Analytical Solution Of Spiral Waves In The Excitable Media

Wave patterns are ubiquitous in various physical, chemical and biological systems which are far away from the equilibrium. Recently, the discovery that the spiral wave patterns play an essential role in cardiac arrhythmia and fibrillation makes the research about the spiral dynamics become a hot area. A lot of study about the spiral activities is done through chemical or biological experiments and numerical simulations. But people have also been working on the theoretical work of the spiral dynamic and have obtained some important results. However, due to the complexity of the nonlinear system and the singularity of the spiral tip, the study about the spiral wave in the excitable system encounters some tough problems. At present, the analytical results that can match the simulation results quantitatively and have a guiding role for the experiment and simulation work are relatively less.In this paper, we will give a brief statement about some essential properties of the reaction-diffusion system and the excitable system, and then the basic dynamic theory of the spiral wave in the excitable system. Next, we will introduce a new phenomenon about the influence of external field on the spiral wave, that is, when a suitable circularly polarized electric field is imposed on a subexcitable system, the original retracting excitation wave can be supported in the medium and transfer into a rigidly rotating spiral wave. What’s more, the frequency of the spiral is identical to the one of the circularly polarized electric field. In the frame of Hakim and Karma’s theory, we offer a semi-analytical interpretation for this phenomenon. And the result got from the semi-analytical theory agrees with the one got from the simulation quantitatively. Since the circularly polarized electric field has been realized in the Belousov-Zhabotinski chemical reaction, thus the above phenomenon is hopeful to be observed in this kind of chemical experiment.Next, we conclude from the numerical simulation that, the system variables on the grid point in the core region do harmonic oscillations but not relaxation ones, and their amplitudes grow linearly with the distance from the rotation center. An analytical core solution is proposed to describe the spiral dynamics around the core. Quantitative comparisons are performed between the simulation result and the analytical solution. These two sets of results turns out to be in good agreement with each other, which shows that the core solution can work well in describing the dynamics of rigidly rotating spirals in highly excitable media.In summary, we explore the spiral dynamics through numerical simulation, and try to offer a quantitative theoretical explanation for it. All these may provide a basic evidence for the experiment, and be instructive and meaningful to understand and control the dynamics of the spiral waves.