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

2026-01-20

AI Summary

This work proposes a geometric state-space neural network that integrates state-space modeling with Riemannian geometry to capture the intrinsic dynamics of brain functional connectivity. Existing approaches often neglect the Riemannian manifold structure of symmetric positive-definite (SPD) functional connectivity matrices, thereby failing to characterize the geometric nature of the brain as a self-organizing system. By directly modeling the temporal evolution of functional connectivity on the SPD manifold, the proposed method establishes a manifold-aware recurrent framework that enables smooth and geometrically consistent tracking of high-dimensional brain state trajectories. Experiments demonstrate that the model accurately captures task-evoked brain state transitions and effectively identifies early biomarkers for Alzheimer’s disease, Parkinson’s disease, and autism spectrum disorder. Furthermore, it exhibits strong generalization performance across multiple action recognition benchmarks.

Technology Category

• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

State-space models (SSMs) have become a cornerstone for unraveling brain dynamics, revealing how latent neural states evolve over time and give rise to observed signals. By combining the flexibility of deep learning with the principled dynamical structure of SSMs, recent studies have achieved powerful fits to functional neuroimaging data. However, most existing approaches still view the brain as a set of loosely connected regions or impose oversimplified network priors, falling short of a truly holistic and self-organized dynamical system perspective. Brain functional connectivity (FC) at each time point naturally forms a symmetric positive definite (SPD) matrix, which resides 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 GeoDynamics, a geometric state-space neural network that tracks latent brain-state trajectories directly on the high-dimensional SPD manifold. GeoDynamics embeds each connectivity matrix into a manifold-aware recurrent framework, learning smooth and geometry-respecting transitions that reveal task-driven state changes and early markers of Alzheimer's disease, Parkinson's disease, and autism. Beyond neuroscience, we validate GeoDynamics on human action recognition benchmarks (UTKinect, Florence, HDM05), demonstrating its scalability and robustness in modeling complex spatiotemporal dynamics across diverse domains.

Problem

• brain dynamics
• Riemannian manifold
• functional connectivity
• state-space models
• SPD matrices

Innovation

• Riemannian manifold
• state-space model
• symmetric positive definite matrix
• geometric deep learning
• brain dynamics