The noise that helps a weak signal through

In 1981 two groups of climate scientists, Roberto Benzi, Alfonso Sutera and Angelo Vulpiani in Rome and Catherine Nicolis in Brussels, were puzzling over the ice ages. The record of the last million years showed the Earth swinging between glacial and warm states about every hundred thousand years, and the only astronomical rhythm with that period, a slow wobble in the eccentricity of the Earth’s orbit, changes the sunlight reaching the Earth by a fraction of a per cent — far too little, on any simple accounting, to tip the climate from one state to the other. Their proposal was that the climate is a system with two stable states, that the weak orbital push could not switch it on its own, but that the random year-to-year variability of the weather, added to the push, could. And that there would be an amount of variability for which the switching followed the push most faithfully.

Whether that is the right account of the ice ages is still argued. But the mechanism they described turned out to be general, to be demonstrable on a laboratory bench within two years, and to appear in electronics, lasers and the nervous systems of animals. They called it stochastic resonance.

A push too weak to tip #

The simplest system that shows it is a particle in a double well: two valleys separated by a hill, such as $V(x) = -x^2/2 + x^4/4$, whose valleys sit at $x = \pm 1$ with a barrier a quarter of a unit high between them. The particle is heavily damped, so it does not oscillate; it simply slides downhill into whichever valley it is in and rests there. Now push it back and forth with a weak periodic force, $A\cos\Omega t$. The force tilts the landscape, deepening one valley and raising the other, and half a cycle later the reverse.

If the push is strong enough, it tilts the landscape so far that one valley disappears altogether and the particle is forced over into the other, every half-cycle; that happens for a push above $2/3\sqrt3 \approx 0.385$. The push drawn here is less than a third of that. It lowers the barrier out of one well from 0.25 to 0.14 and raises it to 0.38 half a cycle later, but the barrier never vanishes, and a particle in either valley simply rocks a little and stays. A detector watching which valley the particle is in sees nothing — the push is invisible.

Too little noise, the right amount, too much #

Now add noise: random kicks of intensity $D$, as a small particle in a fluid receives from the molecules around it. The jiggle that proved atoms followed such kicks in a single well; in a double well they do something new, because occasionally a run of kicks in the same direction carries the particle over the barrier into the other valley. How often that happens depends exponentially on the ratio of the barrier to the noise. Kramers worked out the rate in 1940 for exactly this geometry,

$$r = \frac{\sqrt{2}}{2\pi}\, e^{-\Delta V/D},$$

an Arrhenius law of the kind the exponential that decides everything found governing every barrier-crossing from chemistry to diffusion in solids. Its exponential is the key to everything that follows: because the push changes the barrier, it changes the rate, and it changes it exponentially. Lowering the barrier from 0.25 to 0.14 at a noise of 0.085 raises the rate of hopping out of that valley about thirty-fold.

The three traces show the whole effect. With very little noise, the particle never hops; the push is still invisible. With a lot of noise, the particle hops constantly, many times in each half-cycle, and the push is lost again in the random hopping. With an intermediate amount — here an intensity of 0.085, a third of the barrier — the particle hops about once every half-cycle, almost always in the half-cycle in which the push is lowering the barrier in front of it. The output has become a clean square wave that follows a push too weak to cause a single hop. The noise supplies the energy for each crossing; the push decides when the crossings happen.

A resonance in the noise #

The name is a deliberate analogy. The frequency that gets an answer found an oscillator responding most strongly when the drive’s frequency matches its own; here the system responds most strongly when its own timescale, set by the noise, matches the drive’s. The hopping rate depends on the noise, and the response is largest when the mean time between noise-induced hops is about half the push’s period — when the particle is, on average, ready to hop just as the push makes hopping easy.

Bruce McNamara and Kurt Wiesenfeld gave the theory its standard form in 1989 by treating the particle as simply in one well or the other, hopping between them at Kramers rates modulated by the push. The response at the push’s frequency then has a closed form: proportional to the push, inversely proportional to the noise, and multiplied by a factor $2r/\sqrt{4r^2 + \Omega^2}$ that rises from nothing as the noise grows and the hopping rate $r$ climbs towards the drive frequency $\Omega$. The product of a factor that falls with noise and one that rises with it has a maximum, here at a noise of 0.083 for a push of period 400. A slower push peaks at less noise, because slower hopping suffices to keep up with it, and higher, because more of each half-cycle is available; a faster push needs more noise and gets less.

