  Balanced colorings of networks classify robust synchrony patterns -- those
that are defined by subspaces that are flow-invariant for all admissible ODEs.
In symmetric networks the obvious balanced colorings are orbit colorings, where
colors correspond to orbits of a subgroup of the symmetry group. All other
balanced colorings are said to be exotic. We analyze balanced colorings for two
closely related types of network encountered in applications: trained Wilson
networks, which occur in models of binocular rivalry, and opinion networks,
which occur in models of decision making. We give two examples of exotic
colorings which apply to both types of network, and prove that Wilson networks
with at most two learned patterns have no exotic colorings. We discuss how
exotic colorings affect the existence and stability of branches for
bifurcations of the corresponding model ODEs.
