Information Technology Reference
In-Depth Information
[
41
,
45
] on the basis of the assumption that 1/
f
noise is produced by the renewal events.
Historically the connection could not have been made because 1/
f
noise required the
autocorrelation function to be stationary. The stationary condition is no longer neces-
sary when the statistics are of renewal type. According to this theory of non-stationary
processes the spectral index
α
μ
of (
7.184
) is related to the inverse power-law index
of
the hyperbolic distribution by
α
=
3
−
μ.
(7.185)
We see, therefore, that the ideal condition of 1/
f
noise,
2.
On the other hand, as we mentioned earlier, the hyperbolic form of the event probability
density has a diverging mean time
α
=
1, corresponds to
μ
=
t
for
μ
≤
2, thereby generating ergodicity break-
down [
68
] with
1. Thus, the ideal condition of 1/
f
noise is the boundary between
the ergodic and non-ergodic regimes.
Thus, there is always habituation of simple stimuli. In the four experiments con-
ducted by Kello
et al
.[
36
] the spectral index was determined to be in the range
0
α
≥
.
53
≤
α
≤
0
.
66, indicating from (
7.185
) that the probability index is in the interval
2
Consequently, these experiments determine that the intrinsic fluctua-
tions of cognitive function are ergodic, assuming that the underlying process is renewal.
Note also that this range of
.
47
≥
μ
≥
2
.
34
.
values is consistent with the Gerstein-Mandelbrot finding
of 1.5 for the power-law index of the survival probability for single neurons. However,
these values do not exhaust the full range of inverse power-law indices.
μ
7.4.2
Information resonance and dishabituation
w
eff
changes in time as a consequence of
the stimulus. The abrupt switching on of the stimulus at
t
It is important to point out that the variable
=
0 triggers the occurrence
of an event, which, in turn, activates the
R
process, with the resulting habituation. It
has recently been shown [
33
] that a web of interacting neurons generates a sequence of
renewal events corresponding to an age-dependent rate
g
(
t
)
(
t
)
that is a decreasing function
of time and can yield a virtually infinite mean time
. The phenomenon of habituation
to a periodic perturbation depends on the fact that the abrupt switching on of the per-
turbation triggers the occurrence of an event. The time of occurrence of the ensuing
events is only slightly influenced by the external stimulus and weakly departs from the
condition of natural occurrence. There are preliminary indications [
33
] that, in the spe-
cial case in which the stimulus contains renewal events with the same complexity as the
network, habituation is overcome and dishabituation can occur.
Aquino
et al
.[
6
] consider two complex networks denoted by S and P, with hyper-
bolic distributions having indices
t
μ
P
, respectively, that are interacting with one
another. The indices for both webs are in the interval 1
μ
S
and
3 and it is considered
that web P perturbs web S. They determined that there is a singularity at the condition
μ
S
=
μ
P
=
<μ<
2, see the center of Figure
7.10
, where the value of the asymptotic cross-
correlation function jumps from zero to one. If the perturbing web P is non-ergodic,
meaning that 1
<μ
P
<
2, and the web being perturbed, S, is ergodic, meaning that
2
<μ
S
<
3
,
then asymptotically the signals generated by the two webs are completely
