Phase independent resetting in relaxation and bursting oscillators

Relaxation oscillators that depend on one slow variable, such as the Fitzhugh-Nagumo oscillator, reset in a phase-dependent manner. A complete oscillation can be divided into two parts, the "plateau" and "trough", and a prematurely induced plateau or trough is significantly shorter than normal. The class of square-wave bursting oscillators can be viewed as relaxation oscillators with rapid spikes during the plateau, and reset similarly when modeled with one slow variable. However, it has been reported that a physiological bursting oscillator, the membrane potential of the pancreatic beta-cell, resets in a phase-independent manner, such that a prematurely induced plateau/trough has normal length. A possible model for such an oscillator requires two slow variables, one to control the length of the plateau and the other the length of the trough. Here, we explore the geometric solution structure of two such models, which exhibit the desired resetting. One is a generalization of the Fitzhugh-Nagumo equations, and the other is a bursting oscillator using known beta-cell electrical currents with an additional hypothetical slow outward current.