Detecting unitary events without discretization of time

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Abstract

In earlier studies we developed the ‘Unitary Events’ analysis (Grün S. Unitary Joint-Events in Multiple-Neuron Spiking Activity: Detection, Significance and Interpretation. Reihe Physik, Band 60. Thun, Frankfurt/Main: Verlag Harri Deutsch, 1996.) to detect the presence of conspicuous spike coincidences in multiple single unit recordings and to evaluate their statistical significance. The method enabled us to study the relation between spike synchronization and behavioral events (Riehle A, Grün S, Diesmann M, Aertsen A. Spike synchronization and rate modulation differentially involved in motor cortical function. Science 1997;278:1950–1953.). There is recent experimental evidence that the timing accuracy of coincident spiking events, which might be relevant for higher brain function, may be in the range of 1–5 ms. To detect coincidences on that time scale, we sectioned the observation interval into short disjunct time slices (‘bins’). Unitary Events analysis of this discretized process demonstrated that coincident events can indeed be reliably detected. However, the method looses sensitivity for higher temporal jitter of the events constituting the coincidences (Grün S. Unitary Joint-Events in Multiple-Neuron Spiking Activity: Detection, Significance and Interpretation. Reihe Physik, Band 60. Thun, Frankfurt/Main: Verlag Harri Deutsch, 1996.). Here we present a new approach, the ‘multiple shift’ method (MS), which overcomes the need for binning and treats the data in their (original) high time resolution (typically 1 ms, or better). Technically, coincidences are detected by shifting the spike trains against each other over the range of allowed coincidence width and integrating the number of exact coincidences (on the time resolution of the data) over all shifts. We found that the new method enhances the sensitivity for coincidences with temporal jitter. Both methods are outlined and compared on the basis of their analytical description and their application on simulated data. The performance on experimental data is illustrated.

Introduction

It is now generally accepted that both perceptual and motor functions are based on joint processing in neuronal networks which are widely distributed over various brain structures. However, it is much less clear, how these networks organize dynamically in space and time to cope with momentary computational demands. The concept emerged that computational processes in the brain could rely on the relative timing of spike discharges among neurons within such functional groups (von der Malsburg, 1981, Abeles, 1982, Abeles, 1991, Gerstein et al., 1989, Palm, 1990, Singer, 1993), commonly called cell assemblies (Hebb, 1949). In this view, changes of the cooperative interplay among neurons within an assembly, induced by sensory and behavioral events, should be reflected in systematic and rapid modulations of precise timing of spike occurrences in the participating neurons. An essential ingredient of the notion of coordinated ensemble activity is its flexibility and dynamic nature. To critically test if such a temporal scheme is actually implemented in the central nervous system, it is necessary to simultaneously observe the activities of many neurons, and to analyze these activities for signs of temporal coordination. The opportunity to decipher the functional cooperativity among neurons was entranced by the recent development of new technologies for recording multiple single-neuron activities in brain structures of behaving animals. Associated with this development, new computational tools were designed to analyze and interpret the large amount of information in such multichannel recordings.

In the conceptual framework of distributed networks, it is particularly intriguing to trace the temporal evolution of cooperative neuronal activity within such networks. For that purpose, the Joint-peri-stimulus-time histogram (JPSTH, Aertsen et al., 1989), Gravitational Clustering (Gerstein and Aertsen, 1985), and various flavours of Hidden Markov Models (HMM, Abeles et al., 1995, Gat et al., 1997) have been developed. However, although the dynamics of synchronicity can be observed as a function of time by averaging over trials, it has so far not been possible to analyze individual spike coincidences on a trial by trial basis. With this goal in mind, we recently developed the ‘Unitary Events’ analysis (Grün, 1996, Grün et al., 1999a, Grün and Aertsen, 1999b) for detecting the presence of conspicuous spike coincidences in multiple single neuron recordings and evaluating their statistical significance. Basically, this technique allows one to determine those spike coincidences which violate the assumption of independence of the participating neurons and insofar are an expression of the activation of a functional cell assembly (Aertsen et al., 1991). The statistical null-hypothesis is formulated on the basis of the individual firing probabilities of the participating neurons. By means of this null-hypothesis, it is possible to calculate the number of expected coincidences. As a result of calculating the statistical significance of the difference between expected and measured coincidences, one obtains both the amount and the moment in time of the significant excess coincident spiking activities (‘Unitary Events’; for technical details, see Appendix B). To account for the dynamics of synchronized activity as well as to deal with non-stationarities in the firing rate of the neurons, synchronicity is estimated on the basis of small time segments, by sliding a boxcar window in steps along the data. This technique allows one to describe a detailed relationship between spike synchronization, rate variations and behaviorally relevant events (Riehle et al., 1997). Effectively, UE-analysis is strongly related to evaluating the dynamics and significance of the diagonal trace of the JPSTH-matrix (Aertsen et al., 1989). Also the significance measure used (the modified surprise function) is very similar. UE-analysis deviates from JPSTH-analysis, however, in that it is not satisfied with detecting significant dynamic correlation per se, but makes a first step towards recovering the actual events that constitute this dynamic correlation.

