C. Diekman, M. Golubitsky, T. McMillen and Y. Wang
Reduction and Dynamics of a Generalized Rivalry Network with Two Learned Patterns
SIAM J. Appl. Dynam. Sys. 11 (4) (2012) 1270-1309.
We use the theory of coupled cell systems to analyze a neuronal network model for generalized rivalry posed by Hugh Wilson. We focus on the case of rivalry between two patterns and show how large networks of n attributes and m intensity levels can reduce to a model consisting of two or three cells depending on whether or not the patterns have any attribute levels in common. The two-cell reductions are equivalent to certain recent models of binocular rivalry. Notably, these reductions can lead to large recurrent excitation in the reduced network even though the individual cells in the original network may have none. We also show that symmetry-breaking Takens-Bogdanov bifurcations occur in the reduced network, and this allows us to further reduce much of the dynamics to a planar system and deduce the co-existence of states of rivalry and winner-take-all behavior. We analyze the dynamics of the of the two-cell model near the Takens-Bogdanov singularity, discussing how variation of the system input parameter I organizes the dynamics. We also discuss how the network structure acts recurrent excitation in the reduced networks, and the consequences for the dynamics.