Vol. 277, Issue 2, H452-H458, August 1999
Ventricular response in atrial fibrillation: random or
deterministic?
Kenneth M.
Stein1,
Jeff
Walden1,
Neal
Lippman2, and
Bruce B.
Lerman1
1 Division of Cardiology,
Department of Medicine, New York Hospital-Cornell Medical Center,
New York, New York 10021; and
2 Division of Cardiology,
Department of Medicine, University of Connecticut, Farmington,
Connecticut 06030
 |
ABSTRACT |
The ventricular
response in atrial fibrillation is often described as "chaotic,"
but this has not been demonstrated in the strict mathematical sense. A
defining feature of chaotic systems is sensitive dependence on initial
conditions: similar sequences evolve similarly in the short term but
then diverge exponentially. We developed a nonlinear predictive
forecasting algorithm to search for predictability and sensitive
dependence on initial conditions in the ventricular response during
atrial fibrillation. The algorithm was tested for simulated R-R
intervals from a linear oscillator with and without superimposed white
noise, a chaotic signal (the logistic map) with and without
superimposed white noise, and a pseudorandom signal and was then
applied to R-R intervals from 16 chronic atrial fibrillation patients.
Short-term predictability was demonstrated for the linear oscillators,
without loss of predictive ability farther into the future. The chaotic
system demonstrated high short-term predictability that declined
rapidly further into the future. The pseudorandom signal was
unpredictable. The ventricular response in atrial fibrillation was
weakly predictable (statistically significant predictability in 8 of 16 patients), without sensitive dependence on initial conditions. Although
the R-R interval sequence is not completely unpredictable, a
low-dimensional chaotic attractor does not govern the irregular
ventricular response during atrial fibrillation.
heart rate; nonlinear dynamics
| |
INTRODUCTION |
CHAOS THEORY, referring to the overlapping mathematical
disciplines of fractal geometry and nonlinear dynamics, has been used to provide insight into the pathophysiology and prognostic importance of cardiac arrhythmias (2-5, 8, 22, 23). In the
mathematical sense, "chaos" refers to a system that is aperiodic
but deterministic (nonrandom). A defining characteristic of chaotic
behavior is sensitive dependence on initial conditions; i.e., similar
sequences evolve similarly in the near future but then diverge
exponentially. Because of sensitive dependence on initial conditions,
if the dynamics of a chaotic system are sufficiently described, the
behavior of the system can be predicted in the short term but, due to
inherent imprecision in measurement, not in the long term. In contrast, a linear system is equally predictable in both short and long terms,
whereas a random system is equally unpredictable in both short and long terms.
Techniques of nonlinear dynamics have been applied to the analysis of
heart rate variability during sinus rhythm. Although the issue of
whether sinus rhythm is chaotic remains controversial (16), these techniques have been used successfully for
risk stratification (18, 19). In contrast, although the irregular ventricular intervals in atrial fibrillation are often described as
chaotic, it has yet to be shown that this represents chaos in the
mathematical sense. Thus, although reduced heart rate variability during atrial fibrillation conveys an increased cardiovascular risk
(20), the underlying dynamics of the ventricular response during atrial
fibrillation are incompletely understood.
To better understand whether the ventricular response in atrial
fibrillation is chaotic, we developed an algorithm [based on an
original proposal by Sugihara and May (25)] that uses nonlinear
predictive forecasting to search for evidence of sensitive dependence
on initial conditions in a time series. We applied the algorithm to a
set of simulated R-R intervals ("test sets") and to the analysis
of R-R interval sequences obtained in 16 patients during chronic atrial
fibrillation. A similar algorithm previously has been demonstrated to
be useful for interpolating to correct for the effects of ectopy in
calculations of heart rate variability during sinus rhythm (9, 10).
| |
METHODS |
Nonlinear predictive forecasting.
The behavior of every nonrandom (deterministic) system, no matter how
complex, can be considered to occur within a "phase space," which
can be represented by a Cartesian coordinate system with one axis for
each of the variables necessary to completely describe the state of the
system at any given time. For most biologic systems, the nature of
these variables, or even their number, is unknown. However, a
topological equivalent of phase space can be constructed from empirical
data using the technique of "lags" (26). With the use of this
technique, an embedding dimension (E) is chosen and the points that
constitute a phase space are represented as the vectors
|
(1)
|
where
RRn is the
nth R-R interval in the time series
and L, the lag, is a positive integer.
