Stochastic resonance (sensory neurobiology)

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https://en.wikipedia.org/wiki/Stochastic_resonance_(sensory_neurobiology)

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.

The three criteria that must be met for stochastic resonance to occur are:

1. Nonlinear device or system: the input-output relationship must be nonlinear
2. Weak, periodic signal of interest: the input signal must be below threshold of measurement device and recur periodically
3. Added input noise: there must be random, uncorrelated variation added to signal of interest

Stochastic resonance occurs when these conditions combine in such a way that a certain average noise intensity results in maximized information transfer. A time-averaged (or, equivalently, low-pass filtered) output due to signal of interest plus noise will yield an even better measurement of the signal compared to the system's response without noise in terms of SNR.

The idea of adding noise to a system in order to improve the quality of measurements is counter-intuitive. Measurement systems are usually constructed or evolved to reduce noise as much as possible and thereby provide the most precise measurement of the signal of interest. Numerous experiments have demonstrated that, in both biological and non-biological systems, the addition of noise can actually improve the probability of detecting the signal; this is stochastic resonance. The systems in which stochastic resonance occur are always nonlinear systems. The addition of noise to a linear system will always decrease the information transfer rate.

History

Stochastic resonance was first discovered in a study of the periodic recurrence of Earth's ice ages. The theory developed out of an effort to understand how the Earth's climate oscillates periodically between two relatively stable global temperature states, one "normal" and the other an "ice age" state. The conventional explanation was that variations in the eccentricity of Earth's orbital path occurred with a period of about 100,000 years and caused the average temperature to shift dramatically. The measured variation in the eccentricity had a relatively small amplitude compared to the dramatic temperature change, however, and stochastic resonance was developed to show that the temperature change due to the weak eccentricity oscillation and added stochastic variation due to the unpredictable energy output of the Sun (known as the solar constant) could cause the temperature to move in a nonlinear fashion between two stable dynamic states.

Images

The above 256-grayscale-level images of the Arc result when the original is modified by the addition of noise and carrying out a nonlinear threshold operation; each panel shows a different level of noise variance, with a standard deviation of 10 grayscale levels in the top left, 50 levels in the top right, 100 in the bottom left and 150 in the bottom right. Different panels allow the best detection of different features; for example, the designs on the pillars are best seen in the top right, while the full outline of the arc is best seen in the bottom left. The appearance of features also changes with the size of the image as a result of averaging of the image; this can be observed by viewing the image at different distances.

As an example of stochastic resonance, consider the following demonstration after Simonotto et al.

The Arc de Triomphe

The image to the left shows an original picture of the Arc de Triomphe in Paris. If this image is passed through a nonlinear threshold filter in which each pixel detects light intensity as above or below a given threshold, a representation of the image is obtained as in the images to the right. It can be hard to discern the objects in the filtered image in the top left because of the reduced amount of information present. The addition of noise before the threshold operation can result in a more recognizable output. The image below shows four versions of the image after the threshold operation with different levels of noise variance; the image in the top right hand corner appears to have the optimal level of noise allowing the Arc to be recognized, but other noise variances reveal different features.

The quality of the image resulting from stochastic resonance can be improved further by blurring, or subjecting the image to low-pass spatial filtering. This can be approximated in the visual system by squinting one's eyes or moving away from the image. This allows the observer's visual system to average the pixel intensities over areas, which is in effect a low-pass filter. The resonance breaks up the harmonic distortion due to the threshold operation by spreading the distortion across the spectrum, and the low-pass filter eliminates much of the noise that has been pushed into higher spatial frequencies.

A similar output could be achieved by examining multiple threshold levels, so in a sense the addition of noise creates a new effective threshold for the measurement device.

Animal physiology

Cuticular mechanoreceptors in crayfish

Evidence for stochastic resonance in a sensory system was first found in nerve signals from the mechanoreceptors located on the tail fan of the crayfish (Procambarus clarkii). An appendage from the tail fan was mechanically stimulated to trigger the cuticular hairs that the crayfish uses to detect pressure waves in water. The stimulus consisted of sinusoidal motion at 55.2 Hz with random Gaussian noise at varying levels of average intensity. Spikes along the nerve root of the terminal abdominal ganglion were recorded extracellularly for 11 cells and analyzed to determine the SNR.

