SYNCHRONIZED PATTERNS IN HIERARCHICAL NETWORKS OF NEURONAL OSCILLATORS WITH D-3 X D-3 SYMMETRY

The spatiotemporal patterns generated by systems of nine coupled nonli near oscillators which are equivariant under the permutation symmetry group D-3 x D-3 are determined. This system can be interpreted as a hi erarchically organized network composed of three interacting systems e ach of which consists of three coupled oscillators. We determine gener ic synchronized oscillation patterns and transitions between these ana lytically, by numerical simulations, and experimentally with an electr onic analog-network. In the theoretical analysis the representative no nlinear ordinary differential equations are reduced to the normal form equations for coupled Hopf bifurcations in an eight-dimensional cente r eigenspace, whose generic states have been classified previously. Th e results are applied to a specific model system in which the network is formed by a class of oscillators, each composed of two asymmetrical ly coupled Hopfield neurons. Experiments performed on an analog-electr onic network of such nonlinear oscillators show that most of the state s predicted by the theory of the Hopf bifurcation with D-3 x D-3-symme try appear in a stable way. We find a great variety of periodic and qu asiperiodic oscillation patterns of maximal and submaximal symmetry wh ich can be classified in a two-level pattern hierarchy. In addition to these states we find in simulations homoclinic cycles within the same isotropy class as well as heteroclinic switchings between such cycles . Copyright (C) 1998 Elsevier Science B.V.