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Thus, we replace (
7.172
) with
t
0
ψ(
t
)ξ
p
(
t
)
dt
,
w
eff
(
t
)
=
t
,
(7.175)
with no need for the thermodynamic temperature. Consequently, the effective synaptic
strength is an average of the stimulus over an aged ensemble of realizations describing
the rich dynamics of the neuron web. We now examine the properties of the probabil-
ity density consistent with experiment and necessary to describe the phenomenon of
habituation.
7.4.1
Simple stimulus and renewal theory
In the present context we now have that the probability density for a complex web of
neurons is given by the observed inter-event-interval density [
63
] having the hyperbolic
form. This form of probability density for neuronal webs is consistent with the experi-
mental observation of 1/
f
noise in human cognition made from the errors in the recall
of intervals [
31
] and in the variability of implicit measures of bias [
22
], both of which
were described by a phenomenological theory of 1/
f
noise in human cognition [
33
].
The theory developed in the previous section can now be used to provide the effec-
tive synaptic weight of the response of the neuronal network to an external stimulus
in terms of an average over the aged event probability density. This is done for both a
single frequency stimulus and a sequence of Gaussian-shaped pulses (set as a problem).
To demonstrate the phenomenon of habituation using this expression, we assume a
periodic signal of frequency
t
)
=
ε
t
)
ω
with small amplitude
ε<
1
,ξ
p
(
cos
(ω
, yielding
t
dt
ψ(
t
)
t
).
w
eff
(
t
)
=
ε
t
,
cos
(ω
(7.176)
0
Thesolutionto(
7.176
) is obtained from the real part of the Laplace transform [
8
]:
Re
E
w
eff
(
u
)
=
ε
(
u
)
,
(7.177)
E
where
is the Laplace transform of the partially Fourier-transformed aged distri-
bution density
(
u
)
t
0
ψ(
e
−
i
ω
t
dt
.
t
)
E
(
t
)
≡
t
,
(7.178)
t
)
The exact expression for
allows us to make exact predictions for the behavioral
response when the first term on the rhs in the renewal process (
7.156
) is a known func-
tion
ψ(
t
,
ψ
1
(
t
)
=
ψ(
t
)
. Following Barbi
et al
.[
8
]aswellasWest
et al
.[
68
] and using
ψ (
t
)
to determine the aged waiting-time distribution density, we can write
ψ(
ω)
−
ψ(
i
ω
u
+
i
u
)
E
(
u
)
=
,
(7.179)
−
ψ(
1
u
+
i
ω)
