Convolution Generated Motion as aLink Between Cellular

Cellular automata have been used to model the formation and dynamics of patterns in a variety of chemical, biological and ecological systems. However, for patterns in which sharp interfaces form and propagate, automata simulations can exhibit undesirable properties, including spurious anisotropy and poor representation of interface curvature eeects. These simulations are also prohibitively slow when high accuracy is required, even in two dimensions. Also, the highly discrete nature of automata models makes theoretical analysis diicult. In this paper, we present a method for generating interface motions that is similar to the threshold dynamics type cellular automata, but based on continuous convolutions rather than discrete sums. These convolution generated motions naturally achieve the ne-grid limit of the corresponding automata, and they are also well suited to numerical and theoretical analysis. Because of this, the desired pattern dynamics can be computed accurately and eeciently using adaptive resolution and Fast Fourier Transform techniques, and for a large class of convolutions the limiting interface motion laws can be derived analytically. Thus convolution generated motion provides a numerically and analytically tractable link between cellular automata models and the smooth features of pattern dynamics. This is useful both as a means of describing the continuum limits of automata, and also as an independent foundation for expressing models for pattern dynamics. In this latter role, it also has a number of beneets over the traditional reaction-diiusion/Ginzburg-Landau continuum PDE models of pattern formation, which yield true moving interfaces only as singular limits. We illustrate the power of this approach with convolution generated motion models for pattern dynamics in developmental biology and excitable media.