Cheers, Ken Hanly
Les Schaffer wrote:
>
>
> 1.) first, what is a (non-)commutative mathematical operation?
>
> if someone gives you three cookies for sweeping floors and later two
> cookies for making your bed, you have 3 + 2 = 5 cookies.
>
> if another scenario gives you two cookies for sweeping floors and
> later three for making your bed, you have 2 + 3 = 5 cookies, still.
>
> pretty elementary, right? 2 + 3 = 3 + 2 = 5 is the commutative
> property of addition of numbers.
>
> and if you need six 2x6's for some construction project, and one
> lumber yard sends 2x6's in sets of three's, and another in pairs, then
> one yard will deliver two sets of three 2x6's and the other will
> deliver three sets of two 2x6's, and you still get six 2x6's because
> 2 x 3 = 3 x 2 = 6.
>
> still boring, eh?
>
> well, there's more to things than addition and multiplication of
> numbers, right?
>
> one of the first undergrad examples we teach of a non-commutative
> algebra is the mathematics of rotations in 3 dimensions.
>
> take a book, imagine there are two axes attached to the book as you
> are looking at its cover: one along the binding (A), and one
> perpendicular to the binding and lying in the plane of the cover
> (B). (a chalk-board would be useful here).
>
> rotate the book about the A axis by 90 degrees (A-90), then rotate it
> about the B axis by 90 degrees (B-90).
>
> you wind up with one orientation for the book in space.
>
> now reverse the order of the rotations. do B-90 followed by A-90.
>
> what's that? different orientation of the book in space, eh?
>
> a natural representation of rotation in 3 dimensions is matrix
> multiplication, which is non-commutative.
>
> means: the 3x3 matrix representing the
> rotation operation B-90
> ||||||
> VVVVVV
> matrix(A-90) * matrix(B-90) != matrix(B-90) * matrix(A-90)
> ^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^
> do B-90 followed by A-90 != do A-90 followed by B-90
>
> the symbol "!=" means "does not equal"
>
> 2.) here is the abstract and the first 3 andthe 6-th paragraphs of the
> paper:
>
> ***
> Nature 399, 261 - 263 (1999) © Macmillan Publishers Ltd.
>
> Non-commutativity in the brain
>
> DOUGLAS B. TWEED*, THOMAS P. HASLWANTER?, VERA HAPPE? & MICHAEL FETTER?
>
> * Departments of Physiology and Medicine, 1 King's College Circle,
> University of Toronto, Toronto M5S 1A8, Canada
>
> ? Department of Neurology, University Hospital, Frauenklinikstrasse
> 26, Zurich 8091, and Department of Physics, ETH Hoenggerberg, Zurich
> 8093, Switzerland
>
> ? Department of Neurology, University of Tbingen, Hoppe-Seyler-Strasse
> 3, Tbingen 72076, Germany
>
> Correspondence and requests for materials should be addressed to D.B.T.
>
> [Abstract] In non-commutative algebra, order makes a difference to
> multiplication, so that a x b != b x a (refs 1, 2). This feature is
> necessary for computing rotary motion, because order makes a
> difference to the combined effect of two rotations[3-6]. 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 limbs[4,5,7-15], and sensory circuits
> that handle spatial information[12,15]. This idea is
> controversial[12,13,16-21]: studies of eye and head control have
> revealed behaviours that are consistent with non-commutativity in the
> brain[7-9,12-15], but none that clearly rules out all commutative
> models[17-20]. 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.
>
> The vestibulo-ocular reflex (VOR) is perhaps the simplest of the many
> neural responses to rotary motion: sensors in the inner ear measure
> head velocity and send commands to the eye muscles, moving the eyes in
> the opposite direction when the head turns, so as to keep the eyeballs
> from rotating relative to space[6]. By stabilizing our retinal images,
> the VOR allows us to see while moving.
>
> An ideal VOR must be capable of non-commutative operations. This point
> has been argued before[5,13], but it is perhaps more clearly
> illustrated by the thought experiment in Fig. 1, which shows that the
> VOR, if it is to keep the gaze on target, must compute different final
> eye-position commands when a person undergoes the same two rotations
> in different orders. Starting by looking at a space-fixed target 30°
> to the left, a subject who turns first 10° counterclockwise (CCW) and
> then 60° left must end up looking 30° right and 5° up to stay on
> target (Fig. 1a). A subject who turns first 60° left and then 10° CCW
> must end up looking to the right and down (Fig. 1b).
>
> Surprisingly, this ideal behaviour is not predicted by most models of
> the neural circuitry underlying the VOR. For example, Fig. 2 shows a
> computer simulation of an influential theory[19]. This model elegantly
> explains many features of the VOR, but as the simulation shows, it
> fails to keep the eye on target when put through the rotations from
> Fig. 1. Regardless of the order of rotations, it leaves the eye in the
> same final position relative to the head. Relative to space, then, the
> eye is incorrectly positioned, pointing above or below the target.
>
> ...[snip]...
>
> It has been proposed that non-commutative effects in ocular control
> need not imply non-commutative computations in the brain, because some
> of these effects can be explained by the geometry of the eye muscles,
> in particular the fact that the muscles run through pulleys in the
> orbital wall[22,23]. Pulleys do have major implications for eye
> control[12,19-23], but neither they nor any other muscle geometry can
> explain the non-commutativity in Fig. 3. In fact, the model in Fig. 2
> incorporates pulleys, but it remains commutative. Whatever the muscle
> geometry, there is always a correspondence between final, steady-state
> eye position and motor-neuron firing: to hold different eye positions,
> you need different firing rates[6,24]. Even if the pulleys themselves
> are adjusted by a separate set of motor neurons, final eye position
> will still be determined by the total motor-neuron pool. The distinct
> final eye positions in Fig. 3 therefore imply that the final
> activities of the motor neurons differ, depending on the order of head
> rotations. There must be non-commutative processing between the inner
> ear and the motor endplate.
>
> ***
>
> 3.) full article for those interested in reading more.
>
> les schaffer