Dynamic and low-dimensional modeling of brain functional connectivity on Riemannian manifolds - Mingyu Wang; Yueming Wang; Yuxiao Yang (NeuroImage)
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.