© 2000 American Heart Association, Inc.
Cellular Biology |
Power-Law Behavior of Beat-Rate Variability in Monolayer Cultures of Neonatal Rat Ventricular Myocytes
From the Department of Physiology (J.P.K., S.R.), University of Bern, and Institute of Microsystems (M.O.H., P.R.), Swiss Federal Institute of Technology, Lausanne, Switzerland.
Correspondence to Jan P. Kucera, MD, Physiologisches Institut, Bühlplatz 5, CH-3012 Bern, Switzerland. E-mail kucera{at}pyl.unibe.ch
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Abstract |
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Abstract—It is known that extracardiac factors (nervous, humoral, and hemodynamic) participate in the power-law behavior of heart-rate variability. To assess whether intrinsic properties of cardiac tissue might also be involved, beat-rate variability was studied in spontaneously beating cell cultures devoid of extracardiac influences. Extracellular electrograms were recorded from monolayer cultures of neonatal rat ventricular myocytes under stable incubating conditions for up to 9 hours. The beat-rate time series of these recordings were examined in terms of their Fourier spectra and their Hurst scaling exponents. A non-0 Hurst exponent was found in 21 of 22 preparations (0.29±0.09; range, 0.11 to 0.45), indicating the presence of fractal self-similarity in the beat-rate time series. The same preparations exhibited power-law behavior of the power spectra with a power-law exponent of -1.36±0.24 (range, -1.04 to -1.96) in the frequency range of 0.001 to 1 Hz. Furthermore, it was found that the power-law exponent was nonstationary over time. These results indicate that the power-law behavior of heart-rate variability is determined not only by extracardiac influences but also by components intrinsic to cardiac tissue. Furthermore, the presence of power-law behavior in monolayer cultures of cardiomyocytes suggests that beat-rate variability might be determined by the complex nonlinear dynamics of processes occurring at the level of the cellular network, eg, interactions among a large number of cell oscillators or metabolic regulatory systems.
Key Words: heart-rate variability • cardiac cell cultures • physiology • extracellular recording
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Introduction |
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Beat-to-beat variations in cardiac cycle length have received increasing attention, because their characteristics may be used for the assessment and follow-up of various cardiovascular derangements and for risk stratification in the course of cardiac disease, including prediction of arrhythmias in cardiac patients.1 2
Spectral analysis of beat-rate time series reveals periodic components in heart-rate variability (HRV).3 In humans, 2 major components are present at frequencies around 0.1 Hz (low-frequency [LF] component) and around 0.25 Hz (high-frequency [HF] component).4 It is well established that these periodic components are linked to breathing and blood pressure control and that they are mediated by the autonomic nervous system5 6 and by endocrine influences.3 However, the analysis of long-term recordings (ie, 24-hour Holter electrocardiograms) indicates that >95% of the spectral power of HRV is concentrated at frequencies below LF.1 At these very low frequencies, the spectrum follows a power-law behavior; ie, the intensity of the power spectrum is a power function of frequency. The exponent of this power-law is close to -1 in healthy human subjects.7 Several studies showed that this exponent is different after myocardial infarction8 9 or cardiac transplantation9 and that the characterization of power-law behavior can yield potent estimates for risk stratification.9 10
Although the physiological basis of HRV is established for LF and HF components, the mechanisms underlying the power-law relationship are still not known. Specifically, numerous studies focused on the characterization of HRV in vivo; however, none investigated whether HRV depends on factors intrinsic to cardiac tissue, ie, whether fluctuations in cycle length also occur in the absence of extrinsic influences (nervous, hemodynamic, or endocrine).
It was the goal of the present study to characterize beat-rate variability in spontaneously beating cultures of cardiac cells under stable experimental conditions, because these preparations are devoid of extrinsic influences and therefore permit the investigation of factors governing HRV that are intrinsic to cardiac tissue. The experiments showed that the beat-rate variability of cardiac cell monolayer cultures was characterized by fractal properties and that their power spectra exhibited power-law behavior. This suggests that complex nonlinear dynamics of processes occurring at the level of cellular oscillators are present. Because beat-rate variability found in these preparations and HRV observed in vivo shared similar characteristics, it is likely that the power-law behavior of HRV is determined not only by extracardiac influences but by factors intrinsic to cardiac tissue as well.
