Figure 1.14. The input-output characteristic (hysteresis loop) of
the Schmitt trigger.
and period T 0, for example, to V+, we periodically modulate the critical level.
After adding a random noise at the input, the system is able to jump between the
two states ±V . As in the example of glacial cycles we can consider a discontinuous
modulation, for instance given by V+(t) = a sign(sin (
)). The whole picture is
now similar to the one in (1.3). Here the periodic modulation of the reference
voltage corresponds to the tilting of the potential wells.
Fauve and Heslot [38] studied the power spectrum of the system and, as in the
glacial cycle example, established that the energy carried by the spectral component
of Y at a given driving frequency has a local maximum for a certain intensity of
the input noise.
The Schmitt trigger provides another interpretation to the phenomenon of sto-
chastic resonance. A system displaying stochastic resonance can be considered as
a random amplifier. The weak periodic signal which cannot be detected in the
absence of noise, can be successfully recovered if the system (the Schmitt trigger
or (1.3)) is appropriately tuned. In other words, the weak underlying periodicity is
exhibited at appropriately chosen non-zero levels of noise, and gets lost if noise is
either too small or too large.
To date, the most important application area of threshold models is neural
dynamics (see Bulsara et al. [17], Douglass et al. [29], Patel and Kosko [85]) and
transmission of information (see Neiman et al. [81], Simonotto et al. [99], Stocks
[101], Moss et al. [79]). The recent book [73] by McDonnell et al. gives a very
complete account on the theory of non-dynamic or threshold stochastic resonance.
1.5.3. The paddlefish. In this well known and frequently discussed example
stochastic resonance appears in the noise-enhanced feeding behavior of the pad-
dlefish Polyodon spathula (see Greenwood et al. [47], Russel et al. [95], Freund et
al. [42]). This species of fish lives in the Midwest of the United States and in the
Yangtze River in China, and feeds on the zooplankton Daphnia. To detect its prey
animals under limited visibility conditions at river bottoms, the paddlefish uses the
long rostrum in front of its mouth as an electrosensory antenna. The frequency
range of sensitivity of the rostrum’s electroreceptors well overlaps with the range
of frequencies produced by the prey. Roughly, the capture probability is observed
as a function of the position of the prey relative to the rostrum. In experiments,
external noise was generated by electrodes connected to an electric noise genera-
tor. It was observed that the spatial distribution and number of strike locations
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