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An application

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 $l=21$. 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] $W$ to 21. We used $e=3$ in equation (10) to reduce its variance. The scaling region was $]0.061,0.198]$ so $r_u/r_l$ 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 $d=11$ and $e=9$ are $\hat{\nu} = 2.6 \pm 0.1$ and $\hat{K_2} = 9.7 \pm 0.8$ [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 $W$ does hardly change our results. To create a time series that mimicks the fibrillation data we used the $x$-component of the Rössler system (see Wolf et al., 1985). For parameter values $L = 4000$, $\Delta t = 0.06$, $l=W=22$, $]r_l,r_u] = ]0.029,0.102]$, $d=11$ and $e=9$, we found for the ``double'' correlation dimension and entropy $\hat{\nu} = 2.2 \pm 0.1$ and $\hat{K_2} = 0.10 \pm 0.01$ [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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Next: Conclusions Up: mlbmb Previous: Coverage frequencies
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