Continuum limit of cellular automata

Question

Is there any function defined for say the plane, that has interesting nontrivial behaviour similar to Conway's Game Of Life, but where every point's on/off status is decided by something like the density of on points in some finite neighbourhood of the point in the plane. Alternatively the status is decided by the average status weighted by 1/distance. Preferably every point should be part of a continuous curve for easier visualization.