Published in
Physiol
.
Meas
. 22, 551–564 (2001).
Modelling couplings among the oscillators of the
cardiovascular system
Aneta Stefanovska‡k, Dmitrii Luchinsky† and Peter V E
McClintock†
‡ Group of Nonlinear Dynamics and Synergetics, Faculty of Electrical Engineer ing,
University of Ljubljana, Trˇzaˇska 25, 1000 Ljubljana, Slovenia. (aneta@osc.fe.uni-lj.si)
† Department of Physics, University of Lancaster, Lancaster LA1 4 YB, UK .
(d.luchinsky@lancaster.ac .uk, p.v.e.mcclintock@lancaster .ac.uk)
Abstract. A mathematical model of the cardiovascular system is simulated
numerically. The basic unit in the model is an oscillator that possesses a structural
stability and robustness motivated by physiological understanding and by the a nalysis
of measured time series. Oscillators with linea r couplings are found to reproduce the
main characteristic features of the experimentally obtained s pec tra. To explain the
variability o f cardiac and respir atory frequencies , however, it is essential to take into
account the rest of the system, i.e. to consider the effect of noise. It is found that the
addition of noise also results in epochs of synchronization, a s observed experimentally.
Preliminary analysis suggests that there is a mixture of linear and parametric couplings,
but that the line ar coupling see ms to dominate.
PACS numbers: 87.10.+e, 5.45.Xt, 5.40.Ca , 87.19.Hh
k To whom correspondence should be addressed.
Modelling couplings among the oscillators of the cardiovascular system
1. Introduction
Coupled oscillators are ubiquitous in nature. They have been studied intensively over
the years in many a r eas of science and technology including e.g. physics (Haken 1983,
Strogatz 1994), chemistry (Kuramoto 1984) and biology (Winfree 1980). However, mo st
theoretical work on coupled oscillators refers either to a small number of oscillators (e.g.
2, or at most 3), or to large ensembles that can be treated statistically; only in a few
cases is the effect of ra ndo m fluctuations included explicitly.
It has long been known that t he heart of a healthy human subject in repose does
not beat regularly. The rhythmic va r iation in the heart rate occurring at the frequency
of respiration, known as respiratory sinus arrhythmia, has been the most studied cardio-
respiratory interaction since Hales described changes in blood pressure related to t he
respiratory pattern in a horse ( Hales 1 733). Since then there ha s been much significant
work on the cardio-respiratory interaction (e.g. Angelone and Coulter 1964, Davies and
Nielson 1966, Hirsh a nd Bishop 1981). However, studies of the interactions between the
other processes involved in the dynamics of the cardiovascular system (CVS) are still
in their infancy. It has recently been shown (Stefanovska and Braˇciˇc 1999, Stefanovska
et al 1999a, 1999b, Braˇciˇc et al 2000, Braˇciˇc Lotriˇc et al 2 000, Stefanovska et al this
issue) that , on the time scale of one average circulatio n period, the cardiovascular system
behaves in many ways as a set of 5 coupled, autonomous, nonlinear oscillators o f widely
differing frequencies.
There are many different approaches to the problem of modelling cardiovascular
dynamics including e.g. modelling of the blo od flow through the system of pipes, or
modelling of the coupling mechanisms as a system of differential equations with delay
(Hyndman et al 1 971, Kitney et al 1985 , deBoer et al 1987, Baselli et al 19 88, Saul et
al 1991, Starkee and Westerhof 1993, TenVoorde et al 1995, Seidel and Herzel 1995,
Cavalcanti and Belardinelli 1996 ) . The majority of models proposed interpret the
observed oscillations in the heart rate and blood pressure in terms of nonlinearities and
time delays. They concentrate mainly on short-time blood pressure control mechanisms,
including respiratory oscillations and oscillations with period around 10s. Although
these models provide deep insight into par ticular aspects of the CVS dynamics, their
range of application is restricted. In particular, there are many characteristics revealed
by time series analysis of the blood flow that t hey do not reproduce.
