Summary
The phase shift of an oscillator in response to an external stimulus has been used to characterise, understand and predict biological behaviours. Such phase shifts are studied mathematically by first modelling the oscillator as an attracting periodic orbit, and then considering its isochrons. Each isochron is an (n − 1)-dimensional submanifold defined as the set of points in the basin of attraction whose trajectories converge to the oscillator with the same phase. While methods exist to compute one- and even two-dimensional isochrons, the dimension of isochrons of biological models that motivate their computation are often much higher. One approach to this problem is to restrict the space of interest to points perturbed away from the periodic orbit for a perturbation with fixed form but where the amplitude and phase of application may vary. The slices of isochron that exist in this two-dimensional subspace are generally one-dimensional curves. We present our method to compute isochron slices via the continuation of a two-point boundary-value problem that relates the phase of a point resulting from a perturbation, with the amplitude of the perturbation and the phase in the cycle at which it is applied. We demonstrate our approach by computing isochrons of the four-dimensional Hodgkin–Huxley model, where we are able to resolve intricate geometrical detail.
