Horizontal eye-position with one degree of freedom is controlled by the lateral and medial rectus muscles, which have two. The rule used by the oculomotor control system to solve this redundancy problem is not known. The effectiveness of a plausible candidate rule, namely minimisation of post-movement retinal slip, was tested by modelling an adaptive controller for horizontal eye position that used retinal slip as an error signal.

The adaptive controller was represented by two linear nets, one for each horizontal rectus muscle. Premotor neurones formed the first layer of each net, and oculomotor neurones (OMN's) the second. The OMN pool for an individual muscle contained 100 neurones, each connected to a motor unit: motor unit strengths were derived from the measurements of Meredith and Goldberg (1986). The pools received (i) uniform excitatory input from a single premotor neurone with the same ON-direction, corresponding to position-vestibular-pause cells; and (ii) individual inhibitory input from a single premotor neurone with the opposite ON-direction, corresponding to cells in the nucleus prepositus hypoglossi (the putative integrator for horizontal eye position). On each trial, a random horizontal eye-position was used to generate the firing rates of the premotor neurones. Retinal slip was estimated from the discrepancy between this position and the one adopted by the model, then used as an error signal for a gradient-descent learning rule that adjusted the inhibitory weights. After learning was complete, the resultant simulated firing rates and muscle forces were compared with those observed experimentally.

The main findings were:

1. The firing-rate slopes for the simulated OMN's increased with their thresholds in accordance with electrophysiological findings. This was a consequence of the push-pull arrangement of premotor inputs: higher threshold units received more powerful inhibitory input.
2. With noise-free OMN firing-rates the model learnt accurate position control, but the muscle forces used were dependent on the initial conditions, and in general were unrealistically high.
3. When a noise term, consistent with experimental data (Goldstein and Robinson 1986), was added to OMN firing-rates the model learnt position control and exhibited the following features:
   1. the values for muscle forces were realistic;
   2. firing-rate threshold increased with motor-unit strength (the size principle);
   3. the system displayed minimum-norm (i.e. pseudo-inverse) control over the central ±30 deg of the oculomotor range as is found experimentally (Dean et al. 1999).

It appeared that the learning rule suppressed those units making the largest contribution to the noise-related error, causing the strongest units to have the highest thresholds. Because firing-rate threshold was linked to slope, this in turn meant stronger units had higher firing rate slopes, as required by minimum-norm control. These findings suggest that a learning rule based on image slip applied to a simple architecture is able to generate important properties of OMN's and their motor units.