Wave propagation in filamental cellular automata

Motivated by questions in biology and distributed computing, we investigate the behaviour of particular cellular automata, modelled as one-dimensional arrays of identical finite automata. We investigate what sort of self-stabilising cooperative behaviour these can induce in terms of waves of cellular state changes along a filament of cells. We discover what the minimum requirements are, in terms of numbers of states and the range of communication between automata, to observe this for individual filaments. We also discover that populations of growing filaments may have useful features that the individual filament does not have, and we give the results of numerical simulations.