Bifurcations of a neural network model with symmetry

TL;DR

This paper investigates how permutation symmetries in neural network models lead to different bifurcation structures, revealing distinct behaviors in all-to-all and clustered topologies as coupling strength varies.

Contribution

It provides a detailed analysis of bifurcation structures caused by symmetries in neural networks with different topologies, highlighting qualitative differences.

Findings

All-to-all network exhibits Hopf bifurcations and stable periodic orbits at high coupling.

Clustered network has no Hopf bifurcations and maintains stable fixed points at high coupling.

Symmetry considerations determine the qualitative bifurcation behavior of the networks.

Abstract

We analyze a family of clustered excitatory-inhibitory neural networks and the underlying bifurcation structures that arise because of permutation symmetries in the network as the global coupling strength $g$ is varied. We primarily consider two network topologies: an all-to-all connected network which excludes self-connections, and a network in which the excitatory cells are broken into clusters of equal size. Although in both cases the bifurcation structure is determined by symmetries in the system, the behavior of the two systems is qualitatively different. In the all-to-all connected network, the system undergoes Hopf bifurcations leading to periodic orbit solutions; notably, for large $g$, there is a single, stable periodic orbit solution and no stable fixed points. By contrast, in the clustered network, there are no Hopf bifurcations, and there is a family of stable fixed points for large $g$.