# Boolean analysis of lateral inhibition

## TL;DR

This paper analyzes Boolean network models of lateral inhibition, focusing on fixed points, reachability, and robustness, providing insights into pattern formation in biological systems and guiding future simulations.

## Contribution

It offers a novel Boolean framework for analyzing lateral inhibition, characterizing fixed points, trap spaces, and robustness in asynchronous dynamics.

## Key findings

- All attractors are fixed points regardless of neighbor relations.
- Characterization of trap spaces and basins of attraction.
- Patterns are robust to certain perturbations.

## Abstract

We study Boolean networks which are simple spatial models of the highly conserved Delta-Notch system. The models assume the inhibition of Delta in each cell by Notch in the same cell, and the activation of Notch in presence of Delta in surrounding cells. We consider fully asynchronous dynamics over undirected graphs representing the neighbour relation between cells. In this framework, one can show that all attractors are fixed points for the system, independently of the neighbour relation, for instance by using known properties of simplified versions of the models, where only one species per cell is defined. The fixed points correspond to the so-called fine-grained "patterns" that emerge in discrete and continuous modelling of lateral inhibition. We study the reachability of fixed points, giving a characterisation of the trap spaces and the basins of attraction for both the full and the simplified models. In addition, we use a characterisation of the trap spaces to investigate the robustness of patterns to perturbations. The results of this qualitative analysis can complement and guide simulation-based approaches, and serve as a basis for the investigation of more complex mechanisms.