We investigate the dynamics of a one-dimensional network of spiking neurons with spatially modulated excitatory and inhibitory interactions through extensive numerical simulations. We find that the network displays a rich repertoire of dynamical states as a function of the interaction parameters, including homogeneous oscillations, oscillatory bumps, traveling waves, lurching waves, standing waves, quasi-periodic and chaotic states as well as regimes of multistability. Combining analytical calculations and simulations we show that similar dynamics are found in a reduced rate model provided that the interactions are delayed.