Fast-slow analysis of the Integrated Oscillator Model for pancreatic β-cells

Insulin-secreting pancreatic β-cells are electrically excitable cells that are unusual because their electrical activity is influenced directly by metabolism via ATP-sensitive K⁺ channels. At the same time, changes in the intracellular Ca²⁺concentration that result from the cell's electrical activity influence metabolism in several ways. Thus, there is bidirectional coupling between the electrical dynamics and the metabolic dynamics in β-cells. A mathematical model has been previously developed, called the Integrated. Oscillator Model (IOM), to highlight the bidirectional coupling involved in the oscillation mechanism. In this study, we show how this coupling can produce oscillations in β-cell activity. These oscillations have period similar to that of insulin secretion pulses observed in rats, mice, dogs, and humans, which has been shown to facilitate the action of the liver in maintaining glucose homeostasis. In a companion paper we show that the IOM can produce oscillations using two distinct mechanisms, depending on the values of electrical and metabolic parameters. In the present article, we use fast-slow analysis to understand the mechanisms underlying each of these oscillations. In particular, we show why a key variable in the glycolytic pathway generates a pulsatile time course in one type of oscillation, while it generates a sawtooth time course in the other type. The significance of these patterns is that the time course is a reflection of whether an intrinsic glycolytic oscillator is active, or whether the oscillations are a direct consequence of Ca²⁺ feedback onto glycolysis.