Non-commutativity in the brain

In non-commutative algebra, order makes a difference to multiplication, so that a x b not equal 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 noncommutative computation in the vestibule-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.