The usual time resolution of the data aquisition in electrophysiological recordings is less then or equal to 1 ms. There is recent experimental evidence that the timing accuracy of spikes, which might be relevant for higher brain functions, can be as precise as 1–5 ms (Abeles et al., 1993, Riehle et al., 1997). To detect synchronous spikes on a particular time scale, we sectioned the observation interval into short disjunct time slices (‘bins’) (disjunct binning, DB). After such binning, binary processes were constructed from each spike train by assigning a ‘1’ to time slices in which one or more spikes occurred (‘clipping’) and ‘0’ to time slices in which no spike occurred. Although coincident spiking events can reliably be detected by using such discretized process, the method looses sensitivity for higher temporal jitter of the coincident events (Grün, 1996). This is mainly due to the non-linear effect of binning and clipping of the single spike trains, on the one hand, and the application of the same binning grid over multiple spike trains, on the other.

Here we present an alternative approach, the ‘multiple shift’ method (MS). This method overcomes the need for binning, and thereby treats the data in their (original) high time resolution. Technically, coincidences are detected by shifting the spike trains against each other over the range of allowed coincidence width and integrating the number of exact coincidences (on the time resolution of the data) over all shifts.

We first present the analytical descriptions for both methods (a list of symbols used is given in Appendix A). We then compare the two methods using surrogate data sets. Conceptually, we separate spike trains in ‘background’ spikes, i.e. uncorrelated spikes, and spikes being involved in coincidences. Thus, in our simulations we first generate independent spike trains with a given background rate, and then ‘inject’ coincident spikes of a given coincidence width (tolerance) into both trains. The firing rate levels are chosen in physiologically plausible ranges. Our analytical descriptions are constrained to low coincidence rate levels, such that interactions of injected coincidences are neglectable.

In a second step, we compare the two methods for their reliability to detect near-coincidences. It turns out, that MS is more sensitive to detect near-coincidences than DB. To illustrate the performance of our method, we apply MS to a particular experimental data set. Based on the result of this analysis, we set up a simple model for the composition of the coincident spiking activity. Using our analytical description, we estimated the parameters of our model from the experimental result and verified our assumptions by simulations.

For simplicity, we only discuss two parallel processes, but this work serves as a basis for an expansion to M parallel processes.

Section snippets

Simulation experiment

In order to calibrate and test the analysis methods, we used simulated spike trains in which we could control the firing rates of the ‘neurons’ and the temporal precision of the spike coincidences (the coincidence width). Here, firing rates are composed of both background activity and coincident activity (see Fig. 1).

In a first step, spike trains of time duration T were generated independently as Poisson spike trains, simulating uncorrelated background activity. At each instant of time a random

Detection of near-coincidences

To enable our analysis to detect coincident spike events in simultaneously recorded spike trains with a certain tolerance regarding coincidence precision, we generate a new process on a less restrictive time scale by sectioning the observation interval T intoNb=Tb·hdisjunct time segments (bins) of width b (in units of the original time resolution h). In order to treat this process as a binary process (possible outcomes within a bin ϵ{0, 1}) data are clipped to 1 in the case of more than one

Detection of near-coincidences

In the alternative method of multiple shifts (cf. Fig. 6), the simultaneously recorded spike trains are analyzed for coincident events on their ‘recording’ time resolution h. Spikes that occur at the same time in the parallel spike trains are counted as coincident events. To account for near-coincidences, the second spike train is shifted against the first in steps of h up to ±b′. For each shift, the ‘exact’ coincidences are counted, the sum over all shifts yields the observed coincidence count

Comparison of methods: sensitivity for near-coincidences

The performance of the disjunct binning (DB) and the multiple shifts (MS) method is compared in relation to their ability to detect excess coincidences with respect to various coincidence widths of injected synchronous activity. The joint-surprise S is used as a measure for evaluating the sensitivity for excess coincidences. It compares the observed number of coincidences with the expected number of coincidences, based on the assumption of independent processes. Thus, the two methods are

Conclusions

We examined two alternative methods, the disjunct binning and the multiple shifts, to detect excessive coincidences and their coincidence width. The performance of these methods was tested using surrogate data sets. In the case of disjunct binning, the number of detected coincidences is reduced considerably (for small bin sizes) by splitting coincidences due to the application of the binning grid. This effect decreases with increasing bin size. Binning followed by clipping leads to an increase

Acknowledgements

We thank Stefan Rotter for stimulating discussions and Wolf Singer for constructive comments. Funding was received from the German-Israel Foundation for Research and Development (GIF: MD, AA), the Deutsche Forschungsgemeinschaft (DFG: MD, AA), the French ‘MENRT’ (FG), ‘Mobilité Internationale’ (FG) and ‘Aires culturelles’ programs (FG).

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