SOLITARY WAVES OF INTEGRATE-AND-FIRE NEURAL FIELDS

Arrays of interacting identical neurons can develop coherent firing pa tterns, such as moving stripes that have been suggested as possible ex planations 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. One of these variables distinguishes between the two different states of refractor iness and depolarization and acquires topological meaning when it is t urned into a field. Hence, it leads to a topologic characterization of the ensuing solitary waves, or excitons. They are limited to pointlik e excitations on a line and linear excitations, including all the exam ples noted above, on a two-dimensional surface. A moving patch of firi ng activity is not an allowed solitary wave on our neural surface. Onl y the presence of strong inhomogeneity that destroys the neural field continuity allows for the appearance of patchy incoherent firing patte rns driven by excitatory interactions.