The Bursting Mechanism Of Map-based Mean Field Coupling Neurons

Neurons play a key role in processing, generation and transmission of in-formation in the central nervous system and reflect rich nonlinear characteristics. Stud-ies have shown that neurons can produce different firing patters such as period bursting and spiking, chaotic bursting and spiking. In the whole nervous system, the transmis-sion of the nerve impulses needs at least two neurons to accomplish through the way of coupling. Therefore, the coupled neuron system is a very complex high dimensional nonlinear dynamic system. Studying the nonlinear characteristics of neural systems has become an important issue especially in providing references for experiments.In this paper, we first introduce the single Rulkov neuron model, and then discuss the nonlinear dynamic mechanism for tapered bursting, square bursting and spiking, especially the mechanism of tapered bursting. We point out that the key to tapered bursting is the saddle-node bifurcation and the flip bifurcation, and it belongs to the fold/flip type bursting. The duration of the tapered bursting increases asαincreases. We also give the parameter ranges of the single neuron corresponding to different bursting patterns.Then, we discuss the mechanism for the mean field coupling neurons in the same and different initial conditions. And we find that the mean field coupling neurons would not only produce tapered bursting, square bursting and spiking, but also will produce two kinds of triangle burstings. The mechanism of the two kinds of triangle bursting seems to be given for the first time. We get the conditions for each kind of bursting and further analyze its stability using the master stability function. When each neuron has different initial values, it will produce different burstings and synchronizations. When the coupling strength is positive, it will take place in-phase synchronous square bursting, and when the coupling strength is negative, it will give rise to anti-phase synchronous elliptic bursting.Finally, we summarize the whole contents of this paper, offering a good foundation for a more comprehensive and complete discussion of the rich dynamic characteristic of the discrete neuron models.