| Summary: | Pancreatic β-cells are responsible for producing and secreting insulin, a hormone which is essential in the regulation of blood glucose level. In the presence of glucose, β-cells exhibit periodic bursting electrical activity (BEA) consisting of active and silent phases. During the active phase, the membrane potential of these cells oscillates rapidly, whereas the membrane potential changes slowly during the silent phase. The plateau fraction, which is the ratio of the active phase duration to the period of the bursting phenomenon, has been correlated to the insulin-glucose response of these cells.
There are a number of mathematical models describing BEA in pancreatic β-cells.
In the first part of this thesis, the class of first-generation models which consist of three coupled first-order ordinary differential equations is analyzed. Numerical and analytical techniques are presented for determining the approximate plateau fractions from the model equations as a function of the glucose-dependent parameter.
A consistent nondimensionalization of the model equations is proposed which permits
all of the models to be written in a standard form. The equations then are separated into
a second-order fast subsystem and a first-order slow subsystem. A bifurcation analysis of the fast subsystem in which the slow variable is treated as the bifurcation parameter reveals that the transition from the active phase to the silent phase occurs near a homo-clinic bifurcation. The use of several numerical and analytical methods is demonstrated for the determination of the approximate location of this bifurcation, which is needed in the computation of the approximate silent and active phase durations and, subsequently, the approximate plateau fractions. Leading-order problems for the silent and active phases are obtained using a combination of singular perturbation and multiple scales analyses and averaging techniques. The corresponding leading-order silent and active phase durations and the resulting leading order plateau fractions are reduced to quadrature. Finally, the approximations are compared with the respective exact (numerically computed) silent and active phase durations
and plateau fractions over a range of the glucose-dependent parameter for which the models exhibit BEA.
In the remainder of this thesis, a detailed study of a polynomial analog model of BEA
in pancreatic β-cells is carried out. Depending on the values of the model parameters, the model exhibits a wide variety of oscillatory solution behaviour, including types of bursting observed in excitable cells other than pancreatic β-cells.
A bifurcation map in the fast subsystem parameter space is produced by computing
curves which represent codimension-2 bifurcations. These curves bound regions on the map within which bifurcation diagrams of the fast subsystem are qualitatively the same. The importance of the bifurcation map is that it shows the relationships between the various types of oscillatory behaviour and, hence, provides a basis for an extension of the classification of bursting oscillations. Since the analog model consists of polynomial functions, the curves on the bifurcation map can be derived analytically.
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