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Atrial fibrillation is a commonly encountered arrhythmia of the heart.
Understanding its nature is an important issue in clinical
cardiology.
To study the underlying dynamics,
recordings of induced fibrillation were collected from chronically
instrumented conscious dogs with 30 unipolar electrodes sutured to the
atria (see Rensma et al., 1988, for experimental details).
Time series consisting
of approximately 4000 points were obtained using an 8-bit analog-to-digital
converter at a sampling frequency of 1000 Hz.
The time lag for the reconstruction of phase space was
determined by the mutual information criterion [Fraser and Swinney, 1986],
which yielded . See
figure 2a
for a plot of the time series and
figure 2b
for a phase portrait.
The correlation dimension and
entropy were estimated from 40000 distances
(from randomly chosen vector indices).
We set the Theiler correction parameter [Theiler, 1986] to 21.
We used in equation (10)
to reduce its variance. The scaling region was so
is about 3.
The results are presented in
figure 3a
and
figure 3b.
We see that both
The ``double'' correlation dimension
and the correlation entropy for and
are
and
[nats/s].
These
results suggest that this particular episode of
fibrillation may be characterized
by a low-dimensional chaotic process.
To check whether the saturation is not caused by autocorrelated
or coloured noise, we randomized the phases of the signal [Theiler, 1990a].
For this ``randomized'' signal, the estimated correlation
dimension does not saturate with increasing embedding dimension.
Increasing the Theiler correction parameter
does hardly change our results.
To create a time series that mimicks the fibrillation data we used
the -component of
the Rössler system (see Wolf et al., 1985).
For parameter values
,
, ,
,
and , we found for the ``double'' correlation dimension and entropy
and
[nats/s] which is
in reasonable agreement with the literature values.
We remark that the Rössler time series was only corrupted
by a small amount of ``integration noise''.
Work is in progress with longer experimental electrograms and
time series from model systems to study both the effects of noise
and methods of noise reduction [Kostelich and Yorke, 1990] on the estimated
correlation dimension and entropy.

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