Sparse system identification for discovering brain connectivity from fMRI time series

This paper presents a convex framework for problems of fitting multivariate autoregressive (AR) models that cooperate Granger causality constraints to fMRI time series which describe the dynamics of the human brain activity level. The Granger causality characterizes a relationship structure of variables in the system and can be explained from a common zero pattern of AR coefficients. We present two important model estimation problems that can be expressed as constrained least-squares and ℓ-type regularized least-squares formulations. We show that the first problem has a closed-form solution, while the second one admits a group lasso formulation which can be solved efficiently by a convex optimization technique. In combination with model selection criteria, these two problems produce a sparse AR model whose coefficients´s sparsity can reveal a Granger causal inference illustrated as a graphical model for fMRI time series. We verify the proposed method on simulated data sets and find that a desirable performance is obtained if the graph underlying the true model is sparse. The experiment result on an fMRI data set shows that this method leads to a reasonable graphical model of brain dynamics which can be a useful guideline for further studies in neuroscience.