| Waves of excitation occur naturally in biological, chemical and physical systems. Some examples of excitable media include biological media, including heart muscle and nerve tissue, and chemical media like the unstirred Belousov-Zhabotinsky reagent.; In the biological context these waves serve to communicate information rapidly over long distances and to coordinate the activity of tissues and organs [1]. The Belousov-Zhabotinsky reagent is a chemical excitable medium which mimics the excitability of biological media. Russian mathematicians and biophysicists trying to understand flutter and fibrillation of the heart had studied spiral waves of excitation in mathematically idealized excitable media years before the chemical medium was discovered [2].; We study the formation and existence of spiral excitation wave patterns in orientable 2-dimensional and 3-dimensional excitable media, and non-orientable 2-dimensional media. The mathematical model for excitation waves in excitable media is based on a phase map that assigns at a fixed time a reaction phase to spatial locations, and computes the obstruction to extending waves observed on the boundary of a manifold to waves on the entire manifold.; [1] J. J. Tyson, S. H. Strogatz, the Differential Geometry of Scroll Waves, International Journal of Bifurcation and Chaos 1(4) (1991), 82-95. [2] A. T. Winfree, Rotating Chemical Reactions , Scientific American 230 (1974), 82-95. |
