Optimality of position commands to horizontal eye muscles: A test of the minimum-norm rule

Six muscles control the position of the eye, which has three degrees of freedom. Daunicht proposed an optimization rule for solving this redundancy problem, whereby small changes in eye position are maintained by the minimum possible change in motor commands to the eye (the minimum-norm rule). The present study sought to test this proposal for the simplified one-dimensional case of small changes in conjugate eye position in the horizontal plane. Assuming such changes involve only the horizontal recti, Daunicht's hypothesis predicts reciprocal innervation with the size of the change in command matched to the strength of the recipient muscle at every starting position of the eye. If the motor command to a muscle is interpreted as the summed firing rate of its oculomotor neuron (OMN) pool, the minimum-norm prediction can be tested by comparing OMN firing rates with forces in the horizontal recti. The comparison showed 1) for the OMN firing rates given by Van Gisbergen and Van Opstal and the muscle forces given by Robinson, there was good agreement between the minimum-norm prediction and experimental observation over about a +/-30 degrees range of eye positions. This fit was robust with respect to variations in muscle stiffness and in methods of calculating muscle innervation. 2) Other data sets gave different estimates for the range of eye-positions within which the minimum-norm prediction held. The main sources of variation appeared to be disagreement about the proportion of OMNs with very low firing-rate thresholds (i.e., less than approximately 35 degrees in the OFF direction) and uncertainty about eye-muscle behavior for extreme (>30 degrees ) positions of the eye. 3) For all data sets, the range of eye positions over which the minimum-norm rule applied was determined by the pattern of motor-unit recruitment inferred for those data. It corresponded to the range of eye positions over which the size principle of recruitment was obeyed by both agonist and antagonist muscles. It is argued that the current best estimate of the oculomotor range over which minimum-norm control could be used for conjugate horizontal eye position is approximately +/-30 degrees. The uncertainty associated with this estimate would be reduced by obtaining unbiased samples of OMN firing rates. Minimum-norm control may result from reduction of the image movement produced by noise in OMN firing rates.