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Materials and Methods |
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Microelectrode Arrays (MEAs)
Extracellular indium-tin oxide MEAs were fabricated using standard photolithographic procedures.11 The MEAs were coated with collagen (human placenta, type VI) to support cell attachment and were mounted in custom-made cylindrical culture chambers (16 mm in diameter, 20 mm high). Dissociated cells from neonatal rat ventricular tissue (Wistar) were prepared according to previously published procedures12 13 and were seeded on MEAs at a density of 2700/mm2. This resulted in dense monolayer cultures, as shown in Figure 1A
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Recording of Extracellular Electrograms
Using a reference indium-tin oxide electrode inserted into the
culture medium, unipolar extracellular electrograms were recorded
from up to 4 sites. The signals were amplified (gain, 1000x;
bandwidth, 5 kHz) and digitized at 10 kHz with 12-bit resolution by a
PC-based data-acquisition system.
Experimental Protocol
Recordings were performed on 2- to 7-day-old cultures in
a water-jacketed CO2 incubator (Jouan), thus
ensuring stable temperature (36°C) and CO2
concentration (0.8%) during the entire duration of the experiments.
Because medium exchanges themselves tend to transiently alter
spontaneous beat rates,14 they were halted 24 hours before
the experiment. Spontaneous sustained activity of the cell cultures was
induced by adding 0.5 to 1 µmol/L of the L-type calcium channel
agonist BayK8644 from a stock solution to the culture medium (M199 with
HBSS, GIBCO). An electrode yielding a high signal-to-noise ratio
was selected, and data acquisition was started after an equilibration
period of 15 to 20 minutes. Data acquisition was continued as long as
permitted by data storage resources or until instabilities in the
electrogram baseline caused missing data as a result of saturation of
the amplifier.
Offline Data Analysis
Activation times were derived from extracellular action
potentials as the time of occurrence of the steepest point of their
downward deflection (see Figure 1C
). The raw traces were
visually inspected to ensure detection of all action potentials and to
remove detection artifacts. To convert the resulting series of
activation times into a data representation suitable for
spectral analysis (beat-rate time series: frequency values
spaced evenly in time), instantaneous frequencies were calculated at
0.5-second intervals (
t) along the entire recording from the
linearly interpolated successive points of the series of activation
times.
Hurst Exponent
The Hurst scaling exponent (H) characterizes the
shape of self-similar signals and ranges from 0 to 1. A self-similar
signal with H
0 resembles white noise with spiky
oscillations. A signal with H0.5 shows
brownian noise-like oscillations, whereas signals with
H1 exhibit smooth oscillations. H
of the beat-rate (BR) time series was calculated as the slope of linear
regression of log{SD[BR(t+k ·
t)–BR(t)]} versus log(k) for
k=1, 2, 4, 8,... ,2n up to the
maximal possible power of 2.15 Fractal
self-similarity was assumed to be present if the correlation
coefficient (r) was >0.85.
Spectral Analysis
The power spectral density (PSD) of beat-rate variability was
computed by a discrete fast-Fourier transform with a Hann window. The
exponent (ß) of the power-law relationship was computed as the slope
of the linear regression of log(PSD) versus log(frequency). The
discrete PSD points were given a weight inversely proportional to their
density on the log(frequency) abscissa. A power-law was assumed to be
present if r>0.85.
Statistics
Data are given as mean±SD. Statistical differences were
assessed by the 2-tailed Student t test.
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Results |
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The experimental data consist of 18 short recordings (23 minutes to 2 hours, 42 minutes) and 4 long recordings (4 hours, 28 minutes to 9 hours, 5 minutes).
