Reduction and Dynamics of a Generalized Rivalry Network with Two Learned Patterns

We use the theory of coupled cell systems to analyze a neuronal network model for generalized rivalry posed by H. Wilson. We focus on the case of rivalry between two patterns and identify conditions under which 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 reduction is 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 (TB) bifurcations occur in the reduced networks, and this allows us to further reduce much of the dynamics to a planar system. We analyze the dynamics of the quotient systems near the TB singularity, discussing how variation of the input parameter I organizes the dynamics. This variation leads to a degenerate path through the unfolding of the TB point. We also discuss how the network structure affects recurrent excitation in the reduced networks, and the consequences for the dynamics.