Pancreatic $β-$Cell Dynamics with Three-Time-Scale Systems

TL;DR

This work analyzes a $3$-time-scale network model of pancreatic $β$-cell dynamics to understand ATP-driven bursting and islet synchronization. Using Geometric Singular Perturbation Theory and a blow-up desingularization near a pseudo-singular point, it reveals canard dynamics that govern the quiescent phase and its glucose dependence. Numerical results show that increasing glucose $G$ shortens the bursting period and reduces linger time, while synchronization across cells requires coupling strength that scales with linger time. The study links rigorous slow–fast mathematics to beta-cell physiology, offering a framework for exploring diabetes-related dysregulation and potential therapeutic strategies for glucose homeostasis.

Abstract

Pancreatic $β-$cells regulate insulin secretion through complex oscillations, which are vital for glucose control and diabetes research. In this paper, an existing mathematical model of $β-$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.

Figures (10)

• Figure 1: Bursting oscillation of simplified $\beta-$cell model (\ref{['ScaledMarinelli']}) of Marinelli et al. model marinelli2022oscillations, for $G=13\ mM$.
• Figure 2: Critical manifold of the system (\ref{['ScaledMarinelli']}), (a) in ($v,x,y$) and (b) in ($v,x,z$). The color bar for $z$ in (a) shows how the critical manifold $\mathcal{C}_1$ (\ref{['cm:eps0delta0']}) embedded on $\mathcal{C}$ (\ref{['cm:eps0']}).
• Figure 3: Critical manifold ($\varepsilon$=0 and $\delta=0$) and the trajectory ($t\in[0,30]$ minutes) of the full system (blue) for $G=8\ mM$, and other parameters remain the same as Table \ref{['tab:ScalePara']}. When the trajectory reaches the fold curve $\mathcal{L}$ (\ref{['cm:eps0foldcurve']}) (magenta) of the critical manifold $\mathcal{C}$, it leaves the stable branch.
• Figure 4: Eigenvalue at the pseudo singular point of the system (\ref{['eq:PSPatzero4D']}) for glucose concentration, $G$ ($mM$), in $[6,15]$.
• Figure 5: In the invariant manifold $r_3=0$, the special solution (red) and the solution of the system (\ref{['K3systembetacell']}) (blue) in $(v_3, x_3)$.

Theorems & Definitions (12)

• Definition 1.1: Limit Cycle
• Definition 1.2: Poincaré Map wiggins2003intro
• Definition 1.3: Canard solution tchizawa2007generic
• Remark 2.1
• Definition 3.1: Pseudo-singular point
• Definition 3.2: Pseudo-singular saddle/node
• Theorem 3.3: Benoit Theorem benoit1990canardstchizawa2007generic
• Remark 3.5
• Definition 3.6
• Definition 3.7: Directional Blow-Up