  Motivated by a model for the perception of textures by the visual cortex in
primates, we analyse the bifurcation of periodic patterns for nonlinear
equations describing the state of a system defined on the space of structure
tensors, when these equations are further invariant with respect to the
isometries of this space. We show that the problem reduces to a bifurcation
problem in the hyperbolic plane D (Poincar\'e disc). We make use of the concept
of periodic lattice in D to further reduce the problem to one on a compact
Riemann surface D/T, where T is a cocompact, torsion-free Fuchsian group. The
knowledge of the symmetry group of this surface allows to carry out the
machinery of equivariant bifurcation theory. Solutions which generically
bifurcate are called "H-planforms", by analogy with the "planforms" introduced
for pattern formation in Euclidean space. This concept is applied to the case
of an octagonal periodic pattern, where we are able to classify all possible
H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These
patterns are however not straightforward to compute, even numerically, and in
the last section we describe a method for computation illustrated with a
selection of images of octagonal H-planforms.
