The Synchronization Of Spiral Waves And Pattern Formation In Two-layers Coupled Excitable Systems

Spiral wave is a most common pattern in nonlinear reaction diffusion systems. Spiral waves have been observed in diverse systems, such as physical systems, chemical systems, biologic systems (i.e., cardiac tissue, frog oocyte, neocortex) and so on. The study found that spiral wave occurred in neocortex can organize and modulate cortical population activity. It may affect normal cortical processing and lead to pathological pattern of activity such as those found in epilepsy. The electrical spiral waves in cardiac tissues can lead to tachycardia. The breakup of spiral waves will results in the transition from ventricular tachycardia to ventricular fibrillation. So far the dynamics of spiral waves in reaction-diffusion systems have extensively been studied by scientists via experimental and theoretical methods. The harm brought to heart by spiral waves has deeply been understood. Many control methods of spiral waves are proposed.It is well known that the brain network has the small world feature, in which there are hub neurons (super-connected neurons). The coupling among neurons is either excitatory coupling or inhibitory coupling. Both excitatory coupling and inhibitory coupling can promote synchronization of neurons. However, the inhibitory coupling as well as network coupling with hub nod has not been considered in the investigations of the pattern formation in reaction-diffusion systems so far. Heart is the excitable system, which consists of various cell types, mainly myocytes and fibroblasts. A myocyte can electrically couple to some myocytes and fibroblasts by gap junction channels. The coupling between myocyte and fibroblast is either excitatory coupling or inhibitory coupling. Furthermore, the couplings of cells form complex network structure. Thus there may be hub cells in cardiac tissue. So it is necessary to extend the investigation of spiral wave dynamics in reaction-diffusion systems from uniformity to more complex construction. In this thesis, we adopt the Bar-Eiswirth model to study the dynamical behaviors of spiral waves in tow-layer media coupled by average field method or inhibitory and excitatory scheme. The thesis is organized as follows.The first and second chapters are the overviews of the thesis. The first chapter introduces the two types of reaction-diffusion systems, i.e., oscillation system and excitable system. Some models of excitable media are presented. The second chapter introduces some types of spiral waves, the formation and breakup of spiral waves as well as the synchronization and control of spiral waves.The dynamics of spiral waves in a two-layer coupled excitable media is studied in the third chapter. The two-layer media connect via network, i.e., a excitable unit is selected in each column of an excitable medium as a central point, and all excitable units in the same column of a layer medium connect only with the corresponding central point and its eight neighbors in the opposite medium. The numerical results show that when the coupling strength is suitably small, the two coupled spiral waves via local coupling can achieve synchronization. Increasing coupling strength will induce meandering and drifting of spiral waves, leading to desynchrony of the coupled spiral waves. The coexistences of spiral wave with the resting state, low frequency plan wave and irregular pattern are observed. The disappearance of the coupled spiral waves via the transition from spiral wave to synchronous plane waves is also seen if the coupling strength is properly chosen.The effect of the inhibitory and excitatory asymmetric coupling on spiral waves in a two-layer excitable medium is studied in the fourth chapter. The numerical results show that the excitatory asymmetric coupling can promote the frequency-locking of two spiral waves with different frequencies. The two spiral waves can achieve frequency-locking even if the frequency difference between them is large; When the coupling strength and control parameters are chosen appropriately, the inhibitory and excitatory asymmetric coupling can keep spiral wave unchanged in one medium and result in the transition from spiral wave to the resting state or target wave with low-frequency in the other; The coupling also induces the meandering of spiral waves or leads to the transition from two spiral waves to two target waves in the two-layer medium. The generated target waves finally either disappear or develop the plane-wave-liked oscillation patterns. Furthermore the oscillation of patterns is anti-phase. In addition, the locally intermittent frequency-locking of two spiral waves is observed The fifth chapter is the summary of our works and outlook.