We study the propagation of traveling solitary pulses in one-dimensional cortical networks with two types of one-dimensional architectures: networks of excitatory neurons and networks composed of both excitatory and inhibitory neurons. Each neuron is represented by the integrate-and-fire model, and is allowed to fire only one spike. The velocity and stability of propagating, continuous pulses are calculated analytically. For excitatory-only networks, two continuous pulses with different velocities exist if the synaptic coupling is larger than a minimal value; the pulse with the lower velocity is always unstable. Above a certain critical value of the constant delay, continuous pulses lose stability via a Hopf bifurcation, and lurching pulses with spatio-temporal periodicity emerge. The parameter regime for which lurching occurs is strongly affected by the synaptic footprint (connectivity) shape. Two types of stable propagating pulses are observed in networks of excitatory and inhibitory neurons. During fast pulses, inhibitory neurons fire a short time before or after the excitatory neurons. During slow pulses, inhibitory cells fire well before neighboring excitatory cells, and potentials of excitatory cells become negative and then positive before they fire. This work shows that simple models of spiking neurons exhibit a large variety of propagating pulses with various spatio-temporal properties.