Apparent discontinuities in the phase-resetting response of cardiac pacemakers

Injection of a brief stimulus pulse resets the spontaneous periodic activity of a sinoatrial node cell: a stimulus delivered early in the cycle generally delays the time of occurrence of the next action potential, while the same stimulus delivered later causes an advance. We investigate resetting in two models, one with a slow upstroke velocity and the other with a fast upstroke velocity, representing central and peripheral nodal cells, respectively. We first formulate each of these models as a classic Hodgkin-Huxley type of model and then as a model representing a population of single channels. In the Hodgkin-Huxley-type model of the slow-upstroke cell the transition from delay to advance is steep but continuous. In the corresponding single-channel model, due to the channel noise then present, repeated resetting runs at a fixed stimulus timing within the transitional range of coupling intervals lead to responses that span a range of advances and delays. In contrast, in the fast-upstroke model the transition from advance to delay is very abrupt in both classes of model, as it is in experiments on some cardiac preparations ("all-or-none" depolarization). We reduce the fast-upstroke model from the original seven-dimensional system to a three-dimensional system. The abrupt transition occurs in this reduced model when a stimulus transports the state point to one side or the other of the stable manifold of the trajectory corresponding to the eigendirection associated with the smaller of two positive eigenvalues. This stable manifold is close to the slow manifold, and so canard trajectories are seen. Our results demonstrate that the resetting response is fundamentally continuous, but extremely delicate, and thus suggest one way in which one can account for experimental discontinuities in the resetting response of a nonlinear oscillator.