We study the propagation of traveling solitary pulses in one-dimensional networks of excitatory and inhibitory integrate-and-fire neurons. Slow pulses, during which inhibitory cells fire well before neighboring excitatory cells, can propagate along the network at intermediate inhibition levels. At higher levels, they destabilize via a Hopf bifurcation. There is a bistable parameter regime in which both fast and slow pulses can propagate. Lurching pulses with spatiotemporal periodicity can propagate in regimes for which continuous pulses do not exist.