# The Azimuth Project Blog - The stochastic resonance program (part 1) (Rev #57, changes)

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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 software take 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 informed inform its it. creation. So we will also 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 . Stochastic Here resonance I is will a outline phenomenon some in of which, under certain conditions, a noise signal may amplify the effect math and science behind the program, and next time dissect the program. By way of introduction, I am a weak software input developer to with a research signal training detector. This concept was originally used in computer science, so this is a hypothesis new about field the timing of ice-age investigation cycles, for but me. has Any since amendments been or applied clarifications to are a welcome! wide range of phenomena, including neuronal detection mechanisms, and patterns of traffic congestion.

You see, there are a lot of interesting currents that come together in the field of scientific programming. In this report I outline some of the math and science behind this program. I myself am a software developer with research training in computer science, so let’s think of this as a beta-test of the documentation for s scientific program – any corrections or amendments will be welcomed!

### The concept of stochastic resonance

Suppose Stochastic that resonance we is have a phenomenon in which, under certain conditions, a noise signal detector may whose amplify internal state is driven by an input signal. Furthermore, suppose that the internal effect states of are divided into “on” states and “off” states; this digital state is an abstraction from the concrete states. With a light weak switch, signal. we This could concept treat was originally used in a hypothesis about the force timing as of the ice-age input cycles, signal, but the has angle since as been the applied concrete to state, a wide range of phenomena, including neuronal detection mechanisms, and the patterns up or down direction of the traffic angle congestion. as the digital state.

Let’s Suppose consider that the we effect have of a periodic input signal on detector the whose digital internal state. state Suppose is driven by an input signal. Also, suppose that the wave internal amplitude states is are not divided large into enough “on” to states change and the “off” states; this digital state, state but that it is sufficiently an large abstraction to from drive the detector’s concrete state close to the boundary between the digital states. Then, With a bit light of switch, random we noise, could occurring treat near the peak force of as an the an input cycle, signal, may give a sufficient “tap” to cross the system angle over as to the other concrete digital state, state. and We the will up/down therefore classification see a phase-dependent probability of transitions between digital states. In this relationship, which is catalyzed by the noise, angle we as see the input digital frequency state. being transmitted through to the output signal albeit in complex and stochastic manner. The noise hasamplified the input signal.

One Let’s place consider where stochastic resonance has been found is in the signal effect detection mechanisms of neurons. a There periodic are, input for signal example, on cells in the tails digital of state. crayfish Suppose which that are the tuned wave amplitude is not big enough to respond change the digital state, yet large enough to low-frequency drive signals in the movement detector’s concrete state close to the digital state boundary. Then, a bit of random noise, occurring near the water, peak generated by the motions of predators. an These input signals cycle, alone may are “tap” not over strong enough to raise the neurons system over to the firing other threshold, digital but state. with So the there right will amount be a phase-dependent probability of transitions between digital states. In this relationship, which is catalyzed by the noise, we see the neurons input do frequency respond is being “stochastically transmitted” through to the signal. output. The noise hasamplified the input signal.

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 reach the firing threshold for the neurons, but with the right amount of noise, the neurons do respond to the signal.

Stochastic resonance, Azimuth Library

### Bistable stochastic resonance

Stochastic resonance was originally defined with for the additional special assumption case of a bistable system – where each digital state is the basin of attraction for a stable point of equilibrium. In fact, the first application of stochastic resonance was to a hypothesis within the bistable theory of climate dynamics. Although the stochastic resonance hypothesis has not been confirmed, it is of interest at least historically, and because the underlying “Milankovitch” theory on which it is built is part of the state-of-the-art attempts to explain the timing of the ice ages.

In The first application of stochastic resonance was to a hypothesis concerning the bistable cycling model of the ice-age ice-ages cycles, within the two framework of bistable climate states dynamics. are Although a this cold, hypothesis “snowball” has Earth not and been a confirmed, hot, iceless Earth. The snowball Earth is stable because it is remains white, of and interest hence (1) reflects historically, solar (2) energy, which keeps it frozen. The iceless Earth is stable because it the is the dark, timing of the ice-age cycles remains an open problem, and hence (3) absorbs because solar the energy, Milankovitch theory upon which keeps it melted. rests is an active part of the current scientific attempts to explain the timing of the ice ages.

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

The Milankovitch hypothesis states that the drivers behind climate state change are the long-duration cycles in the solar energy received in the northern latitudes, which in turn 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 (ovalness) of the Earth’s elliptical orbit, with a period of 100 thousand years

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

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

In the stochastic resonance hypothesis, the Milankovitch signal is amplified by random events to produce changes in the state of the climate. More recent forms of Milankovitch theories invoked 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. energy received in the north (called the insolation).

Milankovitch cycle, Azimuth Library

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

### A mathematical model of stochastic resonance

The Azimuth demo program defines the behavior of the output signal through a stochastic differential equation (SDE), which specifies the derivative as a function of time, the current signal value, and a random noise process.

The program’s SDE sets the deterministic component of the derivative to a bistable function of the current signal value plus a sine function of time:

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

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

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

Now let’s analyze the bistable polynomial, which has roots at -1, 0 and 1. The root at 0 is an unstable equilibrium, and -1 and 1 are points of stable equilibrium. The basin of attraction for -1 is all the negative numbers, the basin for 1 is the positive numbers, and the basins are naturally separated by the unstable equilibrium.

We will treat each basin as one of the digital states of a bistable system.

Now let’s put the sine wave and the bistable polynomial together. If the wave is 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, it will cause the system to oscillate between the two basins.

Now let’s include the noise. Suppose the sine wave was large enough at its peaks to pull the system close to zero, but not enough to cross it over. Then a noise event occurring near the peak of the sine wave could push the system over to the other side. Stochastic resonance will occur.

### 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 develop new programs.

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