Traveling waves in pancreatic islets
Moreland, Heather Lyn.
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In response to an increase in blood glucose levels, insulin is released into the bloodstream by the pancreatic islets of Langerhans. As a result of this influx of glucose, the islets start what are called bursting oscillations of the membrane potential and the intracellular calcium concentration. Time delays of several seconds in the activity of distant cells in the islets have been observed, indicating the presence of traveling waves through the islets. By considering a robust model of a pancreatic islet in one dimension, we study the relationship between the wave speed and the model parameters for the existence of traveling wave fronts and traveling wave pulses. After a systematic reduction of the model equations, the wave fronts (or heteroclinic connection) are studied. Using the bi-stable equation, for which an exact expression of the heteroclinic connection can be computed, we use a homotopy parameter to move from this equation to an islet model. A relationship between the wave speed and the conductance of the ATP-modulated potassium channel is constructed. Upon the inclusion of the slow gating variable back into the model equations, we observe the presence of a traveling wave pulse (or homoclinic connection). Using a high period periodic orbit to approximate the homoclinic orbit, a similar relationship between these two parameters is constructed. We observe that the heteroclinic connection is a good approximation for a portion of the homoclinic connection. Comparisons of the speed of the wave traveling through the islet in the partial differential equation model and the model in traveling coordinates is carried out. Since pancreatic islets are roughly circular, a two-dimensional model of an islet is also simulated on the unit disk. Due to the higher dimensionality of the problem, the numerics become more expensive. An alternating direction implicit (ADI) method for nonlinear parabolic differential equations is adapted for polar coordinates and mixed (Robin) boundary conditions. Using this method, we observe that the larger the portion of the boundary of the islet exposed to a stimulatory glucose concentration, the more rapidly the wave activity reaches the interior of the islet.