The combinatorial code and the graph rules of Dale networks

TL;DR

This paper characterizes the combinatorial codes of Dale networks, showing how network connectivity and synaptic strength influence neural equilibrium states, with implications for understanding sensory processing.

Contribution

It provides a mathematical framework linking network structure and synaptic properties to neural coding patterns, including conditions for convex codes and a learning rule for sublattices.

Findings

In weak coupling, the code depends only on network connectivity.

In strong coupling, the code is intersection-complete and convex.

A learning rule for encoding sublattices in weakly coupled networks.

Abstract

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.