Slow waves in mutually inhibitory neuronal networks

A variety of experimental and modeling studies have been performed to investigate wave propagation in networks of thalamic neurons and their relationship to spindle sleep rhythms. It is believed that spindle oscillations result from the reciprocal interaction between thalamocortical (TC) and thalamic reticular (RE) neurons. We consider a network of TC and RE cells reduced to a one-layer network model and represented by a system of singularly perturbed integral-differential equations. Geometric singular perturbation methods are used to prove the existence of a locally unique slow wave pulse that propagates along the network. By seeking a slow pulse solution, we reformulate the problem to finding a heteroclinic orbit in a 3D system of ODEs with two additional constraints on the location of the orbit at two distinct points in time. In proving the persistence of the singular heteroclinic orbit, difficulties arising from the solution passing near points where normal hyperbolicity is lost on a 2D critical manifold are overcome by employing results by Wechselberger [Singularly perturbed folds and canards in R3, Thesis, TU-Wien, 1998].