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In this paper we use the parameterization method to provide a completedescription of the dynamics of an $n$-dimensional oscillator beyond theclassical phase reduction. The parameterization method allows, via efficientalgorithms, to obtain a parameterization of the attracting invariant manifoldof the limit cycle in terms of the phase-amplitude variables. The method hasseveral advantages. It provides analytically a Fourier-Taylor expansion of theparameterization up to any order, as well as a simplification of the dynamicsthat allows for a numerical globalization of the manifolds. Thus, one canobtain the local and global isochrons and isostables, including the slowattracting manifold, up to high accuracy, which offer a geometrical portrait ofthe oscillatory dynamics. Furthermore, it provides straightforwardly theinfinitesimal Phase and Amplitude Response Functions, that is, the extendedinfinitesimal Phase and Amplitude Response Curves, which monitor the phase andamplitude shifts beyond the asymptotic state. Thus, the methodology presentedyields an accurate description of the phase dynamics for perturbations notrestricted 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. Weillustrate our methods by applying them to different three dimensional singleneuron and neural population models in neuroscience.

Global phase-amplitude description of oscillatory dynamics via the parameterization method