Estimating brain connectivity using copula Gaussian graphical models

Electroencephalogram (EEG) has been widely used to study cortical connectivity during acquisition of motor skills. Previous studies using graphical models to estimate sparse brain networks focused on time-domain dependency. This paper introduces graphical models in the spectral domain to characterize dependence in oscillatory activity between EEG channels. We first apply a transformation based on a copula Gaussian graphical model to deal with non-Gaussianity in the data. To obtain a simple and robust representation of brain connectivity that explains most variation in the data, we propose a framework based on maximizing penalized likelihood with Lasso regularization utilizing the cross-spectral density matrix to search for a sparse precision matrix. To solve the optimization problem, we developed modified versions of graphical Lasso, Ledoit-Wolf (LW) and the majorize-minimize sparse covariance estimation (SPCOV) algorithms. Simulations show benefits of the proposed algorithms in terms of robustness and accurate estimation under non-Gaussianity and different structures of high-dimensional sparse networks. On EEG data of a motor skill task, the modified graphical Lasso and LW algorithms reveal sparse connectivity pattern among cortices in consistency with previous findings. In addition, our results suggest regions over different frequency bands yield distinct impacts on motor skill learning.