Tensorial computer model of gaze—I. Oculomotor activity is expressed in non-orthogonal natural coordinates

Abstract

The central nervous system expresses its function in natural frames of reference. A most conspicuous feature of such frames is their non-orthogonality. Gaze stabilization and, in particular, the sensorimotor transformations performed by the vestibulo-ocular reflex, are prime examples of such general coordinate transformations between and within multidimensional non-orthogonal frames. Since such operations can be described by tensor formalisms in an abstract manner, this methodology is applied here to develop a tensorial computer model of gaze stabilization. The representation of sensorimotor transformations by a reference-frame independent method obviates the necessity to simplify the intrinsic coordinate systems either by a reduction of the dimensionality or by a presumption of orthogonality. The frames of reference intrinsic to vestibulo-ocular reflex transformation (the vestibular semicircular canals and extraocular muscles) as well as the covariant character of the sensory input and the contravariant character of the motor output are physically obvious. A model built on these intrinsic systems of coordinates first serves to quantitate the degree of non-orthogonality in the extraocular muscle system, and thus to demonstrate both the necessity and the applicability of representing them by a formalism suitable for non-orthogonal systems, such as tensor network theory. The actual non-orthogonality of the gaze-stabilization system can be quantitated on the basis of the difference of covariant and contravariant expressions as follows. Tensor network theory describes sensorimotor transformations by employing a covariant embedding procedure. This, however, yields a covariant intention-type motor vector. If the central nervous system were to transmit these sensory-type components directly to the extraocular muscle motor mechanism, an error-angle would occur since covariants do not physically compose the intended movement. The error in every direction of gaze would be zero only if the extraocular muscle system would constitute an orthogonal set of rotation axes. Otherwise, the error, called refraction angle, is a measure of non-orthogonality. The complexity of the quantitation of non-orthogonality is compounded by the fact that these rotation axes change with the moving eye. Calculation of eye movements, executed both by covariant and contravariant vectors from primary and secondary eye positions, is based on the simplest assumption that the central nervous system establishes the covariant-contravariant transformation in the retinal tangent plane. The results quantitate (1) the significant degree of non-orthogonality in the oculomotor system, (2) the degree of sensitivity of non-orthogonality to the direction of eye movement and to the initial eye position. With the degree of non-orthogonality made explicit, the errors inherent in remanent orthogonal treatments are quantitated. Presenting the differences of covariants and contravariants in the form of eye movement trajectories provides experimentally testable predictions.