Hopf bifurcation control of the M–L neuron model with type I

Abstract

Many neurological diseases are known to be caused by bifurcations induced by a change in the values of one or more regulating parameter of nervous systems. The bifurcation control may have potential applications in the diagnosis and therapy of these dynamical diseases. In this paper, a washout filter-aided dynamic feedback controller composed of the linear term and the nonlinear cubic term is employed to control the onset of Hopf bifurcation in the Morris–Lecar (M–L) neuron model with type I. It is shown that the linear term determines the location of the Hopf bifurcation, while the nonlinear cubic term regulates the criticality of the Hopf bifurcation, preventing it from occurring in a certain range of the externally applied current. The relationships among the externally applied current, the linear control gain and the reciprocal of the filter time constant are further systematically analyzed, which help to make the best choice from the feasible parameter space to achieve our control task. Simulation results are provided to illustrate the effectiveness of the proposed methods.