Geometric singular perturbation analysis of the active metabolic oscillator in pancreatic \b{eta}-cells

Abstract

Pancreatic \b{eta}-cells secrete insulin in response to blood sugar levels to maintain glucose homeostasis. This vital insulin exocytosis is controlled by the cell's bursting behaviours, which are regulated by tight bidirectional coupling of inherent electrical and metabolic oscillators. The Integrated Oscillator Model suggests that slower metabolic oscillations are mediated either by glycolytic oscillations-through an independent active metabolic oscillator (AMO)-or by Ca2+ effects on ATP consumption via a passive metabolic oscillator (PMO). By clamping the Ca2+ and ATP dynamics, our study focuses on the decoupled AMO which is the driver of pulsatile dynamics. Using appropriate reference scales, we first non-dimensionalise the model to identify small parameters and processes evolving on different timescales. We show that the AMO can be recast as a surrogate relaxation oscillator, a more general class of multiple timescale problems involving oscillation cycles comprising fast and slow segments, which are amenable to rigorous analysis using the machinery of geometric singular perturbation theory. Using the parametrisation method to identify invariant manifolds and blow-up analysis to desingularise degenerate vector fields, we fully characterise the hierarchy of timescales and the complex singular geometry constituting the metabolic oscillations. Our work considerably extends the `fast-slow' analysis of glycolytic oscillators and is a stepping stone towards understanding how the slower metabolic system temporally patterns the faster electrical bursting dynamics.