The a pproach discussed here – modelling the CVS as a system of coupled oscillators
– is qualitatively different. It concentrates on the oscillatory nature of the dynamics on
the time-scale of one average circulation period. It aims to develop the simplest possible
model that is able to reflect the oscillatory character of the CVS dynamics and is capable
of reproducing the main statistical and dynamical characteristics of the measured time
series, including the form of the spectra, the variability of the heart rat e and respiration,
as well as of the blood flow and blood pressure, a nd the main experimentally observed
features of synchronization phenomena (Hildebrandt 1987, Raschke 1987, Sch¨afer et al
1998, 1999, Braˇciˇc Lotr iˇc and Stefanovska 2000, Mrovka et al 2000, Stefanovska et al
Modelling couplings among the oscillators of the cardiovascular system
2000).
In this paper, we present the results of numerical simulations of the proposed model,
focusing on the type of couplings and the effect of noise on the dynamics of t he oscillators’
interaction. In section 2 we summarise very briefly the experimental observations that
have been made to date, and which a satisfactory model may be expected to encompass.
Section 3 discusses how the model is constructed, and Section 4 reports the results
obtained from it under different assumptions about the coupling constants and the
presence or absence of random fluctuations (noise) representing external influences and
parts of the CVS that are not included explicitly in the model. Section 5 summarises
the results and draws conclusions.
2. Background
The propo sed model is based on the experimental observation that, on a time scale
of minutes, there are five almost periodic oscillatory subsystems contributing to the
regulation of blood flow (Stefanovska and Kroˇselj 1997, Braˇciˇc and Stefanovska, 1998a,
1998b, Stefanovska and Braˇciˇc 1999, Stefanovska et al 1999a, Kvernmo et al 1998, 1999,
Braˇciˇc et al 2000, Braˇciˇc Lotriˇc et al 2000). Complex and multiple control mechanisms
regulate both the frequency and the volume of the blood expelled by the heart in each
cycle, allowing the organism to respond to the metabolic demand of different parts of
the body, to adjust the pressure and the capacity of the CVS, and react to external
excitations sensed by the autonomous nervous system. The experimental o bservations
that, taken together as a whole, point unequivocally to the dynamical nature o f the
rhythmic activity of the CVS, can be summarized as follows –
• Timescales. Five t imescales are involved, and they differ substantially from each
other (average frequencies of ∼ 1.0, 0.2, 0.1, 0.03 and 0.01 Hz). The physiological
origin of the different frequency components are o bvious in t he case of 1 Hz (heart
beat), and 0.2 Hz (respiration), reasonably well established in the case of 0.1 Hz
(intrinsic myogenic activity of smooth muscles), but not yet known with certainty
in the cases of 0.03 Hz (connected to the autonomous nervous system’s control
over the car diovascular system) or 0 .01 Hz (apparently associated with endothelial
activity). However, they all seems to be characterized by autonomous oscillatory
dynamics.
• Signal and site invariance. All 5 basic frequencies are o bserved in all measured
signals characterizing cardiovascular dynamics, including: heart-rate variability
(where, however, the heart -rate is by definition missing), blood flow, blood pressure,
respiration and ECG.
• Time variations. Continuous wavelet a nalysis of the time series r eveals that the
characteristic f r equencies and amplitudes of the oscillatory processes are themselves
of a slowly-varying periodic nature.
Modelling couplings among the oscillators of the cardiovascular system
b
• Spectral width. In the amplitude-frequency plane each of the characteristic peaks
averaged over time is relatively broad, and they seems to be superimposed on a
broad (noisy?) spectral ba ckground. This suggests that stochastic (or possibly
chaotic) components also play a role in the regulation of the cardiovascular
dynamics.
• Combin ation f requencies. In cases where the characteristic or basic frequencies were
almost constant (e.g. in the case of a patient in coma), their linear combinations
were clearly demonstrated.
• Synchronization. Recent analysis of cardiorespiratory synchrograms (Sch¨afer et al
1998, 1999, Braˇciˇc Lotriˇc and Stefanovska 2 000, Mrovka et al 2000, Stefanovska
et al 2000) show that cardiac and respiratory oscillators can become synchronized
in different n:m synchronization regimes. Furthermore, it is now established that
synchronisation phenomena can also arise between some of the o t her oscillators
(Stefanovska and Hoˇziˇc 2000).
• Zero Lyapunov exponent. In the blood flow of healthy subjects, one of t he Lypunov
exp onents is always found to be equal to zero within experimental and numerical
error. A zero value of one of t he Lyapunov exponents suggests (Abarbanel et al,
1993) that the source is governed by differential equations, or by a finite time map.
Based on this inference, we may take the zero exponent in the blood flow of healthy
resting humans as implying the dominantly deterministic nature of t he CVS signals.
