The Santa Fe Institute Colloquium Series is pleased to 

PDE and CA Modeling of Three-Dimensional 
Excitable Media
Presented by:
Chris Henze
Virginia Tech, Department of Biology


Excitable media support undamped propagating waves of activity, such as the slowly moving waves of cAMP release in lawns of Dictyostelium, or the rapidly spreading waves of electrical depolarization in heart tissue. Such waves provide striking and important examples of spatiotemporal self-organization, and are of particular interest when they form spirals (in 2D) or scrolls (in 3D). Although excitable media typically possess a globally attracting steady-state, spiral and scroll waves are self-perpetuating, and can keep the medium indefinitely in a nonuniform and continuously evolving state far from equilibrium. Spiral waves typically drive the system at near-maximal frequency and are highly stable to outside perturbations -- this may be a good thing in the case of Dictyostelium, but is decidedly a bad thing in large mammalian hearts, where rapidly rotating scroll waves are thought to produce ventricular fibrillation. Traditional (non-computer) experiments and analytical approaches to the problems of understanding scroll wave dynamics are hard and have met with limited success. I will present the results of many large-scale computer simulations of 3D excitable media, which relied upon numerical solution of the underlying reaction-diffusion partial differential equations. These simulations uncovered a half-dozen or so new periodic solutions to the PDE's, and revealed several important qualifications of the theoretically derived notion that the scroll dynamics are determined by the local differential geometry of the singularity. Secondly, I will discuss my recent attempts at more efficient modeling of 3D excitable media with cellular automata. The CA update rules are derived directly from the reaction kinetics, allowing comparison of the CA behavior against a meaningful standard. In both the PDE and CA methodologies, making sense of the simulations requires extensive graphical and geometric analyses, which will also be described.