Dynamical Analysis In Brain Nerve Model With Delayed Feedback

Recent years, the study of synchrony in a globally coupled ensemble is an intensively developing branch in nonlinear science. Relevant to many problems of physics, chemistry, and life sciences, in particular, to neuroscience. Numerous experiment researches express that synchronization of individual neurons is believed to play the crucial role in the emergence of pathological rhythmic brain activity such as in Parkinson's disease, and there are many models to describe that, e.g. Rulkov Model, Hindmarsh-Rose model and Hodgkin-Huxley model, etc. Though the study of these models, we can get some useful results for the deeply brain stimulation procedure.In this paper, we consider an important kind of amplitude equation, introducing three kinds of control scheme: direct control, differential control and multiple-delay control. We begin with the stability research of direct control scheme, we find that the stability switch occurs when delay varies, the system undergoes Hopf bifurcation when delay passes through a sequence of critical values, and a bifurcation set is drawn in appropriate parameters plane. All the same method can be used to deal with the differential control scheme. In order to study the multiple delay control scheme, we get some results in the view point from neutral functional differential equation. Moreover, we study the direction of the bifurcation and stability of bifurcating solution by the center manifold theorem and normal form method. Finally, certain numerical illustrations are performed to support our results.