On the low dimensional dynamics of structured random networks

TL;DR

This paper investigates how the structure of random neural networks influences their low-dimensional chaotic dynamics, revealing a phase transition and linking network connectivity to functional behavior.

Contribution

It extends a mean-field approach to connect network structure with low-dimensional dynamics and derives the critical point for chaos in structured random networks.

Findings

Networks undergo a phase transition from silent to chaotic states.

Autocorrelation functions are confined to a low-dimensional subspace.

Derived the spectrum support of connectivity matrices with heterogeneous degrees.

Abstract

Using a generalized random recurrent neural network model, and by extending our recently developed mean-field approach [J. Aljadeff, M. Stern, T. Sharpee, Phys. Rev. Lett. 114, 088101 (2015)], we study the relationship between the network connectivity structure and its low dimensional dynamics. Each connection in the network is a random number with mean 0 and variance that depends on pre- and post-synaptic neurons through a sufficiently smooth function $g$ of their identities. We find that these networks undergo a phase transition from a silent to a chaotic state at a critical point we derive as a function of $g$. Above the critical point, although unit activation levels are chaotic, their autocorrelation functions are restricted to a low dimensional subspace. This provides a direct link between the network's structure and some of its functional characteristics. We discuss example applications of the general results to neuroscience where we derive the support of the spectrum of connectivity matrices with heterogeneous and possibly correlated degree distributions, and to ecology where we study the stability of the cascade model for food web structure.