Heterogeneous Multiscale Multivariate Autoregressive Model: existence, sparse estimation and application to...

In neuroscience, functional connectivity is a concept that can be mathematically formulated as a graph of interactions between oscillatory brain rhythms and individual neuronal activity. Typically, each graph corresponds to a cognitive state and provides insight into higher cognitive processes, such as learning. However, a model or method to assess directed interactions within and across brain oscillations and individual neuronal activity is lacking. In this article, we propose a new model called HM-MVAR (Heterogeneous Multiscale Multivariate Autoregressive) that represents linear combinations of neural interaction patterns. These patterns include phase-locking, where the amplitude of one signal is modulated by the phase of another, and power-triggered phenomena, where the amplitude of one signal is modulated by the amplitude of another. Due to the multiscale structure, we use a block version of stationarity to exhibit conditions under which the process obeying the HM-MVAR model exists and is stationary. We also propose a data-driven weighted LASSO estimator based on martingale exponential deviation inequalities, which may have intrinsic interest in probability theory. We prove that our estimator satisfies an oracle inequality and present its strong performance in realistic simulations. Finally, applying our model and method on a publicly available multiscale electrophysiological data set, we recover interactions described in the literature but also uncover new phenomena of potential interest.