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Blog - The stochastic resonance program (part 1) (Rev #66)

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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 software takes the form of interactive web pages, which 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 inform it. So we will be writing articles to explain the science, the math, and the programming behind these models.

In this two-part series, I will cover the Azimuth stochastic resonance example program, by Allan Erskine and Glyn Adgie. Today we’ll look at some of the math and science behind the program, and next time we’ll dissect the program. By way of introduction, I am a software developer with research training in computer science, so this is a new area for me. Any amendments or clarifications are welcome!

The concept of stochastic resonance

Stochastic resonance is a phenomenon in which, under certain conditions, a noise source may amplify the effect of a weak signal. This concept was used in an early hypothesis about the timing of ice-age cycles, and that has been applied to a wide range of phenomena, including neuronal detection mechanisms, and patterns of traffic congestion.

Suppose that we have a signal detector whose internal, analog state is driven by an input signal, and suppose the analog states are partitioned into two regions, called the “on” and “off” state – so we have a digital state which is abstracted from the analog state. With a light switch, we could take the force as the input signal, the angle as the analog state, and the up/down classification of the angle as the digital state.

Let’s consider the effect of a periodic input signal on the digital state. Suppose that the wave amplitude is not big enough to change the digital state, yet large enough to drive the detector’s analog state close to the digital state boundary. Then, a bit of random noise, occurring near the peak of an input cycle, may “tap” the system over to the other digital state. We will therefore see a probability of state-transitions that is synchronized with the input signal. In a complex way, the noise has amplified the input signal.

But it’s a pretty funky amplifier! Here is a picture from the Azimuth library article on stochastic resonance:

Stochastic resonance has been found in the signal detection mechanisms of neurons. There are, for example, cells in the tails of crayfish that are tuned to low-frequency signals in the water arising from the motions of predators. These signals are too weak to cross the firing threshold for the neurons, but with the right amount of noise, the neurons are triggered by these signals.

See:

Stochastic resonance, Azimuth Library

Stochastic resonance in neurobiology, David Lyttle

Bistable stochastic resonance and Milankovitch theories of ice-age cycles

Stochastic resonance was originally defined for systems that are bistable – where each digital state is the basin of attraction for a point of stable equilibrium.

An early application of stochastic resonance was to a hypothesis, within the framework of bistable climate dynamics, about the timing of the ice-age cycles. Although it has not been confirmed, it remains of interest (1) historically, (2) because the timing of ice-age cycles remains an open problem, and (3) because the Milankovitch hypothesis upon which it rests is an active part of the current research.

In the bistable model, the climate states are a cold, “snowball” Earth and a hot, iceless Earth. The snowball Earth is stable because it is white, and hence reflects solar energy, which keeps it frozen. The iceless Earth is stable because it is dark, and hence absorbs solar energy, which keeps it melted.

The Milankovitch hypothesis states that the drivers of climate state change are long-duration cycles in the insolation – the solar energy received in the northern latitudes – caused by periodic changes in the Earth’s orbital parameters. The north is significant because that is where the glaciers are concentrated, and so a sufficient “pulse” in northern temperatures could initiate a state change.

Three such astronomical cycles have been identified:

• Changing of the eccentricity of the Earth’s elliptical orbit, with a period of 100 kiloyears

• Changing of the obliquity (tilt) of the Earth’s axis, with a period of 41 kiloyears

• Precession (swiveling) of the Earth’s axis, with a period of 23 kiloyears

In the stochastic resonance hypothesis, the Milankovitch signal is amplified by random events to produce climate state changes. In more recent Milankovitch theories, invoke a deterministic forcing mechanism is used. In a theory by Didier Paillard, the climate is modeled with three states, called interglacial, mild glacial and full glacial, and the state changes depend on the volume of ice as well as the insolation.

See:

Milankovitch cycle, Azimuth Library

Mathematics of the environment (part 10), John Baez. This gives an exposition of Paillard’s theory.

Increasing the Signal-to-Noise Ratio with More Noise, Glyn Adgie and Tim van Beek, Azimuth Blog. Subtitle: Are the Milankovitch Cycles Causing the Ice Ages?

Bistable systems defined by a potential function

Any smooth function with two local minima can be used to define a bistable system. For instance, consider the function V(x)=x 4/4x 2/2V(x) = x^4/4 - x^2/2:

To define a bistable system, construct a differential equation where the time derivative of x is set to the negative of the derivative of the potential at x:

dx/dt=V(x)=x 3+x=x(1x 2)dx/dt = -V'(x) = -x^3 + x = x(1 - x^2)

So, for instance, where the potential graph is sloping upward as x increases, -V’(x) is negative, which sends X(t) “downhill” towards the minimum.

The roots of V’(x) yield stable equilibria at 1 and -1, and an unstable equilibrium at 0. The unstable equilibrium separates the basins of attraction for the stable equilibria.

Discrete stochastic resonance

Here we describe a discrete-time model that exhibits stochastic resonance – this is what is used in the Azimuth demo program.

We construct a discrete-time derivative, by combining the negative of the derivative of the potential function, a sampled sine wave, and a normally distributed random number:

ΔX t=V(X t)*Δt+SineWave(t)+RandomSample(t)=X t(1X t 2)Δt+α*sin(ωt)+β*GaussianSample(t)\Delta X_t = -V'(X_t) *\Delta t + SineWave(t) + RandomSample(t) = X_t (1 - X_t^2) \Delta t + \alpha * sin(\omega t) + \beta * GaussianSample(t)

where Δt\Delta t is a constant and tt is restricted to multiples of Δt\Delta t.

Note that this difference equation is the discrete-time counterpart to a corresponding Stochastic differential equation.

Next time we will take a look at the Azimuth demo program: how to use it, how it works, and how to change it to make new programs.

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