Dynamical Analysis Of Hopf Bifurcation Of Ca²⁺ Oscillation And Calcium-induced Neuronal Hyper-excitation

This thesis applies the theory of characterization and stability of differential equations,the central manifold theory and bifurcation theory to investigate the nonlinear dynamical behaviour of the calcium oscillation system proposed by Marhl et al.In this thesis,the nonlinear dynamical behaviour of the calcium oscillation system proposed by Marhl et al.is investigated,including the existence,type and stability of the equilibrium of the system.The Hopf bifurcation of the system is critically analyzed from a numerical point of view.The bifurcation diagrams,time courses and phase diagrams of the system are presented to verify the results of the theoretical analysis and numerical calculations,and the saddle-node bifurcation,torus bifurcation and period doubling bifurcation of the limit cycle are found in the bifurcation diagrams.In addition,a mathematical model for the coupling of neurons and astrocytes was constructed.In addition,a coupled mathematical model of neuronal-astrocyte was constructed and numerical simulations were carried out with suitable and controllable system parameters to analyze the IP3(inositol triphosphate)production rate and the effect of astrocyte feedback currents on the firing pattern of neuron.This thesis is divided into six parts.The first section contains an introduction to the background,purpose and significance of the research,including the current status and significance of the nonlinear dynamic system,the calcium oscillation system and the communication network of neurons and astrocytes.In the second section,the mathematical theories and basic concepts applied to the research are presented,including central manifold theory and bifurcation theory.In the third section,the calcium oscillation model proposed by Marhl et al.is used to calculate and simulate the dynamics of the system with suitable parameters.Numerical calculations and numerical simulations are carried out.The main contents are: the existence,number,type and stability of the equilibrium of the system for different values of the parameters.The supercritical Hopf bifurcation and subcritical Hopf bifurcation of the system reveal the reasons for the generation and disappearance of calcium oscillations.The numerical simulations further verify the validity of the theoretical analysis and reveal more complex calcium oscillations.In the fourth section,the existence,number,type and stability of equilibrium of the system with different parameters are discussed,and the conditions required for the bifurcation of the equilibrium and the determination of the type of bifurcation are discussed using the central manifold theory and bifurcation theory.In this section,the type of equilibrium is determined by the characteristic root discriminant method,which is different from the Hurwitz criterion in section 3.In addition to the theoretical and numerical analysis of the dynamic behaviour of the system,numerical simulations of the system are also presented,including time courses diagrams,phase diagrams and single-parameter bifurcation diagrams.In the fifth section,a simple mathematical model of bidirectional information exchange in a coupled neuron-astrocyte system is constructed.This model enables us to better understand the dynamic mechanism of seizure in the brain.To test the hypothesis that neuronal hyper-excitation is influenced by glutamate,we introduced a relationship between astrocyte calcium concentration and inward currents loaded onto neighboring neurons,and investigated the effect of glutamate receptor expression levels on abnormal neuronal firing after external electrical stimulation of neuron and astrocyte,respectively.In addition,we also investigated the effect of astrocytes on neighboring neurons.Finally,we conclude the thesis with a summary and a vision for future work.