Synchronization Analysis Of Rulkov Model Coupling By Chemical Synapses

The neurons are believed to be the key elements in the signal processing of neural systems. The generation and transmitting of neural information are nonlinear, so the dynamical performances of individual or coupled neurons receive much attention in the field of neuroscience research. In this thesis, a system consisting of two identical Rulkov map-based neurons coupled by chemical synapses is discussed. The parametric conditions for stabilities of fixed points are derived, along with the stable regions of fixed points. Three cases are taken into account in order to identify what conditions are satisfied when the system takes place in-phase and antiphase synchronizations, respectively.The layout of this thesis is as follows.In Chapter 1, we introduce a brief review concerning dynamical theory, such as the chaos theory, synchronization theory and cross-correlation function, as well as the background of Rulkov map-based neuron model.In Chapter 2, a system consisting of two identical Rulkov model coupled by chemical synapses is concerned. When the two fast variables of the system are all larger than the presynaptic threshold (θ), the system is divided into two subsystems. When the chemical synapses are inhibitory (v=-2), the system takes place in-phase synchronization, while the excitatory (v= 1) synapses give rise to anti-phase synchronization.In Chapter 3, we discuss the two fast variables of the system, in which one is larger than 9, and the other smaller thanθ. The Cross-correlation function is applied to investigate the relationship between the two neurons. When synapses are inhibitory (v=-2), the system synchronizes mostly in-phase, while excitatory (v=1) synapses occur mostly anti-phase synchronization.In Chapter 4, a brief analysis is given when the relationship between the two fast variables and the presynaptic thresholdθis in the random state. The research has shown that the system has strong synchronization behaviors. In particular, the result is totally different with the above two cases. When synapses are inhibitory (v=-2), the system synchronizes anti-phase, while excitatory (v=1) synapses support in-phase synchronization.In Chapter 5, we briefly conclude the thesis.