Two separate measurements were used to estimate the signal-to-noise ratio of the neural response. The first was based on the Fourier power spectrum of the spike time series response. The power spectra from the averaged spike data for three different noise intensities all showed a clear peak at the 55.2 Hz component with different average levels of broadband noise. The relatively low- and mid-level added noise conditions also show a second harmonic component at about 110 Hz. The mid-level noise condition clearly shows a stronger component at the signal of interest than either low- or high-level noise, and the harmonic component is greatly reduced at mid-level noise and not present in the high-level noise. A standard measure of the SNR as a function of noise variance shows a clear peak at the mid-level noise condition. The other measure used for SNR was based on the inter-spike interval histogram instead of the power spectrum. A similar peak was found on a plot of SNR as a function of noise variance for mid-level noise, although it was slightly different from that found using the power spectrum measurement.

These data support the claim that noise can enhance detection at the single neuron level but are not enough to establish that noise helps the crayfish detect weak signals in a natural setting. Experiments performed after this at a slightly higher level of analysis establish behavioral effects of stochastic resonance in other organisms; these are described below.

Cercal mechanoreceptors in crickets

A similar experiment was performed on the cricket (Acheta domestica), an arthropod like the crayfish. The cercal system in the cricket senses the displacement of particles due to air currents utilizing filiform hairs covering the cerci, the two antenna-like appendages extending from the posterior section of the abdomen. Sensory interneurons in terminal abdominal ganglion carry information about intensity and direction of pressure perturbations. Crickets were presented with signal plus noise stimuli and the spikes from cercal interneurons due to this input were recorded.

Two types of measurements of stochastic resonance were conducted. The first, like the crayfish experiment, consisted of a pure tone pressure signal at 23 Hz in a broadband noise background of varying intensities. A power spectrum analysis of the signals yielded maximum SNR for a noise intensity equal to 25 times the signal stimulus resulting in a maximum increase of 600% in SNR. 14 cells in 12 animals were tested, and all showed an increased SNR at a particular level of noise, meeting the requirements for the occurrence of stochastic resonance.

The other measurement consisted of the rate of mutual information transfer between the nerve signal and a broadband stimulus combined with varying levels of broadband noise uncorrelated with the signal. The power spectrum SNR could not be calculated in the same manner as before because there were signal and noise components present at the same frequencies. Mutual information measures the degree to which one signal predicts another; independent signals carry no mutual information, while perfectly identical signals carry maximal mutual information. For varying low amplitudes of signal, stochastic resonance peaks were found in plots of mutual information transfer rate as a function of input noise with a maximum increase in information transfer rate of 150%. For stronger signal amplitudes that stimulated the interneurons in the presence of no noise, however, the addition of noise always decreased the mutual information transfer demonstrating that stochastic resonance only works in the presence of low-intensity signals. The information carried in each spike at different levels of input noise was also calculated. At the optimum level of noise, the cells were more likely to spike, resulting in spikes with more information and more precise temporal coherence with the stimulus.

Stochastic resonance is a possible cause of escape behavior in crickets to attacks from predators that cause pressure waves in the tested frequency range at very low amplitudes, like the wasp Liris niger. Similar effects have also been noted in cockroaches.^[6]

Cutaneous mechanoreceptors in rats

Another investigation of stochastic resonance in broadband (or, equivalently, aperiodic) signals was conducted by probing cutaneous mechanoreceptors in the rat.^[7] A patch of skin from the thigh and its corresponding section of the saphenous nerve were removed, mounted on a test stand immersed in interstitial fluid. Slowly adapting type 1 (SA1) mechanoreceptors output signals in response to mechanical vibrations below 500 Hz.

The skin was mechanically stimulated with a broadband pressure signal with varying amounts of broadband noise using the up-and-down motion of a cylindrical probe. The intensity of the pressure signal was tested without noise and then set at a near sub-threshold intensity that would evoke 10 action potentials over a 60-second stimulation time. Several trials were then conducted with noise of increasing amplitude variance. Extracellular recordings were made of the mechanoreceptor response from the extracted nerve.

