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Published: 11 June 2013

Could maths help solve epilepsy equation?

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

The mystery dates back to 1665, when Dutch mathematician, astronomer, and physicist Christiaan Huygens, 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 produced by both clocks, affecting their motions.

The effect, now referred to by scientists as 'indirect coupling,' was not mathematically analysed 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.

Seizures triggered by neuronal activity

The researchers believe the formula could lead towards 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 external cues to synchronise growth.

To apply their formula to an epilepsy model, the team assumed that neurons oscillate, or turn off and on, in a regular fashion. G. Bard Ermentrout, University Professor of Computational Biology and professor in Pitt’s Department of Mathematics compares this to Southeast Asian fireflies that flash rhythmically, encouraging synchronisation.

'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 neurons to fire all out of phase with each other.'

Further applications

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. Researcher Jonathan E. Rubin illustrates the idea by using a predator-prey system, such as rabbits and foxes.

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.