Analysis And Application Of Hysteresis Bifurcation In Time-Delay Fitzhugh-Nagumo Neural Network

The hysteresis nonlinear phenomenon is common in all fields of biology,but the time to study the hysteresis nonlinear phenomenon in biology by mathematical method is not long.The limitation of experimental method and the difficulty to capture the potential mechanism of this nonlinear process make the mathematical modeling and simulation become particularly important.At present,most of the researches on the hysteresis phenomenon in the biological field identify and model biological processes by inserting the hysteresis operator into the mathematical equation.However,there are still various biological models without explicitly embedding the hysteresis operator,but the hysteresis phenomenon is clearly shown in the bifurcation diagram of these models.In the neural network with time delay,because of the existence of time delay,the behavior of the system will change,resulting in various forms of bifurcations.In the study of bifurcations,Hopf bifurcation is a common dynamic bifurcation,in which the subcritical Hopf bifurcation is called catastrophic bifurcation,which will produce the phenomenon of hysteresis bifurcation.At present,the research on Hopf bifurcation of neural networks with time-delay is usually tedious and only analyzes the direction and stability of the bifurcation,but seldom analyzes the hysteresis phenomenon.Therefore,in order to reveal the mechanism of the hysteresis bifurcation in the time-delay neural network,better understand the law of neural activity,and reduce the cost of experiment and numerical analysis,this paper uses a weak nonlinear analysis method based on multi-scale to analyze the hysteresis bifurcation of the time-delay neural network,and applies it to the famous FitzhoughNagumo(FHN)neural network model.The main research contents of this article are as follows:(1)Based on the principle of multi-scale and micro disturbance,combined with bifurcation theory,this paper analyzes the hysteresis of Hopf bifurcation in time-delay system.This method transforms the original nonlinear differential equation into a series of linear differential equations,and then solves these linear differential equations,which can not only analyze the direction and stability of Hopf bifurcation,but also obtain the bistable region of hysteresis bifurcation caused by subcritical Hopf bifurcation and the analytical expression of the amplitude of stability and unstability limit cycle.And this method can reduce the cost of experiment and numerical calculation.(2)On the basis of(1),a general model for the analysis of Hopf hysteresis bifurcation of neural networks with time-delay is proposed,and it is applied to a simple neural network,namely the Fitzhugh-Nagumo neural network with two neurons.The direction and stability of Hopf bifurcation of the neural network are analyzed with ? as the bifurcation parameter.The bistable region of the hysteresis bifurcation and the analytical expression of the amplitude of the limit cycle are obtained,and the results are verified by simulation.(3)On the basis of(2),the Hopf hysteresis bifurcation of Fitzhugh-Nagumo neural network with N neurons is analyzed with the time-delay ? as the bifurcation parameter.The theoretical analysis is applied to the case of N = 6,that is,Hopf bifurcation and hysteresis bifurcation of Fitzhugh-Nagumo neural network with six neurons,and the results are verified by simulation.