192
E. Gengel and A. Pikovsky
12.1 Introduction and Overview
This chapter deals with the art of phase reconstruction. We focus on Hilbert transforms, however, much of the introduced methodology is not bound to Hilbert transforms alone.
In general, three approaches to signal analysis of oscillatory signals can be identified. The first approach applies statistical methods to extract information from
observations assuming no further model [2, 22, 34]. The second approach takes
the theory of dynamical systems into account and analyses the signals in terms
of the phase and the amplitude notions provided in this theory [1, 7, 9, 17–19, 24,
29, 30, 37]. In an intermediate methodology, a phase and an amplitude are extracted
from the data and then analysed in terms of statistical quantities. These methods
may or may not take an underlying theory into account [15, 23, 25, 28, 36, 42, 43]
Alternatively, one applies machine learning techniques to obtain equations of motion
directly from observations [6, 41].
Here we focus on signal analysis approaches suitable for oscillating systems. The
basic assumption is that the signal originates from a dynamical oscillating system,
interacting with other systems and/or with the environment, and the goal is to understand the dynamics. This task is especially important and challenging in life science,
where a theoretic description of the oscillators is in many cases lacking, because the
underlying mechanisms are not clear. On the other hand, measurements of the full
phase space dynamics are impossible, or would destroy the system itself. The latter
aspect introduces the common setting where measurements of the systems are passive, i.e., an observer collects data from the free running system and may only apply
weak perturbations to prevent damage. For such passive observations, we pursue here
the approach inspired by the dynamical system theory: we try to extract the phases
from the signals, with the aim to build models as close to theoretical descriptions as
possible.
The ideas of the phase dynamics reconstruction has been widely used in physics,
chemistry, biology, medicine and other areas [4, 20, 31, 35] to understand properties
of oscillators and coupling between them (see also Chaps. 2, 3 and 11 of this book).
The reason for this, as we discuss below, is that the phase is sensitive to interactions
and external perturbations. In particular, many studies apply Hilbert transforms to
reconstruct the phase from data (for example see [3, 14, 38, 43] and references
therein). However, several fundamental issues in the process of phase reconstruction are unresolved, long standing and mostly omitted in the community. One issue
deals with the role of the amplitudes [8, 21]. And from the view point of pure
signal processing: how to deal with phase-amplitude mixing in Hilbert transforms
[10, 13]. The latter issue will be discussed in particular here and we describe a
solution by virtue of iterative Hilbert transform embeddings (IHTE) [12].
First, we describe the theoretical concepts. We then discuss the art of phase reconstruction with a focus on IHTE. We illustrate this method by presenting results for
a Stuart-Landau non-linear oscillator, including reconstruction of the infinitesimal
phase response curve (iPRC). Finally, we discuss difficulties of application in case
of noisy oscillations.
E. Gengel and A. Pikovsky
12.1 Introduction and Overview
This chapter deals with the art of phase reconstruction. We focus on Hilbert transforms, however, much of the introduced methodology is not bound to Hilbert transforms alone.
In general, three approaches to signal analysis of oscillatory signals can be identified. The first approach applies statistical methods to extract information from
observations assuming no further model [2, 22, 34]. The second approach takes
the theory of dynamical systems into account and analyses the signals in terms
of the phase and the amplitude notions provided in this theory [1, 7, 9, 17–19, 24,
29, 30, 37]. In an intermediate methodology, a phase and an amplitude are extracted
from the data and then analysed in terms of statistical quantities. These methods
may or may not take an underlying theory into account [15, 23, 25, 28, 36, 42, 43]
Alternatively, one applies machine learning techniques to obtain equations of motion
directly from observations [6, 41].
Here we focus on signal analysis approaches suitable for oscillating systems. The
basic assumption is that the signal originates from a dynamical oscillating system,
interacting with other systems and/or with the environment, and the goal is to understand the dynamics. This task is especially important and challenging in life science,
where a theoretic description of the oscillators is in many cases lacking, because the
underlying mechanisms are not clear. On the other hand, measurements of the full
phase space dynamics are impossible, or would destroy the system itself. The latter
aspect introduces the common setting where measurements of the systems are passive, i.e., an observer collects data from the free running system and may only apply
weak perturbations to prevent damage. For such passive observations, we pursue here
the approach inspired by the dynamical system theory: we try to extract the phases
from the signals, with the aim to build models as close to theoretical descriptions as
possible.
The ideas of the phase dynamics reconstruction has been widely used in physics,
chemistry, biology, medicine and other areas [4, 20, 31, 35] to understand properties
of oscillators and coupling between them (see also Chaps. 2, 3 and 11 of this book).
The reason for this, as we discuss below, is that the phase is sensitive to interactions
and external perturbations. In particular, many studies apply Hilbert transforms to
reconstruct the phase from data (for example see [3, 14, 38, 43] and references
therein). However, several fundamental issues in the process of phase reconstruction are unresolved, long standing and mostly omitted in the community. One issue
deals with the role of the amplitudes [8, 21]. And from the view point of pure
signal processing: how to deal with phase-amplitude mixing in Hilbert transforms
[10, 13]. The latter issue will be discussed in particular here and we describe a
solution by virtue of iterative Hilbert transform embeddings (IHTE) [12].
First, we describe the theoretical concepts. We then discuss the art of phase reconstruction with a focus on IHTE. We illustrate this method by presenting results for
a Stuart-Landau non-linear oscillator, including reconstruction of the infinitesimal
phase response curve (iPRC). Finally, we discuss difficulties of application in case
of noisy oscillations.