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in linear single-channel webs. Consequently, the time scales in these complex neuron
webs are hierarchally coupled in such a way as to promote their interdependence and
synchronization over long times [
68
]. An excitation at a given time scale generates a
cascade of perturbing multiple time scales through the multiple-channel coupling, and,
because the time scale of the stimulus is only one of many, the response at that time scale
is suppressed over time. However, whether the asymptotic state to which the stimulus is
habituated is zero or some constant value depends on the particular kind of 1/
f
scaling
of that region of the brain. Consequently, depending on whether we hear, taste, touch
or smell the stimulus determines the ergodic nature of the complex neuronal webs and
consequently the form of habituation.
The complexity of the time series measured herein in terms of the power-law index
has an interesting interpretation. As the index approaches infinity the statistics of the
process become Poisson and the arrival of new events is more or less evenly spaced in
time. However, in the region of psychological interest considered here, 1
3, the
critical events are intermittent, occurring in clusters and having clusters within clusters,
manifesting a fractal statistical structure in time. Consequently, there is no characteristic
time scale in the propagation of information in neuron webs. This needs to be compared
with the critical branching processes used by Beggs
et al
.[
11
,
53
] to explain the “neu-
ronal avalanches” measured in cortical neurons. The former effect is in time and the
latter is in space. Preliminary calculations extending the decision-making model [
64
]to
include spatial effects reveal localized “communities” with an inverse power-law dis-
tribution of sizes with slope
<μ<
5 obtained by Beggs
and Plenz [
11
]. These results suggest that the “neuronal avalanches,” a spatial fractal
effect, may be explained using either the critical branching processes [
11
] or the critical
events introduced herein [
33
,
64
]. However, determining the compatibility of the two
approaches is an area for future investigation and is outside the scope of this topic.
It should be pointed out that the habituation damping factor (
7.169
) is obtained using
a generalization of LRT [
4
,
7
,
8
], which is one of the most robust techniques of statistical
physics and follows from the fluctuation-dissipation theorem. Other researchers [
35
,
59
,
60
,
67
] have adopted an approach closely related to that of our research group [
4
,
7
,
8
]
and their approach is based on expressing the linear response function as
−
1, compared with the value
−
1
.
t
)
=
χ(
t
,
t
)/
dt
t
)
=−
t
)/
d
dt
. It is possible to show that either
choice generates habituation, although it seems that networks of interacting neurons
adopt the choice we made (
7.173
). The asymptotic convergence to zero of the response
to a periodic input signal has been interpreted by physicists [
35
,
59
,
60
,
67
] as the failure
of LRT and in fact has been called the “death of linear response.” It was expected that
because the stimulus remains constant in amplitude the response should do the same
rather than asymptotically vanish; when this did not occur the theory was thought to
be “dead.” LRT is as applicable to neural phenomena as it is to complex physical webs
when suitably generalized and the so-called death of linear response is a consequence
of the web's complexity.
In the critical-branching theory [
11
] information transfer in the web is determined by
the magnitude of the branching parameter, independently of the properties of the stim-
ulus. By contrast, in SHM the relative complexity of the web dynamics and excitation
(
t
,
rather than
χ(
t
,
d
(
t
,
