Research On Mixed Mode Oscillations Of Neuron Models Based On Singular Perturbation Theory

The neuron model is the theoretical basis of neuroscience,which is a kind of model that has the rich nonlinear dynamic phenomena in the nervous system,and the action potential characteristics of the model show the encoding of nerve conduction.Based on the singular perturbation theory(GSPT)and dimensionality reduction method,this thesis reveals the mixed-mode oscillations(MMOs)phenomenon in the neuron model,and gives a theoretical analysis of its generation mechanism.The main work is as follows:Chapter 1,as an introduction,expounds the research status and progress of mixedmode oscillations under singular perturbation theory,dimensionality reduction methods,bifurcation theory,computational singular perturbation methods,and summarizes the research work of this paper.Chapter 2 elaborates the study of the relationship between ion conductance and ion channels in the classical Hodgkin-Huxley(HH)model.This chapter finds for the first time that the ion channels in the HH model can be characterized by the bivariate Taylor formula for ion conductance,which corresponds to the fourth-order Taylor formula coefficients of the ion conductance near the origin,therefore the 28 species HH-like models were constructed.Then,the conductance curves(peak difference)under different types of HH models were compared by numerical simulation and we also observe different firing patterns(periodic spiking,bursting,mixed-mode oscillations patterns,etc.).Chapter 3 deepens the research on the dynamics of the high-dimensional pyramidal cells(PCs)model.In this chapter,the central manifold theorem is used to simplify the six-dimensional neuron model to three-dimensional.First,based on the canard theory,the existence of 1s(s>0)-type MMOs in the 3D model and the generation mechanism of MMOs induced by canard are proved.Then,using the singular perturbation theory(GSPT),taking two-slow/one-fast and one-slow/two-fast systems as examples,we perform slow-fast dynamics analysis on the 3D model,and obtain the trajectory of the slow-fast system which restricted to the critical manifold.Finally,based on bifurcation theory,the chapter discusses one-parameter and two-parameter bifurcation curves.Chapter 4 considers the MMOs phenomenon in the three-dimensional cardiomyocyte action potential model.In this chapter,for the improved model,we prove the existence of MMOs.First,by adding a pacemaking stimulus current mechanism to this model,we found a special MMOs phenomenon,that is,early afterdepolarizations(early afterdepolarizations,EADs).Then,based on the canard theory,the existence of 1s(s>0)-type MMOs is proved,and we conclude that there exists two singular canards(at the folded singular saddle point)will perturb to the maximum canard.Finally,using the singular perturbation theory,we also obtain the critical manifold and singular periodic orbits of the three-dimensional system with two slow variables.