Predicting patterns in lateral inhibition systems via graph partitioning

Abstract

We analyze pattern formation on a network of cells where each cell inhibits its neighbors through cell-to-cell contact signaling. The network is modeled as a graph where each identical individual cell is a vertex and where neighboring cells are connected by an edge. We search for steady-state patterns by partitioning the graph vertices into disjoint classes, where the cells in the same class have the same final fate. To prove the existence of steady-states with this structure, we use results from monotone systems theory. Finally, we analyze the stability of these patterns by relying on a block decomposition that is based on the graph partition.