Perturbation Techniques for Models of Bursting Electrical Activity in Pancreatic $\beta $-Cells

Abstract

Pancreatic $\beta $-cells exhibit periodic bursting electrical activity consisting of active and silent phases. Experimentally, the ratio, $\rho _f $, of the active phase duration to the overall period is correlated to the insulin response of these cells to glucose concentration. Several different mathematical models of the $\beta $-cell have been developed to describe changes in the intracellular ionic concentrations and the ionic flow through the cellular membrane. The membrane potential in each of these models exhibits bursting patterns similar to those observed experimentally. Values of the plateau fraction, $\rho _f $, can be computed from these models. The Sherman–Rinzel–Keizer (SRK) model of this phenomenon consists of three, coupled, first-order nonlinear differential equations that describe the dynamics of the membrane potential, the activation parameter for the voltage-gated potassium channel, and the intracellular calcium concentration. These equations are transformed into a Liénard differential equation coupled to a single first-order differential equation for the slowly changing nondimensional calcium concentration. Leading-order perturbation problems for the silent phase and the transition regions are reduced to quadrature. The solution of the leading-order active phase problem is a limit cycle which depends on the value of the intracellular calcium concentration. Since the active phase equations exhibit weak damping, Melnikov’s method can be applied to determine the bifurcation point of these equations. Thus, an explicit expression for the active phase duration is obtained. Together with the silent phase analysis, an approximation of the plateau fraction $\rho _f $ is derived and its value compared to the plateau fraction numerically obtained from the SRK model.