Stable Particle-Like Solutions to the Nonlinear Wave Equations of Three-Dimensional Excitable Media

So-called "excitable" media support nonlinear waves that propagate as pulses, e.g., the action potential in nerve fibers and heart muscle, and the oxidation pulse in certain chemical reactions. In two-dimensional media the right stimulus will also create rotating vortices which radiate such pulses periodically in every direction. Their investigation in diverse excitable media has dramatically expanded in the USA and USSR since their discovery in both countries in 1970. In three-dimensional media the pivot of the rotating structure is not a point but a curved line, generically a closed ring. Rings linked together and knotted constitute topologically distinctive organizing centers for periodic activity throughout the medium. It is of interest to understand how stimuli create organizing centers, and how organizing centers persist or decay. The numerically discovered stable organizing centers exhibit enough symmetries to encourage mathematical attention. This paper illustrates organizing centers by computer graphics, describes their taxonomy and anatomy in terms of topological theorems about surfaces and about phase maps, demonstrates their dynamics by vectorized numerical integration of partial differential equations of reaction and diffusion, poses questions about the results in terms of differential geometry, and illustrates their detection in chemically excitable media and in cardiac muscle.