Mixed-Mode Oscillations in Chemical and Physiological Models

Chemical and physiological processes are usually modeled by systems of nonlinear differential equations. One of the various patterns in the dynamics of such systems are Mixed-Mode oscillations (MMOs). They owe their name to their alternating large and small amplitudes. The large oscillations usually consist of fast and slow phases while the small oscillations are always slow. The dynamical systems approach to such slow-fast systems is by means of geometric singular perturbation theory. By this method it is possible to find a mechanism generating the MMOs. There exists a special solution which is a so called canard solution separating the small from the large oscillations. This fact allows to construct a one-dimensional map accounting for the complex behavior of MMOs. The first part of the project will be to apply the above concept to neuronal model displaying a dynamical pattern similar to MMOs. The second part regards technical questions concerning the mathematics of canard solutions. They are a recent discovery in dynamical systems and not completely understood, yet.