Pendulum swings back on 350-year-old mathematical mystery

June 10, 2013 by B. Rose Huber

( —A 350-year-old mathematical mystery could lead toward a better understanding of medical conditions like epilepsy or even the behavior of predator-prey systems in the wild, University of Pittsburgh researchers report.

The mystery dates back to 1665, when Dutch mathematician, astronomer, and physicist , inventor of the pendulum clock, first observed that two pendulum clocks mounted together could swing in opposite directions. The cause was tiny vibrations in the beam caused by both clocks, affecting their motions.

The effect, now referred to by scientists as "indirect coupling," was not mathematically analyzed until nearly 350 years later, and deriving a formula that explains it remains a challenge to mathematicians still. Now, Pitt professors apply this principle to measure the interaction of "units"—such as neurons, for example—that turn "off" and "on" repeatedly. Their findings are highlighted in the latest issue of Physical Review Letters.

"We have developed a to better understanding the 'ingredients' in a system that affect in a number of medical and ecological conditions," said Jonathan E. Rubin, coauthor of the study and professor in Pitt's Department of Mathematics within the Kenneth P. Dietrich School of Arts and Sciences. "Researchers can use our ideas to generate predictions that can be tested through experiments."

More specifically, the researchers believe the formula could lead toward a better understanding of conditions like epilepsy, in which neurons become overly active and fail to turn off, ultimately leading to seizures. Likewise, it could have applications in other areas of biology, such as understanding how bacteria use to synchronize growth.

Together with G. Bard Ermentrout, University Professor of and professor in Pitt's Department of Mathematics, and Jonathan J. Rubin, an undergraduate mathematics major, Jonathan E. Rubin examined these forms of indirect communication that are not typically included in most mathematical studies owing to their complicated elements. In addition to studying neurons, the Pitt researchers applied their methods to a model of artificial gene networks in bacteria, which are used by experimentalists to better understand how genes function.

"In the model we studied, the genes turn off and on rhythmically. While on, they lead to production of proteins and a substance called an autoinducer, which promotes the genes turning on," said Jonathan E. Rubin. "Past research claimed that this rhythm would occur simultaneously in all the cells. But we show that, depending on the speed of communication, the cells will either go together or become completely out of synch with each another."

To apply their formula to an epilepsy model, the team assumed that neurons oscillate, or turn off and on in a regular fashion. Ermentrout compares this to Southeast Asian fireflies that flash rhythmically, encouraging synchronization.

"For neurons, we have shown that the slow nature of these interactions encouraged 'asynchrony,' or firing at different parts of the cycle," Ermentrout said. "In these seizure-like states, the slow dynamics that couple the neurons together are such that they encourage the to fire all out of phase with each other."

The Pitt researchers believe this approach may extend beyond medical applications into ecology—for example, a situation in which two independent animal groups in a common environment communicate indirectly. Jonathan E. Rubin illustrates the idea by using a predator-prey system, such as rabbits and foxes.

"With an increase in rabbits will come an increase in foxes, as they'll have plenty of prey," said Jonathan E. Rubin. "More rabbits will get eaten, but eventually the foxes won't have enough to eat and will die off, allowing the rabbit numbers to surge again. Voila, it's an oscillation. So, if we have a fox-rabbit oscillation and a wolf-sheep oscillation in the same field, the two oscillations could affect each other indirectly because now rabbits and sheep are both competing for the same grass to eat."

Explore further: Pitt receives $2.5 million to simulate and analyze brain, immune system activity

More information: The paper, "Analysis of synchronization in a slowly changing environment: how slow coupling becomes fast weak coupling," was first published online May 13 in Physical Review Letters.

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4 / 5 (4) Jun 10, 2013
More specifically, the researchers believe the formula could lead toward a better understanding of conditions like epilepsy, in which neurons become overly active and fail to turn off, ultimately leading to seizures.

