Distributed Control Of Spiral Waves In Excitable Media

In this paper we studied the dynamic characteristics of spiral wave and the distributed control-Lyapunov method of spiral wave and the distributed optimal control method of spiral wave.Spiral waves are characteristic structures of excitable media far away from equilibrium that have been observed in systems as different as catalytic surface oxidation, the Belousov-Zhabotinsky chemical reaction, aggregating colonies of slime mold, and heart tissue, etc. Many studies have shown that Spiral waves and spatiotemporal chaos usually are harmful and need to be suppressed. For example, many important evidences proved that the appearance of spiral waves is linked to one kind of cardiac arrhythmia. The instability of spiral waves will induce a rapid death of heart, named as ventricular fibrillation (VF). Therefore, it is important to find effective schemes to prevent the appearance of spiral waves and spatiotemporal chaos in many domains.There are many schemes to control or eliminate the existing spiral waves. first, Using a FitzHugh-Nagumo model as an example, we studied the stabilities of spiral waves in two-dimensional small-world network. It was found that there exists a critical rewiring probability due to the small-world connectivity. It was also confirmed that this spiral instability is different with the Eckhaus instability and the Doppler instability. next, we studied the effect of Gaussian colored noise on the formation and instability of spiral waves described by one class of the modified FitzHugh-Nagumo equation. It was found that Gaussian colored noise plays a constructive role in formation, transition and instability of spiral wave. In a certain noise level, spiral wave is maintained in medium, where spiral wave cannot be observed in the absence of the noise. It is difficult to make a stable spiral wave into instable state by Gaussian colored noise, unless the noise level is very strong. Lastly, we discussed the effect of periodic forcing on the fluctuation of propagating period of spiral waves in the light-sensitive Belousov-Zhabotinsky (BZ) reaction described by Oregonator model. It is found that the amplitude and period of the external forcing are decisive factors in the development and period fluctuation of spiral wave. Too strong external amplitude could make against the formation and propagation of spiral waves.Since spiral wave is a kind of spatial and temporal distributed structure, it usually isn't full controllable by a concentrating forces. Therefore, we developed the distributed control theory and methods in the reaction-diffusion system and applied them to the prevention and control of spiral wave.We applied the distributed predictive control strategies of spiral wave in cardiac excitable media. The modified FitzHugh-Nagumo model was used to approximately express the cardiac excitable media. Based on the distributed control-Lyapunov theory and the distributed optimal control theory developed here, we obtained their distributed control equations, respectively, The feasibility of the strategies were demonstrated via the illustrative examples, in which the spiral wave were prevented successfully and the possibility for inducing ventricular fibrillation was eliminated. The method s are helpful for designing various cardiac devices. the distributed optimal control method is more efficient and robust than the distributed control-Lyapunov method and other methods. In addition, our spatiotemporal control, is suitable to the quick-response control systems, especially distributed optimal control theory, can be applied to other cardiac models, even to other reaction-diffusion systems.