Modelling morphogenetic processes in excitable media using novel cellular automata

As an efficient alternative to partial differential equations, cellular automata have been proposed for the simulation of excitable media. However, the automata proposed so far are anisotropic since the shapes of the propagating waves are related to the shapes of the cells (e.g. squares or hexagons). This problem is solved in the present work by automata based on a random distribution of excitable elements. The following results were obtained using a (minimal) model with only three parameters: a) Target patterns (concentric circular waves) and spirals in two dimensions in agreement with measurements in the Belousov-Zhabotinskii (BZ) reagent and in heart muscle during ventricular fibrillation. b) Waves rotating around a hole in agreement with observations on the cerebral cortex. c) A linear eikonal equation relating the normal velocity to the curvature of the wavefront in agreement with measurements in the BZ reagent and in the aggregating slime mold D. discoideum. d) Turbulences similar to those observed in experiments. e) Three-dimensional waves (scroll waves), some of which have been observed in thick heart muscle and in the BZ reagent. An extension of the model including two more parameters yields a dispersion relation comparable to that obtained from measurements and from an analysis of the relevant partial differential equations.