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.

考虑到电磁感应，与保持信息传输的信息编码有关的神经激发活动在质量上有所不同。单个神经元的放电可以通过Hopf分叉分析进行研究。在本文中，借助中心流形理论和代数不变式方法，获得了忆阻FitzHugh-Nagumo（FHN）和Hindmarsh-Rose（HR）模型中Hopf分叉的存在和稳定性的一般参数条件。通过研究特征多项式的根，已经研究了导致系统发生振荡的参数范围。借助于李雅普诺夫函数，对于忆阻FHN模型获得了第一李雅普诺夫系数的一般形式。通过使用中心流形理论和代数不变量方法，忆阻型HR模型简化为更易于管理的模型，该模型继承了原始模型的局部动力学。对于一般参数条件，研究了电磁感应对神经振荡的影响。通过数值模拟说明了模型的振荡爆破点火方式。