From Univariate to Multivariate Coupling Between Continuous Signals and Point Processes: A Mathematical Framework

Author:

Safavi Shervin1,Logothetis Nikos K.2,Besserve Michel3

Affiliation:

1. MPI for Biological Cybernetics, and IMPRS for Cognitive and Systems Neuroscience, University of Tübingen, 72076 Tübingen, Germany shervin.safavi@tuebingen.mpg.de

2. MPI for Biological Cybernetics, 72076 Tübingen, Germany; International Center for Primate Brain Research, Songjiang, Shanghai 200031, China; and University of Manchester, Manchester M13 9PL, U.K. nikos.logothetis@tuebingen.mpg.de

3. MPI for Biological Cybernetics and MPI for Intelligent Systems, 72076 Tübingen, Germany michel.besserve@tuebingen.mpg.de

Abstract

Abstract Time series data sets often contain heterogeneous signals, composed of both continuously changing quantities and discretely occurring events. The coupling between these measurements may provide insights into key underlying mechanisms of the systems under study. To better extract this information, we investigate the asymptotic statistical properties of coupling measures between continuous signals and point processes. We first introduce martingale stochastic integration theory as a mathematical model for a family of statistical quantities that include the phase locking value, a classical coupling measure to characterize complex dynamics. Based on the martingale central limit theorem, we can then derive the asymptotic gaussian distribution of estimates of such coupling measure that can be exploited for statistical testing. Second, based on multivariate extensions of this result and random matrix theory, we establish a principled way to analyze the low-rank coupling between a large number of point processes and continuous signals. For a null hypothesis of no coupling, we establish sufficient conditions for the empirical distribution of squared singular values of the matrix to converge, as the number of measured signals increases, to the well-known Marchenko-Pastur (MP) law, and the largest squared singular value converges to the upper end of the MP support. This justifies a simple thresholding approach to assess the significance of multivariate coupling. Finally, we illustrate with simulations the relevance of our univariate and multivariate results in the context of neural time series, addressing how to reliably quantify the interplay between multichannel local field potential signals and the spiking activity of a large population of neurons.

Publisher

MIT Press - Journals

Subject

Cognitive Neuroscience,Arts and Humanities (miscellaneous)

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