# Stochastic resonance

Stochastic resonance

Stochastic resonance (also known as SR) is observed when noise added to a system improves the systems performance in some fashion. More technically, SR occurs if the signal-to-noise ratio 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. Quite surprisingly, 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. [Benzi R, Parisi G, Sutera A, and Vulpiani A, "Stochastic resonance in climatic change." Tellus 34:10-16. 1982.] 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.

Neuroscience & Biology

Stochastic resonance has been observed in the neural tissue of the sensory systems of several organisms [Kosko B, "Noise". 2006. New York:Viking.] Computationally, neurons exhibit SR because of non-linear in their processing. SR has yet to be fully explained in biological systems, but neural synchrony in the brain (specifically in the Gamma wave frequency [ Ward LM, Doesburg SM, Kitajo K, MacLean SE, Roggeveen AB. [http://www.ncbi.nlm.nih.gov/sites/entrez?Db=pubmed&Cmd=ShowDetailView&TermToSearch=17285879&ordinalpos=1&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVDocSum] "Neural synchrony in stochastic resonance, attention, and consciousness." Can J Exp Psychol. 2006 Dec;60(4):319-26.] ) have been suggested as an possible neural mechanism for SR by researchers who have investigated the perception of "subconscious" visual sensation. [ Lucia Melloni, Carlos Molina, Marcela Pena, David Torres, Wolf Singer and Eugenio Rodriguez [http://www.jneurosci.org/cgi/pmidlookup?view=long&pmid=17360907 Final proof of role of neural coherence in consciousness?] The Journal of Neuroscience, March 14, 2007, 27(11):2858-2865; doi:10.1523/JNEUROSCI.4623-06.2007.]

Medicine

SR-based techniques has been used to create a novel class of medical devices (such as vibrating insoles) for enhancing sensory and motor function in the elderly, patients with diabetic neuropathy, and patients with stroke.

See the Review of Modern Physics [ [http://www.physik.uni-augsburg.de/theo1/hanggi/Papers/195.pdf L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Review of Modern Physics] , Vol. 70 (1), p.223-287 (1998)] article below for a comprehensive overview of stochastic resonance.

ignal Analysis

A related phenomenon is dithering applied to analog signals before analog-to-digital conversion. [ [http://nipslab.fisica.unipg.it/files/PRE1995-SR-and-dithering-p4691_1.pdf] L. Gammaitoni, Stochastic resonance and the dithering effect in threshold physical systems Phys. Rev. E 52, 4691 - 4698 (1995)] . 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. [ [http://www.opticsinfobase.org/abstract.cfm?id=36963] A. Palonpon, J. Amistoso, J. Holdsworth, W. Garcia, and C. Saloma, Measurement of weak transmittances by stochastic resonance Optics Letters 23, Issue 18, 1480-1482 (1998)]

ee also

*"Stochastic Resonance" (book)
*Neural oscillations
*(Link: Stochastic resonance on Scholarpedia) http://www.scholarpedia.org/article/Stochastic_resonance
*Stochastic

References

* Hänggi P., " Stochastic resonance in biology: how noise can enhance detection of weak signals and help improve biological information processing." ChemPhysChem 2002; 3: 285–290. [http://www.physik.uni-augsburg.de/theo1/hanggi/Papers/282.pdf ]
* Moss F, Ward LM, Sannita WG. Stochastic resonance and sensory information processing: a tutorial and review of application. Clin Neurophys 2004;27:677– 682.
* Wiesenfeld K, Moss F. Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs. Nature 1995;373:33–36.
* A. Bulsara, L. Gammaitoni, Tuning in to noise, Physics Today, 03/1996, Volume 49, Issue 3, p.39-45, (1996) [ [http://nipslab.fisica.unipg.it/files/vol49no3p39-45.pdf] A. Bulsara, L. Gammaitoni, Tuning in to noise, Physics Today, 03/1996, Volume 49, Issue 3, p.39-45, (1996)]
*Priplata A. et al. [http://www.bu.edu/abl/files/fulltext.pdf Noise-Enhanced Balance Control in Patients with Diabetes and Patients with Stroke.] Ann Neurol 2006;59:4–12. PMID 16287079.

;Articles:
* IEEE [http://www.spectrum.ieee.org/apr05/1294 Spectrum] , April 2005
* Newsweek [http://www.freedman.com/newsweekmess.html] "Being messy, both at home and in foreign policy, may have its own advantages";Conferences:
* [http://www.stochastic-resonance.org Stochastic Resonance Conference] 1998-2008 ten years of continuous growth. 17-21 Aug. 2008, Perugia (Italy)

References

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