  In this paper we use the parameterization method to provide a complete
description of the dynamics of an $n$-dimensional oscillator beyond the
classical phase reduction. The parameterization method allows, via efficient
algorithms, to obtain a parameterization of the attracting invariant manifold
of the limit cycle in terms of the phase-amplitude variables. The method has
several advantages. It provides analytically a Fourier-Taylor expansion of the
parameterization up to any order, as well as a simplification of the dynamics
that allows for a numerical globalization of the manifolds. Thus, one can
obtain the local and global isochrons and isostables, including the slow
attracting manifold, up to high accuracy, which offer a geometrical portrait of
the oscillatory dynamics. Furthermore, it provides straightforwardly the
infinitesimal Phase and Amplitude Response Functions, that is, the extended
infinitesimal Phase and Amplitude Response Curves, which monitor the phase and
amplitude shifts beyond the asymptotic state. Thus, the methodology presented
yields an accurate description of the phase dynamics for perturbations not
restricted to the limit cycle but to its attracting invariant manifold.
Finally, we explore some strategies to reduce the dimension of the dynamics,
including the reduction of the dynamics to the slow stable submanifold. We
illustrate our methods by applying them to different three dimensional single
neuron and neural population models in neuroscience.
