Pancreatic $\beta-$Cell Dynamics with Three-Time-Scale Systems

TL;DR

This paper analyzes pancreatic beta-cell oscillations using a three-time-scale mathematical model, applying advanced geometric methods to understand glucose-dependent dynamics and their implications for diabetes research.

Contribution

It introduces a three-time-scale framework and blow-up analysis to study beta-cell oscillations, revealing new insights into bursting behavior and glucose regulation mechanisms.

Findings

Identifies the role of ATP in oscillatory dynamics

Defines linger time at pseudo-singular points

Provides conditions for bursting synchronization

Abstract

Pancreatic $\beta-$cells regulate insulin secretion through complex oscillations, which are vital for glucose control and diabetes research. In this paper, an existing mathematical model of $\beta-$cell dynamics is analyzed using a three-time-scale framework to study interactions among fast, intermediate, and slow variables. Through Geometric Singular Perturbation Theory (GSPT), the influence of ATP on oscillatory dynamics via membrane potential is explored. At the non-hyperbolic point, where standard methods fail, blow-up analysis is applied to investigate canard dynamics shaped by intermediate and slow variables. Numerical simulations with varied parameters reveal the glucose-dependent oscillations linked to slow dynamics near the pseudo-singular points. By leveraging the pseudo-singular point, the linger time is defined, and simulated results for the coupling strength needed for bursting initiation synchronization are presented as a sufficient condition. This study links mathematics and biology, offering insights into diabetic studies.