The Geometry of Bursting in the Dual Oscillator Model of Pancreatic $\beta$-cells

Abstract

We describe an extended fast-slow analysis for the dual oscillator model (DOM) for bursting oscillations in pancreatic beta-cells, which occur on a wide range of time scales, from seconds to minutes. This wide dynamic range has been suggested to result from the interactions of a very slow metabolic, possibly glycolytic, oscillator and a faster electrical oscillator, itself containing several negative feedback mechanisms with a range of time scales. Although the high dimensionality of the slow subsystem would defeat a straightforward fast-slow analysis, we show that an approximate geometrical analysis that exploits particular features of the DOM and is based on superimposing the bifurcation diagrams of the component oscillators leads to new insights into the functioning of the system.