Dynamics of Two Mutually Coupled Slow Inhibitory Neurons

Inhibition in oscillatory networks of neurons can have apparently paradoxical e ects, sometimes creating dispersion of phases, sometimes fostering synchrony in the network. We analyze a pair of biophysically modeled neurons and show how the rates of onset and decay of inhibition interact with the time scales of the intrinsic oscillators to determine when stable synchrony is possible. We show that there are two di erent regimes in parameter space in which di erent combinations of the time constants and other parameters regulate whether the synchronous state is stable. We also discuss the construction and stability of non-synchronous solutions, and the implications of the analysis for larger networks. The analysis uses geometric techniques of singular perturbation theory that allow one to combine estimates from slow ows and fast jumps.