GeoDynamics: A Geometric State‑Space Neural Network for Understanding Brain Dynamics on Riemannian Manifolds

State space models (SSMs) have become a cornerstone for unraveling brain dynamics, capturing how latent neural states evolve over time and give rise to observed signals. By combining deep learning's flexibility with SSMs' principled dynamical structure, recent studies have achieved powerful fits to functional neuroimaging data. However, most approaches still view the brain as a set of loosely connected regions or impose oversimplified network priors, falling short of a truly holistic, self organized dynamical system perspective. Brain functional connectivity (FC) at each time point naturally forms a symmetric positive definite (SPD) matrix, which lives on a curved Riemannian manifold rather than in Euclidean space. Capturing the trajectories of these SPD matrices is key to understanding how coordinated networks support cognition and behavior. To this end, we introduce, a geometric state space neural network that tracks latent brain state trajectories directly on the high dimensional SPD manifold.