Analytical study of mixed-mode oscillations in human β-cells

β-cells are specific cells located in pancreatic islets that produce and secrete insulin. The secretion of insulin happens as a consequence of a electrical activity: in the present thesis we analyse a mathematical model regulating that phenomenon. This model depends on eight variables, which are recognized having different time scales and hence classified as slow, medium or fast. By varying some parameters of the model we remark the generation of mixedmode oscillations: small amplitude oscillations with a global return mechanism. In order to explain it, a reduction of the model is performed and a 3d-model is obtained. The study of the singularities of this system let us notice the appearance of mathematical objects called canards in their neighborhood. The strong canard determines the funnel where simulations have to enter in order to begin oscillating around the weak canard. Finally secondary canards induce a discretization of the space and determine the number of small amplitude oscillations of the solution of the system. We find good agreement between our analytical studies and numerical simulations.