The encoding of the pressure stimulus in the neural signal was measured by the coherence of the stimulus and response. The coherence was found to be maximized by a particular level of input Gaussian noise, consistent with the occurrence of stochastic resonance.

Electroreceptors in paddlefish

The paddlefish (Polyodon spathula) hunts plankton using thousands of tiny passive electroreceptors located on its extended snout, or rostrum. The paddlefish is able to detect electric fields that oscillate at 0.5–20 Hz, and large groups of plankton generate this type of signal.

Due to the small magnitude of the generated fields, plankton are usually caught by the paddlefish when they are within 40 mm of the fish's rostrum. An experiment was performed to test the hunting ability of the paddlefish in environments with different levels of background noise.^[8] It was found that the paddlefish had a wider distance range of successful strikes in an electrical background with a low level of noise than in the absence of noise. In other words, there was a peak noise level, implying effects of stochastic resonance.

In the absence of noise, the distribution of successful strikes has greater variance in the horizontal direction than in the vertical direction. With the optimal level of noise, the variance in the vertical direction increased relative to the horizontal direction and also shifted to a peak slightly below center, although the horizontal variance did not increase.

Another measure of the increase in accuracy due to the optimal noise background is the number of plankton captured per unit time. For four paddlefish tested, two showed no increase in capture rate, while the other two showed a 50% increase in capture rate.

Separate observations of the paddlefish hunting in the wild have provided evidence that the background noise generated by plankton increase the paddlefish's hunting abilities. Each individual organism generates a particular electrical signal; these individual signals cause massed groups of plankton to emit what amounts to a noisy background signal. It has been found that the paddlefish does not respond to only noise without signals from nearby individual organisms, so it uses the strong individual signals of nearby plankton to acquire specific targets, and the background electrical noise provides a cue to their presence. For these reasons, it is likely that the paddlefish takes advantage of stochastic resonance to improve its sensitivity to prey.

Individual model neurons

Stochastic resonance was demonstrated in a high-level mathematical model of a single neuron using a dynamical systems approach.^[9] The model neuron was composed of a bi-stable potential energy function treated as a dynamical system that was set up to fire spikes in response to a pure tonal input with broadband noise and the SNR is calculated from the power spectrum of the potential energy function, which loosely corresponds to an actual neuron's spike-rate output. The characteristic peak on a plot of the SNR as a function of noise variance was apparent, demonstrating the occurrence of stochastic resonance.

Inverse stochastic resonance

Another phenomenon closely related to stochastic resonance is inverse stochastic resonance. It happens in the bistable dynamical systems having the limit cycle and stable fixed point solutions. In this case the noise of particular variance could efficiently inhibit spiking activity by moving the trajectory to the stable fixed point. It has been initially found in single neuron models, including classical Hodgkin-Huxley system.^[10][11] Later inverse stochastic resonance has been confirmed in Purkinje cells of cerebellum,^[12] where it could play the role for generation of pauses of spiking activity in vivo.

Multi-unit systems of model neurons

An aspect of stochastic resonance that is not entirely understood has to do with the relative magnitude of stimuli and the threshold for triggering the sensory neurons that measure them. If the stimuli are generally of a certain magnitude, it seems that it would be more evolutionarily advantageous for the threshold of the neuron to match that of the stimuli. In systems with noise, however, tuning thresholds for taking advantage of stochastic resonance may be the best strategy.

A theoretical account of how a large model network (up to 1000) of summed FitzHugh–Nagumo neurons could adjust the threshold of the system based on the noise level present in the environment was devised.^[13][14] This can be equivalently conceived of as the system lowering its threshold, and this is accomplished such that the ability to detect suprathreshold signals is not degraded.

Stochastic resonance in large-scale physiological systems of neurons (above the single-neuron level but below the behavioral level) has been investigated experimentally by Winterer und colleagues: In an electrophysiological study (EEG) of healthy subjects they found that a somewhat increased noise levbel in brain acitivity improves information processing as measured by reaction time (Winterer et al 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). In subsequent studies, Winterer et al also showed that very high levels of noise as measured by EEG and functional MRI are found in schizophrenia patients who usually present severe thought disorders. This work constituted the clinical-experimental basis for a computational dynamic attractor model of schizophrenia (Edmund T Rolls, Marco Loh, Gustavo Deco, Georg Winterer. Computational models of schizophrenia and dopamine modulation in the prefrontal cortex.

