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

TL;DR

This paper introduces a modified cellular automaton model that incorporates activation and inhibition regions with specific geometries to simulate pattern formation on anisotropic substrates with biaxial dependence, enhancing stability and directional sensitivity.

Contribution

The model extends Young's cellular automaton by integrating geometrically shaped activation/inhibition regions and a transition zone, allowing for better simulation of anisotropic patterning processes.

Findings

Directional sensitivity detected via correlation functions.

Pattern evolution depends on activation/inhibition geometry.

Model demonstrates two-parameter influence on final cell concentration.

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.