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beats. In another example, that of neurons, either absolutely regular pacemaker activ-
ity or highly explosive global neuronal firing patterns would develop in the absence
of neural time-series fluctuations. The erratic component of the neural signal acts to
maintain a functional independence of different parts of the nervous system. A complex
web whose individual elements act more or less independently is more adaptable than
one in which the separate elements are phase-locked. This same randomness-induced
decoupling effect can be found in the immune network and in population dynamics.
One might also find cross-level effects in the context of population dynamics and in
particular the significance of such effects for genetic structure and evolution. The basis
for this argument is computer models of ecosystem dynamics in which organisms feed
and reproduce in the modeling context, with certain phenotypic traits arising from genes
surviving through interactions among the individual organisms by Darwinian variation
and selection.
Finally, the disturbance of a biological web may relax back to regular behavior by
means of a fluctuating trajectory absorbing the perturbation. In a formal context both
fluctuations and dissipation in a network arise from the same source. The many degrees
of freedom traditionally thought necessary to produce fluctuations and dissipation in
physical networks may be effectively replaced by low-dimensional dynamical webs that
are chaotic. If the web dynamics can be described by a
strange attractor
then all trajec-
tories on such an attractor are equivalent and any disturbance of a trajectory is quickly
absorbed through the
sensitive dependence of the trajectory on initial conditions
and
a perturbed trajectory becomes indistinguishable from an unperturbed trajectory with
different initial conditions. In this way chaos provides an effective mechanism for dissi-
pating disturbances. The disturbance is truly lost in the noise but without changing the
functionality of the strange attractor [
16
].
4.6
Problems
4.1 The Gaussian distribution
Use the probability density given by (
4.8
) to calculate the average location of
the walker as well as her mean-square displacement. Use Stirling's approximation
n
n
ln
n
to the factorial to derive the Gaussian distribution from this
equation as well.
!≈
n
ln
nn
!≈
4.2 Asymptotic properties of gamma functions
The asymptotic form of the expansion coefficient in the solution to the fractional random
walk can be determined using Stirling's approximation for the factorial representation
of the gamma functions. Prove that (
4.55
) follows from (
4.54
) in the asymptotic limit.
4.3 Anomalous diffusion
Physical webs whose second moment increases as a power law in time, for example,
Q
2
t
t
2
H
;
∝