The simulated dots, from integrating the particle’s equation of motion with random kicks for 120 cycles at each noise level, put the peak where the theory does but fall below it near the top. The two-state picture treats every hop as instantaneous and ignores the particle’s jiggling within its well, and both approximations overstate the response near the optimum. The qualitative claim — a peak at a nonzero noise, set by matching rates — is what the simulation confirms.

Why the best noise is a fraction of the barrier #

The location of the peak can be estimated without the full theory. The response is best when the hopping rate is about twice the drive frequency — one hop per half-cycle — so the optimal noise satisfies, roughly, $r(D) \approx \Omega/\pi$. Since $r$ depends on $D$ through $e^{-\Delta V/D}$, solving for $D$ gives

$$D_{\text{opt}} \approx \frac{\Delta V}{\ln\!\left(\sqrt{2}/2\Omega\right)},$$

a fraction of the barrier set by a logarithm of the drive period. For the period of 400 used here the logarithm is about 4, which puts the optimum at about a quarter of the barrier — close to the third the full theory finds. The logarithm is why the effect is robust: a tenfold change in the drive’s period shifts the optimal noise by only a modest factor, so a system need not be finely tuned to benefit.

Between hops, the particle spends its time near the bottom of one well or the other, distributed in each as the Boltzmann factor $e^{-V/D}$ dictates — the picture of the potential is where the wanderers stop, with the noise intensity playing the part of a temperature. The push tilts the wells, so for half a cycle one of them is deeper and the equilibrium occupation favours it. Stochastic resonance is what happens when the wanderers are allowed just enough time to follow the equilibrium as it swings from side to side. With too little noise they cannot reach it in half a cycle; with too much, the wells are too shallow, compared with the noise, for the tilt to matter.

When the hops happen #

The amplitude of the response is one way to see the effect. A sharper one is to record how long the particle stays in each valley before it hops.

Without the push, the residence times would follow an exponential distribution, most of them short — hops are a memoryless process, and the next one is equally likely at any moment. With the push, the distribution becomes a series of peaks. The tallest is near half a period: the particle arrives in a valley as the barrier behind it rises, waits through the unfavourable half-cycle, and leaves during the next favourable one. A second, smaller peak sits near one and a half periods, where the particle missed its first opportunity and took the next, and a third near two and a half. The peaks decay geometrically, each a missed chance.

The figure is drawn at a noise of 0.05, below the 0.083 that maximised the response amplitude, and the difference is instructive. At the higher noise, hops come often enough that the particle sometimes hops back within the same favourable window, and the peaks blur together. The noise that makes the hops most regular and the noise that transmits the most power at the push’s frequency are not the same, and which one is “the” stochastic resonance depends on what the system downstream is measuring. Most of the arguments over whether a given system shows the effect have been arguments about which measure to use.

Noise as a clock-setter #

The residence peaks suggest another way to describe the effect: the noise-driven hopping is a crude oscillator, with its own average rate, and the weak push locks it. The clock that is pulled into step found a self-sustained oscillator drawn onto a weak external rhythm when the two frequencies were close enough, inside a tongue of locking that widens as the coupling grows. The noisy double well has no self-sustained oscillation, only a rate of random hopping, but the same picture applies approximately: when the mean hopping rate is near twice the drive frequency, the hops are drawn into step with it. That is why the effect has been described as noise-induced synchronisation, and why it appears in the same systems — lasers, electronic circuits, neurons — in which synchronisation is studied.

The first laboratory demonstrations were of exactly that kind. In 1983 Stéphan Fauve and François Heslot fed a noisy, weak periodic signal into a Schmitt trigger, an electronic circuit with two stable output states and a switching threshold, and saw the output’s spectral peak at the signal frequency rise and fall with the added noise. In 1988 McNamara, Wiesenfeld and Rajarshi Roy did it with a ring laser that could lase clockwise or anticlockwise: modulating the losses weakly and adding noise to the pump, they watched the laser’s direction switch in step with the modulation at the optimal noise. These were the experiments that turned a speculation about the ice ages into a recognised phenomenon.

A threshold is enough #

The double well is the cleanest model, but the effect does not need wells or barriers or any dynamics at all. It needs only a threshold.

