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 very rich phase diagram as a function of the interaction parameters. Dynamical states include homogeneous oscillations, oscillatory bumps, traveling waves, lurching waves, standing waves arising via a period doubling bifurcation, aperiodic regimes and regimes of multistability. We show that a similar diversity is found in the framework of a reduced rate model in which the interactions are delayed. A great deal of this study can be performed analytically. Our work suggests that the delay in the response of the neurons to their synaptic inputs due to the non-linearity underlying spike generation have important consequences for the collective dynamics of large neuronal systems.