Study On The Dynamics Of Spiral Wave In The Nonlinearly Coupled Excitable Media

In recent decades, the investigation of spatiotemporal pattern dynamics has become a hot topic of nonlinear science. The investigation can help us to understand the phenomena that occur in nature and human society. Spiral wave is the important spatiotemporal pattern, which has been observed in some physical, chemical and biological systems, such as cardiac tissues, the calcium wave in the frog egg cell, the Belousov-Zhabotinsky chemical reaction system, and so on. In some cases, spiral waves and spatiotemporal chaos is harmful to humans. For example, spiral wave in myocardial tissue would lead to tachycardia and the breakup of it into spatiotemporal chaos will lead to fibrillation, endangering the lives of patients if the fibrillation does not timely and effectively treated. Therefore, study on the control of spiral wave and spatiotemporal chaos is of great value of application. However, it relies on our thorough understanding of the dynamics of the spatiotemporal pattern to solve completely the suppression of spiral wave and spatiotemporal in cardiac tissues. The systems, in which spiral waves can be generated, are the reaction diffusion systems, such as the excitable system. The biological neurons, cardiac cells are the typical excitable systems. The dynamics of spiral wave and other spatiotemporal pattern in single-layer medium has been widely studied, and a lot of results have been obtained so far. However, the related investigations in the coupled excitable media still focus on linearly coupled media, and nonlinearly coupled media are less applied. The ventricular wall in the real cardiac system is composed of myocardium, epicardium and endocardium, and the cells in three layers media have different electrophysiological properties. In order to better understand the phenomenon occurring in cardiac system, the pattern dynamics in the coupled systems need to be studied. So we investigate dynamics of spiral waves in two-layer coupled excitable media based on Bar model. The two results that we obtain are respectively introduced as follows:The first chapter is the overview section of this article. The some basic properties of spatiotemporal chaos, reaction diffusion system Bar model, as well as the generation of spiral wave in excitable medium, the movement of spiral wave’s tip are briefly introduced. Synchronization and control of spatiotemporal pattern are briefly introduced too.In Chapter2we investigate the evolution of spiral waves in a system of two coupled excitable media by using Bar model. The drive-response coupling schemes with or without constraint condition are proposed. The response and drive subsystems are respectively in the state of spiral wave and spatiotemporal chaos before the coupling turns on. We find that spiral wave exhibits different dynamical behaviors for different parameters. When the coupling strength is weak, the dynamical behavior of spiral wave in response system almost remains unchanged. When the coupling strength is strong, the coupling without constraint condition always leads to the breakup of spiral wave. When the related parameters are properly chosen, spatiotemporal chaos not only can enhance the excitation of the forced medium but also reduce it. In addition, it can induce chaotic meander or drift of a stable or meandering spiral wave, and even causes spiral wave to move out of system. It can greatly delay the breakup of an unstable spiral wave. Specially, it causes an unstable spiral wave become a stable or chaotically meandering spiral wave under the coupling with constraint condition.In Chapter3we investigate the evolution of spiral waves in nonlinearly coupled excitable media. We find that the evolutions of coupled spiral waves depend on phase difference, coupling strength and the distance between spiral wave’s tips. When the two subsystems have same parameters, the synchronization and anti-synchronization of spiral waves, the occurrence of the transition from spiral wave states to the different steady states or the resting state, meandering of spiral wave, and the movements of spiral wave’s tips in opposite direction are observed if coupling strength and phase difference are properly chosen. Furthermore, the excitation-synchronization of the spiral waves in subsystems is found for the first time. The reason why two spiral waves do not achieve synchronization is that the high-frequency oscillation arises during the peak stage of the wave. Moreover, oscillations at space points with mutual coupling form anti-phase. The dynamic behavior of local medium is similar to the spiking of neuron. Most of above-mentioned phenomena will also appear when two subsystems have different parameters, in which two subsystems can not achieve synchronization and anti-synchronization. However, phase and anti-phase synchronizations of two subsystems can be observed for the correct choice of phase difference and coupling strength. These results show that our proposed coupling methods can help one to understand the dynamical behavior of neuronal systems, and also help one to understand the effect of ventricular structure on wave propagation.