Human perception

Psychophysical experiments testing the thresholds of sensory systems have also been performed in humans across sensory modalities and have yielded evidence that our systems make use of stochastic resonance as well.

Vision

The above demonstration using the Arc de Triomphe photo is a simplified version of an earlier experiment. A photo of a clocktower was made into a video by adding noise with a particular variance a number of times to create successive frames. This was done for different levels of noise variance, and a particularly optimal level was found for discerning the appearance of the clocktower. Similar experiments also demonstrated an increased level of contrast sensitivity to sine wave gratings.

Tactility

Human subjects who undergo mechanical stimulation of a fingertip are able to detect a subthreshold impulse signal in the presence of a noisy mechanical vibration. The percentage of correct detections of the presence of the signal was maximized for a particular value of noise.

Audition

The auditory intensity detection thresholds of a number of human subjects were tested in the presence of noise. The subjects include four people with normal hearing, two with cochlear implants and one with an auditory brainstem implant.

The normal subjects were presented with two sound samples, one with a pure tone plus white noise and one with just white noise, and asked which one contained the pure tone. The level of noise which optimized the detection threshold in all four subjects was found to be between -15 and -20 dB relative to the pure tone, showing evidence for stochastic resonance in normal human hearing.

A similar test in the subjects with cochlear implants only found improved detection thresholds for pure tones below 300 Hz, while improvements were found at frequencies greater than 60 Hz in the brainstem implant subject. The reason for the limited range of resonance effects are unknown. Additionally, the addition of noise to cochlear implant signals improved the threshold for frequency discrimination. The authors recommend that some type of white noise addition to cochlear implant signals could well improve the utility of such devices.

Stochastic resonance

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Stochastic_resonance 

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. Phenomena attributable to stochastic resonance have also been observed in other types of physical systems, such as chemical reactions, 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.

Stochastic resonance was discovered and proposed for the first time in 1981 to explain the periodic recurrence of ice ages. Nowadays stochastic resonance is commonly invoked when noise and nonlinearity concur to determine an increase of order in a system's response.

See the Review of Modern Physics article "Stochastic resonance" 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, in order to explain how small variations in the motion of the Earth (the so-called Milankovitch cycles) can cause large variations in the Earth's climate (in particular, the transition from glacial periods to interglacial periods, and vice versa). According to Parisi's account, the name "stochastic resonance" was coined by Benzi during a conference. At the same time, a very similar explanation was also proposed by the Belgian physicist Catherine Nicolis.

An initial experimental verification was found as early as 1983 in a bistable electronic circuit, 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 ${\displaystyle x}$) 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

Generic shape of the potential ${\displaystyle U(x)}$

Below is a didactic toy model capable of capturing the essential aspects of the stochastic resonance mechanism.

Starting equation

Consider a time-dependent physical variable ${\displaystyle x(t)}$ (which, in the original climate papers, was the mean temperature of the Earth), described by a stochastic differential equation of the Langevin equation type:

$${\displaystyle \frac{dx}{dt}=-\frac{\mathrm{\partial }U}{\mathrm{\partial }x}+f(t)+ϵ\cos (\omega t),}$$

where ${\displaystyle U(x)}$ is a potential function determining the system dynamics, ${\displaystyle f(t)}$ is a stochastic forcing corresponding to background noise, and ${\displaystyle ϵ\cos (\omega t)}$ is the external periodic perturbation, of small amplitude (${\displaystyle ϵ\ll 1}$), which in the climate studies corresponded to the Milankovitch cycles. It is assumed that ${\displaystyle U(x)}$ has a double-well shape (describable by a fourth-degree polynomial), i.e., it possesses two stable minima ${\displaystyle x_{-}}$ and ${\displaystyle x_{+}}$ (glacial and interglacial periods) separated by an unstable maximum ${\displaystyle x_0}$, corresponding to a potential barrier ${\displaystyle \mathrm{\Delta }U}$ (for simplicity it is assumed that the two minima lie at the same level ${\displaystyle U(x_{\pm })}$). For ${\displaystyle f(t)}$ the simplest possible form is considered, namely that of Gaussian noise, with zero mean and Dirac delta correlation: ${\displaystyle ⟨f(t)⟩=0}$ and ${\displaystyle ⟨f(t_1)f(t_2)⟩=2D\delta (t_2-t_1)}$.

