Euclidean shift-twist symmetry in population models of self-aligning objects

We consider the symmetry properties of a general class of nonlocal population models describing the aggregation and alignment of oriented objects in two dimensions. Such objects could be at the level of molecules, cells, or whole organisms. We show that the underlying interaction kernel is invariant under the so-called shift-twist action of the Euclidean group acting on the space R-2 x S-1. This group action was previously studied within the context of a continuum model of primary visual cortex. We use perturbation methods to solve the eigenvalue problem arising from linearization about a homogeneous state, and then use equivariant bifurcation theory to identify the various types of doubly periodic patterns that are expected to arise when the homogeneous state becomes unstable. We thus establish that two distinct forms of spatio-angular order can occur, corresponding to scalar and pseudoscalar representations of the Euclidean group.