Exact Neural Fields Incorporating Gap Junctions

Abstract

We consider networks of quadratic integrate-and-fire neurons coupled via both chemical synapses and gap junctions. After transforming to “theta neuron” coordinates, a network's governing equations are of a form amenable to the use of the Ott--Antonsen ansatz. This ansatz allows us to derive an exact description of a network's dynamics in the limit of an infinite number of neurons. For an all-to-all connected network we derive a single (complex) ordinary differential equation, while for spatially extended networks we derive neural field equations (nonlocal partial differential equations). We perform extensive numerical analysis of the resulting equations, showing how the presence of gap junctional coupling can destroy certain spatiotemporal patterns, such as stationary “bump” solutions, and create others, such as traveling waves and spatiotemporal chaos. Our results provide significant insight into the effects of gap junctions on the dynamics of networks of Type I neurons.