Bifurcation Analysis Of Rulkov Neuron Model Coupled By Mean Field

In this thesis, the bifurcation analysis of a map-based Rulkov neuron model coupled by mean field is considered. At first, the existence condition of the fixed point and its stability are investigated by taking into account the fast subsystem. Second, the study shows that, according to the qualitative theory and bifurcation analysis, the map-based Rulkov neuron model includes rich bifurcation phenomena such as saddle-node bifurcation, period-doubling bifurcation, External (internal) crisis bifurcation. Attention is paid on the mathematical mechanism related to bursting phenomenon. Finally, the effect of coupling strength on the in-phase (or anti-phase) synchronization of the coupling Rulkov neuron network is also taken into account.The layout of this thesis is as follows.In Chapter 1, a brief review concerning dynamical theory is introduced, such as the theory of local bifurcation, the map-based neuron models, and the classical Hodgkin-Huxley neuron models.In Chapter 2, the single-cell Rulkov neuron model is presented.In Chapter 3, the condition of the existence of fixed point and its stability of Rulkov neuron model are investigated by using the qualitative theory and bifurcation analysis. Some specific bifurcations are analyzed including saddle-node bifurcation, period-doubling bifurcation, and External (internal) crisis bifurcation. Finally, the effect of the coupling strength on the synchronization of the Rulkov neuron network are taken into account.In Chaper 4, we briefly conclude the thesis.