This chapter is the first of a pair that examines motor coordinate VOR models based on the Kinematic VOR Model of chapter 3. The Simplified Motor Coordinate VOR Model is presented first and the Full Motor Coordinate VOR Model is presented in chapter 6. Motor coordinates here means that the eye velocity and acceleration equations of the Kinematic VOR Model are written for an eye position vector whose components are expressed in the muscle coordinate system of the eyes. In contrast, the Global VOR Model in chapter 3 used the velocity equation of the Kinematic VOR Model written for an angular eye velocity. The model is simplified in the sense that a simplified muscle coordinate system of the eye in two dimensions was used in the calculations. In the Full Motor Coordinate model, a three-dimensional coordinate system defined by the six extraocular muscles was used in the calculations.
Emphasis is placed on the relevance and precision of simple models derived to explain and reproduce the psychophysical monkey VOR data of Snyder and King (Snyder92b; Snyder personal communication).
Two simple models are presented and their results compared to VOR data recorded from three monkeys. They differ from the Global VOR Model of chapter 3 in many respects. First, the Global VOR Model described the VOR calculations using an angular eye velocity representation while the two simplified models use the velocity of the coordinates of an eye position vector. This has two consequences: 1) it simplifies the VOR computation, and 2) it is closer to the neurobiological representation of eye position. Second, the Global VOR Model only used the velocity equation of the Kinematic VOR Model while the Motor Coordinate models used both the velocity and acceleration equations to model the VOR. In the Global VOR Model, the acceleration signals were determined by seven independent free parameters, which were fitted manually. In the Motor Coordinate models the velocity and acceleration signals were weighted by only two parameters, which were determined by linear regression. Third, the results from the Global VOR model were only compared to the data from one monkey and only the velocity was examined. The Motor Coordinate models, in contrast, are compared to the data from three monkeys in many more conditions and both velocity and acceleration results were examined.
The acceleration equation of the Kinematic VOR Model describes the geometrical relationship of angular and linear head accelerations to the eye position vector acceleration to stabilize targets on the retina. The mathematical expression derived for the acceleration turns out to be much more complex than the velocity equation presented in chapter 3. Nevertheless, it is shown that the only terms that contribute significantly to the eye acceleration responses are the terms homologous to the velocity equation of the Kinematic VOR Model. This may explain why
the encoding of head velocity and acceleration information by the vestibular canals occur together. Indeed, since head velocity and acceleration must obey the same transformation rules to result in an ideal VOR, there is no evolutionary pressure to differentiate between velocity and acceleration information in the encoding by the canals and they are, therefore, encoded together ([Lisberger1994]) .
In addition to the acceleration equation, the mathematical expressions for the VOR retinal slip velocity and acceleration are derived. These expressions determine the velocity and acceleration of a target as they would be observed at the retina during incomplete VOR stabilization. These expressions will be useful in future cerebellar models of VOR adaptation since evidence suggests that VOR adaptation is driven by velocity and acceleration of target visual slip on the retina. This information is reported in the VOR pathways by the climbing fibers of the inferior olive innervating the Purkinje cells in the cerebellum ([Simpson and Alley1974]).
The first Simplified Motor Coordinate model explains 99.3% of the variability of the monkey psychophysical data with only four free parameters (two delays and two regression coefficients). The second Simplified Motor Coordinate model, with nine free parameters (five delays, one gain, one gating factor and two regression coefficients) explains about the same variability in the data, but yields a better qualitative fit than the first model.
The second Simplified Motor Coordinate model,although mathematically more intensive and exact in its description, is closely related to the model outlined by Snyder & King (Snyder96). The main differences are: 1) the presence in our model of a fast eye response with a very short delay, 2) the presence of a modulation with viewing direction, and 3) the absence of a fourth stage between 45 and 100 ms after rotation onset, which, in their model, fully compensated for eye translation. Our precise mathematical description is more complete and allowed us to study finer details of VOR sensorimotor interaction which must be verified in future experiments.
In this chapter, the models were kept simple to get at the most relevant features in the data and in this way avoid overfitting naturally occurring variations present in the data. In retrospect, undesireable oscillations in the head inputs were quite useful in establishing the parameters in the models. This suggests that simple head velocity input signals might not be suitable to construct a complete model of the VOR. Using persistence of excitation as a criterion to estimate whether an open-loop experiment on a linear system is informative ([Ljung1987]), I concluded that the experiment was not fully informative, and that the head acceleration, in contrast to head position and velocity, was the most informative. Thus, more complex time-varying head inputs need to be used in the future to better characterize the VOR dynamics.