Multivariate Autoregressive Modeling and Granger Causality Analysis of Multiple Spike Trains

Algorithm 1

Algorithm outline.

(1) Estimate mean rates 𝐸[𝜆_𝑖(𝑡)]=𝐸[Δ𝑁_𝑖] and pairwise auto- and cross-correlation matrices 𝑅_{𝜆(𝑖,𝑗)}(𝑘)≜𝐸[𝜆_𝑖(𝑛)𝜆_𝑗(𝑛+𝑘)]=𝐸[Δ𝑁_𝑖(𝑛)Δ𝑁_𝑗(𝑛+𝑘)] from spike trains Δ𝑁_𝑖(𝑡)≜lim_Δ𝑡→0𝑁_𝑖(𝑡+Δ𝑡)−𝑁_𝑖(𝑡). Note that, for zero-lag autocorrelations (𝑖=𝑗;𝑘=0): 𝐸[𝜆²_𝑖]=𝐸[Δ𝑁²_𝑖]−𝐸[Δ𝑁_𝑖]/Δ𝑡. A smoothing spline can optionally be used to remove variance from the noisy spike-train correlation estimates.

(2) Estimate the nonlinearity parameters and predistort the correlation (see Table 1).

(3) Solve Yule-Walker equations (3). This step can be performed by directly inverting a Toeplitz matrix, or more efficiently and robustly using the Levinson-Wiggins-Robinson (LWR) algorithm [21].