Homogenization of Heterogeneously Coupled Bistable ODE's-Applied to Excitation Waves in Pancreatic Islets of Langerhans.

We consider a lattice of coupled identical differential equations. The coupling is between nearest neighbors and of resistance type, but the strength of coupling varies from site to site. Such a lattice can, for example, model an islet of Langerhans, where the sites in the lattice model individual but identical β-cells, and the coupling between cells is made of gap junctions.By using a homogenization technique we approximate the coupled discrete equations by a PDE, basically a nonlinear heat equation (a Fisher equation). For appropriate parameters this equation is known to have wave-solutions. Of importance is the fact, that the resulting diffusion coefficient does not only depend on the average of the coupling, but also on the variance of the strength. This means that the heterogeneity of the coupling strength influences the wave velocity-the greater the variance, the slower is the wave. This result is illustrated by simulations, both of a simple prototype equation, and for a full model of coupled beta-cells in both one and two dimensions, and implies that the natural heterogeneity in the islets of Langerhans should be taken into account.