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

Tingting Dan, Jiaqi Ding, Guorong Wu

TL;DR

GeoDynamics is introduced, a geometric state-space neural network that tracks latent brain-state trajectories directly on the high-dimensional SPD manifold, demonstrating its scalability and robustness in modeling complex spatiotemporal dynamics across diverse domains.

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.

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

TL;DR

GeoDynamics is introduced, a geometric state-space neural network that tracks latent brain-state trajectories directly on the high-dimensional SPD manifold, demonstrating its scalability and robustness in modeling complex spatiotemporal dynamics across diverse domains.

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.
Paper Structure (37 sections, 38 equations, 5 figures, 7 tables)

This paper contains 37 sections, 38 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Classical State Space Model
  4. Riemannian Geometry on the SPD Manifold
  5. Group action and isometric "translation".
  6. Weighted Fréchet mean (wFM).
  7. SPD convolution on $\mathcal{M}$.
  8. Method
  9. State Update and Observation on the SPD Manifold
  10. Weighted Fréchet mean aggregation.
  11. Group action translation.
  12. SPD-Preserving Attention
  13. Summary of the GeoDynamics Framework
  14. Experiments
  15. Experimental Setup
  16. ...and 22 more sections

Figures (5)

  • Figure 1: The architecture of RNNs (a) typically relies on a Multi‐Layer Perceptron (MLP) to project the system state space into the output space, where various downstream tasks are then performed. These models operate entirely within Euclidean space. In contrast, vanilla SSMs (b, black solid box) incorporate two ordinary differential equations (ODEs), the state equation (upper) and observation equation (lower), which can directly perform downstream tasks through the inferred observed output, also within Euclidean space, focusing primarily on temporal information. Our geometric deep model of SSM (b, purple dashed box) extends this approach by capturing both temporal and spatial information, operating on a manifold space.
  • Figure 2: The construction of SPD matrices for HBC (a) and HAR (b) datasets. Learning the system dynamics on manifold space as illustrated in (c).
  • Figure 3: Evaluation performance for different methods across HBC datasets. The best performance is highlighted in bold, while the second-best is underlined.
  • Figure 4: Critical connections from SPD-preserving attention map on HBC datasets.
  • Figure 5: Results on HAR dataset.