Dynamic Analysis Of Mammalian Cortical Neuron Model

The discharge activity of neurons is closely related to the expression of biological in-formation.Different functional neurons will exhibit different dynamic behaviors in their discharge activities.Therefore,the study of neuron dynamics is conducive to under-standing the expression and feedback mechanism of neuronal bioinformatics.This study focuses on a three dimensional neuron model,which express the characteristics of mam-mals new cortex.It is a neuron model that obtained by combining the Wilson model and the Hindmarsh-Rose(HR)model,and its fast variables obey Ohm's law.In this paper,we combine the bifurcation theory of dynamical system with the fast-slow dynamics analysis method,to study the dynamic properties of the model through theoretical analysis and numerical simulation.The main work is as follows:In the third chapter,based on qualitative theory and bifurcation theory of differential equations,we adopt(gR,gH)as the control variables to study the number of equilibrium points and the stability of the equilibrium point.Then the slow variable H is taken as the parameter of the fast subsystem,the first-order Lyapunov coefficient corresponding to the Hopf bifurcation point is theoretically calculated,and the direction of the Hopf bifurcation is determined,then the stability of the periodic orbit is determined.Next,we use(gR,gH)as the control parameter,and make the two-parameter bifurcation diagram of the system.By numerical calculation,we obtain the generalized Hopf bifurcation,Cusp bifurcation and Bogdanov-Takens bifurcation on the parameter plane(gR,gH).Finally,we study the Bogdanov-Takens bifurcation of the model,and obtain the saddle-node bifurcation curve,the non-degenerate Hopf bifurcation curve,and a saddle homoclinic bifurcation curve.In the fourth chapter,the slow variable H in the mammalian cortical neuron model is used as the bifurcation parameter.The three distinct types of bursting are obtained using the fast-slow dynamics analysis methodby adjusting the control variables gR and gH,which are respectively the“fold/homoclinic”bursting via“fold/homoclinic”hysteresis loop,the“fold/homoclinic”bursting via“subHopf/homoclinic”hysteresis loop and the“fold/fold”hysteresis loop bursting,and the three types of bursting are analyzed.Finally,the model is electrically coupled with the Morris-Lecar(ML)neuron model to obtain a new six-dimensional neuron model.By adding direct current and alternating current stimulation to the new model,the coupling strength and the influence of external alternating current stimulation on the coupled system are analyzed and discussed.