The combinatorial code and the graph rules of Dale networks

Relating connectivity features to the function of a recurrent neuronal network is a longstanding
problem that still remains at large. To this end we describe the combinatorics of equilibria
and steady states of neurons in recurrent networks that satisfy the Dale’s law. It turns out that
the combinatorial code of a Dale network can be characterized in terms of two properties: (i) the
network connectivity graph, and (ii) a spectral condition on the synaptic matrix. We find that in the
weak coupling regime the combinatorial code depends only on the connectivity graph, and not on the
particulars of the synaptic strengths. Moreover, we prove that the combinatorial code of a weakly
coupled network is a sublattice, and we provide a learning rule for learning a sublattice in such a
network.
Neural activity in sensory areas of the brain is shaped both by the stimulus and by the internal
neural dynamics. Receptive fields of individual neurons are convex in a number of brain regions (such
as the hippocampus, and the visual cortex). Not any combinatorial code is compatible with convex
receptive fields. This raises a natural question: how do recurrent networks produce convex codes?
Surprisingly, we find that the architecture of a Dale network “enforces” convex code output, both in
weak and strong coupling regimes.