As proposed by Sugihara and May (25), the future behavior of a system
may be predicted ("forecasted") by observing other sufficiently
similar trajectories in phase space. Importantly, and in distinction
from other methods of forecasting such as autoregressive modeling, this
scheme does not require any assumptions about the underlying
mathematical relationships of the system other than the initial
assumption that the system is deterministic (or, more precisely, that
its behavior can be described within a phase space of finite dimension).
The nonlinear predictive forecasting algorithm therefore consists of
the following steps. 1)
The technique of lags is used to reconstruct 8 different phase spaces
with embedding dimensions from 3 to 10. 2) For each R-R interval in the time
series (RRi), the three nearest
neighbors in each phase space, determined using the Cartesian distance
metric 
i,j = 
are chosen from every other point
(RRj) in the same time series.
The predicted next R-R interval
(RRi+1) is
determined as the average of the R-R intervals that follow (RRj+1)
the three nearest neighbors (each weighted by the inverse of its
i,j).
3) The Pearson correlation coefficient (r) between predicted
and actual R-R intervals is computed for all intervals in the
time series for each of the eight phase spaces.
4) The phase space with the
embedding dimension yielding the highest correlation coefficient is
determined. Within that embedding dimension, the 3 nearest neighbors
of each R-R interval trajectory in phase space are used to
predict the evolution of that trajectory from 1 to 10 steps into
the future using a method analogous to that described in
the second step. For a system of dimension
n, and for a sufficiently large data
set, the correlation coefficient should be approximately the
same for all embedding dimensions
>n. Therefore, the
embedding dimension giving the maximum correlation coefficient does not
strictly correspond to the dimension of the system but
suffices for the practical purpose of nonlinear prediction.
5) For each distance into the
future, the correlation coefficient between predicted and actual R-R
intervals is computed for all intervals in the time series.
The algorithm can be used to test for the hallmark of chaotic systems:
sensitive dependence on initial conditions. In linear systems, nearby
trajectories tend to remain nearby, and therefore these systems are
equally predictable in the far future as well as in the near future.
Random sequences, in contrast, are equally unpredictable in both the
near future and far future. In chaotic systems, however, nearby
trajectories tend to diverge exponentially. Therefore, they are
predictable in the near future but not the far future. For example, the
weather is easily predictable 5 min into the future, modestly
predictable 5 h into the future, and virtually unpredictable 5 days
into the future. By computing the (Pearson) correlations between values
predicted from 1 to 10 steps into the future using the nonlinear
predictive forecasting algorithm, we can therefore gain insight into
whether the system behaves more like a linear system, a random system,
or a chaotic system.
Simulated R-R intervals.
We tested the above algorithm by applying it to computer-generated
sequences of simulated R-R intervals (test sets) of varying lengths
(50-2,000 intervals) and with varying amounts of added noise to
assess the impact of sequence length and measurement error on the
performance of the algorithm. The following test sets were
created. 1) A linear
oscillator was constructed (chosen to simulate the dominant low- and
high-frequency oscillations in human heart period during sinus rhythm)
2)
A linear oscillator with a high signal-to-noise ratio was constructed
[using a computerized pseudorandom number (Rand) generator]
3)
A linear oscillator with a low signal-to-noise ratio was constructed
4)
A chaotic system was constructed using the logistic map, a known
chaotic system. This system is iteratively determined according to the
equations
5)
A chaotic system with a high signal-to-noise ratio was constructed by
adding uniformly random numbers scaled over the interval (
25,
+25) to the values
X(t)
from the logistic map in test set 4.
6) A chaotic system with a low
signal-to-noise ratio was constructed by adding uniformly random
numbers scaled over the interval (100, +100) to the values
X(t)
from the logistic map. 7) Finally, a
random system consisting of uniformly random numbers scaled over the
interval (600, 1,000) was constructed.
Atrial fibrillation.