Self-Similarity of Beat-Rate Time Series
The beat-rate variability of a cell culture is depicted in
Figure 2A. The trace represents a
plot of the resampled beat-rate time series as a function of time.
Typically, the trace exhibits oscillations that occur in an
apparently erratic manner. The self-similar property of this trace is
revealed by the close-up shown in Figure 2B. The rescaled
segment exhibits fluctuations that look highly similar to those on the
original trace. The self-similarity of the graphical trace suggests
that beat rate does not exhibit any periodic oscillations
but that it comprises all frequencies such that details look similar
after rescaling.
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To characterize and quantify this fractal self-similarity, the Hurst exponent (H) and the corresponding correlation coefficient (r) were computed for each beat-rate time series. In the 18 short experiments, 17 were characterized by self-similar beat-rate time series (r=0.95±0.04; range, 0.87 to 0.99) with H=0.29±0.08 (range, 0.11 to 0.45). The beat-rate time series of the remaining short experiment was not self-similar (r=0.44). In the 4 long experiments, all beat-rate time series were self-similar (r=0.97±0.03; range, 0.93 to 0.99) with H=0.29±0.12 (range, 0.14 to 0.42).
Spectral Analysis and Power-Law Behavior
The recording durations of the 18 short experiments
permitted the examination of the power spectrum over 3 decades ranging
from 0.001 to 1 Hz. The power spectra of the beat-rate variations of 2
cell cultures are shown in Figure 3. The
spectra did not exhibit any distinct peak and followed a descending
line. This linear dependence indicates that the PSD is a power-law of
frequency. In the 17 experiments in which the beat-rate time series
were self-similar, the spectra of beat-rate variability were
characterized by a power-law with an exponent ß (corresponding to the
slope of the linear regression over the frequency range 0.001 to 1 Hz)
of -1.31±0.20 (range, -1.04 to -1.74). In the single experiment in
which the beat-rate time series was not self-similar, no power-law was
present (ß=0.07, r=0.09). As judged by visual
inspection, the spectrum illustrated in Figure 3A closely
follows the regression line. Such a close fit was present in 8 of
the 17 short experiments (r=0.97±0.03, n=8). In the
remaining cases, as illustrated by the example in Figure 3B, the
spectrum displayed 2 scaling regions. Accordingly, the correlation
coefficients of these data (r=0.94±0.04, n=11) were
significantly lower (P<0.05).
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The 4 long electrograms permitted the examination of the power spectrum
over 4 decades, from 0.0001 to 1 Hz, as illustrated in Figure 4. When assessed over the entire
frequency range (0.0001 to 1 Hz), ß amounted to -1.48±0.38 (range,
-1.04 to -1.84). To allow comparison with short recordings
and to ensure that different recording durations had no
influence on the quantification of ß, this exponent was also computed
over the range used for short experiments (0.001 to 1 Hz). These
calculations yielded an average ß of -1.56±0.32 (range, -1.18 to
-1.96), which was not statistically different from ß of the short
recordings. When assessed over frequencies ranging from 0.0001
to 0.01 (the range used in clinical studies), ß amounted to
-1.35±0.72 (range, -0.76 to -2.24).
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No correlation was found between the exponents characterizing the
fractal properties of the beat-rate time series (H and ß)
and the mean beat rate of the preparations or the age of the
preparations. Also, the exponents of the experiments characterized by a
close linear fit (eg, Figure 3A) and those of the experiments
characterized by 2 scaling domains (eg, Figure 3B) were not
significantly different.
Temporal Variations of the Power-Law Exponent
The broad distribution range of ß indicates that this exponent
exhibits a large variability from preparation to preparation. Moreover,
this parameter was subject to fluctuations in the course of
a given experiment. To quantify these fluctuations, ß was computed in
a short time window of 2048 seconds (frequency range, 0.001 to 1 Hz),
which was moved along the entire beat-rate time series of a 9-hour-long
recording. As illustrated in Figure 5, "local" ßs calculated according
to this scheme displayed temporal variations ranging from -1.1 to
-1.8, whereas ß of the entire recording amounted to -1.6. A
similar range of variability was found in the other 3 long
recordings. This suggests that temporal nonstationarities of
ß contributed to its broad distribution range.
