Patterns are macroscopic structures which are derived from the self-organization of a system consisting of macroscopic interacting entities. The formation of spatial patterns was originally based on the competition between fast diffusing chemicals, the inhibitors, and the slow diffusing activators. This theory originally assumed a continuous space, but we now know that if one looks closely, many systems are discrete, and can be abstracted as a network. One such system is the brain, which can be described as a network of connected neurons. There has been a wealth of macroscopic patterns of activity observed in brain networks. In this work we have developed a mathematical theory which explains the crucial role of the topology of the underlying networks. Using the spectrum of the Laplacian of a network, we show that modularity can induce the formation of patterns.
Thus, our main result is that modular networks can exhibit Turing instabilities for cases where the ratio between the activator and inhibitor diffusion coefficients is close to one. Turing patterns can arise in nonlinear systems of reaction-diffusion equations involving at least two species. They form when a spatially homogeneous steady state is linearly unstable to an inhomogeneous perturbation. This can normally only occur when there is a large difference between the diffusion constants of the activator and inhibitor.
Turing instabilities are studied by linearizing about the homogeneous steady state, and calculating the dispersion relation; in the networks case this relates to the eigenvalues of the Laplacian (which represent the different spatial patterns of instability via their eigenvectors) and the eigenvalues of the Jacobian (which represent the speed at which a small perturbation will grow or shrink). For modular networks, the eigenvectors can force macroscopic patterns, as reported in the brain.