Bifurcation Analysis of Nonlinear Oscillations in the Electrical Activity of Pancreatic β‐Cells

ABSTRACT Cell biological systems are characterized by complex relationships and nonlinear processes. The modeling of these processes improves the understanding, and represents a significant enrichment of the experimental investigation. An example of such a system is the regulation of blood glucose concentration by pancreatic ‐cells through the secretion of insulin. ‐cells are electrically active and insulin secretion is regulated by an interaction of metabolic and electrophysiological components, resulting in a change in membrane potential between the silent and active burst phases. Mathematically, this behavior can be described by a set of ordinary nonlinear differential equations. Because metabolic and electrical activities occur on different timescales, they are treated separately in this paper. In the electrical system, different types of bifurcations occur as the ATP concentration varies, which links the metabolic and electrical activity. The state of the system changes from a stable equilibrium to a limit cycle and back again. The corresponding transition is characterized by an increase in period duration, which is due to the type of bifurcation, the merging of a limit cycle with a saddle point. Phase plane analysis is used to characterise the occurring bifurcations and properties related to the higher frequency variables, that is, the electrical system. The equilibrium points and their type and stability, as well as the limit cycles, are determined and shown in bifurcation diagrams. The change in the period duration is approximated by the eigenvalue analysis of the saddle point.