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Prior to presenting our results it is worthwhile to note that both the RBN and
CA models can be seen as limiting cases of the present model in the absence of
noise: A RBN model corresponds to a completely random network with dierent
Boolean rules for the units, while a CA model corresponds to k
e
= 0 and all units
evolving according to the same rule.
In our systematic analysis, we rst calculated the exponent for systems whose
units are randomly assigned a Boolean function from the set of 256 functions of
three inputs | as in the classical RBN model. We found white-noise dynamics for
essentially any pair of values of k
e
and within the ranges considered, suggesting
that a system of random Boolean functions cannot generate complex dynamics.
This result is not unexpected, since the random collection of Boolean functions
comprising the system prevents the development of any order or predictability in
its dynamics.
In order to prevent this fact, we considered systems whose units all evolve ac-
cording to the same rule, as in CA models (but noisy dynamics in our case). We
systematically studied the 24 dierent Boolean functions of three inputs which do
display uctuations, for dierent pairs of values of k
e
and .
In our previous works [Amaral et al. (2004); Diaz-Guilera et al. (2007)] we
showed the phase spaces for dierent rules displaying dierent behaviors. We clas-
sied them in three dierent sets depending on the diversity of ranges of dynamics.
In the rst and more interesting set, shown in Fig. 5.5, three dierent types of dy-
namical behavior are displayed, depending on both the intensity of the noise and
on the excess connectivity. In a second set, complex uctuations are only obtained
for small intensities of the noise without dependence on the excess connectivity.
Finally, there is a collection of rules for which uctuations are always uncorrelated.
Only the rst case is then worth of consideration. In particular, rule 232 (the ma-
jority rule) which is encountered in many biological or physical models, plays the
most important role in our further analysis. In our case the most interesting result
is the dependence of the exponent on the topology for a xed noise intensity, as
has been discussed along the previous paragraphs. This simple model (rule 232 in
Fig. 5.5) can hence explain the experimentally observed dependence of the scaling
properties of some physiological signals on age and failure. Age and failure can
be related, respectively, to missing physiological \links" and to misscommunication
among some physiological subunits [Lipsitz (2002); Goldberger et al. (2002)].
5.4.5. Robustness
In order to determine the generality of the results presented above, one needs to
address the questions of how these ndings are aected by (i) changes in the topology
of the network or (ii) \errors" in the units' implementation of the rules. Since from
both a biological and a physical point of view, the majority rule has a meaningful
interpretation we take it as a starting point, but it is worth to note that it could be
performed for any of the \complex" rules represented in Fig. 5.5 (rules 19 and 1).
