Dynamical Transitions Of Two Identical Rulkov Neurons Coupling With Electrical Synapses

Neurons play a key role in the central nervous system, information processing, the generation and transmission of information. Stability of fixed points of neuronal systems may be changed because of the different values of the parameters. Active and resting states of neurons are mainly determined by external stimuli and internal parameters, which can be predicted by bifurcation values based on mathematical models of different neuron systems. Different states of neurons may be transmitted when at least two or more coupling neurons are concerned, so coupling neuronal systems are usually high-dimensional complex nonlinear dynamical systems, whose stability and bifurcation analyses can provide a theoretical reference and physiological and medical basis.First, the first chapter introduces the background associated with this paper. In the second chapter, according to different forms of nonlinear functions in a single Rulkov neuron model, the coupled neuronal network system is classified two different models, namely, a regular Rulkov neuron model and a chaotic neuron model. Numerical simulations are used to analyze their properties of the regular and chaotic neuron model, respectively. The neuronal system is divided into a fast subsystem and a slow subsystem, respectively. Attention is focused on motions and correlations of two coupled neurons. By using master stability functions, necessary and sufficient conditions of stability of the neuronal system are analyzed. Also, when the coupling strength is greater than0, less than0, equal to0, respectively, different dynamical behaviors and synchronizations are observed. These results are verified by numerical simulations.Secondly, in order to investigate the chaos and bifurcations of the coupling Rulkov neuron systems, the largest Lyapunov exponents are computed. The stability domains and the correlation coefficients of coupling Rulkov neuron models are discussed when various parameters are chosen with the evolution of time. These researches indicate that there is a certain correlation between the fast subsystem and the slow subsystem. Numerical simulations are used to identify these mathematical results corresponding to different parameters.Finally, the Chapter three summarizes this paper including problems and future work.