In neurophysiology

Neurophysiology

Neurophysiology is a part of physiology. Neurophysiology is the study of nervous system function...

, commutation is the process of how the brain's neural circuits

Biological neural network

In neuroscience, a biological neural network describes a population of physically interconnected neurons or a group of disparate neurons whose inputs or signalling targets define a recognizable circuit. Communication between neurons often involves an electrochemical process...

 exhibit non-commutativity

Commutativity

In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

.
Physiologist Douglas B. Tweed and coworkers consider whether certain neural circuits in the brain

Brain

The brain is the center of the nervous system in all vertebrate and most invertebrate animals—only a few primitive invertebrates such as sponges, jellyfish, sea squirts and starfishes do not have one. It is located in the head, usually close to primary sensory apparatus such as vision, hearing,...

 exhibit noncommutativity and state:

In noncommutative algebra, order makes a difference to multiplication, so that . This feature is necessary for computing rotary

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

 motion, because order makes a difference to the combined effect of two rotations. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor system

Motor system

The motor system is the part of the central nervous system that is involved with movement. It consists of the pyramidal and extrapyramidal system....

 circuits that steer the eye

Human eye

The human eye is an organ which reacts to light for several purposes. As a conscious sense organ, the eye allows vision. Rod and cone cells in the retina allow conscious light perception and vision including color differentiation and the perception of depth...

s, head and limbs, and sensory system

Sensory system

A sensory system is a part of the nervous system responsible for processing sensory information. A sensory system consists of sensory receptors, neural pathways, and parts of the brain involved in sensory perception. Commonly recognized sensory systems are those for vision, hearing, somatic...

 circuits that handle spatial information. This idea is controversial: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain, but none that clearly rules out all commutative models.

Tweed goes on to demonstrate non-commutative computation in the vestibulo-ocular reflex

Vestibulo-ocular reflex

The vestibulo-ocular reflex is a reflex eye movement that stabilizes images on the retina during head movement by producing an eye movement in the direction opposite to head movement, thus preserving the image on the center of the visual field. For example, when the head moves to the right, the...

by showing 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.