Stochastic resonance
Encyclopedia
Stochastic resonance is a phenomenon that occurs in a threshold measurement system (e.g. a man-made instrument or device; a natural cell, organ or organism) when an appropriate measure of information transfer (signal-to-noise ratio
, mutual information
, coherence
, d, etc.) is maximized in the presence of a non-zero level of stochastic
input noise
thereby lowering the response threshold; the system resonates at a particular noise level.
of a nonlinear system or device increases for moderate values of noise
intensity
. It often occurs in bistable systems or in systems with a sensory threshold
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 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
of signal-to-noise ratio as a function of noise intensity shows a '∩' shape.
Strictly speaking, stochastic resonance occurs in bistable systems, when a small periodic (sinusoidal
) 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 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 time scales: the period of the sinusoid (the deterministic time scale) and the Kramers rate (i.e., the inverse of the average switch rate induced by the sole noise: 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.
benefit in a nonlinear system. Unlike most of the nonlinear systems where stochastic resonance occurs, suprathreshold stochastic resonance occurs not only when the strength of the fluctuations is small relative to that of an input signal, but occurs even for the smallest amount of random noise. Furthermore, it is not restricted to a subthreshold signal, hence the qualifier, suprathreshold, in suprathreshold stochastic resonance.
frequency) has been suggested as a possible neural mechanism for SR by researchers who have investigated the perception of "subconscious" visual sensation.
) for enhancing sensory and motor function in the elderly, patients with diabetic neuropathy, and patients with stroke.
See the Review of Modern Physics article for a comprehensive overview of stochastic resonance.
ing applied to analog signals before analog-to-digital conversion
. Stochastic resonance can be used to measure transmittance amplitudes below an instrument's detection limit. If Gaussian
noise
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.
Signal-to-noise ratio
Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. It is defined as the ratio of signal power to the noise power. A ratio higher than 1:1 indicates more signal than noise...
, mutual information
Mutual information
In probability theory and information theory, the mutual information of two random variables is a quantity that measures the mutual dependence of the two random variables...
, coherence
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....
, d, etc.) is maximized in the presence of a non-zero level of stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
input noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
thereby lowering the response threshold; the system resonates at a particular noise level.
Definition
Stochastic resonance 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 ratioSignal-to-noise ratio
Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. It is defined as the ratio of signal power to the noise power. A ratio higher than 1:1 indicates more signal than noise...
of a nonlinear system or device increases for moderate values of noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
intensity
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...
. It often occurs in bistable systems or in systems with a sensory threshold
Sensory threshold
Sensory threshold is a theoretical concept used in psychophysics. A stimulus that is less intense than the sensory threshold will not elicit any sensation...
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 is passed through it. For large noise intensities, the output
Output
Output is the term denoting either an exit or changes which exit a system and which activate/modify a process. It is an abstract concept, used in the modeling, system design and system exploitation.-In control theory:...
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
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
of signal-to-noise ratio as a function of noise intensity shows a '∩' shape.
Strictly speaking, stochastic resonance occurs in bistable systems, when a small periodic (sinusoidal
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...
) 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 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 time scales: the period of the sinusoid (the deterministic time scale) and the Kramers rate (i.e., the inverse of the average switch rate induced by the sole noise: 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 stochastic resonance
Suprathreshold stochastic resonance is a particular form of stochastic resonance. It is the phenomenon where random fluctuations, or noise, provide a signal processingSignal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
benefit in a nonlinear system. Unlike most of the nonlinear systems where stochastic resonance occurs, suprathreshold stochastic resonance occurs not only when the strength of the fluctuations is small relative to that of an input signal, but occurs even for the smallest amount of random noise. Furthermore, it is not restricted to a subthreshold signal, hence the qualifier, suprathreshold, in suprathreshold stochastic resonance.
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 waveGamma wave
A gamma wave is a pattern of neural oscillation in humans with a frequency between 25 to 100 Hz, though 40 Hz is prototypical.According to a popular theory, gamma waves may be implicated in creating the unity of conscious perception...
frequency) has been suggested as a possible neural mechanism for SR by researchers who have investigated the perception of "subconscious" visual sensation.
Medicine
SR-based techniques has been used to create a novel class of medical devices (such as vibrating insolesJames Collins (Boston University)
James J. Collins is an American bioengineer, Professor of Biomedical Engineering at Boston University, and a Howard Hughes Medical Institute Investigator...
) for enhancing sensory and motor function in the elderly, patients with diabetic neuropathy, and patients with stroke.
See the Review of Modern Physics article for a comprehensive overview of stochastic resonance.
Signal analysis
A related phenomenon is ditherDither
Dither is an intentionally applied form of noise used to randomize quantization error, preventing large-scale patterns such as color banding in images...
ing applied to analog signals before analog-to-digital conversion
Analog-to-digital converter
An analog-to-digital converter is a device that converts a continuous quantity to a discrete time digital representation. An ADC may also provide an isolated measurement...
. Stochastic resonance can be used to measure transmittance amplitudes below an instrument's detection limit. If Gaussian
GAUSSIAN
Gaussian is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70. It has been continuously updated since then...
noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
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
- Neural oscillationsNeural oscillationsNeural oscillation is rhythmic or repetitive neural activity in the central nervous system. Neural tissue can generate oscillatory activity in many ways, driven either by mechanisms localized within individual neurons or by interactions between neurons...
- Stochastic resonance on Scholarpedia
- StochasticStochasticStochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
- Stochastic resonance (sensory neurobiology)Stochastic resonance (sensory neurobiology)Stochastic resonance is a phenomenon that occurs in a threshold measurement system when an appropriate measure of information transfer is maximized in the presence of a non-zero level of stochastic input noise thereby lowering the response threshold; the...
External links
- NewsweekNewsweekNewsweek is an American weekly news magazine published in New York City. It is distributed throughout the United States and internationally. It is the second-largest news weekly magazine in the U.S., having trailed Time in circulation and advertising revenue for most of its existence...
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, http://prl.aps.org/abstract/PRL/v102/i22/e226802