Clas Blomberg - Physics of life-Elsevier Science (2007)

312 Part VII. Non-linearity
or contributes to a more favourable signal transmission. There is no place here to go further
into these models. For general accounts (see Dykman et al., 1995; Wiesenfeld and Moss,
1995). Variants, also without thresholds, have been discussed (see Bezrukov and Vodyanov,
1997), but there is no place here to go deeply into that. One point is that if signals below a
certain threshold are cut out, noise would help signals to cross the threshold and be transmitted.
A medical application of this, which has been proposed, is for hearing aids. There,
the threshold for hearing may for some reason becomes too high for appropriate hearing. It
may then be sufficient and helpful, just to add noise to pass the barrier.
As said in Chapter 23, an interesting starting application of this is to periodicities of
ice ages (see Nicolis, 1993). A typical example concerns laser beams in a circular apparatus
(McNavara, 1988). The beams can go in either direction, and can change due to fluctuating
disturbances. With a relatively small periodic force, not in itself sufficient to accomplish
changes of the wave directions, stochastic resonance can yield synchronised changes in the
beam directions.
It is suggested that some animals, in particular, water-borne ones which look for small
signals from possible predators or prey for their own food can make use of existing noise
everywhere in the surroundings to amplify signal effects (see e.g. Moss and Wiesenfeld,
1995; Dykman and McClintock, 1998).
The brain may use irregular signals for seeking previous memories (as computers use
random effects for similar purposes). There may in such cases be optimal effects, closely
related to stochastic resonance. Such a proposal has been proposed and studied in artificial
neural networks (see Liljenström and Wu, 1995).
This kind of model has been generalised in various directions and different types of
models have been proposed. Normally, they involve a threshold effect, but not necessarily
several steps and barrier passages. In another model, it is proposed that simple noise in the
form of irregular spikes can be modulated by a signal and then accomplish a response.
29C
Non-linear stochastic equations
We had previously Brownian motion as a archetype of stochastic motion, and let us now use
that approach also for more complex situations and more general forces. Then, situations are
not quite straightforward.
General methods are always possible as long as the basic equation is linear. This is also the
case for a harmonic oscillator, oscillating influenced by a force proportional to the deviation
from a certain stability point. With damping, the particle may go to the resting position with
lowest potential energy, but with the fluctuation force, there is always some activity and
oscillations.
However, there are no simple ways of treating this kind of equations for more general
forces. What always is possible to do is to make numerical calculations where the irregular
force is treated just as it is, a random variable. For each integration step, its influence is
represented by a suitable selected random number. For the pure Brownian motion, the
basis of this is clear as the influence at subsequent times is completely independent of
each other.
Before going further, we can give an outline of various generalisations of the basic equation
and some of their difficulties.