Periodic bursting and boundary equilibrium bifurcation in a simplified McKean neuron model

Abstract To facilitate mathematical description and simplify circuit implementation, a simplified McKean neuron model is proposed by utilizing a simple piecewise linear term to substitute the original piecewise linear term. The simple piecewise linear term, which is symmetric about the origin, can be succinctly expressed and does not require extra voltage sources when implementing the circuit. Chaotic dynamics, period-adding bifurcation behavior, and periodic bursting and quasi-periodic spiking modes are disclosed by employing numerically simulated methods, and then bifurcation mechanism of periodic bursting modes is elaborated by solving the equilibrium locus with stability evolution. Therefore, the mode transition between the spike and rest states is demonstrated, and the periodic bursting modes caused by boundary equilibrium bifurcations are thereby identified. In brief, the stability transition of the equilibrium locus triggers the boundary equilibrium bifurcation, resulting in periodic bursting modes. Finally, the simplest analog circuit of the simplified model is designed, and the periodic bursting modes are well verified by circuit simulations and physical measurements.