Scalar and pseudoscalar bifurcations motivated by pattern formation on the visual cortex

Bosch Vivancos, Chossat and Melbourne showed that two types of steady-state bifurcations are possible from trivial states when Euclidean equivariant systems are restricted to a planar lattice-scalar and pseudoscalar-and began the study of pseudoscalar bifurcations. The scalar bifurcations have been well studied since they appear in planar reaction-diffusion systems and in plane layer convection problems. Bressloff, Cowan, Golubitsky, Thomas and Wiener showed that bifurcations in models of the visual cortex naturally contain both scalar and pseudoscalar bifurcations, due to a different action of the Euclidean group in that application. In this paper, we review the symmetry discussion in Bressloff et al and we continue the study of pseudoscalar bifurcations. Our analysis furthers the study of pseudoscalar bifurcations in three ways. (a) We complete the classification of axial subgroups on the hexagonal lattice in the shortest wavevector case proving the existence of one new planform-a solution with triangular D-3 symmetry. (b) We derive bifurcation diagrams for generic bifurcations giving, in particular, the stability of solutions to perturbations in the hexagonal lattice. For the simplest (codimension zero) bifurcations, these bifurcation diagrams are identical to those derived by Golubitsky, Swift and Knobloch in the case of Benard convection when there is a midplane reflection-though the details in the analysis are more complicated. (c) We discuss the types of secondary states that can appear in codimension-one bifurcations tone parameter in addition to the bifurcation parameter), which include time periodic states from roll and hexagon solutions and drifting solutions from triangles (though the drifting solutions are always unstable near codimension-one bifurcations). The essential difference between scalar and pseudoscalar bifurcations appears in this discussion.