Hindawi Publishing Corporation Computational Intelligence and Neuroscience Volume 2010, Article ID 752428, 9 pages doi:10.1155/2010/752428 Research Article Multivariate Autoregressive Modeling and Granger Causality Analysis of Multiple Spike Trains Michael Krumin and Shy Shoham Faculty of Biomedical Engineering, Technion—Israel Institute of Technology, 32000 Haifa, Israel Correspondence should be addressed to Shy Shoham, sshoham@bm.technion.ac.il Received 3 April 2009; Revised 13 August 2009; Accepted 11 January 2010 Academic Editor: Andrzej Cichocki Copyright © 2010 M. Krumin and S. Shoham. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recent years have seen the emergence of microelectrode arrays and optical methods allowing simultaneous recording of spiking activity from populations of neurons in various parts of the nervous system. The analysis of multiple neural spike train data could benefit significantly from existing methods for multivariate time-series analysis which have proven to be very powerful in the modeling and analysis of continuous neural signals like EEG signals. However, those methods have not generally been well adapted to point processes. Here, we use our recent results on correlation distortions in multivariate Linear-Nonlinear-Poisson spiking neuron models to derive generalized Yule-Walker-type equations for fitting “hidden” Multivariate Autoregressive models. We use this new framework to perform Granger causality analysis in order to extract the directed information flow pattern in networks of simulated spiking neurons. We discuss the relative merits and limitations of the new method. 1. Introduction The analysis of multivariate neurophysiological signals at the cellular (spike trains) and population scales (EEG/MEG, LFP, and ECOG) has developed almost independently, largely due to the mathematical differences between continuous and point-process signals. The analysis of multiple neural spike train data [1] has gained tremendous relevance recently with the advent and widespread application of arrays of microelectrodes in both basic and applied Neurosciences. Furthermore, emerging optical methods for network activity imaging [2] and control [3] are likely to further compound this growth. Currently, the analysis of multichannel spike trains is still largely limited to single-channel analyses, to bivariate cross-correlation and metric-space analyses [4], and to spike train filtering (“decoding”). In contrast, much of EEG/MEG time series analysis has revolved around linear and nonlinear models and analyses that are essentially multivariate, most prominently the multivariate autoregressive (MVAR) model. The MVAR framework is associated with a powerful set of time- and frequency-domain statistical tools for inferring directional and causal information flow based on Granger’s framework [5], including linear and nonlinear Granger causality, directed transfer function, directed coherence, and partial directed coherence (see [6–8] for reviews). Scattered attempts at applying this general framework to neural spike trains have relied on smoothing the spike trains to obtain a continuous process that can be fit with an MVAR model [9– 12]. This approach has the clear disadvantage of being highly kernel dependent and of introducing unwanted distortions. The inability to estimate multivariate autoregressive models for spike trains has recently motivated Nedungadi et al. [13] to develop an alternative nonparametric procedure for computing Granger causality based on spectral matrix factorization (without fitting the data with an autoregressive model). The purpose of this paper is to bridge this divide in neurophysiological data analysis by introducing a correlation-distortion-based framework for applying multi- variate autoregressive models to multichannel spike trains. The primary aim of making this connection is to enable direct identification of causal information flow among populations of neurons using the powerful Granger causality