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A Case With Different Dynamics
As mentioned above, one preparation produced a beat-rate time
series that was not self-similar and that did not exhibit power-law
scaling behavior. This preparation had a mean beat rate of 6.6 Hz,
which was at the upper limit of the observed values (3.3±1.4 Hz;
range, 0.9 to 6.6 Hz). The beat-rate time series of this preparation is
illustrated in Figure 6B, top trace, in
comparison with another series exhibiting fractal properties (Figure 6A, top trace). Compared with the series with scaling behavior,
in which no repeating pattern was present, this series exhibited
repeating patterns at irregular intervals, thus indicating that this
series was less complex. A simple method to untangle a deterministic
behavior in a beat-rate time series consists of using phase portraits.
In this data representation, a given data point is plotted as a
function of the previous data point. If, in such a
representation, data form an organized pattern around a
so-called attractor, it is likely that the system is deterministic. As
indicated by the phase portraits shown in the lower panels of Figure 6, the data points of the preparation with power-law behavior
formed a normally distributed "cloud" filling the phase space. In
contrast, the data points and interconnecting lines of the preparation
without power-law behavior formed a cluster along an organized
attractor that did not fill the phase space. This indicates determinism
in the dynamics of the system and suggests that the mechanisms
underlying the variations of the beat rate in this particular case were
different from those present in the remaining experiments.
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Discussion |
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It is well known that aperiodic components of human HRV in vivo exhibit a broad-band spectrum in the very low frequencies (<0.01 Hz), which is characterized by a power-law relationship between PSD and frequency.7 8 9 It is generally accepted that this power-law behavior of HRV in vivo depends on extracardiac influences consisting of the nonlinear interactions of the sympathetic and the parasympathetic nervous systems occurring over a broad range of time scales.16 17 18 The present study shows that spontaneously beating monolayer cultures of cardiomyocytes are capable of displaying similar types of beat-rate variability in vitro. Their beat rate follows a fractal pattern exhibiting a power-law behavior, whereas periodic fluctuations are absent. Because these preparations are completely devoid of neurohumoral components, this finding suggests that the power-law behavior of HRV might, in addition to extracardiac influences, be determined by factors intrinsic to cardiac tissue.
Spontaneous Activity in Monolayer Cultures of
Cardiomyocytes
It is well known that cardiomyocytes in culture
can form synchronously contracting cell monolayers, and it is generally
assumed that these networks are driven by a focal pacemaker. Like the
sinus node, the preparations consist of a network of coupled excitable
elements exhibiting specific spatiotemporal activation patterns. It was
shown previously on the basis of computer simulations of networks of
pacemaker cells with different intrinsic cycle lengths that, once the
cells are coupled, their cycle lengths synchronize, giving rise to
focal excitation of the entire network.19 20 The presence
of focal pacemaking regions in dense monolayer cultures of cultured
cardiomyocytes has recently been verified experimentally in
monolayer cultures.21 In contrast to previous studies,
which found spontaneous activity to be a ubiquitous phenomenon in
monolayer cultures of cardiomyocytes,14 the
preparations used in the present study were generally only active
after enhancing L-type Ca2+ currents with
BayK8644. This is in accordance with an earlier study showing a
Ca2+ current dependence of the pacemaking
mechanism in these preparations.22 The failure to beat
spontaneously was not due to abnormalities in the electrophysiology of
the cells, as extracellular stimulation resulted in rapid activation of
the entire monolayers. Rather, it might be speculated that, in the
absence of BayK8644, the putative pacemaking regions within the
preparations were clamped to resting potential by the surrounding
well-coupled and -polarized cardiomyocytes of
ventricular origin and that only an increase in L-type
Ca2+ currents permitted these cells to overcome
this impedance mismatch.13 23
Spontaneous Activity Exhibiting Power-Law Behavior
In 21 of 22 preparations, the beat-rate times series was
characterized by a power-law behavior. The short recordings
permitted assessment of the power-law exponent ß over the range 0.001
to 1 Hz and yielded a value of -1.31±0.20, whereas the long
recordings, in which ß could be assessed over an additional
decade (0.0001 to 1 Hz), yielded a value of -1.48±0.38. When examined
over an identical frequency range (0.001 to 1 Hz), ß of long
recordings was -1.56±0.32 and was not significantly different
from that of short recordings. Thus, the different
recording lengths had no impact on ß determined in the range
of 0.001 to 1 Hz. Also, power-law behavior was not dependent on the
presence of BayK8644; in 3 experiments performed without this
substance, beat-rate variabilities assessed over the frequency range of
0.001 to 1 Hz exhibited a power-law with ß (-1.60±0.21; range,
-1.39 to -1.80) being not significantly different from that observed
in the preparations exposed to BayK8644.
