Spatio-temporal autoregressive models defined over brain manifolds

Multivariate Autoregressive time series models (MAR) are an increasingly used tool for exploring functional connectivity in Neuroimaging. They provide the framework for analyzing the Granger Causality of a given brain region on others. In this article, we shall limit our attention to linear MAR models, in which a set of matrices of autoregressive coefficients A(k) (k = 1, ..., p) describe the dependence of present values of the image on lagged values of its past. Methods for estimating the A(k) and determining which elements that are zero are well-known and are the basis for directed measures of influence. However, to date, MAR models are limited in the number of time series they can handle, forcing the a priori selection of a (small) number of voxels or regions of interest for analysis. This ignores the full spatio-temporal nature of functional brain data which are, in fact, collections of time series sampled over an underlying continuous spatial manifold-the brain. A fully spatio-temporal MAR model (ST-MAR) is developed within the framework of functional data analysis. For spatial data, each row of a matrix A(k) is the influence field of a given voxel. A Bayesian ST-MAR model is specified in which the influence fields for all voxels are required to vary smoothly over space. This requirement is enforced by penalizing the spatial roughness of the influence fields. This roughness is calculated with a discrete version of the spatial Laplacian operator. A massive reduction in dimensionality of computations is achieved via the singular value decomposition, making an interactive exploration of the model feasible. Use of the model is illustrated with an fMRI time series that was gathered concurrently with EEG in order to analyze the origin of resting brain rhythms.