The Study On Dynamics Of Spiral Waves In Excitable Media Using Cellular Automata Models

The excitable media exist widespread in the nature. It can produce all kinds of pattern configuration and a variety of walking waves in space. The traveling wave, the target wave and the spiral wave are the typical nonlinear waves in excited media. People study on excitable media through nonlinear waves mainly. The excitable media arouse the interest of different researchers because its importance for the science and application.In this paper, we have studied the dynamics of spiral waves in excitable media using Greenberg-Hastings cellular automata model(G-H model). The paper content is as follows:The first chapter is the introduction. In the first section of this chapter, the excitable medium, the significance, the common models and the main method on study of excitable media are introduced. In the second section, the production of the spirals, the meander, the breakup and the control of spiral waves are mainly introduced. Finally, we introduce the main parameters and the basic model of complex networks simply.In the second chapter, we have studied the influence of the rewiring probability p of the directed small-world network on dynamical behaviors of spiral waves using the G-H model. The computer simulation results show that as p is small enough, the stable spiral wave under the regular networks keeps its stability unchanged, and when p increases, the phenomenon such as meandering, breakup and disappearance of the spiral waves appears. The relation between the excitability and p turns to the conclusion that those above phenomenon occurring base on the reducing of the excitability. On the other hand, the period of cell is also related with p.In the third chapter, we have studied the influence on the dynamical behaviors of spiral waves caused by the heterogeneous cells. We find that the instability of spiral wave is related to the refractory period of the heterogeneous cells, the number of heterogeneous cells and the excitability of the media. The numerical simulation results show that:(1) As the refractory period is shorten, for the system which has a single excited state, the breakup of spiral wave is due to the early excitation of many heterogeneous cells, and it occurs only in small parameter region; for the system which has several excited states, the breakup of spiral wave is related with the uniformly distributed of heterogeneous cells.(2) As the refractory period is increased, the critical density of breakup and disappearance of spiral wave are higher when the system has several excited states. (3) The disappearance of spiral wave can be observed when the refractory period change, but the same phenomena will not appearance as the excited period changed.In the chapter IV, we firstly studied the effect of the regular distributed heterogeneous cells on spiral wave using G-H model. When the heterogeneous cells are added to the stable spiral waves, we observed the phenomenon such as the stable spiral wave following the drifting, the continue drifting spiral and the disappearance of spiral wave. The influence of the location and the refractory of the heterogeneous for the tip of the spiral wave is found. We consider that the tip of the spiral wave changes just because the attraction of the heterogeneous cells. Finally, we control the system with several spiral waves using the attraction of the heterogeneous cells for spiral tip.The summary and outlook is made in the chapter V. The model used here has the strongpoint such as speediness and without truncation error. But the G-H model is too simple to simulate the actual excited media such as the heart. So, the results obtained in this paper are very preliminary, and the further study needs to be done. In addition, in this paper, we just consider a few factors which influence the evolution of spiral waves, in the actual case, more factors should be considered.