In clinical studies, ß is assessed over the range 0.0001 to
0.01 Hz because, at higher frequencies, the power-law is disrupted by
the LF and HF components. In healthy subjects, ß is
-1.0,7 whereas it is -1.2 after myocardial
infarction8 9 10 and -2 after cardiac
transplantation.9 When determined over the same frequency
range, ß in the present study was -1.35±0.72 and is therefore
positioned between values after myocardial infarction and cardiac
transplantation. Even though a direct comparison between ß determined
in vivo and in vitro has to be made with great caution (see Limitations
of the Study section), the value of ß obtained in the cultured
preparations deprived of neuroendocrine influences could lend support
to the hypothesis that, during cardiac disease, a partial breakdown of
autonomic control may unmask components of HRV intrinsic to cardiac
tissue, which display a steeper power-law relationship than that
observed in healthy subjects.
Power-Law With Multiple Scaling Domains
In 9 of 17 experiments, the power spectrum did not closely
follow the regression line but rather consisted of 2 distinct scaling
domains exhibiting a crossover (see Figure 3B). This
double-scaling behavior is also found in Fourier power spectra of
HRV8 and of blood pressure variability24 and
was assessed in more detail in human HRV time series using detrended
fluctuation analysis.17 25 The double-scaling
behavior was especially prominent in data from elderly subjects and
from subjects with congestive heart failure. The authors suggested that
the double-scaling behavior is related to the breakdown of extracardiac
regulatory mechanisms during aging or disease. In the present
study, in which extracardiac regulations were absent, the
double-scaling behavior was observed in about half of the experiments,
whereas the others exhibited only 1 scaling domain. Although this does
not provide a physiological explanation for the
presence or absence of multiple scaling domains, and although the
caveat regarding the comparison of in vivo and in vitro results
applies, this finding suggests that factors different from extracardiac
influences might possibly contribute to the establishment of the true
power-law behavior observed in healthy subjects.
Fluctuations of the Power-Law Exponent
It was observed that the power-law exponent behaved in a
nonstationary and erratic manner in a given preparation over time (see
Figure 5). This intrinsic nonstationarity, with ß changing
drastically within hours, is most likely the main reason for the large
variability of the ß and H exponents observed in the short
experiments. Moreover, the finding that the SD of ß in long
experiments was not smaller than in the short experiments suggests that
the temporal variability of ß does not follow a gaussian behavior and
that the dispersion of ß values most likely would not have been
decreased by performing longer (eg, 24 hours) recordings.
Furthermore, the collected values of ß throughout this study
exhibited a larger SD than that observed in previous clinical
studies.9 10 This suggests that the neuroendocrine
modulation of HRV might have a stabilizing effect on ß. This
hypothesis is further supported by the finding of Bigger et
al9 that the dispersion of ß is smaller in healthy
subjects than in diseased or transplanted patients.
