1. This series of three papers aims to describe the three-dimensional, kinematic input-output relations of the rotational vestibuloocular reflex (VOR) in humans, and to identify the functional advantages of these relations. In this first paper the response to sinusoidal rotation in darkness at 0.3 Hz, maximum speed 37.5°/s, was quantified by the use of the three-dimensional analogue of VOR gain: a 3 x 3 matrix where each element describes the dependence of one component (torsional, vertical, or horizontal) of eye velocity on one component of head velocity. 2. The three matrix elements indicating collinear gains (i.e., dependence of torsional eye velocity on torsional head velocity, vertical on vertical, and horizontal on horizontal) were smaller than the -1's required for optimal retinal image stabilization. Of these three the torsional gain was weakest: -0.37 for rotation about an earth-vertical axis, versus -0.73 and -0.64 for vertical and horizontal gains. Matrix elements indicating cross talk were mostly negligible. There was a tendency to leftward eye rotation in response to clockwise head motion, but this was not statistically significant. 3. VOR responses were compared for rotation about earth-vertical and earth- horizontal axes. The varying otolith input due to the rotation of the gravity vector relative to the head during earth-horizontal axis rotation made no difference to the collinear gains. 4. There were no consistent phase leads or lags except for a torsional phase lead of up to 10°, usually more marked for clockwise head rotation versus counterclockwise, and for oblique axis rotations versus purely torsional. 5. Torsional gain was magnified, averaging -0.52, when the torsional component of head rotation was only a small part of a predominantly vertical or horizontal rotation, i.e., when the axis of head rotation was near the frontal plane. Because most natural head rotations occur about such axes, the torsional VOR is probably somewhat stronger than the response to pure torsion would suggest. 6. The speed of eye rotation in response to a given stimulus varied widely among subjects, but the direction of rotation was much more uniform. For head rotations about oblique axes out of the frontal plane, there was a systematic misalignment of eye and head axes, with eye axes tilted toward the frontal plane. These findings can be explained on the basis of a strategy where the VOR balances the muscular effort of rotating the eyes against the cost of retinal slip.
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