Stochastic resonance
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Stochastic resonance (SR) is a phenomenon in which a signal that is normally too weak to be detected by a sensor, can be boosted by adding
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
to the signal, which contains a wide spectrum of frequencies. The frequencies in the white noise corresponding to the original signal's frequencies will
resonate Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...
with each other, amplifying the original signal while not amplifying the rest of the white noise – thereby increasing the
signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in deci ...
, which makes the original signal more prominent. Further, the added white noise can be enough to be detectable by the sensor, which can then filter it out to effectively detect the original, previously undetectable signal. This phenomenon of boosting undetectable signals by resonating with added white noise extends to many other systems – whether electromagnetic, physical or biological – and is an active area of research. Stochastic resonance was first proposed by the Italian physicists Roberto Benzi, Alfonso Sutera and Angelo Vulpiani in 1981, and the first application they proposed (together with
Giorgio Parisi Giorgio Parisi (born 4 August 1948) is an Italian theoretical physicist, whose research has focused on quantum field theory, statistical mechanics and complex systems. His best known contributions are the QCD evolution equations for parton densit ...
) was in the context of climate dynamics.


Technical description

Stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
resonance (SR) is observed when noise added to a system changes the system's behaviour in some fashion. More technically, SR occurs if the
signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in deci ...
of a
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
system or device increases for moderate values of
noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arise ...
intensity Intensity may refer to: In colloquial use *Strength (disambiguation) *Amplitude * Level (disambiguation) * Magnitude (disambiguation) In physical sciences Physics *Intensity (physics), power per unit area (W/m2) *Field strength of electric, ma ...
. It often occurs in bistable systems or in systems with a
sensory threshold In psychophysics, sensory threshold is the weakest stimulus that an organism can sense. Unless otherwise indicated, it is usually defined as the weakest stimulus that can be detected half the time, for example, as indicated by a point on a probabi ...
and when the input signal to the system is "sub-threshold." For lower noise intensities, the signal does not cause the device to cross threshold, so little
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
is passed through it. For large noise intensities, the output is dominated by the noise, also leading to a low signal-to-noise ratio. For moderate intensities, the noise allows the signal to reach threshold, but the noise intensity is not so large as to swamp it. Thus, a
plot Plot or Plotting may refer to: Art, media and entertainment * Plot (narrative), the story of a piece of fiction Music * ''The Plot'' (album), a 1976 album by jazz trumpeter Enrico Rava * The Plot (band), a band formed in 2003 Other * ''Plot' ...
of signal-to-noise ratio as a function of noise intensity contains a peak. Strictly speaking, stochastic resonance occurs in bistable systems, when a small periodic (
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
) force is applied together with a large wide band stochastic force (noise). The system response is driven by the combination of the two forces that compete/cooperate to make the system switch between the two stable states. The degree of order is related to the amount of
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
that it shows in the system response. When the periodic force is chosen small enough in order to not make the system response switch, the presence of a non-negligible noise is required for it to happen. When the noise is small, very few switches occur, mainly at random with no significant periodicity in the system response. When the noise is very strong, a large number of switches occur for each period of the sinusoid, and the system response does not show remarkable periodicity. Between these two conditions, there exists an optimal value of the noise that cooperatively concurs with the periodic forcing in order to make almost exactly one switch per period (a maximum in the signal-to-noise ratio). Such a favorable condition is quantitatively determined by the matching of two timescales: the period of the sinusoid (the deterministic time scale) and the Kramers rate (i.e., the average switch rate induced by the sole noise: the inverse of the stochastic time scale). Thus the term "stochastic resonance." Stochastic resonance was discovered and proposed for the first time in 1981 to explain the periodic recurrence of ice ages. Since then, the same principle has been applied in a wide variety of systems. Nowadays stochastic resonance is commonly invoked when noise and nonlinearity concur to determine an increase of order in the system response.


Suprathreshold

Suprathreshold stochastic resonance is a particular form of stochastic resonance, in which
random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
fluctuations, or noise, provide a
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
benefit in a
nonlinear system In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
. Unlike most of the nonlinear systems in which stochastic resonance occurs, suprathreshold stochastic resonance occurs when the strength of the fluctuations is small relative to that of an input signal, or even small for
random noise In electronics, noise is an unwanted disturbance in an electrical signal. Noise generated by electronic devices varies greatly as it is produced by several different effects. In particular, noise is inherent in physics, and central to the ...
. It is not restricted to a subthreshold signal, hence the qualifier.


