Some Dynamical Behaviors of an Excitable System Driven by Stable Lévy Processes

Author:Cai Rui

Supervisor:Duan Jinqiao, j ü Rgen Kurths


Degree Year:2018





In general,the interference of noise to a system will produce a variety of phenomena,such as the change of stability,stochastic resonance,noise-induced transition and noise-triggered abrupt change.In order to analyze some phenomena more accurately by using dynamical systems,research on stochastic dynamic systems is particularly important.In the noise simulation,Gaussian white noise is the most common,and its corresponding math-ematical model is Brownian motion.Lévy process is suitable for the kind of noise with jumping and bursty behaviors in real life.Compared to the Brownian motion,the non-Gaussian Lévy process is more universal.In nonlinear systems,there is a special type of system called excitable system.This kind of systems is very sensitive to noise.Most of the existing studies concern with the effect of Gauss’s noise.In this thesis,we consider the effect of symmetricα-stable Lévy noise on an excitable system called FitzHugh-Nagumo model.In the two-dimensional FitzHugh-Nagumo model,the variable x(t)and y(t)represent the fast and slow variables corresponding to the neuron membrane potential and the slow re-cover of the K~+ion-gating channels.Three deterministic indexes are used to describe the dynamic behaviors of the excitable system driven by symmetricα-stable Lévy noise:first escape probability,mean first exit time,the most probable orbit.At the same time,the case of Brownian motion under the same conditions is also considered as a comparison.This thesis is composed of the following parts.In Chapter 1,we briefly introduced the historical background,research status and main contents of this thesis.In Chapter 2,we introduce related basic knowledge about deterministic dynamical sys-tems,Lévy process,and stochastic dynamical systems.The definitions of the Brownian motion and the Lévy process and some theorems are also reviewed.At the same time,we also introduce in detail the stochastic basin of attraction and the three indexes that charac-terize the behavior of the stochastic dynamic systems.In Chapter 3,we consider the escape behavior of the FitzHugh-Nagumo model driven by the symmetricα-stable Lévy process.The main issue is whether the system will produce an impulse response after being subjected to a noise disturbance with jumps.We also con-sider the case that the noise is Brownian motion.We analyze the probability of transition from equilibrium to excitement through the first escape probability,and describe the effect of noise on system stability through the mean exit time.The greater the probability of the first escape,and the more the noise promote the transfer of the state of the system.The longer the mean first exit time,the more stable the system is.The results of the numeri-cal experiments show that the Lévy noise of small jump and the smaller noise intensity are beneficial to the generation of spikes.However,the larger theαindex and the noise inten-sity,the shorter the average escape time.When the noise intensity is fixed,there is a great difference between the Brown motion case and the Lévy case.In order to characterize the impact of the Lévy noise on this system,we calculate the area corresponding to high first escape probability value and first mean exit time value in escape region under different noise intensity and Lévy index respectively.In Chapter 4,we consider the influence of the symmetricα-stable Lévy noise on the so-lution orbit starting from the equilibrium through the change of probability density function with time.By recording the position coordinate of the spatial maximizer of the probability density function at each time(the most probable orbit),we characterize its changes with time.The evolution of the membrane potential and the recovery variable of the most prob-able orbit changes with time are also shown respectively.At the same time,the residence time of the most probable orbit around the equilibrium and in the high potential region are also calculated respectively under differentαand noise intensity.The reason of the jump occurring in the most probable orbits is explained by analyzing the change of the probability density function in a special time interval.In addition,we select two points in the basin of the left and right branches to calculate the most probable orbit starting from them,in order to show the initial direction of the most probable orbit and whether they are continuous.In Chapter 5,we analyze the“ghost”separatrix of the FitzHugh-Nagumo model in the stochastic environment.The related content of this separatrix is briefly introduced.Then we use the most probable orbit to show the existence of the“ghost”separatrix and describe the separatrix in stochastic environment by first escape probability in the escape region.In Chapter 6,we summarize the main contents of this thesis,and point out some future topics.