Bifurcations in a forced Wilson-Cowan neuron pair

We investigate bifurcations of periodic solutions observed in the forced Wilson-Cowan neuron pair by both the brute-force computation and the shooting method. By superimposing the results given by both methods, a detailed topological classification of periodic solutions is achieved that includes tori and chaos attractors in the parameter space is achieved. We thoroughly explore the parameter space composed of threshold values, amplitude, and angular velocity of an external forcing term. Many bifurcation curves that are invisible when using brute-force method are solved by the shooting method. We find out a typical bifurcation structure including Arnold tongue in the angular velocity and the amplitude of the external force parameter plane, and confirm its fractal structure. In addition, the emergence of periodic bursting responses depending on these patterns is explained.

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