Memristor-based oscillatory behavior in the FitzHugh–Nagumo and Hindmarsh–Rose models

The neural firing activities related to information coding maintaining the information transmission vary qualitatively considering the electromagnetic induction. The firing of a single neuron can be investigated by Hopf bifurcation analysis. In this paper, with the help of the center manifold theory and algebraic invariants method, general parameter conditions are obtained for the existence and stability of Hopf bifurcation in the memristive FitzHugh–Nagumo (FHN) and Hindmarsh–Rose (HR) models. By studying the roots of the characteristic polynomial, the parameter ranges that cause the systems to undergo oscillations have been studied. With the help of the Lyapunov functions, the general form of the first Lyapunov coefficient is obtained for the memristive FHN model. By using the center manifold theory and algebraic invariants method, the memristive HR model is reduced to a more manageable model which inherits the local dynamics of the original model. The effects of electromagnetic induction on neural oscillations are studied for general parameter conditions. The oscillatory bursting firing regimes for the models are illustrated by numerical simulations.