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Stochastic resonance is a phenomenon that occurs in a threshold measurement system when an appropriate measure of information transfer (signal-to-noise ratio, mutual information, coherence, d', etc.) is maximized for a non-zero value of stochastic input noise ^[1]; that is to say, the system resonates at a particular noise level.

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

1. Nonlinear measurement device - input compared to threshold, output is ON or OFF
2. Signal of interest - must be weak, periodic and at some points below threshold of measurement device
3. Input noise - 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 in terms of SNR.

The threshold measurement creates harmonic distortion, in which unwanted frequency components of the output signal occur at integer multiples of the fundamental frequency where the amplitude of the harmonic decreases with its distance from the fundamental. Adding noise before measuring with the threshold causes this harmonic distortion to be whitened, or spread across the frequency spectrum, instead of concentrated in particular low frequency harmonic components, which allows it to be better attenuated by averaging or low-pass filtering ^[2].

Stochastic resonance is interesting because it is a seemingly paradoxical idea. 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 show that in certain types of 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.

The following demonstration is after Simonotto et al ^[3].

If an image is put through a threshold filter, it can be hard to discern the objects in the original image 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 thresholding with different levels of noise variance; the image in the top right hand corner appears to have the optimal level of noise allowing the object to be recognized.

The quality of the image resulting from stochastic resonance can be improved further by squinting one's eyes or moving away from the image. This allows one's brain to average the pixel intensities over areas, which is in effect a low-pass filter. The resonance breaks up the harmonic distortion by spreading it 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 various threshold levels, so in a sense the addition of noise creates a new effective threshold for the measurement device.

Stochastic resonance was first discovered in a study of the periodic recurrence of earth’s ice ages ^{[4] [5]}. 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.

The phenomenon was demonstrated in a high-level mathematical model of a single neuron using a dynamical systems approach ^[6]. The model consists of a bi-stable potential energy function treated as a dynamical system. The model is tested using 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 of the SNR for a particular input noise was apparent, demonstrating stochastic resonance.

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)^[7]. 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 measures of SNR were used. 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 show a clear peak at the 55.2 Hz component with different average levels of broadband noise. The low- and mid-level 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 shows a clear peak at the mid-level noise condition. The other measure used for SNR was based on the inter-spike interval histogram. A similar peak was found for mid-level noise, although it was not equal to 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.

A similar experiment was performed on the cricket (Acheta domestica), an arthropod like the crayfish^[8]. The cercal system in the cricket senses pressure waves in the atmosphere utilizing tympanic membranes located on its forelegs. Sensory interneurons in terminal abdominal ganglion carry information about intensity and direction of pressure perturbations. Crickets were presented with signal and noise stimuli and intracellular recordings from cercal interneurons were made.

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 with noise.

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^[8].

Another investigation of stochastic resonance in broadband (or, equivalently, aperiodic) signals was conducted by probing cutaneous mechanoreceptors in the rat ^[9]. 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 tbe 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.

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^[10]. 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.

Observations of the paddlefish hunting in the wild provide evidence that the background noise generated by plankton increase the paddlefish’s hunting abilities. The individual plankton generate particular signals on top of the background signals from a large group, and the paddlefish do not respond to only noise without individual signals. For these reasons, it is likely that the paddlefish takes advantage of stochastic resonance to improve its sensitivity to prey.

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 ^[11]. 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 not yet been investigated experimentally.

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.

The above demonstration using the Arc de Triomphe photo is a simplified version of an earlier experiment^[3]. In that case, the noise input to the image varied over time, and the subjects perceived that the image was most recognizable at a certain level of noise^[3]. Similar experiments also demonstrated an increased level of contrast sensitivity to sine wave gratings^[3].

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. ^[12]

The auditory intensity detection thresholds of a number of human subjects were tested in the presence of noise ^[13]. 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.

• Peter Hanggi. "Stochastic Resonance in Biology", Chemphyschem 3, 285-290. (2002).
• Wiesenfeld, K., Moss, F. "Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs". Nature. (1995).
• Moss, F., Wilkens, L.A. "Mechanoreceptors and Stochastic resonance [14]", Scholarpedia, 2008.