In 16 patients with chronic atrial fibrillation, 24-h ambulatory
electrocardiographic recordings of bipolar leads CM1 and CM5 were obtained using a standard reel-to-reel Holter
monitoring system (Del Mar Avionics, Irvine, CA) and scanned by a
computer-based system (Marquette Laser XP; Marquette Electronics,
Milwaukee, WI) with total visual verification and correction of beat
morphology and timing by one of the authors. The results were digitized
at 128 Hz, yielding an approximate temporal resolution of 8 ms. After scanning was completed, the first series of 
2,000 consecutive R-R
intervals without intervening ventricular ectopic beats or artifact was
extracted for each patient and downloaded to a personal computer for
application of the identical algorithm used for the test sets.
Statistical and power spectral analysis.
The significance of the observed Pearson correlation coefficients was
determined using a standard two-tailed
t-test. A
P value <0.01 was taken as
indicating that a correlation was significantly different from zero
(i.e., was significantly predictable). For data sets of
1,600-2,000 beats this criterion for weak but significant predictability approximately corresponds to a correlation coefficient >0.06. Grouped comparisons of continuous variables were made via the
t-test. Grouped comparisons of
dichotomous variables were made via Fisher's exact test. For these
purposes a P value <0.05 was
required to reject the null hypothesis. Power spectra were computed for
raw heart period data using a 1,024-point fast Fourier transform
algorithm with a Tukey-Hamming window (11 point).
| |
RESULTS |
Test sets.
As long as a minimum of 100 intervals was specified, short-term
predictability was demonstrated for the system of linear oscillators with no loss of predictive ability farther into the future even in the
presence of a large amount of noise (low signal-to-noise ratio) (Fig.
1). For the case of 50 points, short-term
predictability was paradoxically improved by the addition of noise;
this may relate to boundary and/or aliasing effects specific to
sampling 50 points of the function sin
(2
t/25) + sin
(2t/5). In contrast, although the
chaotic system demonstrated high short-term predictability, the
correlation between predicted and actual intervals declined rapidly
with attempts to forecast further into the future. The chaotic signal
was distinctive with as few as 50 intervals to analyze in the presence
of a high signal-to-noise ratio, although not in the presence of a low
signal-to-noise ratio (Fig. 2). As expected, the pseudorandom signal was not predictable.
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Fig. 1.
Correlations between observed and predicted values at varying distances
into the future as a function of sample size. Results are shown for a
system of sinusoidal oscillators (Sin) and for sinusoidal systems with
a low (LN) or high (HN) amount of superimposed random noise. Although
predictability was less for the HN system than other systems, as long
as the sample contained  100 observations, dynamics were somewhat
predictable for all systems and predictability did not deteriorate for
increasing distances into the future.
N, no. of observations.
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Fig. 2.
Correlations between observed and predicted values at varying distances
into the future as a function of sample size. Results are shown for the
logistic map (Log; evaluated in its chaotic domain), for Log with a low
amount of superimposed random noise (Log + LN), and for Log with a high
amount of superimposed random noise (Log + HN). For logistic maps, as
long as the sample contained 100 observations, dynamics were
predictable in short term (future = 1), but predictability deteriorated
significantly with increasing distances into the future.
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Atrial fibrillation.
The group of 16 patients with chronic atrial fibrillation included 9 males and 7 females with a mean age of 57 ± 12 yr. Of the 16 patients, 6 had no underlying cardiovascular disease, 5 had significant
valvular heart disease (mitral regurgitation or mixed rheumatic
valvular disease), 1 had an ischemic cardiomyopathy, and 3 had
hypertension. Two patients had mitral valve prolapse without
significant mitral regurgitation (including 1 patient with coexisting
hypertension). Also, 12 of 15 patients were being treated with digoxin,
4 of 15 were on 
-blockers, 3 of 15 were on calcium-channel blockers,
and 4 of 15 were on antiarrhythmic drug therapy (2 on procainamide, 1 on amiodarone, and 1 on propafenone). One patient's medications could
not be ascertained with certainty. No patient had clinical evidence of
digitalis intoxication. During the recording period, the mean R-R
interval was 804 ± 186 ms (range: 539-1,090 ms). The average
standard deviation of the R-R interval was 178 ± 48 ms (range:
118-254 ms). In contrast, for the uniform random signal tested,
the mean was 797 and the standard deviation was 143.
