Biomedical subjects

C Schnabolk

Publications and source records attributed to C Schnabolk.

3 recordsLinked to original sources

PubMed1994

Modeling three-dimensional velocity-to-position transformation in oculomotor control.

1. A considerable amount of attention has been devoted to understanding the velocity-position transformation that takes place in the control of eye movements in three dimensions. Much of the work has focused on the idea that rotations in three dimensions do not commute and that a "multiplicative quaternion model" of velocity-position integration is necessary to explain eye movements in three dimensions. Our study has indicated that this approach is not consistent with the physiology of the types of signals necessary to rotate the eyes. 2. We developed a three-dimensional dynamical system model for movement of the eye within its surrounding orbital tissue. The main point of the model is that the eye muscles generate torque to rotate the eye. When the eye reaches an orientation such that the restoring torque of the orbital tissue counterbalances the torque applied by the muscles, a unique equilibrium point is reached. The trajectory of the eye to reach equilibrium may follow any path, depending on the starting eye orientation and eye velocity. However, according to Euler's theorem, the equilibrium reached is equivalent to a rotation about a fixed axis through some angle from a primary orientation. This represents the shortest path that the eye could take from the primary orientation in reaching equilibrium. Consequently, it is also the shortest path for returning the eye to the primary orientation. Thus the restoring torque developed by the tissue surrounding the eye was approximated as proportional to the product of this angle and a unit vector along this axis. The relationship between orientation and restoring torque gives a unique torque-orientation relationship. 3. Once the appropriate torque-orientation relationship for eye rotation is established the velocity-position integrator can be modeled as a dynamical system that is a direct extension of the one-dimensional velocity-position integrator. The linear combination of the integrator state and a direct pathway signal is converted to a torque signal that activates the muscles to rotate the eyes. Therefore the output of the integrator is related to a torque signal that positions the eyes. It is not an eye orientation signal. The applied torque signal drives the eye to an equilibrium orientation such that the restoring torque equals the applied torque but in the opposite direction. The eye orientation reached at equilibrium is determined by the unique torque-orientation relation. Because torque signals are vectors, they commute.(ABSTRACT TRUNCATED AT 400 WORDS)

Abducens Nerve↗

PubMed1992

Modeling 3-D slow phase velocity estimation during off-vertical-axis rotation (OVAR).

Off-vertical-axis rotation (OVAR) in darkness generates continuous compensatory eye velocity. No model has yet been presented that defines the signal processing necessary to estimate head velocity in three dimensions for arbitrary rotations during OVAR. The present study develops a model capable of estimating all 3 components of head velocity in space accurately. It shows that processing of two patterns of otolith activation, one delayed with respect to the other, for each plane of eye movement is not sufficient. (A pattern in this context is an array of signals emanating from the otoliths. Each component of the array is a signal corresponding to a class of otolith hair cells with a given polarization vector as described by Tou and Gonzalez in 1974.) The key result is that estimation of head velocity in space can be achieved by processing three temporally displaced patterns, each representing a sampling of gravity as the head rotates. A vector cross product of differences between pairs of the sampled gravity vectors implements the estimation. An interesting property of this model is that the component of velocity about the axis of rotation reduces to that derived previously using the pattern estimator model described by Raphan and Schnabolk in 1988 and Fanelli et al in 1990. This study suggests that the central nervous system (CNS) maintains a current as well as 2 delayed representations of gravity at every head orientation during rotation. It also suggests that computing vector cross products and implementing delays may be fundamental operations in the CNS for generating orientation information associated with motion.

Head↗

PubMed1988

Modeling slow phase velocity generation during off-vertical axis rotation.

1. Rotation about an off-vertical axis (OVAR) causes continuous unidirectional nystagmus in darkness. An analysis of the dynamics of the nystagmus suggests that the continuous slow-phase velocity is generated by a signal that is an estimate of the velocity of a traveling wave pattern associated with the excitation and inhibition of the cells of the otolith maculae. The estimated velocity signal then excites the velocity storage integrator. 2. A mathematical model has been developed that shows how the velocity of the traveling wave might be estimated from patterns of otolith activation related to head position. The estimation of velocity is based on a "template matching" algorithm. It is assumed that the signal arising in each cell of the macula is delayed by a certain time (T). As the head rotates in the gravitational field, a delayed pattern representing a previous position of the head is available as a "template" that can be compared to the pattern associated with the present position of the head. 3. The delayed signal level for each cell is approximated from the present pattern by a spatial extrapolation in pattern space using information from the given cell and an adjacent one. The value of the displacement that minimizes the mean square error between the extrapolated and the delayed signal values over all cells gives a best estimate of head rotation (d) in time T. The estimated head velocity is proportional to the estimated head displacement (d) and inversely proportional to the delay time (T). 4. By using a linear spatial extrapolation function and assuming a uniformly spaced distribution of polarization vectors over 360 degrees, sinusoidal spatial patterns are obtained. The formula for the estimated head velocity (ŵ) reduces to a sinusoidal function of angular head velocity (w) and delay time (T). For T = 0.85 seconds, the model predicts that the steady state estimate of head velocity will rise as a function of stimulus velocity (w) to a peak value at w = 50 deg/sec. The estimate then declines for larger values of stimulus velocity (w). This type of behavior is observed in the slow-phase velocity characteristics of OVAR in monkeys. 5. The model predicts that when animals are tilted after prolonged rotation about a vertical axis, the estimate of head velocity is delayed relative to actual head velocity. This accounts for the delay in the buildup of slow-phase velocity during the initial second.(ABSTRACT TRUNCATED AT 400 WORDS)

Animals↗