We consider a delayed network of two neurons with both self-feedback and interaction described by an all-or-none threshold function. The model describes a combination of analog and digital signal processing in the network and takes the form of a system of delay differential equations with discontinuous nonlinearity. We show that the dynamics of the network can be understood in terms of the iteration of a one-dimensional map, and we obtain simple criteria for the convergence of solutions, the existence, multiplicity and attractivity of periodic solutions.