Interaction Between Excitable Waves And Heterogeneities In Reaction-diffusion Systems

In real systems, heterogeneities are very common, especially in heart system. Among kinds of heterogeneities, the type of no-flux boundary condition and the type of heterogeneous excitability are most common. Since these two types are corresponding to anatomical structure, blood vessel, tissue damage and weaker excitability due to local ischemia, respectively.Due to heterogeneities, nearby excitable waves are influenced and even absorbed. Thus in real systems which are full of heterogeneities, excitable waves are more likely interacting with heterogeneities.Therefore, it is more practical and important to study the interaction between excitable waves and heterogeneities, and their various characterizations of phenomena.For the interaction between excitable waves with the heterogeneities by no-flux boundary condition, although there are a lot of theoretical analysis, numerical simulation and experimental studies, a theoretical analysis results which can quantitatively coincide with experimental results are still to be improved. In the second chapter, we combine the nonlinear eikonal relation, dispersion relation and kinematical model, and present a theoretical analysis method which quantitatively matches numerical simulation results, a good explanation of the dynamics of anchored spiral waves driven by periodic pacing waves, and their instability points corresponding to the unpinning. Not only can a good explanation reveal the nature of unpinning, it can also be used to explain the initiation of free spirals by heterogeneities and multi-armed anchored spirals. Meanwhile, in order to get a more accurate simulation results, we use the "phase-field method"(or "SBM") and "natural stimulation". Detailed explanations can be found in Appendix I and Appendix II.For the interaction between excitable waves and the heterogeneities by heterogeneous excitability, we, for the first time, find the widely-expecting inwardly rotating spiral waves in a heterogeneous excitable medium, which is consisted by a high excitable annular region surrounding a weakly excitable circular region (we call it "disk" for short). We use the dispersion relation to predict existence conditions of inwardly rotating spirals. The result fits well with numerical simulation on the border of inwardly and outwardly rotating spirals.Furthermore, in order to give a quantitative theory about the dynamics of inwardly rotating spirals affected by heterogeneous excitable obstacles, we simplify the above situation as follow:similarly to the unfolded "disk" medium, we put a strip of high excitable region side by side with a strip of weak excitable region to construct a "color bar" like medium. We measured all the parameters of a wave segment spreading through the medium. Then we use a linear eikonal relation, dispersion relation and kinematical model to give and the existence condition and the dynamics of wave segment in this heterogeneous excitable medium. Comparing the results with numerical simulation ones, we verify the theoretical analyze in the quantitative consistence.In summary, we combine the eikonal relation, the dispersion relation and kinematical model to give a theoretical explanation, which can explain well the interaction between excitable waves with heterogeneities, especially in the type of no-flux boundary condition and of heterogeneous excitability. Thus it is capable of presenting some suggestions to predict or prevent a variety of common phenomena in real systems, and of indicating a new phenomenon which may occur widely in real systems. And may it be helpful to further research of interactions between excitable waves with heterogeneities in reaction-diffusion systems, by providing more quantitative theoretical analysis, and more accurate numerical simulation methods.