1.1. Calcium Oscillation in Pancreatic β-cells.

Pancreatic β-cells are a type of cell found in the pancreatic islets of the pancreas, whose primary function is to store and release insulin. The following figure shows a hierarchical structure of pancreatic β-cells within our body:

Beta Cell

In pancreatic β-cells, glucose-stimulated insulin secretion is mediated by an elevated cytosolic free calcium concentration, and bursting behavior of the potential action is observed. Below is an experiment on pancreatic β-cell pacemakers (Link)

Beta Cell

By solving a system of 5 ODEs using XPP, I got following nice animations (with some simple descriptions in the animations):

Beta Cell
Beta Cell

And the corresponding XPP code is available HERE.

How to explain the bursting phenomena? Actually we can use bifurcation theory. Let's draw a bifurcation diagram of action potential over Calcium density:

Beta Cell

If we project the oscillation of action potential (w.r.t Calcium concentration) onto the above bifurcation diagram, we get the following fitting:

Beta Cell

Which matches with our bifurcation analysis quite well. We see that "bursting" happens when Calcium concentration is relatively low because of a Hopf bifurcation, and it ends when Calcium concentration is relatively high because the limit cycle produced by Hopf bifurcation will coincide with a saddle node, making period infinitely long if Calcium concentration continuously grows.