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M. Lucas et al.
and internal toxins, there would be no sustained life. The idea that one needs to
model living systems as open has indeed been expressed by E. Schrödinger in his
book “What is life?” [51].
Traditionally, the effect of external influences on a system is modelled either by
a very simple deterministic input—i.e. constant or strictly periodic—or for more
complicated-looking cases, by noise defined in terms of stochastic processes. Sometimes, external influences are themselves modelled by an autonomous dynamical
system, such that the system with its external influence can be modelled together
as one autonomous dynamical system. However, here, we demonstrate what can be
achieved by using deterministic non-autonomous models [25] to describe a system
subject to external influences. Such models have been argued to be especially useful
to describe living systems [24, 30], but have been used in other areas such as climate
dynamics [11, 16, 17].
Living systems exhibit two properties of particular interest. Firstly, they maintain
overall stability over long timescales (their lives) despite the input from their environment that is ever-changing in a non-trivial fashion. Secondly, due to this input, their
shorter-timescale macroscopic properties, e.g. the moment-by-moment characteristics of the response to perturbation, vary in time. An example of such time-variability
can be seen when the heartbeat is recorded over time—even for healthy subjects in
repose. The frequency of the heartbeat varies continuously, modulated by the oscillations corresponding to breathing [8] and other low-frequency oscillatory processes
resulting from neurovascular regulation of the heartbeat [52]. We emphasise that, in
fact, dynamics on such shorter timescales can play a crucial role in the functioning
of the system. For example, it has been shown that transitions in cardio-respiratory
synchronization occur during anaesthesia [53], and it has been proposed that, in the
brain, transient synchrony between neurons can improve information routing [43].
Here, we focus on systems of coupled oscillators. Such systems can exhibit synchronisation [49], where all oscillators in the system behave mutually coherently.
The study of synchronisation gained interest with the works of Winfree [61] and
Kuramoto [26] and the many extensions of the Kuramoto model, in particular to
complex network topologies [4].
The effect of non-autonomicity on coupled oscillatory systems has started to
attract attention in recent years. Low-dimensional examples include studies [56,
57], motivated by the cardiovascular system, which coined the term chronotaxicity
to define a class of non-autonomous systems with time-varying frequencies, able to
maintain stability against generic external perturbation. Inverse methods for detection
of chronotaxic dynamics in experimental data were also developed [29]? and applied
successfully to real biological systems [30]. Theoretical phase reduction for nonautonomous oscillators has also been developed [27, 28, 44].
Apart from the chronotaxicity studies [56, 57], a few other theoretical works have
looked into non-autonomous oscillatory systems. In [22], Jensen considered onedimensional non-autonomous Adler equations and described their changing dynamics in the limit of slow variation, and determined how slow the modulation needs to be
for the analysis to hold. In [13], the same system was studied specifically in the case
of periodic forcing, with particular focus on the existence and properties of periodic
M. Lucas et al.
and internal toxins, there would be no sustained life. The idea that one needs to
model living systems as open has indeed been expressed by E. Schrödinger in his
book “What is life?” [51].
Traditionally, the effect of external influences on a system is modelled either by
a very simple deterministic input—i.e. constant or strictly periodic—or for more
complicated-looking cases, by noise defined in terms of stochastic processes. Sometimes, external influences are themselves modelled by an autonomous dynamical
system, such that the system with its external influence can be modelled together
as one autonomous dynamical system. However, here, we demonstrate what can be
achieved by using deterministic non-autonomous models [25] to describe a system
subject to external influences. Such models have been argued to be especially useful
to describe living systems [24, 30], but have been used in other areas such as climate
dynamics [11, 16, 17].
Living systems exhibit two properties of particular interest. Firstly, they maintain
overall stability over long timescales (their lives) despite the input from their environment that is ever-changing in a non-trivial fashion. Secondly, due to this input, their
shorter-timescale macroscopic properties, e.g. the moment-by-moment characteristics of the response to perturbation, vary in time. An example of such time-variability
can be seen when the heartbeat is recorded over time—even for healthy subjects in
repose. The frequency of the heartbeat varies continuously, modulated by the oscillations corresponding to breathing [8] and other low-frequency oscillatory processes
resulting from neurovascular regulation of the heartbeat [52]. We emphasise that, in
fact, dynamics on such shorter timescales can play a crucial role in the functioning
of the system. For example, it has been shown that transitions in cardio-respiratory
synchronization occur during anaesthesia [53], and it has been proposed that, in the
brain, transient synchrony between neurons can improve information routing [43].
Here, we focus on systems of coupled oscillators. Such systems can exhibit synchronisation [49], where all oscillators in the system behave mutually coherently.
The study of synchronisation gained interest with the works of Winfree [61] and
Kuramoto [26] and the many extensions of the Kuramoto model, in particular to
complex network topologies [4].
The effect of non-autonomicity on coupled oscillatory systems has started to
attract attention in recent years. Low-dimensional examples include studies [56,
57], motivated by the cardiovascular system, which coined the term chronotaxicity
to define a class of non-autonomous systems with time-varying frequencies, able to
maintain stability against generic external perturbation. Inverse methods for detection
of chronotaxic dynamics in experimental data were also developed [29]? and applied
successfully to real biological systems [30]. Theoretical phase reduction for nonautonomous oscillators has also been developed [27, 28, 44].
Apart from the chronotaxicity studies [56, 57], a few other theoretical works have
looked into non-autonomous oscillatory systems. In [22], Jensen considered onedimensional non-autonomous Adler equations and described their changing dynamics in the limit of slow variation, and determined how slow the modulation needs to be
for the analysis to hold. In [13], the same system was studied specifically in the case
of periodic forcing, with particular focus on the existence and properties of periodic