Analysis of a Class of Models of Bursting Electrical Activity in Pancreatic $\beta$-Cells

Abstract

Many models of bursting electrical activity (BEA) in pancreatic $\beta$-cells have been proposed. BEA is characterized by a periodic oscillation of the membrane potential consisting of a silent phase during which the membrane potential is varying slowly and an active phase during which the membrane potential is undergoing rapid oscillations. An important experimental observation of BEA is a correlation between the rate of insulin release from $\beta$-cells and the plateau fraction as a function of glucose concentration. The plateau fraction is the ratio of the duration of the active phase to the period of BEA. In [SIAM J. Appl. Math., 52 (1992), pp. 1627--1650], Pernarowski, Miura, and Kevorkian develop analytical techniques to determine the leading-order plateau fraction for one of the models, namely, the Sherman--Rinzel--Keizer (SRK) model [ Biophys. J., 54 (1988), pp. 411--425]. Applicability of these techniques depends critically on the fact that the fast subsystem of the SRK model is an integrable sy...