How high dimensional neural dynamics are confined in phase space

TL;DR

This paper provides an analytic description of how high-dimensional neural dynamics are confined in phase space, revealing an M-shaped region with boundaries that shift with synaptic strength and the emergence of arch-shaped confinement in chaotic regimes.

Contribution

It introduces a novel analytic framework characterizing the geometric shape of neural dynamics confinement in phase space, applicable across different dynamical regimes.

Findings

Confinement region is M-shaped with two sharp boundaries and a flat middle.

Shape remains qualitatively similar despite increasing synaptic strength.

Arch-shaped confinement emerges in deep chaotic regimes.

Abstract

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.