Beyond Wilson-Cowan dynamics: oscillations and chaos without inhibition

TL;DR

This paper extends the classical Wilson-Cowan neural population model by incorporating refractory states, revealing new dynamics such as oscillations and chaos, and providing a more comprehensive understanding of neural activity.

Contribution

It introduces a mean-field model including refractory states derived from a Markov chain, explaining oscillations and chaos not captured by traditional models.

Findings

Refractory states induce new oscillatory dynamics.

The model predicts chaos in minimal neural networks.

Bifurcation analysis explains emergence of periodic solutions.

Abstract

Fifty years ago, Wilson and Cowan developed a mathematical model to describe the activity of neural populations. In this seminal work, they divided the cells in three groups: active, sensitive and refractory, and obtained a dynamical system to describe the evolution of the average firing rates of the populations. In the present work, we investigate the impact of the often neglected refractory state and show that taking it into account can introduce new dynamics. Starting from a continuous-time Markov chain, we perform a rigorous derivation of a mean-field model that includes the refractory fractions of populations as dynamical variables. Then, we perform bifurcation analysis to explain the occurance of periodic solutions in cases where the classical Wilson-Cowan does not predict oscillations. We also show that our mean-field model is able to predict chaotic behavior in the dynamics of networks with as little as two populations.