  Phase resetting is a common experimental approach to investigating the
behaviour of oscillating neurons. Assuming repeated spiking or bursting, a
phase reset amounts to a brief perturbation that causes a shift in the phase of
this periodic motion. The observed effects not only depend on the strength of
the perturbation, but also on the phase at which it is applied. The
relationship between the change in phase after the perturbation and the
unperturbed old phase, the so-called phase resetting curve, provides
information about the type of neuronal behaviour, although not all effects of
the nature of the perturbation are well understood. In this chapter, we present
a numerical method based on the continuation of a multi-segment boundary value
problem that computes phase resetting curves in ODE models. Our method is able
to deal effectively with phase sensitivity of a system, meaning that it is able
to handle extreme variations in the phase resetting curve, including resets
that are seemingly discontinuous. We illustrate the algorithm with two examples
of planar systems, where we also demonstrate how qualitative changes of a phase
resetting curve can be characterised and understood. A seven-dimensional
example emphasises that our method is not restricted to planar systems, and
illustrates how we can also deal with non-instantaneous, time-varying
perturbations.
