  The phase sensitivity curve or phase response curve (PRC) quantifies the
oscillator's reaction to stimulation at a specific phase and is a primary
characteristic of a self-sustained oscillatory unit. Knowledge of this curve
yields a phase dynamics description of the oscillator for arbitrary weak
forcing. Similar, though much less studied characteristic, is the amplitude
response that can be defined either using an ad hoc approach to amplitude
estimation or via the isostable variables. Here, we discuss the problem of the
phase and amplitude response inference from observations using test
stimulation. Although PRC determination for noise-free neuronal-like
oscillators perturbed by narrow pulses is a well-known task, the general case
remains a challenging problem. Even more challenging is the inference of the
amplitude response. This characteristic is crucial, e.g., for controlling the
amplitude of the collective mode in a network of interacting units -- a task
relevant to neuroscience. Here, we compare the performance of different
techniques suitable for inferring the phase and amplitude response,
particularly with application to macroscopic oscillators. We suggest
improvements to these techniques, e.g., demonstrating how to obtain the PRC in
case of stimuli of arbitrary shape. Our main result is a novel technique
denoted by IPID-1, based on the direct reconstruction of the Winfree equation
and the analogous first-order equation for isostable dynamics. The technique
works for signals with or without well-pronounced marker events and pulses of
arbitrary shape; in particular, we consider charge-balanced pulses typical in
neuroscience applications. Moreover, this technique is superior for noisy and
high-dimensional systems. Additionally, we describe an error measure that can
be computed solely from data and complements any inference technique.
