Technical University of Denmark, Denmark

The pancreatic b-cell is the only insulin-producing type of cell in the body and is thus crucial in the regulation of the plasma glucose concentration. Failure of the glucose regulation leads to diabetes mellitus, a medical condition of almost epidemic proportions: According to the WHO, there are presently in the order of 150 million people suffering from diabetes mellitus. A number which is believed to double by the year 2025. The insulin secretion of the b-cell is known to be associated to a distinct pattern of oscillation of the cell membrane potential called bursting. The bursting pattern contains a silent phase which alternates with bursts of rapid oscillations (spikes). Since the mid-80s, a significant number of models of the b-cell electrophysiology have been constructed. In 1987 T. R. Chay proposed a b-cell model, which is based on five ionic currents through the cell membrane: Voltage gated calcium and potassium currents generate the rapid spikes of the active phase, while ATP- and calcium-dependent potassium currents are responsible for the level of activity and the repolarizing of the membrane, respectively. Finally, a leak current stabilizes the dynamics. The model contains three variables. The time constants of the membrane potential, V, and the opening probability of the voltage gated potassium channels, n, are small compared to that of the slow variable, the intracellular free calcium concentration. This difference inspires the separation of the three-dimensional phase space into a plane of rapid dynamics and a slow direction. Using the slow variable as a parameter for the fast subsystem yields the so-called reduced system. It has been shown that the reduced system contains a stable limit cycle which disappears in a homoclinic bifurcation as the system leaves the active phase. The poster will show the homoclinic bifurcation of the reduced system and investigate the dynamics in the three-dimensional phase space in vicinity of the isoclines of the reduced system.