We consider separately evidence derived, first, from the analysis of oscillations in
heart rate variability (HRV), respiration, blood flow and blood pressure and, secondly,
from the phenomenon of synchronization. The reason is that, as we shall see
later, they provide complementary informa tion required to determine the strength
and type of coupling. Furthermore, t he analysis of HRV is of well known clinical
significance (Task Force of t he ESC and the NASPE 1996), while initial investigations
of synchronization between the heart rate and respiration are already demonstrating its
potential importance in clinical measurements.
We emphasize that, if the idea of characterizing CVS dynamics in terms of a system
of coupled oscillators is correct, then one of the general approaches to the analysis of
such a system can indeed be based on an analysis of its synchronization (Rosenblum et
al 2000, Pikovsky et al 2001). We mention here only a few experimental results –
• Cardiorespiratory sync hronization in hea l thy subjects. Only short episodes of
synchronization with a high locking ratio and high variability of the heart and
respiration frequencies were observed in healthy subjects at rest. Their heart beat
is around 1 Hz and the extent of its variability is age-dependent (Kaplan et al 1991,
Jansen-Urstad et al 1997, Stein et al 1997, Yeragani et al 1997, Braˇciˇc Lotriˇc and
Stefanovska 2000).
• A subject in coma. In a sedated subject in a critical state of coma, a very high
heart frequency (1.6 Hz), with no variability and no synchronization, were observed
Modelling couplings among the oscillators of the cardiovascular system
(Stefanovska and Braˇciˇc 1999). Deprived of feedback control from the peripheral
systems, the cardiac frequency resumes its autonomous value, similar to its state
when the heart is taken out of the body.
• Cardiorespiratory synchron i zation in athletes. Athletes are characterized by a
relatively low average heart frequency, high heart rate variability, and longer
episodes of synchronization with lower locking ratios (Kvernmo et al 1998, Sch¨afer
et al 1998, 1999).
• Type II d i abetic subjects. Patients with typ e II diabetes mellitus demonstrate a
high avera ge heart rate, low heart rate variability (Braˇciˇc Lotriˇc et al 2000) and
very long episodes of synchronization (Braˇciˇc L otriˇc e t al unpublished results).
• Anæsthesia. It has recently been shown that during anæsthesia in rats¶ the
cardiorespiratory system passes reversibly through a sequence of different phase-
synchronized states as the anæsthesia level changes (Stefanovska et al 2000). The
synchronisation state seems to provide a direct and objective measure of the depth
of anæsthesia.
• Hea rt rate synchronized b y a weak external forcin g. Cardiorespiratory synchroniza-
tion during paced respiration (Seidel and Herzel 1998) and synchronization of the
heart with external periodic sounds o r light signals (Anishchenko et al 2000) were
also demonstrated.
All these observations demonstrate clearly that different states of the organism may
correspond to different regimes of synchronization which, as already mentioned above,
can be of clinical significance and strongly support the idea of modelling the CVS
dynamics as a system of coupled oscillators.
This list of experimental evidence is far from complete. We mention just one
more important observation. It was found that there is a stat istically significant trend
in the dependence of the time-averaged synchronization index (defined in Tass et al
1998, R osenblum et al 2 000) and the standard deviation of the HRV distribution
(Braˇciˇc Lotriˇc and Stefanovska 2000). Control of the cardiova scular system thus seems
to be maintained by a fine balance between variatio ns of the eigenfrequencies of the
oscillators, resulting in mutual amplitude and frequency modulation and, sometimes,
mutual synchronization between the phases of the oscillators.
Moreover, an inverse relat ionship between the average cardiac frequency and the
extent of its variation seems to exist. In resting subjects, the higher the cardiac frequency
is, the lower is its variability (Tsuji et al 1996). Athletes lie at one extreme, with a low
cardiac frequency (below 1 Hz) and high heart rate variability, whereas a heart taken
out of the body, or disconnected from the rest of the body by anæsthesia or sedatives,
lies at t he other extreme and is characterised by a high frequency (around 1.6 Hz)
and no var iability. Synchronization seems to be involved in this interplay between the
basic, average frequency and the extent of its varia t ion: the more strongly the cardiac
¶ The dy namics of the CVS in rats has be en shown to posses similar features to those in humans,
despite the cardiac and res piratory rhythms in rats being approximately 4 times fa ster than in humans.
