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Synchronized patterns in hierarchical networks of neuronal oscillators with ${𝔻}_{3}×{𝔻}_{3}$ symmetry. (English) Zbl 0939.92005
Summary: The spatiotemporal patterns generated by systems of nine coupled nonlinear oscillators which are equivariant under the permutation symmetry group ${D}_{3}×{D}_{3}$ are determined. This system can be interpreted as a hierarchically organized network composed of three interacting systems each of which consists of three coupled oscillators. We determine generic synchronized oscillation patterns and transitions between these analytically, by numerical simulations, and experimentally with an electronic analog-network. In the theoretical analysis the representative nonlinear ordinary differential equations are reduced to the normal form equations for coupled Hopf bifurcations in an eight-dimensional center eigenspace, whose generic states have been classified previously. The results are applied to a specific model system in which the network is formed by a class of oscillators, each composed of two asymmetrically coupled Hopfield neurons. Experiments performed on an analog-electronic network of such nonlinear oscillators show that most of the states predicted by the theory of the Hopf bifurcation with ${D}_{3}×{D}_{3}$-symmetry appear in a stable way. We find a great variety of periodic and quasiperiodic oscillation patterns of maximal and submaximal symmetry which can be classified in a two-level pattern hierarchy. ln addition to these states we find in simulations homoclinic cycles within the same isotropy class as well as heteroclinic switchings between such cycles.
##### MSC:
 92B20 General theory of neural networks (mathematical biology) 68T05 Learning and adaptive systems 34C23 Bifurcation (ODE) 92C15 Developmental biology, pattern formation