Wave Propagation in Filamental Cellular Automata

1. Introduction

In both the realms of nanotechnology and natural biology there is the need to study how the behaviour of individual cellular components, acting through purely local stimuli, produce patterns of coordinated behaviour in extended systems made up of large numbers of these microscopic cells. Cellular automata have been successfully applied to the study of natural systems (Chopard & Droz, 1998; Deutsch & Dormann, 2005; Ermentrout & Edelstein-Keshet, 1993), and we are particularly interested in the emergence of oscillating wave patterns. Such patterns are central to the study of systems as diverse as cellular pattern formation (Alber, et al., 2002), excitable media (Greenberg & Hastings, 1978) and physiological development (Koch & Meinhardt, 1994).

This paper reports on preliminary studies of particular cellular automata, namely one-dimensional strings (filaments) of identical finite automata (cells). A filament state is simply the string of states of the automata reading, say, from left to right along the filament. The automata take as input states of its neighbours and, depending on its current state, the input determines the next state of the automaton. Working in synchronised cycles this local behaviour determines successive states of the filament. We are then particularly interested in the behaviour, over time, of the filament state. Under what conditions may it exhibit coordination of the individual cellular components? For the domains of interest that we have in mind, we are interested in the simplest of automata, in terms of numbers of states, minimal input and design details that will induce coordination of cellular activity. In particular, we mainly concentrate on automata with no more than three states. Also, we only deal with the simplest of filaments, namely those consisting of identical finite automata.