The evolution of synaptically generated waves in one- and two-dimensional domains

We study the onset and evolution of waves in an integrate-and-fire network of synaptically coupled neurons in one- and two-dimensional domains which are restricted to firing one spike. We determine the critical size of initial excitation of the network necessary for the onset of propagation. We also determine the ignition time. We derive an integro-differential equation for the evolution of the firing time as a function of spatial position. We use the evolution equation to understand propagation failure. We compare the behavior of the integrate-and-fire model with a biophysically based model. Finally, we show that weak heterogeneity has similarly weak effects on the firing times.