Detecting precise firing sequences in experimental data

Abstract

A precise firing sequence (PFS) is defined here as a sequence of three spikes with fixed delays (up to some time accuracy Δ), that repeat excessively. This paper provides guidelines for detecting PFSs, verifying their significance through surrogate spike trains, and identifying existing PFSs. The method is based on constructing a three-fold correlation among spikes, estimating the expected shape of the correlation by smoothing, and detecting points for which the correlations significantly protrude above the expected correlation. Validation is achieved by generating surrogate spike trains in which the time of each of the real spikes is randomly jittered within a small time window. The method is extensively tested through application to simulated spike trains, and the results are illustrated with recordings of single units in the frontal cortex of behaving monkeys. Pitfalls which may cause false detection of PFSs, or loss of existing PFSs, include searching for PFSs in which the same neuron participates more than once, and attempting to produce a surrogate with some fixed statistical property.

Introduction

Since 1981 there have been numerous studies on firing structures which report that neurons fire at precise times after long delays (Klemm and Sherry, 1981, Abeles, 1982, Abeles et al., 1983, Dayhoff and Gerstein, 1983b, Landolt et al., 1985, Legendy and Salcman, 1985, Lestienne and Strehler, 1987, Frostig et al., 1990, Villa and Abeles, 1990, Villa and Fuster, 1992, Lestienne, 1996, Prut et al., 1998, Lestienne et al., 1999, Nadasdy et al., 1999, Villa et al., 1999, Tetko and Villa, 2001b). Some of these findings were the result of a systematic search for such structures, while other studies reported occasional observations. Several methods for detecting such precise firing sequences (PFSs) have been published. Dayhoff and Gerstein (1983a) were the first to suggest such a method, based on long sequences of inter-spike intervals. However, their method was limited to patterns of a single spike train. Abeles and Gerstein (1988) suggested a way of finding such patterns among many neurons recorded simultaneously. However, their method was based on the assumption that all time intervals within a pattern are equally likely. This tends to introduce false positives when pairs of neurons show high correlations and when the same neuron participates twice (or more) in the same pattern. They also pooled together all patterns of the same complexity (say quintuplets), which tends to generate false negatives when only few of the neurons generate PFSs.

This method was improved and tested by numerous simulations of Poissonian spike trains by Tetko and Villa (1997). In this method too, all patterns of a given complexity and repetitions are pulled together.

In parallel recordings of five or more single units, one may find a large number of excessively recurring triplets, a smaller number (with fewer repetitions) of quadruplets and an even smaller number of quintuplets, etc. (Abeles and Gerstein, 1988, Abeles et al., 1993, Tetko and Villa, 1997, Baker and Lemon, 2000). Our experience has been that in order to study the relevance of such sequences, it is advisable to look at triplets: they repeat to a degree which enables examination of their relationship to behavior. Prut et al. (1998) suggested a method for detecting excessively repeating triplets of spikes which overcame the limitations mentioned above, and revealed several unexpected relationships to behavior. However, their method was based on multiple, partially dependent tests of the same data. This may result in an unknown level of confidence, and may introduces false positives. On the other hand, their method of estimating the probability of seeing a triplet by chance suffered from overfitting, which tends to produce false negatives.

A variant on the above methods was used by Tetko and Villa (2001a). They only considered patterns that appeared to be significant in several searching algorithms. They also allowed for variable precision of the various spikes in a pattern. They reported an excess of precise patterns both in the auditory cortex and in the thalamus. The drawback here is that multiple testing is carried out without evaluating the effect of such testing on the probability of obtaining false positives.

Other definitions of precise patterns have also been put forward. Oram et al. (1999) defined a PFS as a sequence of three spikes that repeated twice within a single response. Nadasdy et al. (1999) defined a pattern as a sequence of ordered firing (A followed by B, followed by C, …) disregarding the exact intervals between spikes. This paper is restricted to triplets composed of spikes from three different neurons and repeating with precisely (up to some accuracy Δ) the same intervals.

The methods and findings described above, confronted two closely related lines of critique: first, when deciding whether a certain pattern reoccurs excessively, what is the null hypothesis which must be rejected? Secondly, how does one demonstrate that such patterns do not occur by chance? Oram et al. (1999), recorded single unit activity from monkey V1 and LGN, and found many repeating triplets. However, when they generated surrogate spike trains with the same firing rate fluctuations, the same number of spikes, and the same refractoriness as in the original data, they found just as many repeating triplets. Similar findings for the primary and supplementary motor cortices were reported by Baker and Lemon (2000).

Date et al., 1998, Date et al., 1999 suggested a solution. Their solution may be phrased as follows. The null hypothesis is the assumption that spike timing accuracy does not matter up to some value W (ms). If the null is true, the timing of every spike may be randomly jittered around its true time, without affecting the relevant statistics derived from the spike train. Jittering may be accomplished within a box-car window of width W, or with other window-shapes. This approach will henceforth be referred to as the BDG approach. They further suggested testing the significance of a deviation from the null hypothesis using four steps: The first step involves computing an appropriate statistic from the real data. The second step involves jittering the spike times within a window W and re-computing the same statistic for the jittered spike train. The jittering is then repeated and the statistic computed many times. In the third step, the probability density function of the statistics as derived from the jittered spike train is estimated. Finally, if the statistic derived from the real data is located at the tail of the distribution, where the area under the tail is less than P, we can reject the null hypothesis at confidence level P. Thus the BDG approach solves both problems described above.

The BDG approach does not, however, tell us how to choose an appropriate statistic! The present study concentrates on finding just that. Identification of excessively recurring triplets of spikes is a by-product of this process.