Pattern formation by lateral inhibition: a case study in networked dynamical systems
Abstract:
A common pattern formation mechanism in multi-cellular organisms is lateral inhibition where the cells inhibit their immediate neighbors through a contact signaling mechanism. We will present a broad dynamical model to represent this mechanism and reveal the key properties of the model that are necessary for patterning. The model consists of subsystems representing the biochemical reactions in individual cells, interconnected according to an undirected graph describing which cells are in contact. By making use of input-output properties of the subsystems and the spectral properties of the adjacency matrix for the contact graph, we will present verifiable conditions that determine when the spatially homogeneous steady-state loses its stability and what types of patterns emerge. In particular, we will classify types of graphs using special symmetries and exhibit the associated patterns. Above all, this talk will showcase the merger of a control-theoretic input-output approach with graph-theoretic concepts to infer global behavior of a large-scale and nonlinear networked system.