Analysis of a Class of Models of Bursting Electrical Activity in Pancreatic Β-cells∗

Many models of bursting electrical activity (BEA) in pancreatic β-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 β-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 system to leading order. In this paper, we extend the techniques of Pernarowski, Miura, and Kevorkian to a class of models of BEA, namely, those first-generation models consisting of three first-order ordinary differential equations. We show that the fast subsystem of these models can be reformulated as an integrable system to leading order. The relative ease with which this reformulation can be done depends on a biological property of the models, namely, the value of the integer exponent of the activation variable in the description of the voltage-gated K+ current.