I'm a bit confused why they think this would lead to a better understanding in neurological matters - as it is precisely the understanding of this asynchronicity/synchronicity phenomenon that has lead to the development of brain pacemaker implants to counter severe epilepsy (and also defibrillators)

The bacteria approach looks interesting, though.
1 / 5 (6) Jun 10, 2013
This is another instance of Art Winfree's law of coupled oscillators, which I have described in fifty or sixty prior posts, mostly in physics. Huygens first noticed the phenomenon; Winfree developed the math circa 1967 and applied it to numerous biological systems. Strogatz, Mirollo and Kuramoto, among others, have extended Winfree's work.

Winfree's law applies to any system of "limit-cycle oscillators." The quantum world is a world of periodic oscillations, so Winfree's law should apply to physics and chemistry, in addition to biology. Certain configurations that Winfree identified are stable, or locked. For more info, see "Coupled Oscillators and Biological Synchronization," Scientific American, Dec. 1993, by Strogatz and Ian Stewart, available online at an site. See also "Synchronization of Pulse Coupled Biological Oscillators," by Mirollo, at a site.

In physics, phase transitions, superconductivity, Efimov states: Winfree's law.
5 / 5 (1) Jun 10, 2013
Biophysics and chemical physics have numerous examples of indirect couplings for neuromodulation effects. DBS and pacemakers target, inhibited or suppress the effects.
Of which the underlying causes are poorly understood. They suggest indirect coupling (of neurons) as a underlying source (cause) beyond merely targeting neuromodulation effects gone awry.

All forms of coupling have ubiquitous applications. Scale invariance and coupling are foremost science frontiers - seen in labels such as 'coupling constants'.

Here a neuron model application sample:
Amplitude death in nonlinear oscillators with indirect coupling


Your welcome :)
1 / 5 (4) Jun 10, 2013
This Art Winfree law applies in astronomy too. Take Lagrangian points, as applied to the "Trojan/Greek asteroids" and the Hilda family of asteroids, associated with Jupiter. The Trojan/Greek asteroids occupy Lagrange positions L4 and L5. L4 is the "northern" point on an equilateral triangle that has Jupiter and the sun on the other two points. L5 is the same position as L4, but on the "southern" side. Art Winfree says that in a three oscillator system one stable arrangement is each of the three units one-third out of phase with respect to each other. An equilateral triangle is one way of saying each unit "one-third out of phase."

The Hilda asteroids group in three clusters which are each one third out of phase with the other two--phase in this case being with reference to their orbits around the sun. Astronomers refer to "libration," or balancing. Winfree's model encompasses Lagrangian points and libration. Same principle with our own moon--its spin coupled to its orbit.
2 / 5 (4) Jun 10, 2013
Nothing like the "Art Winfree law" exists, you're living in alternate reality. The gravitational resonance has many different forms (1, 2). We should always distinguish the predictive analogies from postdictive homologies, which have nothing to do with deterministic causality.
1 / 5 (4) Jun 11, 2013
ValeriaT: Good to hear from you, and thanks for posting your links (1,2). Your link 1 shows a system of three periodic oscillators in which each one is one-third out of phase with the other two. That is the Winfree pattern that I mentioned in my post. Your link 2 shows the Trojan/Greek asteroids and the Hilda asteroids, which are depicted as I described. Your links confirm rather than refute my comments, factually.

Your second point, namely that the connection may simply be coincidental, is of course possible. To test that point, consider the recent discovery that merging galaxies align their spin. This was not predicted in advance, but it now seems to be the rule. Spin is a periodic oscillation, so Winfree's law, broadly conceived, suggests that spin alignment between two merging galaxies would be likely. Alternatively, I will consider any other causality that you think likely--even though other experts made no such prediction.
1 / 5 (4) Jun 11, 2013
Moving from the stars above back to the article above, one of the authors, Prof. Ermentrout, is well aware of Art Winfree and his law, and of the work done by Steve Strogatz (Prof. at Cornell, author of Sync) in extending Art's work, for whom Steve worked as a post-doc. You can see various connections online, including Prof. Ermentrout's review of Sync, in which Ermentrout includes comments about Art Winfree (as does Strogatz in his book).

So the article above describes work that is consistent with my expansive view of Art Winfree's work, and my pronouncement that his work amounts to a "law" of coupled oscillators that deserves far greater attention. The article is not just coincidentally consistent with my many posts on PhysOrg--the intellectual roots of the article trace directly to Winfree and Strogatz.

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