Intrinsic Power-Law Behavior: Possible Mechanisms
The presence of a power-law behavior in the absence of
neuroendocrine regulations opens the question of underlying
physiological mechanisms. It was previously
suggested that systems characterized by power-law behavior can consist
of many different regulatory mechanisms operating and interacting over
a broad range of temporal scales.16 17 18 In cardiac tissue,
the intrinsic power-law behavior might therefore be accounted for by
complex interactions of a large number of such processes acting at
different time scales. Possible candidates include ionic channels, in
which fractal patterns of channel gating in single-channel
recordings were observed and mathematical models of channel
gating accounting for this property were formulated.26 27
Moreover, electrophysiological studies
showed the presence of fractal characteristics in the
oscillations of the membrane potential.28
Also, it was shown that, while isolated cardiac cells beat
stochastically, fractal patterns of activity establish when a large
number of cells are electrically coupled and
synchronize.29 Finally, it is feasible that other
processes, such as biochemical reactions, protein synthesis, or gene
expression, contribute to the observed variations in beat rate. Thus,
cardiac tissue possesses a multitude of processes interacting at
different time scales that might, together, be responsible for the
observed tissue inherent power-law behavior.
Spontaneous Activity Exhibiting Low-Dimensional Dynamics
In contrast to all other preparations, a single culture failed to
exhibit a power-law behavior and showed low-dimensional dynamics
instead (see Figure 6). Although the shape of the extracellular
action potentials and the signal amplitudes measured in this culture
were not different from those measured in all other preparations, thus
indicating similar basic
electrophysiological properties, this
culture exhibited a particularly high average beat rate (6.6 Hz versus
3.4±1.4 Hz in the remaining preparations). On the basis of previous
observations that preparations with very high beat rates are usually
driven by functional reentrant excitation,21 the absence
of a power-law behavior in this single preparation might be explained
by the presence of a pacemaking mechanism different from the remaining
preparations (reentrant activation versus focal activation). This
hypothesis is also supported by a theoretical study showing that phase
portraits of interbeat intervals in 2-dimensional networks of cardiac
cells during spiral wave reentry exhibit low-dimensional
properties.30 If indeed the occurrence of low-dimensional
dynamics in cultured cardiomyocytes should be related to
the presence of reentrant excitation, this experimental system might
prove to be useful in screening of pharmacological substances, as the
analysis of the frequency content of signals obtained from a
single point in a culture might disclose the presence of reentry.
Limitations of the Study
When extrapolating the findings of the present study to the
sinus node, several caveats apply, as follows. (1) Cultured cells
differ from nodal cells in situ by the type, distribution and density
of ion channels and gap junctions. (2) The size of the preparations
used is larger than a sinus node. (3) The dimensionality is different
(2-dimensional monolayers versus 3-dimensional sinus node). (4) The
tissue architecture is different (confluent monolayers versus complex
nodal microarchitecture). Nevertheless, cardiomyocytes in
culture represent, in a manner functionally similar to that of
the sinus node, a network of coupled oscillators in which the network
dynamics are essential in the generation of fluctuations in the
discharge rate. However, the ultimate proof that the sinus node itself,
in the absence of neurohumoral influences, is capable of displaying a
power-law behavior will have to await the development of appropriate
experimental models.
Conclusions
The results of the present study suggest that components
intrinsic to cardiac tissue can cause spontaneous activity in cultured
cardiomyocytes to display fractal properties. In this
respect, the results extend earlier findings obtained in vivo that
described fractal properties of HRV to be dependent on nonlinear
interactions of the sympathetic and the parasympathetic nervous
systems. Thus, it might be speculated that the fractal measures of HRV
in vivo, which are correlated with cardiovascular
health and represent an analytical tool for the clinical
assessment of cardiac diseases, might ultimately be the result of an
interplay of factors extrinsic and intrinsic to cardiac tissue.
Obviously, further investigations will be needed to get deeper insights
into the exact nature of this interplay.
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Acknowledgments |
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This work was supported by the Swiss National Science Foundation. We thank Regula Flückiger Labrada for the cell culture work and Dr Stuart Thomas for helpful discussions on the manuscript.
Received March 20, 2000; accepted April 17, 2000.
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