Solitary waves on manifolds of integrate-and-fire neurons

We investigate lattices of interacting integrate-and-fire neurons. Excitatory short-range interactions lead to coherent firing patterns such as moving stripes on a two-dimensional surface. They have been suggested as possible explanations of hallucinatory phenomena. Other known formations include rotating spirals and expanding concentric rings. We obtain all of them using a novel two-variable description of integrate-and-fire neurons that allows for a continuum formulation of neural fields. Within our model we can formulate a topological constraint that explains why all the moving coherent patterns have a dimension lower by one unit from that of the manifold. In the presence of strong inhomogeneity, neural continuity breaks down, and the field approximation no longer holds. In that case, one obtains incoherent firing patterns driven by the excitatory interactions.