Stable Antiphase Oscillations in a Network of Electrically Coupled Model Neurons

Abstract

We consider a model of two identical neurons with electrical coupling and give a rigorous analysis for when the network exhibits stable antiphase behavior. Each neuron is modeled as a relaxation oscillator, and the main technique used in the analysis is geometric singular perturbation theory. Using fast/slow analysis, we derive a one-dimensional map; the antiphase solution corresponds to a fixed point of this map. Detailed analysis of the map, along with its derivative, leads to precise conditions on the duty cycle of an individual cell for when there exists an antiphase solution and when it is stable, for sufficiently weak electrical coupling. The results presented here extend our previous analysis [D. Terman et al., SIAM J. Appl. Dyn. Syst., 10 (2011), pp. 1127--1153] to more biophysical neuronal models.