Hopf Bifurcation Analysis Of FHN And HR Neuron Network Systems With Time Delay

In recent years, with the development of nonlinear science and technology, the studies on complex dynamics of neurobiological systems have received much more attentions.Neuron is the basic unit of the neurobiological system, due to the highly complexity of neuron ?ring activity, it plays a very important role in biological systems. In recent years, to understand information processing in the brain, the FHN model and HR model are commonly used to study neural ?rings because they are the simplest mathematical models describing dynamical behavior of neural activity. It is known that time delays always occur in the signal transmission for real neurons. The existence of delays tends to evolve more complex dynamical properties in the system and bring more di?culties and complexities to our study. Thus, the study of which is very important to understand the system.In this paper, based on FHN model and HR model, we get the coupled FHN neural network system with time delay and the HR neural network system with two time delays.Using the same method, theirs stability and Hopf bifurcation are researched. First,the stability of equilibrium point is analyzed by employing the corresponding characteristic equation. Su?cient conditions for existence of Hopf bifurcation are obtained. Then,increasing of time delay can induce that the stability switches between resting state and periodic activity. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.