Pattern Dynamics In Excitable Media

Pattern dynamics, as an exciting area in nonlinear science, has much application to a wide variety of fields, from physics, chemic, mathematics, biology to engineering economics and medicines. Researches on the pattern dynamics mainly bear a long-sought goal to find "qualitative universality" of those observed patterns. Pattern-forming systems under study historically developed in non-equilibrium fluid systems, such as Rayleigh-Benard convection or Taylor-Couette flow; by far, have been extensively concerned in chemical or biological systems dominated by reaction and diffusion, and nonlinear optics etc.In last five years, the author has persisted continuously in the study of reaction-diffusion systems. This thesis will concentrate, therefore, mainly on the problem of patterns formed in one typical media (excitable media) of reaction diffusion system. Excitable media, which ranges from biological cells and tissues to autocatalytic chemical systems, propagates an excitation as a traveling pulse of activity in one-dimensional (ID) system, or maintain self-organized characteristic structures such as in the form of spiral waves in 2D system and scroll waves in 3D system. By definition, if chemical systems can be provoked by a small stimulus to execute an immediate large excursion followed by a return to nearly the original state, it is called as being "excitable". Target waves and rotating spiral waves are typical examples of such self-excited waves (or autowaves, for short) in 2D excitable media. Spiral waves are often observed in solutions with Belousov-Zhabotinsky reactions, colonies of microorganisms, tissues of the heart muscle and the oxidation reaction of CO in single crystal Pt.A traveling plane-wave of activity in excitable media is followed by a recovery tail, phenomenologically imaged as two curves that specify the front and back of the excited zone propagating with the same constant velocity. While in the case of spiral waves, the front and the back of the excited zone come into contact at a certain point to form a special singular region which is definedly called "spiral tip" (or topological defect). In three dimensions, the analogues of spiral waves and its tip are scroll waves, and the vortex filament (A vortex line about which spiral wave rotates) respectivelyIt has become widely accepted that the most dangerous cardiac arrhythmias are due to reentrant waves, such as the spiral waves. These electrical waves circulate repeatedly through the tissue with higher frequency than the heart's pacemaker, thereby altering the heart's regular function and resulting in inadequate pumping. Therefore, it is important to stop spiral waves and suppress its turbulence to help heart rescue.In detail, contents of the thesis are arranged as following:The first two chapters give a general review about pattern formations with essential concepts and theories in excitable media, which are developed in recent years. We hope that such guidance will make it accessible to the rest of this thesis.In chapter three, front kinematics theory developed by Mikhailov et al, is introduced, but avoids detailed exposition of mathematical steps. The main features of this theory capture the dynamics of spiral wave tips and scroll filaments. Based on its application, it is verified that the straight-line drift frequency of spiral wave under a uniform periodic forcing in 2D system, is not locked to the nature rotation frequency as the forcing amplitude exceeds some characteristic value. In three dimension, both reorientation and accelerated expanding (collapsing) of scroll ring (a particular class of scroll waves whose filament forms a planar ring in 3D space), depending on the forcing parameters, are demonstrated and analyzed using the three-dimensional front kinematics theory.Recently we have derived, directly from the original reaction diffusion equation and its rigidly rotating spiral solution, an approximate formula of the perturbation-induced spiral wave drift velocity. In chapter four, we develop this study for the drift of rigidly rotating spirals under periodic, noisy illuminations and traveling wave modulation, and achieve approximate but explicit formulas of spiral drift velocities.In chapter five, we propose for the first time an approach to suppress scroll waves and its Winfree turbulence in 3D excitable media by applying locally periodic excitation and a traveling-wave modulation in the next sections one after anther. Then in chapter six, another subject is considered, where a close-loop feedback control is imposed locally on the FHN system to suppress the stable spirals and its spatiotemporal chaos according to the principle of self-adaptive coupling interaction.In the last chapter, there come some issues encountered in the recent development of pattern dynamics, which are mostly concerned with mathematical problems and biological applications of spirals and scrolls. The author is also currently working on a subject of how to perform the feedback-mediated control on a pinned spiral and remove it to any referenced point. Finally, a briefly conclusion is made to show an overview of all the problems discussed in this thesis.