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A periodically forced Wilson–Cowan system. (English) Zbl 1035.92007

Summary: A Wilson-Cowan system [H. R. Wilson and J. D. Cowan, Biophys. J. 12, 1–24 (1972)],, which models the interaction between subpopulations of excitatory and inhibitory neurons, is studied for the case in which the inhibitory neurons are receiving external periodic input. If the feedback within the excitatory population is large enough, the response of the system to large amplitude, low frequency input is determined by the relative values of the excitatory threshold ${\theta }_{x}$ and the inhibitory-to-excitatory feedback parameter $b$. Feedback to the inhibitory cells is assumed to be relatively small.

In the parameter range considered, the system has two periodic attractors: a high activity state and a low activity state. It is shown that, depending on the parameter values, periodic input can produce two completely different effects; it can either initiate the high activity state or switch it off. If it is assumed that the threshold ${\theta }_{x}$ increases with increased excitatory activity, there exists a range of $b$ for which periodic input can cause bursting activity in the system.

##### MSC:
 92C20 Neural biology 92B20 General theory of neural networks (mathematical biology) 34C25 Periodic solutions of ODE 37G15 Bifurcations of limit cycles and periodic orbits 34C60 Qualitative investigation and simulation of models (ODE) 37N25 Dynamical systems in biology