# How high dimensional neural dynamics are confined in phase space

arXiv: https://arxiv.org/abs/2410.19348 · HTML: https://mlsanity.com/paper/2410.19348/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Oct 2024.

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

- [Neural Langevin Machine: a local asymmetric learning rule can be creative](https://mlsanity.com/paper/2506.23546/) (2025)

## Abstract (arXiv)

High dimensional dynamics play a vital role in brain function, ecological systems, and neuro-inspired machine learning. Where and how these dynamics are confined in the phase space remains challenging to solve. Here, we provide an analytic argument that the confinement region is an M-shape when the neural dynamics show a diversity, with two sharp boundaries and a flat low-density region in between. Despite increasing synaptic strengths in a neural circuit, the shape remains qualitatively the same, while the left boundary is continuously pushed away. However, in deep chaotic regions, an arch-shaped confinement gradually emerges. Our theory is supported by numerical simulations on finite-sized networks. This analytic theory opens up a geometric route towards addressing fundamental questions about high dimensional non-equilibrium dynamics.