We show how a synchronized pair of integrate-and-fire neural oscillators with non-instantaneous synaptic interactions can destabilize in the strong coupling regime resulting in non-phase-locked behaviour. In the case of symmetric inhibitory coupling, desynchronization produces an inhomogeneous state in which one of the oscillators becomes inactive (oscillator death). On the other hand, for asymmetric excitatory/inhibitory coupling, mode-locking can occur leading to periodic bursting patterns. The consequences for large globally coupled networks is discussed.