Stochastic resonance

Stochastic resonance is an enhancement of a defined response to a weak signal by adding an appropriate amount of noise to a nonlinear system. The response measure and system matter: increasing noise is not a general method for improving every signal. Thresholds or switching between states can allow noise to reveal a modulation that would otherwise produce little observable output.

A primary optical experiment used a bidirectional ring laser. Bruce McNamara, Kurt Wiesenfeld and Rajarshi Roy periodically modulated the asymmetry between counter-rotating lasing modes and injected noise. They observed an improved signal-to-noise ratio relative to the case without externally injected noise.^[ref]

An original threshold model illustrates the competing effects without claiming to reproduce that laser. Let a detector report one when

A cos (ω t) + η > θ

, with

0 < A < θ

and independent Gaussian noise samples

η \sim N (0, σ^{2})

. If

Φ

ϕ

p (ϕ) = 1 - Φ (\frac{θ - A c o s ϕ}{σ}) .

The contrast between the most and least favorable phases is

Δ p = Φ (\frac{θ + A}{σ}) - Φ (\frac{θ - A}{σ}) .

It tends to zero both as

σ \to 0

and as

σ \to \infty

: without noise the subthreshold signal never crosses; overwhelming noise makes the two phases nearly indistinguishable.

For chosen

θ = 1

and

A = 0.5

σ = 0.25

1

and

5

give contrasts about

0.0228

0.2417

and

0.0781

. Moderate noise improves this particular phase-contrast statistic. These numbers are not laser data and are not themselves an output spectral signal-to-noise ratio.

A discriminating experiment holds the weak drive and response definition fixed while sweeping noise intensity, and includes a no-drive comparison. A larger output fluctuation is not enough: one must test whether the output better tracks the imposed signal. Switching rates, noise correlation and observation bandwidth can determine the useful regime in a dynamical system.

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Stochastic resonance is a phenomenon in which the presence of noise in a system can enhance the detection or transmission of weak signals. This counterintuitive effect occurs in various fields, including physics, biology, neuroscience, and engineering. In simple terms, stochastic resonance involves the interplay between a weak signal and random fluctuations or noise. When a weak signal is combined with an appropriate level of noise, the noise can help elevate the signal above a certain threshold, making it easier to detect or respond to.