoretical upper limit on the event density within a spike train. Three cells are shown in Fig. 1 displaying varying degrees of irregularity easily distinguishable by the naked eye. In most excitable cells, other than pacemaker cells, the interval between each spike is highly variable and generally none more so than in neocortical cells (in monkey area V5/MT & striate cortex, Softky and Koch, 1993, Shadlen and Newsome, 1994, Shadlen and Newsome, 1998; in rat sensory neocortex, Stevens and Zador, 1998; in cat area 17, Hu et al., in preparation). At the other extreme, highly regular activity with an occasional drop out or burst occurs in some cortical cells that are less often encountered (Fig. 1C). Moreover, lower motor neurones in the spinal cord of the cat show regular spacing between discharges in the case of tonic muscular contractions (Calvin and Stevens, 1968). Another extreme case of spike train regularity occurs in the brainstem of certain species of electric fish where these oscillations are used for electrolocation and communication (Moortgat et al., 1998). Several factors affect the discharge pattern of spike trains. Discharge patterns are found to differ substantially depending on the type of neurone and its location. Moreover, neurones are subject to neuromodulatory fluctuations that follow, for example the level of arousal in the natural sleep wake cycle (Nakahama et al., 1968, Noda and Adey, 1970) and focal attention (e.g., Vidyasagar, 1998). Spike irregularity, its origin and implications, is still a point of fierce conjecture. In this paper, we examine some of the constraints in measuring the irregularity by classical methods and the rationale and methodology for devising a new measure of irregularity. Recently, similar difficulties have been recognised (Barberini et al., 2001) in assessing variability of neuronal responses to repeated presentations of a stimulus, as measured by Fano factor, but similar considerations have not been given to measuring the degree of irregularity in a spike train, as we do in this study.

We observe that any irregularity or departure in the ISI from its mean level increases the value of the CV (defined in Section 2.1) and it is ordinarily “transportable” across systems. Unfortunately, this is not always the case in biological systems where the mean is not a “stationary” quantity and the variance is large, in which case the CV can become meaningless. Indeed the neural code is frequently described as a rate code where the mean is not stationary and that must impose conditions on the use of the CV in its raw form. As a result, in computing the CV, a number of techniques have been used to select regions of data for matched spike rate. Some of the procedures used included selective sampling in regions of high regularity (Calvin and Stevens, 1968), partitioning trials into matched spike rate categories or binning ISI values (Softky and Koch, 1993), or computing the CV on successive groups of three spikes (Holt et al., 1996) or more spikes (Shinomoto et al., 2002). Such efforts, while purpose specific, have rendered various CV assessments cumbersome to use for comparative purpose. In this paper, we show that the CV exhibits a dependency not only on the refractory period but also on the number of spikes within and the duration of the capture window. We compute a new term, the coefficient of variation proportion of maximum (CVpm) that is transportable and compensates for these three dependencies. Some of our findings have already been presented in abstract form (Hu et al., 2002, Chelvanayagam et al., 2002a, Chelvanayagam et al., 2002b, O’Connor et al., 2004).