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.

翻译：状态空间模型已成为揭示大脑动力学的基石，它阐明了潜在神经状态如何随时间演化并产生观测信号。通过将深度学习的灵活性与状态空间模型的原则性动力学结构相结合，近期研究已能对功能神经影像数据实现强大的拟合。然而，现有方法大多仍将大脑视为一组松散连接的脑区，或施加了过度简化的网络先验，未能真正体现整体且自组织的动力学系统视角。每个时间点的大脑功能连接性自然形成一个对称正定矩阵，该矩阵位于弯曲的黎曼流形上而非欧几里得空间中。捕捉这些对称正定矩阵的轨迹是理解协调网络如何支持认知与行为的关键。为此，我们提出了GeoDynamics——一种在高维对称正定流形上直接追踪潜在大脑状态轨迹的几何状态空间神经网络。GeoDynamics将每个连接矩阵嵌入到具备流形感知能力的循环框架中，学习平滑且遵循几何规律的转移过程，从而揭示任务驱动的状态变化以及阿尔茨海默病、帕金森病和自闭症的早期标志。除神经科学领域外，我们还在人类动作识别基准数据集上验证了GeoDynamics，证明了其在跨领域建模复杂时空动力学方面的可扩展性与鲁棒性。