A modified cellular automaton using activation and inhibition regions geometrically compatible with biaxial anisotropy

Abstract

Young's cellular automaton, recently applied to study the spatiotemporal evolution of binary patterns for favorable/hostile environments, has now been modified from a different point of view. In this model, each differentiated cell (DC) produces two diffusing morphogens: a short-range activator and a long-range inhibitor. Their combination creates the so-called local 'w' field. Undifferentiated cells (UCs) are passive. The question arises how to adapt it to modelling patterning processes in anisotropic substrates with a biaxial dependence of the morphogen diffusion rate. We use activation/inhibition regions with appropriate shape geometry defined by the so-called deformation parameter p. We complement this model by adding a physically reasonable transition zone with controlled local field slope. The patterning process uses the morphogenetic field W calculated separately for each cell, which is the sum of the 'w' values generated by all regional DCs surrounding the cell. It acts as a chemical signal determining the next state of the cell. We also introduce a threshold W* defining the required absolute chemical signal strength. The state of each cell can change depending on the rules for W and W*. This improves the stability of the model evolution and extends its applications. Using two pairs of two-point orthogonal correlation functions, we reveal their directional sensitivity to changes in biaxial anisotropy. Finally, we illustrate the general two-parameter dependence of the average final DC concentration on the geometry of the activation/inhibition regions and on the value of the long-range inhibitor. This facilitates the recognition of characteristic features of the evolution of DC concentration in our model.