Propagation of traveling waves in excitable media.

Traveling waves transmit information through space. Wave patterns arise in phenomena as diverse as the migrating action potentials of neurons (Hodgkin and Huxley 1952), spiraling bands of electrical excitation detected in cardiac muscle (Davidenko et al. 1992), and the movement of calcium within differentiating Xenopus laevis oocytes (Lechleiter et al. 1991). In each of these cases, an excitable medium serves to promote wave propagation. An excitable medium is typically comprised of a continuous set of locally excitable regions, which can be both independently stimulated and inhibited. These media exhibit a sensitivity threshold below which the media persist undisturbed at a stable resting state. While subthreshold perturbations are rapidly diminished, greater-than-threshold signals induce an abrupt local transformation within a portion of the medium. Shortly after this change occurs, the region becomes transiently refractory to further perturbation, after which it relaxes to the resting state (Tyson and Keener 1988}. If a localized medium-transforming signal can bring neighboring regions to threshold before being damped out, a signal wave can spread. The kinetic link between the propagation of a transforming signal and the control of its refractory state underlies the mechanism of movement of many types of traveling waves, including those of a Belousov-Zhabotinsky (BZ) reaction, the oxidation and decarboxylization of malonic acid by bromate, and the developmental waves of Myxococcus xanthus and Dictyostelium cliscoideum cells. In the multicellular prokaryote M. xanthus, traveling waves result from cellcontact-mediated intercellular C-signaling, leading to the movement of cell-cell signaling waves in the absence of bulk cell movement. The C-factor, which is cell bound, acts over a very short range. In contrast, D. discodium ameboid cell migrations are directed in part by diffusing cAMP, which is alternately perceived and produced by relays of aggregating cells. Both chemical autocatalysis and positive autoregulation can direct the formation of waves in excitable media. This review compares these morphogenetic mechanisms and reveals