Euclidean shift-twist symmetry in population models of self-aligning objects
We consider the symmetry properties of a general class of non--local 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 $\R^2 \times S^1$. 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.
University of Utah
| Department of Mathematics
|
bressloff@math.utah.edu
Jan 2004.