In the absence of periodic perturbations (${\displaystyle ϵ=0}$), it is well known from the theory of Markov processes that the variable ${\displaystyle x(t)}$ will fluctuate around the minimum points, with a variance proportional to the Gaussian noise intensity ${\displaystyle D}$. Occasionally the fluctuations will be strong enough to allow crossing of the potential barrier, and therefore the transition from one minimum state to the other. In the limit ${\displaystyle D\ll \mathrm{\Delta }U}$, the mean transition frequency is given by the so-called Kramers formula:

$${\displaystyle r_{\pm }=\frac{1}{2\pi }\sqrt{-U^{{}^{″}}(x_0)U^{{}^{″}}(x_{\pm })}{e}^{-\frac{\mathrm{\Delta }U}{D}},}$$ where ${\displaystyle U^{{}^{″}}(x)}$ is the second derivative of ${\displaystyle U(x)}$.

Stochastic resonance

In the presence of periodic perturbations, the potential ${\displaystyle U}$ is replaced by the generalized potential ${\displaystyle W(x,t)=U(x)-ϵ\cos \omega t}$. This means that the two potential wells vary in height, rising and falling, and therefore the barrier that the system must overcome to jump from one minimum to the other will at some moments be higher and at others lower. However, the oscillations are of small amplitude, so they are not able to completely remove the barrier and thus allow the system to transition between states on their own: the idea of stochastic resonance is that, if the perturbation frequency ${\displaystyle \omega }$ is in some way comparable with the mean transition frequency ${\displaystyle r_{\pm }}$, then the random fluctuations of the system will tend to synchronize with the external oscillations, making the transition more likely.

Using the explicit form of the potential ${\displaystyle U(x)=-\frac{1}{2}{x}^2+\frac{1}{4}{x}^4}$, it can be shown that, at least to first order approximation, the mean value of ${\displaystyle x(t)}$ becomes a periodic function of time with the same frequency as the external forcing:

$${\displaystyle ⟨x(t)⟩=A\cos (\omega t+\varphi ),}$$

where the amplitude ${\displaystyle A}$ and the phase ${\displaystyle \varphi }$ are given by the following relations:

Typical behavior of the amplitude ${\displaystyle A}$ as a function of the noise intensity ${\displaystyle D}$.

$${\displaystyle A=\frac{ϵ⟨x^2{⟩}_0}{D}\frac{2r_{\pm }}{\sqrt{r_{\pm }^2+{\omega }^2/4}},\varphi =-\arctan \left(\frac{\omega }{2r_{\pm }}\right);}$$

with ${\displaystyle ⟨x^2{⟩}_0}$ the variance of ${\displaystyle x}$ in the absence of external perturbations (dependent on the noise intensity ${\displaystyle D}$).

The fundamental aspects are therefore:

• the system oscillations tend, on average, to synchronize with the external perturbations;
• the amplification effect is negligible unless the perturbation frequency ${\displaystyle \omega }$ is comparable with the transition frequency ${\displaystyle r_{\pm }}$ (dependent on the noise intensity), and it is maximal for ${\displaystyle \omega =2r_{\pm }}$;
• for fixed perturbation amplitude ${\displaystyle ϵ}$ and frequency ${\displaystyle \omega }$, the amplitude ${\displaystyle A}$ is, for low noise intensities, a sharply increasing function of ${\displaystyle D}$, reaching a maximum peak at an intermediate noise level, and then becoming a decreasing function of ${\displaystyle D}$ for large noise values.

Suprathreshold

Suprathreshold stochastic resonance is a particular form of stochastic resonance, in which random fluctuations, or noise, provide a signal processing benefit in a nonlinear system. 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. It is not restricted to a subthreshold signal, hence the qualifier.

Neuroscience, psychology and biology

Main article: Stochastic resonance (sensory neurobiology)

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 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. 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.