Date of Award


Degree Type


Degree Name

Doctor of Philosophy


This study develops and tests a mathematical model for the neural circuits controlling saccades, the rapid eye movements that shift the direction of gaze. These movements are three-dimensional eye rotations, but current models for their control are based on the mathematical properties of one-dimensional rotations. Surprisingly, the properties of three-dimensional rotations are very different, and so one-dimensional concepts often do not apply to real eye movements. Thus, a key hypothesis of current models--that the brain computes eye position by integrating eye velocity signals--is incorrect because in three dimensions the integral of velocity is not position. This thesis describes a simple three-dimensional velocity to position transformation. Similarly, in current models saccades are driven by an error signal: desired minus actual eye position. In three dimensions, however, models using subtractive error computation must readjust the axis of eye rotation, and they violate Listing's law (a constraint on eye position). This thesis describes a multiplicative feedback system that generates fixed-axis saccades fitting Listing's law.;Four-component rotational operators called quaternions are used in the model developed in this study. Experiments on three humans and three monkeys, using a new three-dimensional eye movement measuring technique, confirm three predictions of the quaternion model: (1) Listing's law holds during saccades. (2) The rotation axis of the eye is roughly fixed throughout a saccade. (3) These rotation axes are systematically distributed in different planes depending on current eye position.;The quaternion model can be extended to include the superior colliculus, which produces saccades by sending signals coding desired eye rotation to the brainstem. These signals must be translated from the topographic (spatial) representation used in the colliculus to the firing frequency (temporal) code used downstream. The quaternion superior colliculus model yields a spatiotemporal translation with all the experimentally-observed properties, e.g. activation of multiple sites in the colliculus evokes a vector average (weighted by activity levels) of the saccades coded by the individual sites. Since many brain systems use topographic and frequency codes, this quaternion mechanism for spatiotemporal translation may have wide applicability.



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