Stability And Hopf Bifurcation Study On Several Types Of High-dimensional Neural System Dynamical Models

The biological nervous system plays an irreplaceable role in the function adjustment of the body.The nervous system regulates the functions of various organs and systems and makes them form a unified whole.Among them,neurons(also known as nerve cells)are the basic unit of the structure and function of the nervous system,and are mainly responsible for the coding and transmission of bioelectrical information,which involves complex nonlinear activities.Neuroscience reveals the laws and coding mechanisms of the electrical activity of the nervous system as the ultimate research goal,and the neuron model is the basis of research.During the study of the mathematical model of the biological nervous system,extremely complex nonlinear dynamic phenomena were observed.In recent years,the research of nonlinear dynamics theory has provided theoretical support for the construction of mathematical models of nervous system dynamics,and further based on this,we can explore the release mode of nervous system action potentials.In practice,the disorder of the action potential of the nervous system leads to abnormalities and disorders of the underlying physiological mechanism,and further manifests neurological diseases such as epilepsy and Parkinson's disease.From the perspective of nonlinear dynamic science,some of the above-mentioned lesions may be related to the bifurcation phenomenon induced by changes in physiological parameters describing the differential equations of the nervous system.Therefore,further study of the bifurcation of the nervous system may theoretically provide ideas for the treatment of physiological diseases.The main contents of this article are as follows:1.The purpose and research status of nonlinear dynamics and biological neurodynamics are reviewed.Then the nonlinear dynamics theory,basic knowledge of biological nervous system and common neurodynamic models are briefly introduced,including bifurcation concept,stability theory,Hopf bifurcation theory,central manifold theorem,etc.Basic knowledge.Further,the high-dimensional system is transformed into a low-dimensional system by dimensionality reduction,and then the high-dimensional Hopf bifurcation theory is derived.2.Based on the Ghostburster model of a kind of weak electric fish vertebral body neurons,the equilibrium point of the system is obtained by calculating,and its stability is discussed by analyzing the corresponding eigenvalues of the model near the equilibrium point.Then the existence of Hopf bifurcation is discussed by using the theory of the existence of Hopf bifurcation.The direction of Hopf bifurcation,the approximate solution of bifurcation period and the approximate period are obtained by using the method of Hopf bifurcation analysis.Finally,the numerical simulation results of theoretical analysis are given by using MATLAB and other mathematical software.The maximum conductance of potassium ion current in dendritic membrane and the injection current in cell membrane are selected as thecorresponding bifurcation parameters to explore the dynamic phenomena of the system under the change of single parameter.3.Based on the FHN-ML neuron system with electrosynaptic coupling,the number of equilibrium points of the system is determined and the local stability near the equilibrium point is studied.The existence of Hopf bifurcation is proved by using Hassard et al's central manifold dimensionality reduction method.The direction of Hopf bifurcation is determined,and the periodic solution and approximate period are given.Finally,the bifurcation and dynamic behavior of the system under the change of single parameter are studied by using data processing software such as MATLAB and C language.The disturbance effect of external stimuli on the nervous system model was verified.Therefore,controlling physiological parameters within certain range may make the neuron system enter a stable state and avoid unnecessary oscillation.