The dynamical variables represent the change of average firing rate from resting level of activity. The firing rate of the afferents has a tonic component proportional to the velocity and a phasic component proportional to the acceleration of movement. Physiologically, the afferents have a wide range of phasic and tonic amplitudes. This is reflected by a wide selection of parameters in the numerators in the boxes of fig. 3 and fig. 4. The Laplace transform of the integration operator in equation (3) of the eye position coordinates is . Following Robinson (Robinson81), we modeled the neural integrator with a gain and a time constant of 20 seconds. We therefore replaced the pure integrator with in the calculations of eye position. The term
Figure: Comparison between the dynamical model and monkey data. The dotted lines show the effect of viewing distance and location of the axis of rotation on the VOR as recorded by Snyder & King (1992) from monkeys in the dark. The average eye velocity response (of left and right eye) to a sudden change in head velocity is shown for different target distances (left) and rotational axes (right). On the left, the location of the axis of rotation was in the midsagittal plane 12.5 cm behind the eyes (-12.5 cm), and the target distance was varied between 220 cm and 9 cm. On the right, the target distance was kept constant at 9 cm in front of the eye, and the location of the axis of rotation was varied from 14 cm behind to 4 cm in front of the eyes (-14 cm to 4 cm) in the midsagittal plane. The solid lines show the model responses. The model replicates many characteristics of the data. On the left the model captures the eye velocity fluctuations between 20-50 ms, followed by a decrease and an increase which are both modulated with target distance (50-80 ms). The later phase of the response (80-100 ms) is almost exact for 220 cm, and one peak is seen at the appropriate location for the other distances. On the right the closest fits were obtained for the 4 cm and 0 cm locations. The mean values are in good agreement and the waveforms are close, but could be shifted in time for the other locations of the axis of rotations. Finally, the latest peak ( 100 ms) in the data appears in the model for -14 cm and 9 cm location.
The dynamical model is based on the assumption that the cerebellum is required for context modulation, and that because of its architecture, the cerebellum is more likely to implement complex functions of multiple signals than other relevant nuclei. The major contributions of vergence and eye position modulation on the VOR are therefore mediated by the cerebellum. Smaller and more transient contributions from eye position are assumed to be mediated through the vestibular nucleus as shown in fig. 4. The motivation for combining eye position as in fig. 4 are, first, the evidence for eye response oscillations; second, the theoretical consideration that linear movement information () is useless without eye position information for proper VOR.
The parameters in the dynamical model were adjusted by hand after observing the behavior of the different components of the model and noting how these combine to produce the oscillations observed in the data.
Even though the number of parameters in the model is not small, it was not possible to fit any single response in fig. 5 without affecting most of the other eye responses. This puts severe limits on the set of parameters allowed in the model.
The dynamical model suggests that the oscillations present in the data reflect: 1) important acceleration components in the neural signals, both rotational and linear, 2) different time delays between the canal and otolith signal processing, and 3) antagonistic or synergistic action of the canal and otolith signals with different axes of rotation, as described by the two terms in the bracket of equation (2).