  Networks of neurons in some brain areas are flexible enough to encode new
memories quickly. Using a standard firing rate model of recurrent networks, we
develop a theory of flexible memory networks. Our main results characterize
networks having the maximal number of flexible memory patterns, given a
constraint graph on the network's connectivity matrix. Modulo a mild
topological condition, we find a close connection between maximally flexible
networks and rank 1 matrices. The topological condition is H_1(X;Z)=0, where X
is the clique complex associated to the network's constraint graph; this
condition is generically satisfied for large random networks that are not
overly sparse. In order to prove our main results, we develop some
matrix-theoretic tools and present them in a self-contained section independent
of the neuroscience context.
