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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.<\/jats:p>","DOI":"10.1162\/neco_a_01389","type":"journal-article","created":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T23:21:08Z","timestamp":1619133668000},"page":"1751-1817","update-policy":"http:\/\/dx.doi.org\/10.1162\/mitpressjournals.corrections.policy","source":"Crossref","is-referenced-by-count":4,"title":["From Univariate to Multivariate Coupling Between Continuous Signals and Point Processes: A Mathematical Framework"],"prefix":"10.1162","volume":"33","author":[{"given":"Shervin","family":"Safavi","sequence":"first","affiliation":[{"name":"MPI for Biological Cybernetics, and IMPRS for Cognitive and Systems Neuroscience, University of T\u00fcbingen, 72076 T\u00fcbingen, Germany shervin.safavi@tuebingen.mpg.de"}]},{"given":"Nikos K.","family":"Logothetis","sequence":"additional","affiliation":[{"name":"MPI for Biological Cybernetics, 72076 T\u00fcbingen, 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"}]},{"given":"Michel","family":"Besserve","sequence":"additional","affiliation":[{"name":"MPI for Biological Cybernetics and MPI for Intelligent Systems, 72076 T\u00fcbingen, Germany michel.besserve@tuebingen.mpg.de"}]}],"member":"281","published-online":{"date-parts":[[2021,6,11]]},"reference":[{"key":"2021090820494178900_B1","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-68560-1","author":"Aalen","year":"2008","journal-title":"Survival and event history analysis: A process point of view"},{"journal-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","year":"1972","author":"Abramowitz","key":"2021090820494178900_B2"},{"issue":"5","key":"2021090820494178900_B3","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1006934","article-title":"Uncovering functional signature in neural systems via random matrix theory","volume":"15","author":"Almog","year":"2019","journal-title":"PLOS Computational Biology"},{"journal-title":"An introduction to random matrices","year":"2010","author":"Anderson","key":"2021090820494178900_B4"},{"key":"2021090820494178900_B5","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/978-1-4419-5675-0_4","volume-title":"Analysis of parallel spike trains","author":"Ashida","year":"2010"},{"key":"2021090820494178900_B6","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.neuroimage.2013.02.008","article-title":"A note on the phase locking value and its properties","volume":"74","author":"Aydore","year":"2013","journal-title":"NeuroImage"},{"issue":"3","key":"2021090820494178900_B7","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1214\/07-AIHP118","article-title":"Central limit theorems for eigenvalues in a spiked population model.","volume":"44","author":"Bai","year":"2008","journal-title":"Annales de l'Institut Henri Poincar\u00e9, Probabilit\u00e9s et Statistiques"},{"key":"2021090820494178900_B8","first-page":"425","article-title":"Large sample covariance matrices without independence structures in columns","volume":"18","author":"Bai","year":"2008","journal-title":"Statistica Sinica"},{"issue":"7","key":"2021090820494178900_B9","doi-asserted-by":"crossref","first-page":"2700","DOI":"10.1016\/j.spa.2015.01.010","article-title":"On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries","volume":"125","author":"Banna","year":"2015","journal-title":"Stochastic Processes and Their Applications"},{"key":"2021090820494178900_B10","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.jmva.2012.04.019","article-title":"The singular values and vectors of low rank perturbations of large rectangular random matrices","volume":"111","author":"Benaych-Georges","year":"2012","journal-title":"Journal of Multivariate Analysis"},{"issue":"2","key":"2021090820494178900_B11","doi-asserted-by":"crossref","first-page":"598","DOI":"10.1214\/15-AOS1378","article-title":"Large sample behaviour of high dimensional autocovariance matrices","volume":"44","author":"Bhattacharjee","year":"2016","journal-title":"Annals of Statistics"},{"journal-title":"Probability and measure","year":"1995","author":"Billingsley","key":"2021090820494178900_B12"},{"journal-title":"Time series: Data analysis and theory","year":"1981","author":"Brillinger","key":"2021090820494178900_B13"},{"key":"2021090820494178900_B14","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1146\/annurev-statistics-022513-115545","article-title":"High-dimensional statistics with a view toward applications in biology","volume":"1","author":"B\u00fchlmann","year":"2014","journal-title":"Annual Review of Statistics and Its Application"},{"key":"2021090820494178900_B15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2016.10.005","article-title":"Cleaning large correlation matrices: Tools from random matrix theory","volume":"666","author":"Bun","year":"2017","journal-title":"Physics Reports"},{"issue":"5","key":"2021090820494178900_B16","doi-asserted-by":"crossref","first-page":"446","DOI":"10.1038\/nn1233","article-title":"Large-scale recording of neuronal ensembles.","volume":"7","author":"Buzs\u00e1ki","year":"2004","journal-title":"Nature Neuroscience"},{"key":"2021090820494178900_B17","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780195301069.001.0001","author":"Buzsaki","year":"2006","journal-title":"Rhythms of the brain"},{"issue":"6","key":"2021090820494178900_B18","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1038\/nrn3241","article-title":"The origin of extracellular fields and currents\u2013EEG, ECOG, LFP and spikes.","volume":"13","author":"Buzsaki","year":"2012","journal-title":"Nat. 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