The Distribution of Unstable Fixed Points in Chaotic Neural Networks

TL;DR

This paper analytically characterizes the distribution and properties of fixed points in chaotic neural networks, revealing their spatial separation from dynamics and their influence on network behavior.

Contribution

It provides an analytical framework for understanding the distribution, eigenvalues, and influence of fixed points in chaotic neural networks, a novel insight into their structure.

Findings

Fixed points are confined to separate shells from the dynamics.

Eigenvalue spectra of fixed points are determined analytically.

Nearby fixed points influence the network's dynamics as partial attractors.

Abstract

We analytically determine the number and distribution of fixed points in a canonical model of a chaotic neural network. This distribution reveals that fixed points and dynamics are confined to separate shells in phase space. Furthermore, the distribution enables us to determine the eigenvalue spectra of the Jacobian at the fixed points. Despite the radial separation of fixed points and dynamics, we find that nearby fixed points act as partially attracting landmarks for the dynamics.