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5.3. Correlation Functions of the Fluctuations
In discrete models it is usual to dene the state S(t) of the system as the sum of
the states of all the units
X
S(t) =
i
(t) :
(5.2)
i
We start all of our numerical simulations with a random initial conguration and
let the system evolve according to the rules of the individual units. We record the
state of the system at each time step according to (5.2) and quantify the complexity
of the series generated in terms of the auto-correlation function of S(t) [Goldberger
et al. (2002)].
To analyze the character of the uctuations it is useful to apply the detrended
uctuation analysis (DFA) method to the time series generated by the model [Peng
et al. (1995); Taqqu et al. (1995)]. In this method, the rst step is to integrate the
original time series. Then, the integrated time series is divided into boxes of size n
and, for each box, a least-square linear t is performed. Next, the root-mean-square
deviation of the integrated time series from the t, F (n), is calculated. This process
is repeated over dierent box sizes or time scales.
For self-similar signals one nds that F (n) satises a power law relation with
the size of the box n, that is F (n)n
, with the scaling correlation exponent.
This method quanties long-range time-correlations in the dynamical output of a
system by means of a single scaling exponent . The exponent is related to the
exponent of the power spectrum of the uctuations,S(f)1=f
, through the
relation = 21.
Brownian noise yields = 1:5, while uncorrelated white-noise yields = 0:5.
For a number of physiological signals from free-running, healthy, mature systems,
the scaling exponent takes values close to one, so-called 1=f-behavior, which
can be seen as a \trade-o" between the two previous limits [Goldberger et al.
(2002)]. However, dierent types of correlations (given by dierent values of the
exponent ) are encountered in some immature, diseased, or aged physiological
systems [Buchman (2002); Lipsitz (2002)].
5.4. Model Signalling Networks
In the previous section we outlined some results about the complexity of physio-
logical signals measured experimentally. Basically, we highlight the fact that for
healthy mature individuals some of the signals show the 1=f behavior, which in
terms of the exponent of the DFA means that '1 [Goldberger et al. (2002)].
Additional sets of measures showed that this exponent changes with disease and
age. On the other hand, it is also well known [Lipsitz (2002); Buchman (2002)] that
disease and age produce degradation in the communication between subunits in a
physiological system. For this reason we proposed a stylized model in which, with
