Classification of 2-node excitatory–inhibitory networks

Abstract

We classify connected 2-node excitatory–inhibitory networks under various conditions. We assume that, as well as for connections, there are two distinct node-types, excitatory and inhibitory. In our classification we consider four different types of excitatory–inhibitory networks: restricted, partially restricted, unrestricted and completely unrestricted. For each type we give two different classifications. Using results on ODE-equivalence and minimality, we classify the ODE-classes and present a minimal representative for each ODE-class. We also classify all the networks with valence $\le 2$. These classifications are up to renumbering of nodes and the interchange of ‘excitatory’ and ‘inhibitory’ on nodes and arrows. These classifications constitute a first step towards analysing dynamics and bifurcations of excitatory–inhibitory networks. The results have potential applications to biological network models, especially neuronal networks, gene regulatory networks, and synthetic gene networks.