A nonequilibrium-potential approach to competition in neural populations

TL;DR

This paper introduces a nonequilibrium potential (NEP) as a global Lyapunov function for neural network models, accurately predicting dynamics and revealing features like bistability and population bursts, without relying on symmetry constraints.

Contribution

The paper derives a novel energy landscape (NEP) applicable to a broad class of neural rate models, extending the understanding of neural dynamics beyond symmetric systems.

Findings

NEP accurately predicts stable neural dynamics

Models exhibit bistability and population bursts

Limit cycles are proven not to exist under certain conditions

Abstract

Energy landscapes are a useful aid for the understanding of dynamical systems, and a valuable tool for their analysis. For a broad class of rate models of neural networks, we derive a global Lyapunov function which provides an energy landscape without any symmetry constraint. This newly obtained `nonequilibrium potential' (NEP) predicts with high accuracy the outcomes of the dynamics in the globally stable cases studied here. Common features of the models in this class are bistability --with implications for working memory and slow neural oscillations --and `population burst', also relevant in neuroscience. Instead, limit cycles are not found. Their nonexistence can be proven by resort to the Bendixson--Dulac theorem, at least when the NEP remains positive and in the (also generic) singular limit of these models. Hopefully, this NEP will help understand average neural network dynamics from a more formal standpoint, and will also be of help in the description of large heterogeneous neural networks.