Multivariate autoregressive modeling and granger causality analysis of multiple spike trains

Abstract

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.

Figure 1

Structure of simulated networks (a)–(c) of IF neurons. Morphologies of the three networks simulated to provide data for the analysis of realistic local networks (see Figure 4 for the results). Granger causality analysis was performed on the subnetworks marked by dashed boxes. The additional neurons were used to balance excitatory and inhibitory input to the analyzed cells.

Figure 2

Fitting multiple spike trains with an MVAR-Nonlinear-Poisson model. (a) Schematic representation of the model. (b) Rate processes λ(t) and corresponding Poisson spike trains. (c) Correlation structures of the original spike trains and the estimated model.

Figure 3

Granger causality analysis in a 2D case. (a) Schematic of the original model and its observed correlation structure. (b) Granger causality analysis of the spike trains revealed a significant causal connection from cell 1 to cell 2. The numbers represent the pairwise linear Granger causality coefficients for each connection and their P-values (in parenthesis). (c) Correlation structure of the model used as a “null” for the statistical test, with no connection between cell 1 and cell 2.

Figure 4

Granger causality analysis of different 3D models. (a, d) Schematic of the models and their correlation structures. (b, e) Pairwise Granger causality analysis (using (6)) is incapable of distinguishing between direct and indirect connections, and three causal connections are deemed significant. (c, f) Three dimensional Granger causality analysis reveals the connectivity structure that fits the original models. Granger causality coefficients were calculated using (8). The statistical test was performed using surrogate data.

Figure 5

Nonlinearity mismatch has a minor effect on the estimation of the MVAR model. MVAR-N model was estimated from spike trains generated using different nonlinearities (absolute value (dotted line), square (dashed line), or exponential (solid line)). Estimation was done using the exponential nonlinearity in all the cases. The impulse response of the linear MVAR model was affected only slightly by such nonlinearity mismatch. Kernel in row #m and in column #n represents the response of channel #m on impulse input to channel #n.

Figure 6

Granger causality analysis of realistic Izhikevich-type networks of neurons. Granger causality analysis was applied to reconstruct the connectivity structure in three basic architectures—see Figure 1 for the full structure of the simulated populations. (a) Two cells with unidirectional causal connection. (b) Three cells with sequential connection. (c) Three cells with “common input” interaction. Note that in examples (b) and (c) the reduced 2-dimensional models do not reconstruct correctly the 3D causal relations, as expected.

Algorithm 1

Algorithm outline.