The best embedding dimension for prediction one beat into the future
(future = 1) ranged from 3 to 10 (mean: 5.9 ± 2.7). As shown in
Fig. 3 there was no characteristic best
embedding dimension for the group as a whole. With the use of each
individual patient's best embedding dimension, the ventricular
response in atrial fibrillation was only weakly predictable at all time
scales and there was no evidence of sensitive dependence on initial
conditions (Fig. 4).
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Fig. 3.
Correlations (means ± SE) between observed and predicted values in
short term (future = 1) as a function of embedding dimension for
patients with atrial fibrillation. There is no characteristic
"best" embedding dimension and no suggestion that prediction is
improved at higher dimensions.
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Fig. 4.
Correlations (means ± SE) between observed and predicted values at
varying distances into the future for patients with atrial fibrillation
using each patient's individual optimal embedding dimension.
Ventricular response was only weakly predictable at all time scales,
and there was no deterioration of predictability with increasing
distance into the future.
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However, in 8 of 16 patients, the R-R interval series was significantly
(albeit weakly) predictable as evidenced by a correlation coefficient
significantly different from zero (Fig. 5).
These patients were not significantly different from their counterparts in age, gender, presence of structural heart disease, medications, mean
R-R interval, standard deviation of the R-R interval, number of beats
analyzed, or best embedding dimension (all
P > 0.10). For two individual
patients, the ventricular response was moderately predictable
(r values of 0.37 and 0.56, respectively, at future = 1) but did not show evidence of sensitive
dependence on initial conditions. Although the significance is
uncertain, electrocardiograms (ECG) from these patients demonstrated
notably "coarse" fibrillatory activity (Fig.
6). Power spectral analysis revealed
broad-band power spectra for all patients. The power spectrum for the
patient with the highest degree of nonlinear predictability is shown in Fig. 7.
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Fig. 5.
Individual data showing correlations between observed and predicted
values at varying distances into the future for patients with atrial
fibrillation using each patient's individual optimal embedding
dimension. For a subset of patients, ventricular response was
significantly (albeit modestly) predictable without deterioration at
increasing distances into the future.
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Fig. 6.
A 12-lead electrocardiogram from the patient whose ventricular response
demonstrated the greatest degree of predictability. Notably
"coarse" fibrillatory activity is present.
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Fig. 7.
Power spectral analysis of heart period data from the patient whose
ventricular response demonstrated the greatest degree of
predictability. Spectrum is "broad band," with no discrete
high-frequency peaks.
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| |
DISCUSSION |
The principal findings of the present study are
1) that a nonlinear forecasting
algorithm can feasibly be applied to R-R interval data sets and, as
evidenced by the performance when applied to computer-generated test
sets, can reliably discriminate linear, chaotic, and random types of
behavior; 2) that the ventricular response in atrial fibrillation does not appear to represent chaos in
the strict mathematical sense of a deterministic aperiodic system; and
3) that, in a significant number of
patients, the beat-to-beat ventricular response in atrial fibrillation
is not wholly unpredictable.
Nonlinear forecasting.
The results in the test sets demonstrate that nonlinear forecasting is
a practical approach for qualitatively analyzing the dynamics of a
complex system such as that generating the R-R interval data stream.
Nonlinear forecasting was initially proposed by Sugihara and May (25)
for application to biologic time-series data. The primary advantage of
this approach versus linear forecasting systems (e.g., power spectral
analysis or autoregressive modeling) is that it does not require any
assumptions about the underlying mathematics of the system beyond the
hypothesis that the system is deterministic (i.e., predictable).
Other methods that can be used to look for evidence of chaotic behavior
include estimation of the dimension of the attractor for the system to
be analyzed (a measure of complexity/aperiodicity) (8) and estimation
of the largest Lyapunov exponent of the system (a measure of sensitive
dependence on initial conditions). In theory, nonlinear forecasting has
an advantage over the former approach in that it is less dependent on
very long data streams for accurate interpretation (27) and an
advantage over the latter approach in that it is more robust and less
dependent on initial assumptions about the behavior of the system.
