  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.
