Mathematical Biology

This paper builds on the past study of single-spike waves in one-dimensional integrate-and-fire networks to provide a framework for the study of waves with arbitrary (finite or countably infinite) collections of spike times. Based on this framework, we prove an existence theorem for single-spike traveling waves, and we combine analysis and numerics to study two-spike traveling waves, periodic traveling waves, and general infinite spike trains. For a fixed wave speed, finite-spike waves, periodic waves, and other infinite-spike waves may all occur, and we discuss the relationships among them. We also relate the waves considered analytically to waves generated in numerical simulations by the transient application of localized excitation.