Dynamic and low-dimensional modeling of brain functional connectivity on Riemannian manifolds

Modeling brain functional connectivity (FC) is key in investigating brain functions and dysfunctions. FC is typically quantified by symmetric positive definite (SPD) matrices that are located on a Riemannian manifold rather than the regular Euclidean space, whose modeling faces three challenges. First, FC can be time-varying and the temporal dynamics of FC matrix time-series need to be modeled within the constraint of the SPD Riemannian manifold geometry, which remains elusive. Second, the FC matrix time-series exhibits considerable stochasticity, whose probability distribution is difficult to model on the Riemannian manifold. Third, FC matrices are high-dimensional and dimensionality reduction methods for SPD matrix time-series are still lacking. Here, we develop a Riemannian state-space modeling framework to simultaneously address the challenges. First, we construct a new Riemannian state-space model (RSSM) to define a hidden SPD matrix state to achieve dynamic, stochastic, and low-dimensional modeling of FC matrix time-series on the SPD Riemannian manifold. Second, we develop a new Riemannian Particle Filter (RPF) algorithm to estimate the hidden low-dimensional SPD matrix state and predict the FC matrix time-series. Third, we develop a new Riemannian Expectation Maximization (REM) algorithm to fit the RSSM parameters. We evaluate the proposed RSSM, RPF, and REM using simulation and real-world EEG datasets, demonstrating that the RSSM enables accurate prediction of the EEG FC time-series and classification of emotional states, outperforming traditional Euclidean methods. Our results have implications for modeling brain FC on the SPD Riemannian manifold to study various brain functions and dysfunctions.