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5
HRV, slope = 0.68
4
3
HRV shuffled, slope = 0.54
2
1
0
1
2
3
4
5
6
7
Log 2 of time
Figure 2.21.
The entropy or information for the HRV time series calculated using ( 2.83 ) is graphed versus the
logarithm of time. The difference between the slopes of the shuffled and unshuffled data sets is
evident [ 100 ]. Reproduced with permission.
The HRV data are used to construct a histogram for the probability density that can be
found in most texts on data analysis [ 42 ]. For the time being we note that the scaling
parameter
would be 0.5 if the time series were an uncorrelated random process. For
the original data set the probability density is found empirically to scale in time with an
index
δ
68 and when the data are shuffled the statistics do not change but the scal-
ing index drops to
δ =
0
.
A rigorous determination of the latter scaling index would
require doing the shuffling a number of times and taking the average of the resulting
indices. This was not done here because the shuffling was done for illustrative pur-
poses and with the value obtained it is clear that the average would converge to the
uncorrelated value of 0.5.
The long-range correlation implied by the scaling is indicative of an organizing
principle for complex webs that generates fluctuations across a wide range of scales.
Moreover, the lack of a characteristic scale helps prevent excessive mode-locking that
would restrict the functional responsiveness of the cardiovascular network. We propose
later a dynamical model using a fractional Langevin equation that is different from
the random-walk model proposed by Peng et al. [ 58 ], but which does encompass their
interpretation of the functionality of the long-range correlations.
δ =
0
.
54
.
2.4.2
Fractal breaths
Now let us turn our attention to the apparently regular breathing as you sit quietly
reading this topic. We adopt the same perspective as that used to dispel the notion
of “regular sinus rhythm.” The adaptation of the lung by evolution may be closely
tied to the way in which the lung carries out its function. It is not accidental that the
cascading branches of the bronchial tree become smaller and smaller; nor is it good
fortune alone that ties the dynamics of our every breath to this biological structure. We
argue that, like the heart, the lung is made up of fractals, some dynamic, like breathing
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