Mixed-mode Oscillations And Dynamic Research On Neurons

Neural dynamics is a branch of neural computation, one of the directions depicting the electrophysiological activity of a neuron utilizing dynamical system, by constructing the neuron model to examine the dynamical properties, such as, the bifurcation, chaos and nonlinear oscillation.In this paper, the complex nonlinear behaviors, mixed-mode oscillations, as the main object of the research, we investigate the change rules of phenomena in five types of neuron models, and explain the mechanisms of generating mixed-mode oscillations in two of these firing models. We also analyze some issues like the bifurcation of the equilibrium points, bifurcation of the interspike interval sequences and changes of the spike in each burst, and obtain rich results. The paper is divided into four parts:In the opening chapter, we introduce the research progress of mixed-mode oscillation phenomena in neurons. We introduce the basic theories and required concepts starting from the geometric singular perturbation theory and explain the causes of mixed-mode oscillation. Finally, neurons and their models which have the mixed-mode oscillation are given.In the next two chapters, we focus on the generation mechanism of mixed-mode oscillations in a neuron model. In the second chapter, we study the Av-Ron-Parnas-Segel model, which is a four-dimensional model of the lobster cardiac ganglion. First, we analyze the generation mechanism of mixed-mode oscillations in the simplified three-dimensional neuron model by applying the fold node principle in the geometric singular perturbation theory. Then, we explore the change rules of mixed-mode oscillations in the simplified model and the original model, and obtain period-doubling bifurcation diagram and period-adding bifurcation diagram in the interspike interval sequences of the action potential, and the results like the stepped change in the spike counts. Finally, we in-vestigate the effects of parameters in the neuron model on the system. In the interspike intervals bifurcation diagram in, we find that the equilibrium potential and the maximum conductance of a certain ion can change the firing patterns of the system.In the third chapter, we explore the Chay-Keizer model that simulates physiological activities of pancreatic β-cell. First, we divide the system into fast and slow two parts by the fast/slow dynamics analysis method, and analyze the causes of the bursting and mixed-mode oscillations with the slow variable as a parameter. Then, the bifurcation analysis of the equilibrium points in the whole system is carried on and the topology nearby the Bogdanov-Takens bifurcation is given. Finally, we discuss the impact of the parameters of the model. Bifurcation diagrams of the interspike interval sequences and the extreme value, and spike counts during an intra-bursting with two-parameter are obtained.In Chapter 4, firstly, we study a kind of the mathematical model which is a seven-dimensional neuron model with intrinsic bursting. The specific neuron contributes on constructing the thalamic-cortical circuit. On this kind of model we firstly simplify the equations into a three-dimensional model using a dimension reduction method. Then we research the codimension-1 and codimension-2 bifurcations in the three-dimensional mod-el and provide the bifurcation behavior near the Bogdanov-Takens bifurcation. Finally, we explain this cause of the bursting in the system by the fast/slow dynamics and the phase plane analysis. Secondly, we concentrate on the neuron model generating mixed-mode oscillations in the external environment. First, we investigate the change of the Chay model under the external electric field. Mixed-mode oscillations and mixed-mode oscillations nesting phenomena by applying complex periods of the external electric field are exhibited in the Chay model. Then, mixed-mode oscillations in the leech heartbeat network model are observed and their changing processes of spike counts are studied.