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Structural equation models (SEMs) and vector autoregressive models (VARMs) are two broad families of approaches that have been shown useful in effective brain connectivity studies. While VARMs postulate that a given region of interest in the brain is directionally connected to another one by virtue of time-lagged influences, SEMs assert that directed dependencies arise due to instantaneous effects, and may even be adopted when nodal measurements are not necessarily multivariate time series. To unify these complementary perspectives, linear structural vector autoregressive models (SVARMs) that leverage both instantaneous and time-lagged nodal data have recently been put forth. Albeit simple and tractable, linear SVARMs are quite limited since they are incapable of modeling nonlinear dependencies between neuronal time series. To this end, the overarching goal of the present paper is to considerably broaden the span of linear SVARMs by capturing nonlinearities through kernels, which have recently emerged as a powerful nonlinear modeling framework in canonical machine learning tasks, e.g., regression, classification, and dimensionality reduction. The merits of kernel-based methods are extended here to the task of learning the effective brain connectivity, and an efficient regularized estimator is put forth to leverage the edge sparsity inherent to real-world complex networks. Judicious kernel choice from a preselected dictionary of kernels is also addressed using a data-driven approach. Numerical tests on ECoG data captured through a study on epileptic seizures demonstrate that it is possible to unveil previously unknown directed links between brain regions of interest.