Separating manifolds in slow-fast systems

There are many applications that lead to models involving different timescales. For example, this is particularly the case for models of neurons, which involve dynamics of ionic channels across the cell membrane. Due to the slow-fast nature of such models it is difficult to use numerical tools for the investigation of the global behaviour. This paper discusses the computation of global invariant manifolds for slow-fast systems. We explain how the different timescales cause the numerical difficulties and give suggestions on how to deal with these problems. We illustrate the techniques with the computation of separating manifolds in a Hodgkin-Huxley type model of a somatotroph cell; this is an endocrine cell in the anterior pituitary that secretes growth hormone. There are two co-existing attractors in this model and their basins of attraction are separated by global stable manifolds of equilibria or periodic orbits.