Pattern Formation in Excitable Media

The phenomenon of excitability is observed in a wide variety of physical and biological systems. In this work, spatially extended excitable systems are examined from several different perspectives. First, a pedagogical introduction is used to motivate the derivation of the dynamics of one dimensional excitable pulses. In the second part, coupled map techniques for numerical simulation of excitable media and other interfacial systems are described. Examples are given for both excitable media and crystal growth. The third chapter addresses the phenomenon of spiral formation in excitable media. Exact rotating solutions are found for a class of models of excitable media. The solutions consist of two regions: an outer region, consisting of the spiral proper, which exhibits a singularity at its tip, and the core region, obtained by rescaling space in the vicinity of the tip. The tip singularity is resolved in the core region, leading to a consistent solution in all of space. The stability of both the spiral and the core is investigated, with the result that the spiral is found to be stable, and the core unstable. Finally, the stability of excitable waves of the chemical cAMP traveling over aggregating colonies of the slime mold Dictyostelium discoideum is examined by coupling the excitable dynamics of the cAMP signalling system to a simple model of chemotaxis, with result that cellular motion is found to destabilize the waves, causing the initially uniform field of cells to break up into streams.