Blog - The stochastic resonance program (part 1) (Rev #64)

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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. Here I will outline some of the math and science behind the program, and next time I’ll dissect the program. By way of introduction, I am a software developer with research training in computer science, so this is a new field of investigation for me. Any amendments or clarifications are welcome!

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, but has since 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 classified into a region of “on” states and a region of “off” states. This is the digital state, 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. So, there will be a phase-dependent probability of transitions between digital states. This relationship between signal phase and state-transition probabilities bears the stamp of the input frequency. 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 which are tuned to low-frequency signals in the movement of the water, generated by the motions of predators. These signals alone do not cross the firing threshold for the neurons, but with the right amount of noise, the neurons do respond to the signals.

See:

• Stochastic resonance, Azimuth Library

• Stochastic resonance in neurobiology, David Lyttle. An overview, with clear explanations.

Stochastic resonance was originally defined for the special case of a bistable system – where each digital state is the basin of attraction for a stable point of 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 the timing of the ice-age cycles remains an open problem, and (3) because the Milankovitch hypothesis upon which it rests is an active part of the current scientific research agenda.

In the bistable model, the two 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 solar energy received in the northern latitudes (called the “insolation”) that are caused by periodic changes in the Earth’s orbital parameters. The significance of the north is that the glaciers are concentrated there, and so a sufficient “pulse” in the northern temperatures could trigger 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. More recent forms of Milankovitch theories invoke a deterministic forcing mechanism. In a theory by Didier Paillard, the climate is modeled as having 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?

Anytime we have a function that has two local minima, we can use that as a function to define a bistable system. For instance, consider the function V(x) = x^{4}/4 - x^{2}/2:

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

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

So, for instance, at a place where the potential graph is sloping upward as x increases, the time derivative X’(t) is negative, which sends X(t) “downhill” towards the potential minimum. As it approaches the minimum, the slope of the potential graph goes to zero, which means the motion of X(t) slows down. It asymptotically approaches rest at 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.

We now talk about the discrete model, which is implemented in the demo program.

The Azimuth demo program implements a discrete model, which exhibits stochastic resonance.

We use the potential function just described, but using *discrete time*; the derivative is assumed to be constant over the interval between time points. In addition, a random number is added to the discretely sampled derivative. This gives us a discrete-time difference equation:

$\Delta X_t = -V'(X_t) *\Delta t + SineWave(t) + RandomSample(t) = X_t (1 - X_t^2) \Delta t + \alpha * sin(\omega t) + \beta * GaussianSample(t)$

where $\Delta t$ is a constant, $t$ is restricted to multiples of $\Delta t$, and GaussianSample(t) is a sampling from a normal distribution with zero mean and unit variance.

Note that this differential equation is the discrete version of a Stochastic differential equation.

In the next article, we will look into the Azimuth demo program: how to use it, how it works, and how to modify it to develop new programs.

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