Patterns in hierarchical networks of neuronal oscillators with D3×Z3 symmetry

In this paper, we consider a hierarchically organized network composed of three interacting systems each of which consists of three coupled oscillators. This hierarchical network is equivariant under the symmetry group D3×Z3. Using the lattice of isotropy subgroups, we study the reduced equations restricted to invariant fixed-point subspaces and prove that it is possible for the oscillator network to have 4 distinct equilibria or 45 distinct periodic solutions with maximal isotropy subgroups. These are classified and their bifurcation directions are determined in terms of the quadratic coefficients and relevant quantities. A center manifold reduction from the hierarchical network to the normal form equations is then performed in order to investigate the codimension two bifurcations. Using this reduction we find a great variety of equilibria, periodic and quasi-periodic oscillation patterns of maximal and submaximal symmetry which can be classified in a two-level pattern hierarchy.