Independent continuous periodic firing series to chaos in the 3-D Hindmarsh–Rose neuron circuit

Abstract

In order to study the intricate dynamics of the 3-D Hindmarsh–Rose (HR) model under periodic excitation, a semi-analytical method is proposed. This semi-analytical method shows an implicit mapping relationship by discretizing corresponding continuous nonlinear systems. Subsequently, a new phenomenon of independent continuous periodic firing series is discovered for the first time. We prove that saddle–node, periodic-doubling, and Neimark bifurcations occur when the excitation frequency varies to an appropriate value in this system. The generation of saddle–node bifurcation will lead to the change of limit cycles for periodic motion. The number of limit cycles of periodic motions in phase space increases continuously with excitation frequency increasing. Furthermore, the corresponding circuit is implemented to simulate the periodic stimulus effects, which are confirmed precisely by phase planes and harmonic spectrums. The study provides a better understanding for exploring the mechanism of how neurons code information or how they react to complex excitation and serve as guidelines for the analysis and design of biological circuits.