Detecting the existence of higher-order correlations in massively parallel spike trains
Benjamin Staude1, Stefan Rotter2,3 and Sonja Gruen1,4
1Computational Neuroscience Group, RIKEN Brain Science Institute, Waco-Shi, Japan 2Theory and Data Analysis, Institute for Frontier Areas of Psychology and Mental Health, Freiburg, Germany 3Bernstein Center for Computational Neuroscience, Freiburg, Germany 4Bernstein Center for Computational Neuroscience, Berlin, Germany

The cell assembly hypothesis [1] postulates dynamically interacting groups of neurons as building blocks of cortical information processing. Synchronized spiking across large neuronal groups was later suggested as a potential signature for active assemblies [2], predicting higher-order correlations (HOCs) among the spike trains of assembly members. However, the estimation of the necessary parameters of present analysis techniques for HOCs poses serious problems, mainly because their number grows exponentially with the number of recorded neurons [3,4]. As a consequence, most attempts to detect active cell assemblies resort to pairwise correlations. Such pairwise correlations, however, do not reflect potential HOCs and are insensitive for sparse synchronous events [5]. As massively parallel extracellular recordings are becoming more and more available, the limited experimental evidence in favor of the cell assembly hypothesis has to a large extent be assigned to a lack of suitable analysis tools [6].

Here we present a novel procedure to detect HOCs in massively parallel spike trains. Based on estimates of only a few low-order cumulants of the summed activity across all neurons (the 'population histogram') we devise a statistical test for the presence of HOCs among the recorded spike trains. The test exploits the fact that absence of HOCs in a neuronal population also imposes constraints on (population-average) correlations of lower order. The latter can, however, be estimated via the respective cumulants of the the distribution of the entries in the population histogram. Under a compound Poisson assumption, where correlations of various orders are induced by 'inserting' appropriate patterns of near-synchronous spikes [7], the upper bounds for these lower order cumulants in the absence of HOCs can be derived analytically, together with the necessary confidence intervals of the respective k-statistics. This makes the test computationaly very modest and hence applicable to large amounts of data without the need for time consuming bootstrap approaches. Furthermore, the inference of HOCs from cumulants of lower order circumvents the need to estimate large numbers of higher-order parameters, making the test less susceptible to the limited sample sizes typical for in vivo recordings than previous approaches [3,4]. We illustrate the test on data which was simulated using a compound Poisson model, and find that cumulants of third order are already surprisingly sensitive for present HOCs. Furthermore, the proposed test detects HOCs even if their effects on pairwise correlation coefficients c are very small (in the range of c ~ 0.01, compare [5]).

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