Stochastic and coherent dynamics of single and coupled beta cells

Beta cells are responsible for the production of insulin in the pancreas and they are placed in islets, called islets of Langerhans, throughout the pancreas. Each islet holds some few thousands of beta cells. During production and exocytosis of insulin the beta cells conduct an electric bursting phenomenon, modeled by a slow-fast nonlinear system of ordinary differential equations (ODEs). The single cell oscillations are complex as the dynamical behavior is a result of traversing a series of saddle node and homoclinic bifurcations, controlled by the slow variable. We shall present results on the burst period as function of an external applied stochastic term and use a technique for reducing the stochastic differential equations to ODEs for the average and higher order moments. The later method is approximate and we shall discuss the limits of validity. The individual beta cells are coupled through gap-junctions within the islets of Langerhans. Thereby the beta cells form a network of complex oscillators. The network of beta-cells could be viewed as a prototype example of complexity nets and hence constitute an example of broader interest than for biology. We shall present results on coupled beta cells in the simple one dimensional case and show how wave patterns can arise and propagate along the chain. These wave patterns can be blocked by inhomogeneous glucose concentration along the chain, and we shall show how the coupled cell model can be connected to the Fishers equation, which is the simplest reaction-diffusion partial differential equation.