Neuroscience, psychology and biology

Stochastic resonance has been observed in the neural tissue of the sensory systems of several organisms. Computationally, neurons exhibit SR because of non-linearities in their processing. SR has yet to be fully explained in biological systems, but neural synchrony in the brain (specifically in the
gamma wave A gamma wave or gamma Rhythm is a pattern of neural oscillation in humans with a frequency between 25 and 140 Hz, the 40- Hz point being of particular interest. Gamma rhythms are correlated with large scale brain network activity and cognitive ...
frequency) has been suggested as a possible neural mechanism for SR by researchers who have investigated the perception of "subconscious" visual sensation. Single neurons ''in vitro'' including cerebellar
Purkinje cells Purkinje cells, or Purkinje neurons, are a class of GABAergic inhibitory neurons located in the cerebellum. They are named after their discoverer, Czech anatomist Jan Evangelista Purkyně, who characterized the cells in 1839. Structure The ...
and
squid giant axon The squid giant axon is the very large (up to 1.5 mm in diameter; typically around 0.5 mm) axon that controls part of the water jet propulsion system in squid. It was first described by L. W. Williams in 1909, but this discovery was for ...
could also demonstrate the inverse stochastic resonance, when spiking is inhibited by
synaptic noise Synaptic noise refers to the constant bombardment of synaptic activity in neurons. This occurs in the background of a cell when potentials are produced without the nerve stimulation of an action potential, and are due to the inherently random natur ...
of a particular variance.


Medicine

SR-based techniques have been used to create a novel class of medical devices for enhancing sensory and motor functions such as vibrating insoles especially for the elderly, or patients with diabetic neuropathy or stroke. See the ''Review of Modern Physics'' article for a comprehensive overview of stochastic resonance. Stochastic Resonance has found noteworthy application in the field of image processing.


Signal analysis

A related phenomenon is
dither Dither is an intentionally applied form of image noise, noise used to randomize quantization error, preventing large-scale patterns such as color banding in images. Dither is routinely used in processing of both digital audio and digital vide ...
ing applied to analog signals before
analog-to-digital conversion In electronics, an analog-to-digital converter (ADC, A/D, or A-to-D) is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provi ...
. Stochastic resonance can be used to measure transmittance amplitudes below an instrument's detection limit. If
Gaussian noise Gaussian noise, named after Carl Friedrich Gauss, is a term from signal processing theory denoting a kind of signal noise that has a probability density function (pdf) equal to that of the normal distribution (which is also known as the Gaussia ...
is added to a subthreshold (i.e., immeasurable) signal, then it can be brought into a detectable region. After detection, the noise is removed. A fourfold improvement in the detection limit can be obtained.


See also

*
Mutual coherence (linear algebra) In linear algebra, the coherence or mutual coherence of a matrix ''A'' is defined as the maximum absolute value of the cross-correlations between the columns of ''A''. Formally, let a_1, \ldots, a_m\in ^d be the columns of the matrix ''A'', whic ...
*
Signal detection theory Detection theory or signal detection theory is a means to measure the ability to differentiate between information-bearing patterns (called stimulus in living organisms, signal in machines) and random patterns that distract from the information (ca ...
* Stochastic resonance (sensory neurobiology)


References


Bibliography

* * * * * * * * * * * * Hannes Risken ''The Fokker-Planck Equation'', 2nd edition, Springer, 1989


Bibliography for suprathreshold stochastic resonance

* N. G. Stocks, "Suprathreshold stochastic resonance in multilevel threshold systems," ''Physical Review Letters,'' 84, pp. 2310–2313, 2000. * M. D. McDonnell, D. Abbott, and C. E. M. Pearce, "An analysis of noise enhanced information transmission in an array of comparators," ''Microelectronics Journal'' 33, pp. 1079–1089, 2002. * M. D. McDonnell and N. G. Stocks, "Suprathreshold stochastic resonance," ''Scholarpedia'' 4, Article No. 6508, 2009. * M. D. McDonnell, N. G. Stocks, C. E. M. Pearce, D. Abbott, '' Stochastic Resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization'', Cambridge University Press, 2008. *


External links

*
Scholar Google profile on stochastic resonance
* *
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Being messy, both at home and in foreign policy, may have its own advantages
' Retrieved 3 Jan 2011
Stochastic Resonance Conference
1998–2008 ten years of continuous growth. 17-21 Aug. 2008, Perugia (Italy)
Stochastic Resonance - From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization (book)Review of Suprathreshold Stochastic Resonance
*A.S. Samardak, A. Nogaret, N. B. Janson, A. G. Balanov, I. Farrer and D. A. Ritchie. "Noise-Controlled Signal Transmission in a Multithread Semiconductor Neuron" // Phys. Rev. Lett. 102 (2009) 226802

{{DEFAULTSORT:Stochastic Resonance Biophysics Stochastic processes Statistical mechanics Oscillation Signal processing Sensory systems