| The action potentials of individual neurons and their correlations encode a large amount of neural information,and the dynamic study of different firing burst patterns of neurons is helpful to understand the encoding of neural information.The thesis uses the idea of the canards theory,combining with the theoretical analysis and numerical simulation results to give a reasonable dynamics elaboration of the burst discharge mechanism of the three types of neuron models.We first explain the mechanism of spiking,and then prove the mixed-mode oscillation caused by canards and the relationship between canards and transition of burst and spiking.The detailed framework is as follows:Chapter 1 briefly describes the research background,the theory and methods in this thesis.The relationship between the canards and the neuron discharge rhythm as well as the main research content of this thesis is introduced.Chapter 2 studies the dynamical properties of the Purkinje neuron model.First,the dynamical behavior of the reduced Purkinje neuron model and the existence of mixedmode oscillations(MMOs)are mainly explored.A closely potential link between bifurcation phenomenon and the number of spikes per burst as well as mixed-mode oscillations is discovered.Secondly,the generation mechanism of burst in the reduced Purkinje neuron model is illustrated by using the codimension-one bifurcation and slow-fast dynamics analysis.Furthermore,the first Lyapunov coefficient of Hopf bifurcation is computed to determine whether or not it is supercritical,and the bifurcation diagram near the cusp point is obtained by making the codimension-two bifurcation analysis for the fast subsystem.Finally,mixed-mode oscillations is discussed and it is further investigated using the characteristic index that is Devil’s staircase.Chapter 3 studies the mechanism of burst generation and dynamics behavior of the pituitary cell model.First,-type potassium channel and -type potassium channel are added simultaneously to the original model.And its dynamical properties differ from merely adding a fast potassium ion channel to the model.Secondly,the existence of mixed-mode oscillations in the improved pituitary model and its bifurcation behavior are explored via using geometric singular perturbation theory and slow-fast dynamics method,respectively.That is,combining numerical calculation and theoretical analysis to study mixed-mode oscillations and other burst discharge patterns.In addition,the first Lyapunov coefficient of the Hopf bifurcation is computed to determine whether or not it is subcritical to further explain the burst behaviors.Moreover,the codimension-two bifurcation analysis is performed for the full system and great many bifurcation points with abundant behaviors are observed.Finally,we obtain the concrete expression of the pituitary cell model for a saddle-node bifurcation curve,a Hopf bifurcation curve and a saddle homoclinic curve via using center manifold theorem near the Bogdanov-Takens bifurcation point.Chapter 4 studies the mechanism of transition dynamics between burst and spiking of neuron model under electromagnetic induction.First,the dynamics of burst and spiking pattern are demonstrated separately.Secondly,the intermediate state spike of amplitude modulation(AM spike)is found during calculation the spike mode in the neuron model,and the dynamics of AM spike is illustrated using torus canards.Moreover,the discharge patterns of neuron before and after the electromagnetic induction is added in the system are compared,and the effect of changes in the system parameters of electromagnetic induction on the discharge rhythm is discussed.Furthermore,the existence regime of the canards phenomenon changes when the system is under the influence of electromagnetic induction,and the dynamics of burst are also different to the original system.Finally,the function of electromagnetic induction on the dynamical behaviors of neuron burst mode,spiking mode and transition mode between burst and spiking is explained. |
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