Detection of slow-fast limit cycles in a model for electrical activity in the pancreatic beta-cell

In this paper we consider a model for the phenomenon of bursting electrical activity in the pancreatic beta-cell. Consisting of three coupled first-order nonlinear differential equations, the model describes the dynamics of the membrane potential, the activation parameter for the voltage-gated potassium channel, and the intracellular calcium concentration. The bursting electrical activity depends on fast and slow processes with distinctly different time scales which, when taken to the limit of highly diversified dynamics, allow the application of the singular perturbation method. Explicit conditions are derived which differentiate various dynamic behaviours and identify, in particular, the limit cycles composed of alternate slow and fast transitions. Computer simulations are then presented to support our theoretical predictions.