Pattern Formation in Large-Scale Networks with Asymmetric Connections

Two of the most common pattern formation mechanisms are Turing-patterning in reaction-diffusion systems and lateral inhibition of neighboring cells. In this paper, we introduce a broad dynamical model of interconnected cells to study the emergence of patterns, with the above mentioned two mechanisms as special cases. This model comprises modules encapsulating the biochemical reactions in individual cells, and interconnections are captured by a weighted directed graph. Leveraging only the static input/output properties of the subsystems and the spectral properties of the adjacency matrix, we characterize the stability of the homogeneous fixed points as well as sufficient conditions for the emergence of spatially non-homogeneous patterns. To obtain these results, we rely on properties of the graphs (bipartiteness, equitable partitions) together with tools from monotone systems theory. As application example, we consider pattern formation in neural networks to illustrate the practical implications of our results. Our results do not restrict the number of cells or reactants, and do not assume symmetric connections between two connected cells.