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Blog - The stochastic resonance program (part 1) (Rev #47)

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At the Azimuth Code Project, we are aiming to produce educational software that is relevant to the Earth sciences and the study of climate. Our programs take the form of interactive web pages, which are written in javascript and use the JSXGraph library for interactive plotting. They allow you to experiment with the parameters of a model and view its outputs. But in order to fully understand the meaning of a program, we need to know about the concepts and theories that informed its creation. So we will also be writing articles to explain the science, the math, and the programming behind these models.

In this blog and the next, I will cover one of the scientific programs developed at Azimuth, written by Allan Erskine and Glyn Adgie. It demonstrates a phenomenon known as stochastic resonance, in which, under certain conditions, a noise signal may amplify the effect of a weak input to a signal detector. This concept was originally applied in a hypothesis about the timing of ice-age cycles, but has since found widespread applications, including the neuronal detection mechanisms of crickets and crayfish, and patterns of traffic congestion.

The mechanism of stochastic resonance

Suppose that we have an input signal which is fed into a signal detection mechanism. The input signal will drive the state of detection mechanism, in a possibly complex, but deterministic fashion. Further, let’s divide the states of the detector into “on” states and “off” states. In the case of a light switch, the input signal may be the force applied to the switch, the state could be the angular position of the switch, and we could say that the sign of the angle determines whether the switch is on or off.

Let’s input a periodic signal, and examine the conditions under which its frequency will be reflected by the flip-flopping of the digital output signal. In particular, let’s consider the case where the input signal is weak, so that the detector remains in a single digital state. How can we amplify such a signal, so that it becomes present in the output? Well, if the input signal is sufficient to drive the detector’s state close to the boundary between digital states, then a bit of random noise may be enough to get it to cross the boundary, into the other digital state. As the input signal moves towards another phase which would have pulled the state further away from the boundary, the system may then cross back into the state where it started. You can imagine that when the settings are tuned just right, the noise will catalyze the detector output to “stochastically oscillate” at the frequency of the input signal.

Stochastic resonance has been found in a wide range of natural and artificial systems, including the signal detection mechanisms of neurons. For instance, there are cells in the tails of crayfish which are tuned to respond to low-frequency signals in the movement of the water, presumably arising from the motions of predators. On their own, these signals are not strong enough to raise the neurons to the firing threshold. But in the presence of the right amount of noise, these signals do cause the neurons to fire.

Bistable stochastic resonance and Milankovitch theories of ice-age cycles

Stochastic resonance was originally defined in a more restrictive sense, where the two-state system was assumed to be bistable – where each digital state is the basin of attraction for a point of stable equilibrium.

The prototypical application of bistable stochastic resonance was to a form of the Milankovitch theory of the timing of the ice ages.

In the simplest form of Milankovitch theory, the climate is modeled as a bistable system, where one state is a cold, snowball Earth, and the other is a hot, iceless Earth. The snowball Earth is stable because it is white and reflects solar energy, which keeps it cold. The iceless Earth is stable because it is dark and absorbs solar energy, which keeps it hot and melted.

The forcing signals are taken to be the long-duration “Milankovitch” astronomical cycles – tilting of the Earth’s axis, precession of its axis, and variation in the eccentricity of the Earth’s orbit – which vary the amount of solar energy received in the northern latitudes.

The ice-ages are currently running on a roughly 100,000 year schedule, and, intriguingly, the three mentioned Milankovitch cycles have periods of 21, 32 and 100 (FIXME) kiloyears, respectively.

But how would these cycles trigger a change in the state of the Earth’s climate?

In the most recent forms of the Milankovitch hypothesis, a deterministic mechanism is invoked. See, for example Paillard’s paper.

In the original stochastic resonance hypothesis, by Benzi, these Milankovitch cycles produce a signal which in itself is not strong enough to change the state of the climate, but well-timed noise events, at the right phases of a Miloankovitch cycle, could trigger a state change.

The timing of the ice-ages remains a great unsolved problem for modern science.

Stochastic resonance, Azimuth Library

Milankovitch cycle, Azimuth Library

A mathematical model of stochastic resonance

The demo program uses a stochastic differential equation (SDE) to define a bistable system with a sinusoidal driving function and a noise component. In an SDE, the derivative of the output signal is specified as a function of time, the current signal value, and a noise process.

In this differential equation, the deterministic part of the derivative is set to the a time-varying sine wave plus a bistable function of the current signal value:

DerivDeterministic(t, x) = SineWave(t, amplitude, frequency) + Bistable(x),

where Bistable(x) = x (1 - x2).

Now let’s analyze the effects of these terms.

Alone, the sine wave would cause the output signal to vary sinusoidally.

The bistable polynomial has roots at -1, 0 and 1. The root at 0 is an unstable equilibrium, and -1 and 1 are stable equilibria. The basin of attraction for -1 is all the negative numbers, the basin for 1 is the positive numbers, and the point of unstable equilibrium separates the basins.

We view each basin as one of the states of a bistable system.

Now let’s put the sine wave and the bistable polynomial together. If the wave is relatively weak, the system will gravitate towards one of the attractors, and then continue to oscillate around it thereafter – never leaving the basin. But if the wave is large enough, the system will oscillate between the two basins.

Now let’s consider the noise as well. Suppose the sine wave was large enough to periodically pull the system close to zero – but not enough to cross over. Then a well-timed noise event could push the system over to the other side. There will be a phase-dependent probability of the noise triggering a state change, with more noise leading to transitions on more of the cycles. The flip-flopping between states will contain a “stochastic reflection” of the driving sine wave. Even more noise will cause multiple transitions within each cycle, thereby drowning out the signal and turning the output to noise.

This is the stochastic resonance effect: under the right conditions, the noise may amplify the effect of the input signal.

Recap

We’ve talked about the idea of stochastic resonance, touched on some of its applications, and described the mathematical model of stochastic resonance used in the Azimuth demo program. Next time we will look into this program: how to use it, how it works, and how to modify it to make new programs.

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