Synchronization and activation in a model of a network of $\beta$-cells

Islets of pancreatic ß-cells are of utmost importance in the understanding of diabetes mellitus. We consider here a model of a network of such pancreatic ß-cells which are globally coupled via gap junctions. Some of the cells in the islet produce bursting oscillations while other cells are inactive. We prove that the cells in the islet synchronize if the coupling is sufficiently large and all cells are active (or inactive). If the islet consists of both active and inactive cells and the coupling is sufficiently large, an active cluster and an inactive cluster emerge. We show that activity of the islet depends on the coupling strength and the number of active cells compared to the number of inactive cells. If too few cells are active the islet becomes inactiv