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Non-commutativity in the brain

Nature volume 399, pages 261–263 (1999)Cite this article

Abstract

In non-commutative algebra, order makes a difference to multiplication, so that a × b ≠ b × a (refs 1, 2). This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations3,4,5,6. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor circuits that steer the eyes, head and limbs4,5,7,8,9,10,11,12,13,14,15, and sensory circuits that handle spatial information12,15. This idea is controversial12,13,16,17,18,19,20,21: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain7,8,9,12,13,14,15, but none that clearly rules out all commutative models17,18,19,20. Here we demonstrate non-commutative computation in the vestibulo-ocular reflex. We show that subjects rotated in darkness can hold their gaze points stable in space, correctly computing different final eye-position commands when put through the same two rotations in different orders, in a way that is unattainable by any commutative system.

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Figure 1: Why an ideal VOR must be non-commutative.
Figure 2: Computer simulation of a VOR model19 in which all neural processing is commutative.
Figure 3: The actual VOR is non-commutative.

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Acknowledgements

We thank D. M. Broussard, J. D. Crawford, J. Dichgans, C. E. Hawkins, J. A. Sharpe and T. Vilis for comments on the manuscript. This work was supported by the MRC of Canada and the Deutsche Forschungsgemeinschaft.

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Authors and Affiliations

  1. Departments of Physiology and Medicine, 1 King's College Circle, University of Toronto, Toronto, M5S 1A8, Canada

    Douglas B. Tweed

  2. Department of Neurology, and Department of Physics, University Hospital, Frauenklinikstrasse 26, Zurich 8091, ETH Hoenggerberg, Zurich, 8093, Switzerland

    Thomas P. Haslwanter

  3. Department of Neurology, University of Tbingen, Hoppe-Seyler-Strasse 3, Tbingen, 72076, Germany

    Vera Happe & Michael Fetter

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  1. Douglas B. Tweed
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  4. Michael Fetter
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Tweed, D., Haslwanter, T., Happe, V. et al. Non-commutativity in the brain. Nature 399, 261–263 (1999). https://doi.org/10.1038/20441

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