  The Phase Response Curve (PRC) is a tool used in neuroscience that measures
the phase shift experienced by an oscillator due to a perturbation applied at
different phases of the limit cycle. In this paper we present a new approach to
PRCs based on the parameterization method. The underlying idea relies on the
construction of a periodic system whose corresponding stroboscopic map has an
invariant curve. We demonstrate the relationship between the internal dynamics
of this invariant curve and the PRC, which yields a method to numerically
compute the PRCs. Moreover, we link the existence properties of this invariant
curve as the amplitude of the perturbation is increased with changes in the PRC
waveform and with the geometry of isochrons. The invariant curve and its
dynamics will be computed by means of the parameterization method consisting of
solving an invariance equation. We show that the method to compute the PRC can
be extended beyond the breakdown of the curve by means of introducing a
modified invariance equation. The method also computes the amplitude response
functions (ARCs) which provide information on the displacement away from the
oscillator due to the effects of the perturbation. Finally, we apply the method
to several classical models in neuroscience to illustrate how the results
herein extend the framework of computation and interpretation of the PRC and
ARC for perturbations of large amplitude and not necessarily pulsatile.
