An important feature of most physiological systems is that they evolve on multiple scales. For example, the bursting activity of neurons consists of a long interval of quasi steady-state followed by an interval of rapid variation, which is the burst itself. It is the interplay of the dynamics on different temporal or spatial scales that creates complicated rhythms and patterns.

Multiple scales problems of physiological systems are usually modelled by singularly perturbed systems. The geometric theory of multiple scales dynamical systems -- known as Fenichel Theory -- has provided powerful tools for studying singular perturbation problems. In conjunction with the innovative blow-up technique, geometric singular perturbation theory delivers rigorous results on pattern generation in multiple time-scale problems.

As a case study of geometric singular perturbation theory, I will focus on a single neuron model by McCarthy et al. (2008) that looks at the effect of the anesthetic propofol on such a neuron. It is well known that propofol causes paradoxical excitation in low doses. I will show that "canards", exceptional solutions in singular perturbation problems which occur on boundaries of regions corresponding to different dynamic behaviors, provide a possible explanation of the observed paradox.