Formation of Morphogenetic Patterns in Cellular Automata

One of the most important problems in contemporary science, and especially in biology, is to reveal mechanisms of pattern formation. On the level of biological tissues, patterns form due to interactions between cells. These interactions can be long-range if mediated by diffusive molecules or short-range when associated with cell-to-cell contact sites. Mathematical studies of long-range interactions involve models based on differential equations while short-range interactions are modelled using discrete type models. In this paper, we use cellular automata (CA) technique to study formation of patterns due to short-range interactions. Namely, we use von Neumann cellular automata represented by a finite set of lattices whose states evolve according to transition rules. Lattices can be considered as representing biological cells (which, in the simplest case, can only be in one of the two different states) while the transition rules define changes in their states due to the cell-to-cell contact interactions. In this model, we identify rules resulting in the formation of stationary periodic patterns. In our analysis, we distinguish rules which do not destroy preset patterns and those which cause pattern formation from random initial conditions. Also, we check whether the forming patterns are resistant to noise and analyse the time frame for their formation. Transition rules which allow formation of stationary periodic patterns are then discussed in terms of pattern formation in biology.