Information Technology Reference
In-Depth Information
but in order to proceed further we need an explicit expression for
.Weuse
the hyperbolic distribution for the probability density and reduce (
7.179
) to algebraic
manipulation.
The Laplace transform of the hyperbolic distribution is
ψ (
t
)
)
μ
−
1
e
uT
1
ψ(
E
uT
μ
−
u
)
=
(μ
−
1
)(
1
−
μ)(
uT
−
(7.180)
in terms of the generalized exponential
∞
z
n
−
γ
E
z
γ
=
−
γ)
,
(7.181)
(
+
n
1
0
which reduces to the ordinary exponential when
2
the Laplace transform of the network response reduces [
8
,
68
] in the long-time regime
where
u
γ
=
0 and
μ
=
1. In the case
μ<
→
0to
Re
i
ω
1
μ
−
1
u
u
w
eff
(
u
)
=
ε
−
.
(7.182)
+
i
ω
The inverse Laplace transform of (
7.182
) yields the behavioral response to a periodic
input signal as
t
→∞
[
8
,
68
]
cos
(ω
t
−
μπ/
2
)
w
eff
(
t
)
≈
ε
−
μ
.
(7.183)
2
(μ
−
1
)(ω
t
)
This expression clearly shows that the condition
2 suppresses the response to peri-
odic stimulation over time and thus the network habituates. Note that the response does
not distort the stimulus over time; it simply decreases the amplitude of the stimulus as
an inverse power-law in time and shifts its phase.
μ<
Connection with power spectra
We remind the reader that the Wiener-Khintchine theorem proves that the Fourier trans-
form of the autocorrelation function yields the power spectral density, but only under
the condition that the process is stationary. This theorem is generally assumed to be
valid, so that when a spectrum of the inverse power-law form given by
1
f
α
P
(
f
)
∝
(7.184)
with the spectral index within the interval 0
5 is obtained the process
is generally interpreted to have long-time correlations. However, in complex webs
the autocorrelation function can be non-stationary and non-ergodic, invalidating the
traditional Wiener-Khintchine theorem. It has recently been shown that cognitive fluc-
tuations are non-stationary, so the application of the Wiener-Khintchine theorem is
unwarranted, and yet the empirical power spectral density is found to have the 1/
f
noise form. We have addressed this paradox theoretically by replacing the continuous
model of the underlying process with discrete renewal events.
Now we can explain more clearly the connection between the condition of habitua-
tion and 1/
f
noise. In the last few years a new approach to 1/
f
noise has been proposed
.
5
<α<
1
.
