The Leaky Oscillator: Properties of Inhibition-Based Rhythms Revealed through the Singular Phase Response Curve

Oscillatory neural activity has long been a subject of mathematical interest and has been studied with a range of mathematical tools including phase reduction and phase response curves. But many oscillatory systems of biological interest involve strong interactions with dynamics on multiple time scales, and standard mathematical simplifications may not preserve the interesting mathematical structure of such systems. Here we consider the general class of rhythms generated by spiking systems with slow-decaying, fast-resetting feedback inhibition. We use a simple model, the `inhibition-based rhythm” (IBR), to describe the dynamics of these rhythms under strong forcing or coupling. We define a “singular phase response curve” (sPRC), which describes the response of this system to a strong forcing pulse in the limiting case of well-separated time scales of membrane dynamics and inhibitory decay. Using geometric singular perturbation theory, we demonstrate that this function “persists,” i.e., continues to provid...