Stochastic resonance (SR) is a mathematical mechanism and behavior of nonlinear systems (that is, systems in which the change of the output is not proportional to the change of the input) where random (stochastic) fluctuations in the microstate of a system (that is, its specific configuration, including the precise positions and momenta of all its individual particles or components) cause deterministic (that is, non-random) changes in a macrostate (that is, a subset of the system's microstates).

This occurs when the nonlinear nature of the system amplifies certain (resonant) portions of the fluctuations, while not amplifying other portions of the noise. The nonlinear system, immersed in a certain level of stochastic background noise, becomes sensitive to external perturbations that would be too weak to influence it in the absence of such noise.

Originally proposed in the context of climate dynamics, over time it has become important in numerous fields that study a wide variety of systems, particularly in information theory and in neuroscience. quantum systems, and industrial processes. Stochastic resonance is also closely related to the concept of dithering in signal analysis, although how similar or how different the two concepts are depends on the particular definition considered. for a comprehensive overview of stochastic resonance.

History

The mechanism of stochastic resonance was first described in the early 1980s by the Italian physicists Roberto Benzi, Alfonso Sutera, and Angelo Vulpiani, who (with the additional participation of Giorgio Parisi) immediately applied it to climatology, At the same time, a very similar explanation was also proposed by the Belgian physicist Catherine Nicolis. and in 1988 in a laser system. In the early 1990s, the first works appeared in which it was hypothesized that stochastic resonance played an important role in neuronal dynamics, a concept now confirmed.

Climatological interpretation

In the original works of Benzi, Parisi, Sutera, and Vulpiani, the potential depending on the Earth's mean temperature (i.e., the variable corresponding to <math>x</math>) was linked to the albedo of the Earth, that is, the fraction of incoming solar radiation that is reflected back into space rather than absorbed by the planet. It depends on numerous factors closely related to the Earth's climate, the main ones being the extent of the ice sheets and cloud cover. In general it was assumed that the albedo tends to a maximum both for extremely low temperatures (since the planet would be completely covered by highly reflective ice) and for high temperatures (since high temperature is associated with high evaporation, and therefore with extensive cloud cover, which is also reflective), while the two stable states of minimum albedo were associated with glacial periods and interglacial periods.

Information theory

In information theory, SR can be used to reveal weak signals. When a signal that is normally too weak to be detected by a sensor can be boosted by adding white noise 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 with each other, amplifying the original signal while not amplifying the rest of the white noise – thereby increasing the signal-to-noise ratio, 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.

Technical description

Stochastic 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 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 contains a peak.

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 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).

Simplified model

thumb|Generic shape of the potential <math>U(x)</math>

Below is a didactic toy model capable of capturing the essential aspects of the stochastic resonance mechanism. Computationally, neurons exhibit SR because of non-linearities in their processing. SR has yet to be fully explained in biological systems, but neural synchrony or phase coherence in the brain (across a wide spectrum of frequencies including the gamma waveWinterer G, Ziller M, Dorn H, Frick K, Mulert C, Dahhan N, Herrmann WM, Coppola R. Cortical activation, signal-to-noise ratio and stochastic resonance during information processing in man. Clin Neurophysiol 1999 Jul;110(7):1193-203. doi: 10.1016/s1388-2457(99)00059-0. 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 and squid giant axon could also demonstrate the inverse stochastic resonance, when spiking is inhibited by synaptic noise 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.

Stochastic resonance has found noteworthy application in the field of image processing.

Signal analysis

A related phenomenon is dithering, applied to analog signals before analog-to-digital conversion.

See also

  • Mutual coherence (linear algebra)
  • Signal detection theory
  • 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.&nbsp;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.&nbsp;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.
  • [https://scholar.google.com.au/citations?hl=en&user=hug7t0oAAAAJ&view_op=list_works&pagesize=100](https://scholar.google.com.au/citations?hl=en&user=hug7t0oAAAAJ&view_op=list_works&pagesize=100) Scholar Google profile on stochastic resonance]
  • Newsweek Being messy, both at home and in foreign policy, may have its own advantages Retrieved 3 January 2011
  • [http://www.stochastic-resonance.org](http://www.stochastic-resonance.org) Stochastic Resonance Conference] 1998–2008 ten years of continuous growth. 17–21 Aug. 2008, Perugia (Italy)
  • [http://www.cambridge.org/9780521882620](http://www.cambridge.org/9780521882620) Stochastic Resonance - From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization (book)]
  • [http://www.scholarpedia.org/article/Suprathreshold_stochastic_resonance](http://www.scholarpedia.org/article/Suprathreshold_stochastic_resonance) 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](http://prl.aps.org/abstract/PRL/v102/i22/e226802)]