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Mathematics

The restless heart of a spiral

Nature volume 387, pages 655–657 (1997)Cite this article

Rotating spiral waves occur in a variety of two-dimensional systems that all share the property of excitability. A spiral wave has wave-like and particle-like properties — it pervades the medium and has a phase, but it also has a position, given by the coordinates of its tip. The motion of a spiral wave1, defined by motion of its tip, is called meander. This is not just a mathematical curiosity: the potentially lethal re-entrant cardiac arrhythmias of flutter and fibrillation, for example, in which the same wave of excitation repeatedly re-invades the same piece of tissue, can be modelled as spiral waves. Meander is a mechanism by which such arrhythmias can stop; and the artificial control of meander offers a means of low-voltage cardiac defibrillation, by pushing the wave tip out of the heart. At a meeting in May *, an emerging mathematical understanding of meander was described that uses distance-preserving symmetries to simplify the analysis.

If you rotate, reflect or translate a spiral, you still have a spiral. These distance-preserving transformations of the plane are the symmetries of the euclidean group E(2). Based on these symmetries, Barkley2 introduced ad hoc an ordinary differential equation that describes spiral wave motion simply as motions of the tip (instead of variations over the whole plane). Solutions of this equation undergo a ‘Hopf bifurcation’, from a motion with a single frequency (rigid rotation) to periodic motion with two frequencies (rotation plus meander). At resonance — that is, when the two frequencies are equal — the spiral follows a superposition of a linear drift and rigid rotation. The behaviour of this system closely matches that of spiral wave dynamics obtained as solutions of the full partial differential system2, and seen in experiments3. Furthermore, the approach naturally leads to using external periodic perturbations (in the heart, an electrical signal) to induce a controlled movement of the spiral by resonant drift4.

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Figure 1: Visualization of the spiral wave solution of the mammalian ventricular tissue model (reproduced from ref. 11).

References

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  1. Centre for Nonlinear Studies, University of Leeds, LS2 9JT, Leeds, UK

    Arun V. Holden

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Holden, A. The restless heart of a spiral. Nature 387, 655–657 (1997). https://doi.org/10.1038/42597

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