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Fast and slow subsystems for a continuum model of bursting activity in the pancreatic islet. (English) Zbl 0910.34030

Summary: A simple model for the bursting electrical activity of individual pancreatic \(\beta\)-cells is introduced. Using the model, a continuum model for collections (islets) of large numbers of such cells is analyzed, using asymptotic methods, linear stability techniques, and monotone methods. When the coupling strength D is large, fast and slow subsystems for the continuum model are shown to be the same as those for a single cell if the slow variables are replaced by their spatial average. It is demonstrated that fast variables can synchronize while the slow variables simultaneously exhibit nonsynchronous behavior. Last, spatial inhomogeneities in slow subsystem parameters are shown to inactivate islets. It is demonstrated that if a sufficient fraction of cells are inactive, the average value of the slow variables can be decreased significantly.

MSC:

34E13 Multiple scale methods for ordinary differential equations
92C05 Biophysics
34C25 Periodic solutions to ordinary differential equations
34C29 Averaging method for ordinary differential equations
65N40 Method of lines for boundary value problems involving PDEs
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