Diffusion and wave propagation in cellular automaton models of excitable media. (English) Zbl 0744.92002
Excitable media support undamped traveling waves of excitation, such as waves of membrane depolarization in nerve axons, waves of excitation in chemical reaction systems, and waves of star formation in galactic dust clouds. In physical and chemical systems these waves provide striking examples of spontaneous symmetry breaking and spatiotemporal organization, and in biological systems they serve essential roles in cellular communication and organization. Although some of the first models of excitable media were based on the discrete cellular automaton approach, continuous partial differential equation (PDE) models have dominated recent theory of spatiotemporal organization in excitable media. This hegemony of PDEs has come about because the original CA models lacked several essential features of excitable media.
Here (1) We examine general “masks” as discrete approximations to the diffusion equation, showing how to calculate the diffusion coefficient from the elements of the mask. (2) We combine the mask with a thresholding operation to simulate the propagation of waves in excitable media, showing that (for well-chosen masks) the waves obey a linear “speed-curvature” relation with slope given by the predicted diffusion coefficient. (3) We assess the utility of different masks in terms of computational efficiency and adherence to a linear speed-curvature relation.
Here (1) We examine general “masks” as discrete approximations to the diffusion equation, showing how to calculate the diffusion coefficient from the elements of the mask. (2) We combine the mask with a thresholding operation to simulate the propagation of waves in excitable media, showing that (for well-chosen masks) the waves obey a linear “speed-curvature” relation with slope given by the predicted diffusion coefficient. (3) We assess the utility of different masks in terms of computational efficiency and adherence to a linear speed-curvature relation.
MSC:
| 92B05 | General biology and biomathematics |
| 65Z05 | Applications to the sciences |
| 92-08 | Computational methods for problems pertaining to biology |
Keywords:
cellular automaton models of excitable media; finite difference methods; undamped traveling waves of excitation; waves of membrane depolarization; nerve axons; waves of excitation; chemical reaction systems; cellular communication; models of excitable media; discrete approximations to diffusion equations; speed-curvature relationReferences:
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