On the relationship between equilibria and dynamics in large, random neuronal networks

TL;DR

This paper explores the complex relationship between equilibria and chaotic dynamics in large, random neuronal networks, revealing that equilibria are highly correlated and confined, which influences the network's collective behavior.

Contribution

It demonstrates that in large random networks, equilibria are correlated and occupy a small phase space region, linking network structure to chaotic dynamics and collective modes.

Findings

Equilibria are mostly saddle points with low-dimensional unstable manifolds.

Equilibria are strongly correlated despite random connectivity.

Chaotic dynamics are dominated by a small number of collective modes.

Abstract

We investigate the equilibria of a random model network exhibiting extensive chaos. In this regime, a large number of equilibria is present. They are all saddles with low-dimensional unstable manifolds. Surprisingly, despite network's connectivity being completely random, the equilibria are strongly correlated and, as a result, they occupy a very small region in the phase space. The attractor is inside this region. This geometry explains why the collective states sampled by the dynamics are dominated by correlation effects and, hence, why the chaotic dynamics in these models can be described by a fractionally-small number of collective modes.