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We assume without loss of generality that the initial value vanishes,
(
0
)
=
0, and
define the variable
ξ
s
(
t
)
that has the value 1 when the web is in the “on” state and the
value
. Thus, the solution (
7.69
),
which is the average response to the perturbation, can be written
−
1 when the web is in the “off” state to replace
(
t
)
t
dt
χ(
t
)ξ
p
(
t
).
ξ
s
(
t
)
=
ε
t
,
(7.70)
0
In this case the response function is defined by the exponential
ge
−
g
(
t
−
t
)
.
t
)
=
χ(
t
,
(7.71)
We note that the autocorrelation function of the fluctuation response has the form
ξ
s
(
t
)
=
s
(
e
−
g
(
t
−
t
)
,
t
)
=
s
(
t
)
=
t
)ξ
s
(
t
,
t
−
(7.72)
which is a direct consequence of the Poisson statistics. On comparing (
7.72
) with (
7.71
)
it is clear that the response function is the time derivative of the autocorrelation function
t
)
d
s
(
t
−
t
)
=
χ(
t
,
.
(7.73)
dt
The structure of (
7.70
) is the general form of the response to perturbations derived
by Kubo [
48
] when the response function is defined by (
7.73
). A stochastic process
ξ
s
(
, which in the absence of perturbation would vanish, in the presence of a stimulus
with the time dependence
t
)
ξ
p
(
ξ
s
(
)
t
)
generates a non-vanishing average value
t
.The
t
)
χ(
,
kernel
is the time derivative of the unperturbed autocorrelation function. This
prescription from statistical physics holds in general and it is always valid, provided
that the network is at or near equilibrium compatible with the existence of stationarity
for the autocorrelation function
t
s
(
t
)
.
7.2.4
Towards a new LRT
In this section we present a solution to the perturbed GME using an ansatz, that is, we
assume a form of the solution and then verify that it satisfies the GME (
7.64
). Note that
this is the strategy often used to solve differential equations; we assume a general form
for the solution with parameters that are eventually adjusted to satisfy certain specified
conditions. Consider an equation for the assumed response of the two-level GME to a
time-dependent perturbation
ξ
p
(
)
t
:
t
dt
R
t
)(
t
)ξ
p
(
t
).
(
t
)
=
ε
(
t
−
(7.74)
0
The time derivative of (
7.74
)isgivenby
t
t
)
d
(
t
)
d
(
t
−
dt
R
t
)
t
),
=
ε
R
(
t
)(
0
)ξ
p
(
t
)
+
ε
(
ξ
p
(
(7.75)
dt
dt
0
