A dynamical mechanism underlying pattern formation in a spatially extended network of integrate-and-fire oscillators with synaptic interactions is identified. It is shown how in the strong coupling regime the network undergoes a discrete Turing-Hopf bifurcation of the firing-times from a synchronous state to a state with periodic or quasiperiodic variations of the inter-spike intervals on closed orbits. The separation of these orbits in phase space results in a spatially periodic pattern of mean firing-rate across the network that is modulated by deterministic fluctuations of the instantaneous firing-rate.