A detector that fires whenever its input exceeds a fixed level, fed a signal whose peaks stay below that level, never fires and passes nothing. Add noise, and it fires occasionally — more often near the signal’s peaks, where less noise is needed to reach the threshold, and less often near its troughs. The difference between the two firing rates is the signal, carried by the noise. With small noise the detector hardly fires at all; with huge noise it fires almost all the time regardless of the signal. In between, the difference is largest when the noise is about the size of the threshold itself — close to it for every signal amplitude, slightly less for larger ones. The calculation needs only the tail of the Gaussian distribution, the same tail that the second law with a probability attached found deciding how often a fluctuation goes the wrong way.

Engineers knew this long before it had a name. An analogue-to-digital converter rounds its input to the nearest step; a signal smaller than one step is rounded away entirely. Add a little noise — dither — before the conversion, and the rounded output flickers between steps with a probability that follows the signal, and averaging the output recovers a signal smaller than one step. Every digital audio recording and every high-resolution measurement made with a coarse converter uses dither. It is threshold stochastic resonance, chosen deliberately.

Nerves that use noise #

A sensory nerve cell fires an impulse when the voltage across its membrane crosses a threshold, and its membrane is noisy: ion channels open and close at random, and the thermal noise of half a kT in a piece of wire is present in any conductor. That makes nerves candidates for the effect, and in 1993 Frank Moss and his colleagues showed it directly in the mechanoreceptor hairs on the tail fan of a crayfish, which sense water movements. They drove the hairs with a weak periodic water movement and added random water motion of increasing strength; the signal-to-noise ratio of the nerve’s firing rose, peaked and fell, exactly the curve of the figures.

In 1999 the same group showed an animal using it. The paddlefish hunts plankton in murky water with electroreceptors on its long snout that detect the tiny electric fields of the plankton’s muscles. Adding a weak random electric field to the water increased the distance at which the fish could detect and strike at a plankton, up to an optimal noise, and the swarms of plankton themselves supply such a field in nature. In humans, adding a small random vibration to a fingertip lowers the threshold for detecting a weak tap, and vibrating insoles have been tested to improve balance in older people, whose sensation in the soles has declined. Whether nervous systems are tuned to exploit their own noise, or merely tolerate it, is harder to establish than that they can benefit from added noise.

One coordinate, white noise and a two-state shortcut #

The double-well figures use a dimensionless model with the particle heavily damped, so that inertia plays no part. A lightly damped particle can overshoot and recross the barrier, which changes the hopping statistics, and real systems with two stable states — a laser that can lase in either of two directions, a magnetic domain, a climate — have more structure than a single coordinate in a quartic well. The noise is white and Gaussian; coloured noise, correlated over times comparable with the hopping, changes the rate formula and shifts the optimum.

The two-state theory is accurate only when the push is weak and slow and the noise small compared with the barrier; at the optimum it overstates the response, as the simulation shows. None of this is the noise a precision instrument fights: the noise a high Q moves out of the way is about a linear oscillator, where noise only ever adds to the uncertainty and the remedy is to concentrate it in a narrow band away from the signal. The benefit here exists only because the system is not linear. The simulations themselves are single runs of finite length with a fixed seed, and their scatter is part of what they show. The threshold figure assumes the signal varies slowly enough that the detector sees each value independently, which is the opposite limit to the double well; a real nerve, with a refractory period after each impulse and adaptation over longer times, sits somewhere between the two.

Still open: whether the ice ages are an example #

The problem that started the subject is the one least settled. Benzi and Nicolis proposed that the hundred-thousand-year cycle of the last million years is the climate’s two states switched by noise in step with the orbit’s eccentricity. Since then, the record has been refined, and it is clear that the cycles are not regular switches between two fixed states, that the hundred-thousand-year rhythm began only about a million years ago while the eccentricity cycle is much older, and that the timing of terminations correlates at least as well with the 41,000-year cycle of the Earth’s tilt as with eccentricity. Whether stochastic resonance, a deterministic nonlinear oscillation of ice sheets, or some combination best accounts for the record is an open question in palaeoclimate, and the answer may be that the climate’s noise matters without its response being a resonance in the strict sense.

The general lesson does not depend on the ice ages. A system with a threshold or a barrier cannot respond to a push too weak to cross it, and noise lets it — best when the time the noise takes to carry it across matches the time the push spends helping. In linear systems noise only ever obscures a signal. In anything with a threshold, a little of it can be the difference between a signal that is lost and one that comes through.