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considered and the physical observable responds only to a direct or indirect Hamiltonian
interaction with the external stimulus.
Perturbation
. The external stimulus, which is not directly coupled to
ξ
s
, affects the
event dynamics according to the value of the variable
. This is the common situation
in a complex web, where multiple variables interchange information with one another.
The perturbation cascades through the web, thereby generating an indirect effect on
ξ
s
(
t
)
ξ
s
,
and consequently a bias, which is taken to be zero in the absence of the stimulus. We
demonstrate this effect in a number of psychophysical phenomena after reviewing the
results of the physics experiment.
Experimentation
. The experiment on liquid crystals [
5
] supports the following
theoretical prescription for the average web response modeled on (
7.138
):
ξ
s
(
t
)
=
CR
(
t
)
cos
(ω
t
+
φ),
(7.140)
with the amplitude
C
and phase
φ
as fitting parameters; and the power-law index
μ
is determined, as we shall see, by the experiment itself. Equation (
7.140
) admits
an extremely simple and intuitive explanation of the web response fading in time, since
it resembles the response to perturbation of Poisson processes (stochastic resonance
[
42
,
68
]) with the proviso that the cascade of events generated by preparation fades
away with time.
Allegrini
et al
.[
5
] prepare their experiment by bringing the web into a fully devel-
oped turbulent regime corresponding to a Poisson condition and to a high rate of event
production. After a few seconds (20 s) they adopt a control potential to establish cooper-
ative interaction between the web's units (corresponding to defect-mediated turbulence
and
5), and apply a weak perturbation to the web by modulating the AC voltage
amplitude. Unfortunately, the first equilibrium value for transmissivity is different from
the latter, corresponding to the regime of interest. Therefore, the equation they adopt
must take into account also the free regression to equilibrium as well as the component
given by (
7.140
),
μ
=
1
.
ξ
s
(
t
)
=
ξ(
0
)
(
t
)
+
CR
(
t
)
cos
(ω
t
+
φ).
(7.141)
To derive the form of (
7.140
) they perturb the web with cos
(ω
t
)
and
−
cos
(ω
t
)
and
evaluate the sum response
)
=
ξ
s
(
t
)
+
+
ξ
s
(
t
)
−
S
(
t
(7.142)
2
and the difference response
)
=
ξ
s
(
t
)
+
−
ξ
s
(
t
)
−
D
(
t
.
(7.143)
2
Note that
ξ
s
(
t
)
+
and
ξ
s
(
t
)
−
are the responses of the network to cos
(ω
t
)
and
−
, respectively. To obtain the ensemble averages indicated by the angle brack-
ets they repeat the preparation and perturbation procedures approximately 200 times for
each measurement and then average.
D
cos
(ω
t
)
are shown in Figures
7.7
and
7.8
,
respectively. Note that the result shown in Figure
7.8
allows us to determine the power
index
(
t
)
and
S
(
t
)
μ
(
μ
=
1
.
56 in this case) from the slope of the experimental curve. This method
