Properties of a Bursting Model with Two Slow Inhibitory Variables

Models for certain excitable cells, such as the pancreatic $\beta $-cell, must reproduce “bursting” oscillations of the membrane potential. This has previously been done using one slow variable to drive bursts. The dynamics of such models have been analyzed. However, new models for the $\beta $-cell often include additional slow variables, and therefore the previous analysis is extended to two slow variables, using a simplified version of a $\beta $-cell model. Some unusual time courses of this model motivated a geometric singular perturbation analysis and the application of averaging to reduce the dynamics to the slow-variable phase plane. A geometric understanding of the solutionstructure and of transitions between various modes of behavior was then developed. A novel use of the bifurcation code AUTO finds nullclines for the slow variables when the fast variables are periodic by averaging over the fast oscillations. In contrast with the “parabolic” neuronal burster, this model requires bistability in th...