  We describe the combinatorics of equilibria and steady states of neurons in
threshold-linear networks that satisfy Dale's law. The combinatorial code of a
Dale network is characterized in terms of two conditions: (i) a condition on
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 encoding a
sublattice in a weakly coupled excitatory network. In the strong coupling
regime we prove that the combinatorial code of a generic Dale network is
intersection-complete and is therefore a convex code, as is common in some
sensory systems in the brain.
