Dynamics of chance - Complex Webs: Anticipating the Improbable - page 308

principle is of more than theoretical interest and may well explain the fading behavior of
complex webs in response to simple stimuli manifest in multiple phenomena, including
perturbed liquid crystals, as we demonstrate.
The main purpose of Section 7.3 is to determine whether a universal LRT does in fact
exist, even when the stationary condition does not. To support our arguments we adopt
an idealization of reality and study the case when the regression to equilibrium, after
a suitable initial preparation, has an infinitely long duration. We show that in this case
it is also possible to make theoretical predictions on the basis of the LRT structure. If
the non-equilibrium condition is not perennial, the non-stationary LRT has to be valid
both in the initial out-of-equilibrium condition and in the final equilibrium regime. We
shall see that it is possible to satisfy this condition, even if at the present time we are
not aware of any theoretical treatment joining the initial out-of-equilibrium process to
the final equilibrium.
Generalized LRT refers to a condition in which dynamics are determined by events,
and the non-equilibrium condition corresponds, in fact, to making the number of events
per unit time decrease with increasing time. Our arguments do not require a Hamilto-
nian formalism, which is a significant extension of the LRT property from physical to
neurophysiological and sociological webs [ 68 ]. This theory, in the ordinary ergodic and
stationary case, yields results that apparently coincide with the prediction of conven-
tional LRT. We present applications of these ideas to the well-known phenomenon of
habituation in the last part of the chapter. We developed a model to explain the psy-
chophysical phenomena of habituation and dishabituation. Habituation is a ubiquitous
and extremely simple form of learning through which animals, including humans, learn
to disregard stimuli that are no longer novel, thereby allowing them to attend to new
stimuli. As Breed [ 17 ] points out, one of the more interesting aspects of habituation
is that it can occur at different levels of the nervous system. Cohen et al. [ 20 ]show
that sensory networks stop sending signals to the brain in response to repeated stimuli;
for example, such an effect occurs with strong odors. But odor habituation has been
shown in rats to also take place within the brain by Deshmukh and Bhalla [ 25 ], not
just at the sensor level. We hypothesize that 1/ f noise, which is characteristic of com-
plex networks, arising as it does both in single neurons [ 63 ] and in large collections of
neurons [ 22 , 31 ], is the common element that may explain both these observations by
virtue of suppressing signals being transmitted to the brain and inhibiting signals being
transferred within the brain.

7.1 The master equation

There is a variety of ways to introduce the dynamics of discrete probabilities, depend-
ing on the phenomenon being modeled. The chapter on random walks illustrates one
such technique. Another favorite method of statistical physics is the master equation ,
which describes the temporal behavior of singlet probabilities, all multivariate one-time
probabilities for all stochastic processes and the conditional probabilities for Markov
processes [ 51 ]. A singlet probability is the probability that an event occurs at a given