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

arXiv: https://arxiv.org/abs/2601.13570 · HTML: https://mlsanity.com/paper/2601.13570/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 83 references that are in the map.
- First posted on arXiv 20 Jan 2026.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [NTU RGB+D: A Large Scale Dataset for 3D Human Activity Analysis](https://mlsanity.com/paper/1604.02808/) (2016)
- [Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers](https://mlsanity.com/paper/2110.13985/) (2021)
- [On the Parameterization and Initialization of Diagonal State Space Models](https://mlsanity.com/paper/2206.11893/) (2022)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Liquid Time-constant Networks](https://mlsanity.com/paper/2006.04439/) (2020)
- [From Stars to Subgraphs: Uplifting Any GNN with Local Structure Awareness](https://mlsanity.com/paper/2110.03753/) (2021)
- [Brain Network Transformer](https://mlsanity.com/paper/2210.06681/) (2022)
- [Learning Dynamic Graph Representation of Brain Connectome with Spatio-Temporal Attention](https://mlsanity.com/paper/2105.13495/) (2021)
- [NeuroGraph: Benchmarks for Graph Machine Learning in Brain Connectomics](https://mlsanity.com/paper/2306.06202/) (2023)
- [A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1805.11204/) (2018)
- [BrainODE: Dynamic Brain Signal Analysis via Graph-Aided Neural Ordinary Differential Equations](https://mlsanity.com/paper/2405.00077/) (2024)
- [Tensor Representations via Kernel Linearization for Action Recognition from 3D Skeletons (Extended Version)](https://mlsanity.com/paper/1604.00239/) (2016)
- [Deep Efficient Continuous Manifold Learning for Time Series Modeling](https://mlsanity.com/paper/2112.03379/) (2021)
- [Subspace Clustering for Action Recognition with Covariance Representations and Temporal Pruning](https://mlsanity.com/paper/2006.11812/) (2020)
- [Exploring the Enigma of Neural Dynamics Through A Scattering-Transform Mixer Landscape for Riemannian Manifold](https://mlsanity.com/paper/2405.16357/) (2024)
- [Machine Learning on Dynamic Functional Connectivity: Promise, Pitfalls, and Interpretations](https://mlsanity.com/paper/2409.11377/) (2024)
- [Identification of temporal transition of functional states using recurrent neural networks from functional MRI](https://mlsanity.com/paper/1809.05560/) (2018)

## Built on by (most cited first, 1 of 1)

- [BrainLinear: A Linear Model for Brain Network Analysis in Sparse Tangent Subspaces](https://mlsanity.com/paper/2608.15266/) (2026)

## Abstract (arXiv)

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.