Dynamical Transitions Of Globally Coupled Rulkov Neuron Networks

Neurons in human brain play a key role in the processing, generation, and transmission of information. The stability of fixed point in a single neuron can change corresponding to different parameter values, and the active and rest state of neurons can also take place a obvious transition due to the external stimulus. The coupl-ing neuron network is a very complex high dimensional nonlinear dynamical system, its amplitude of oscillation may go to a small range under some circumstances. Similarly, a population of coupled nonlinear oscillators may age when the fraction of non-self-oscillatory elements increases. Likewise, a global oscillation state will turn into a quiescence state as the proportion of inactive elements exceeds a critical value.In this paper, we at first introduce the coupling Rulkov neuron network, and divide the entire network into two clusters with regard to different values of σ. We discuss the single Rulkov neuron model when p=0, k=0, here p is the ratio of inactive oscillators, and k is the coupling strength, and take into account the stability of fixed point in a single Rulkov neuron by using the method of center manifold. We also analyze the Neimark-Sacker bifurcation, and other bifurcations of single Rulkov neuron, gives a boundary between the activity and the inactivity of a single neuron.Then we discuss the coupling Rulkov neuron model when p≠O, here we choose two different values of σ1, σ2. We divide the entire system into two clusters, active and inactive neuron groups. Our study demonstrates that the global oscillation state will turn into a quiescence state at a critical value as the proportion p of inactive elements increases. In particular, we discuss the Aging Transitions between the active and the resting state, and one of very important results is to present the phase diagrams of structure (k, p) in the map-based global coupling neurons.Finally, we summarize the whole contents of this paper. What we have done provides a basic mathematical foundation for a more comprehensive and complete discussion of the rich dynamic characteristic of the discrete neuron models, which could be useful for the biological and medical experiments in the future.