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and visual perception shown by Moss
et al
.[
36
]. Most of the theoretical analyses have
relied on the one-dimensional bistable double-well potential, or a reduced version given
by two-state models, both of which are now classified as dynamic SR. Another version
of SR, which is most commonly found in biological webs, involves the concurrence of
a threshold with a subthreshold signal and noise and is called threshold SR [
36
].
Visual perception is one area in which there have been a great many experiments done
to demonstrate the existence of a threshold. Moss
et al
.[
36
] point out that through the
addition of noise the SR mechanism has been found to be operative in the perception
of gratings, ambiguous figures and letters and can be used to improve the observer's
sensitivity to weak visual signals. The nonlinear auditory network of human hearing
also manifests threshold SR behavior. The absolute threshold for the detection and dis-
crimination of pure tones was shown by Zeng
et al
.[
49
] to be lowered in people with
normal hearing by the addition of noise. They also showed that the same effect could
be achieved for individuals with cholera or brain-stem implants by the addition of an
optimal amount of broad-band noise.
The greatest success of SR has been associated with modeling and simulation of webs
of neurons, where the effects of noise on sensory function match those of experiment.
Consequently, the indications are that the theory of threshold SR explains one of the
fundamental mechanisms by which neurons operate. This does not carry over to the
brain, however, since the SR mechanism cannot be related to a specific function. This
theme will be taken up again later.
3.3
More on classical diffusion
We discussed classical diffusion from the point of view of the trajectory of the Brownian
particle being buffeted by the lighter particles of the background fluid. We again take up
this fundamental physical process but now from a less restrictive vantage point and base
our discussion on two physical concepts: particle transport and the continuity of fluid
motion. The net transport of material across a unit surface is proportional to the gradient
of the material density in a direction perpendicular to the unit area. Particle transport is
modeled by the spatially isotropic form of Fick's law,
j
(
r
,
t
)
≡−
D
∇
n
(
r
,
t
),
(3.102)
where
j
is the particle flux (current) across the unit area,
D
is the diffusion coef-
ficient, which may be a function of position
r
and time
t
, and
n
(
r
,
t
)
is the particle
number density at the location
r
and time
t
. The particle current is defined in terms of
the particle velocity
v
(
r
,
t
)
(
r
,
t
)
by
(
,
)
=
(
,
)
(
,
).
j
r
t
n
r
t
v
r
t
(3.103)
In free space material is conserved so that during movement the diffusing particles are
neither created nor destroyed. This principle is expressed by the continuity equation
∂
n
(
r
,
t
)
+∇·
j
(
r
,
t
)
=
0
.
(3.104)
∂
t