Predictability of atrial fibrillation.
This is the first analysis to discover evidence of significant
(although weak) predictability in the beat-to-beat changes in the
ventricular response in atrial fibrillation in some patients. Subtle
ordering has previously been demonstrated in atrial activation during
atrial fibrillation (7, 17). Significant positive or negative
correlations between successive ventricular beats during atrial
fibrillation have been described in some human subjects (15) as well as
in some animal models (14). However, most analyses have not
demonstrated any systematic order to the ventricular response during
atrial fibrillation in humans, and most authorities describe the pulse
in atrial fibrillation as random (1, 6, 12, 29). It is possible that
these results differed from ours because the analytic approach
(autoregression) depends on an assumption of strict linearity. The
signals that we observed were most compatible with a weak linear system
superimposed on a very noisy background (cf. Figs. 1 and 4). There was
no evidence of sensitive dependence on initial conditions, and our
results are therefore not compatible with the notion that the
ventricular response in atrial fibrillation might represent a form of
low-dimensional chaos.
Although the present analysis shows weak predictability in the
ventricular response during atrial fibrillation, it does not permit us
to identify the physiological causes of that predictability in these
patients. However, van den Berg and colleagues (28) have previously
demonstrated a moderate correlation between heart rate variability in
atrial fibrillation and systemic vagal tone. These observations might
explain the analogy between the prognostic utility of measures of heart
rate variability in atrial fibrillation and the same measures in sinus
rhythm (20, 21). It is possible that the weak predictability
demonstrated in the present study represents the effect of cyclic
oscillations in parasympathetic and/or sympathetic tone at the level of
the atrioventricular node, superimposed on the effectively random
sequence of impulses reaching the distal node due to concealed
penetration of multiple wave fronts (13) engaging the perinodal atrium
from different directions (11, 24). It should be noted that the R-R
interval power spectra did not reveal narrow peaks within ranges
characteristic of parasympathetic or sympathetic oscillations, but it
is possible that this may represent an intrinsic limitation of Fourier
analysis in such a "noisy" system. The two patients with the
highest degrees of predictability had coarse atrial fibrillation on the
surface ECG, but the physiological significance of this observation is uncertain.
Limitations.
It is possible that analysis of longer recordings would have yielded
better predictability because, for a short time series, "nearest"
neighbors in phase space are not necessarily "near" neighbors. We chose a recording length of ~2,000
intervals to maximize stationarity of the underlying system. Likewise,
it is possible that analysis of embedding dimensions >10 would have yielded better predictability. However, there was no evidence suggesting this, and the results would not, in any event, reflect low-dimensional chaos. Similarly, using the three nearest neighbors for
prediction was an empirical choice; others have proposed using the
single nearest neighbor for prediction (16). However, although this may
have had some quantitative effect, it is unlikely that it would
qualitatively affect our results. Finally, to study patients with
stable chronic atrial fibrillation, it was necessary to accept patients
undergoing treatment with a variety of medications including digoxin,
-blockers, calcium-channel blockers, and antiarrhythmic drugs.
Although we observed no important differences accorded to the use of
any of these agents, the possibility that subtle drug effects may have
altered the results cannot be fully excluded.
In summary, despite the above considerations, the present data
demonstrate the potential of nonlinear predictive forecasting as a tool
for qualitative analysis of dynamic systems and demonstrate that atrial
fibrillation is not a chaotic rhythm. Furthermore, although the
pathophysiological significance is not clear, the technique reveals the
existence of a subset of patients with atrial fibrillation whose R-R
interval sequence is not entirely unpredictable.
| |
FOOTNOTES |
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: K. M. Stein,
Div. of Cardiology, Starr-4, New York Hospital-Cornell Medical Center,
525 East 68th St., New York, NY 10021 (E-mail:
kstein{at}mail.med.cornell.edu).
Received 1 September 1998; accepted in final form 12 March 1999.
| |
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Am J Physiol Heart Circ Physiol 277(2):H452-H458
0002-9513/99 $5.00
Copyright © 1999 the American Physiological Society
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