Equilibrium and stability of Wilson and Cowan's time coarse graining model
Résumé
The model proposed by Wilson and Cowan describes the dynamics of two interacting subpopulations of excitatory and inhibitory neurons. It can be used to model neural structures like the olfactory bulb, whisker barrels and the subthalamo-pallidal loop. In the paper, a new necessary and sufficient condition for the existence of multiple equilibria is derived for this model. Then, local asymptotic stability is characterized and related to the existence of multiple equilibria. A sufficient condition for global stability is also proposed. Finally, the case of loop delays is investigated, using a frequency-response analysis. The results obtained in the paper can be used for the stability analysis of the subthalamo-pallidal loop, a deep brain structure involved in Parkinson's disease.