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climate changes. The theory developed to support this conjecture turned out to be very
successful and as of August 2009 the original papers have over a thousand citations.
However, the popularity of stochastic resonance is due not to its intended explanation
of climate change, but instead to its neurophysiologic applications [
66
]. The pioneer-
ingworkofNeiman
et al
.[
48
] established that an external noise can enhance neuron
synchronization and consequently improve the transmission of information when that
information is in the form of a harmonic signal [
52
].
Stochastic resonance has to do with the propagation of a signal in a stochastic envi-
ronment with the surprising result that the signal-to-noise ratio increases with increasing
noise intensity, rather than decreasing as one intuitively expects. In this mechanism the
fluctuations act to facilitate the signal rather than inhibit it under certain conditions. To
establish the essence of this phenomenon, let us consider the equation of motion
dQ
dt
=−
∂
Q
(
Q
)
+
f
(
t
)
+
k
ξ
p
(
t
).
(7.81)
∂
The variable
Q
is the coordinate of a particle (the signal of interest), moving under the
overdamped condition within a double-well potential
(
t
)
. Recall that this equation
can be derived from Newton's force law in the Smoluchosky approximation discussed
in Section 3.2.3. The particle is also under the influence of white noise
f
(
Q
)
(
)
t
and an
ξ
p
(
)
external stimulus
. In the original formulation and most subsequent discussions,
the external stimulus is assumed to have the periodic form
t
ξ
p
(
t
)
=
cos
(ω
t
).
(7.82)
The potential is assigned the double-well form
Q
0
a
4
(
Q
2
a
2
2
(
Q
)
=
−
)
.
(7.83)
To move from
Q
0 the particle must overcome a barrier of intensity
Q
0
separating the two wells. At the bottom of the two wells,
Q
>
0to
Q
<
=±
a
, the potential
U
(
Q
)
vanishes. Owing to the overdamped condition, the function
(
Q
)
and the white noise
f
(
t
)
are the true potential
U
(
Q
)
and the true stochastic force
F
(
t
)
, divided by the fric-
tion
, respectively. The parameter
is the friction coefficient of the phenomenological
equation
d
dt
.
It is straightforward to prove that the time-dependent potential
v(
t
)/
dt
=−
v(
t
)
, with
v
=
dQ
/
U
(
Q
)
(
Q
,
t
)
=
−
kQ
cos
(ω
t
)
(7.84)
generates two wells with different depths, and that their depths decrease or increase with
the unperturbed displacement
Q
0
by the quantity
Q
≈∓
ak
cos
(ω
t
).
(7.85)
Using Kramers' theory [
37
] the transition rates between the wells in the GME can be
written
R
exp
Q
0
D
(
g
±
(
t
)
=
−
1
∓
ka
cos
(ω
t
))
.
(7.86)
