Symmetry and Pattern Formation in Coupled Cell Networks

We describe some basic concepts and techniques from symmetric bifurcation theory in the context of coupled systems of cells (‘oscillator networks’). These include criteria for the existence of symmetry-breaking branches of steady and periodic states. We emphasize the role of symmetry as a general framework for such analyses. As well as overt symmetries of the network we discuss internal symmetries of the cells, ‘hidden’ symmetries related to Neumann boundary conditions, and spatio-temporal symmetries of periodic states. The methods are applied to a model central pattern generator for legged animal locomotion.