Douglas Hanes

Variables Contributing to the Coordination of Rapid Eye/Head Gaze Shifts

In this article results of several published studies are synthesized in order to address the neural system for the determination of eye and head movement amplitudes of horizontal eye/head gaze shifts with arbitrary initial head and eye positions. Target position, initial head position, and initial eye position span the space of physical parameters for a planned eye/head gaze saccade. The principal result is that a functional mechanism for determining the amplitudes of the component eye and head movements must use the entire space of variables. Moreover, it is shown that amplitudes cannot be determined additively by summing contributions from single variables. Many earlier models calculate amplitudes as a function of one or two variables and/or restrict consideration to best-fit linear formulae. Our analysis systematically eliminates such models as candidates for a system that can generate appropriate movements for all possible initial conditions. The results of this study are stated in terms of properties of the response system. Certain axiom sets for the intrinsic organization of the response system obey these properties. We briefly provide one example of such an axiomatic model. The results presented in this article help to characterize the actual neural system for the control of rapid eye/head gaze shifts by showing that, in order to account for behavioral data, certain physical quantities must be represented in and used by the neural system. Our theoretical analysis generates predictions and identifies gaps in the data. We suggest needed experiments.

Motion parallax contribution to perception of self-motion and depth

The object of this study is to mathematically specify important characteristics of visual flow during translation of the eye for the perception of depth and self-motion. We address various strategies by which the central nervous system may estimate self-motion and depth from motion parallax, using equations for the visual velocity field generated by translation of the eye through space. Our results focus on information provided by the movement and deformation of three-dimensional objects and on local flow behavior around a fixated point. All of these issues are addressed mathematically in terms of definite equations for the optic flow. This formal characterization of the visual information presented to the observer is then considered in parallel with other sensory cues to self-motion in order to see how these contribute to the effective use of visual motion parallax, and how parallactic flow can, conversely, contribute to the sense of self-motion. This article will focus on a central case, for understanding of motion parallax in spacious real-world environments, of monocular visual cues observable during pure horizontal translation of the eye through a stationary environment. We suggest that the global optokinetic stimulus associated with visual motion parallax must converge in significant fashion with vestibular and proprioceptive pathways that carry signals related to self-motion. Suggestions of experiments to test some of the predictions of this study are made.

Notes on the Hilbert–Kunz function

Abstract We note certain properties of the Hilbert–Kunz function and Hilbert–Kunz multiplicity, including a strengthened inequality between Hilbert–Kunz and Hilbert–Samuel multiplicities and a characterization of a mixed Hilbert function incorporating both ordinary and Frobenius powers of ideals.

Spatial symmetries in vestibular projections to the uvula-nodulus

The discharge of secondary vestibular neurons relays the activity of the vestibular endorgans, occasioned by movements in three-dimensional physical space. At a slightly higher level of analysis, the discharge of each secondary vestibular neuron participates in a multifiber projection or pathway from primary afferents via the secondary neurons to another neuronal population. The logical organization of this projection determines whether characteristics of physical space are retained or lost.The logical structure of physical space is standardly expressed in terms of the mathematics of group theory. The logical organization of a projection can be compared to that of physical space by evaluating its symmetry group. The direct projection from the semicircular canal nerves via the vestibular nuclei to neck motor neurons has a full three-dimensional symmetry group, allowing it to maintain a three-dimensional coordinate frame. However, a projection may embed only a subgroup of the symmetry group of physical space, which incompletely mirrors the properties of physical space. The major visual and vestibular projections in the rabbit via the inferior olive to the uvula-nodulus carry three degrees of freedom—rotations about one vertical and two horizontal axes—but do not have full three dimensional symmetry. Instead, the vestibulo-olivo-nodular projection has symmetries corresponding to a product of two-dimensional vestibular and one-dimensional optokinetic spaces. This combination of projection symmetries provides the foundation for distinguishing horizontal from vertical rotations within a three dimensional space.In this study, we evaluate the symmetry group given by the physiological organization of the vestibulo-olivo-nodular projection. Although it acts on the same sets of elements and mirrors the rotations that occur in physical space, the physiological transformation group is distinct from the spatial group. We identify symmetries as products of physiological and spatial transformations. The symmetry group shapes the information the projection conveys to the uvula-nodulus; this shaping may depend on a physiological choice of generators, in the same way that function depends on the physiological choice of coordinates. We discuss the implications of the symmetry group for uvula-nodulus function, evolution, and functions of the vestibular system in general.

On the Cohen-Macaulay modules of graded subrings

We give several characterizations for the linearity property for a maximal Cohen-Macaulay module over a local or graded ring, as well as proofs of existence in some new cases. In particular, we prove that the existence of such modules is preserved when taking Segre products, as well as when passing to Veronese subrings in low dimensions. The former result even yields new results on the existence of finitely generated maximal Cohen-Macaulay modules over non-Cohen-Macaulay rings.

BOUNDS ON MULTIPLICITIES OF LOCAL RINGS

ABSTRACT We consider an inequality of C. Lech on the Samuel multiplicities of ideals primary to the maximal ideal of a local ring. We give an analogue of the inequality for Hilbert-Kunz multiplicities in prime characteristic, as well as some improvements on Lechs result for Samuel multiplicities of rings of equal characteristic.

Mathematical requirements of visual–vestibular integration

This article addresses the intersection between perceptual estimates of head motion based on purely vestibular and purely visual sensation, by considering how nonvisual (e.g. vestibular and proprioceptive) sensory signals for head and eye motion can be combined with visual signals available from a single landmark to generate a complete perception of self-motion. In order to do this, mathematical dimensions of sensory signals and perceptual parameterizations of self-motion are evaluated, and equations for the sensory-to-perceptual transition are derived. With constant velocity translation and vision of a single point, it is shown that visual sensation allows only for the externalization, to the frame of reference given by the landmark, of an inertial self-motion estimate from nonvisual signals. However, it is also shown that, with nonzero translational acceleration, use of simple visual signals provides a biologically plausible strategy for integration of inertial acceleration sensation, to recover translational velocity. A dimension argument proves similar results for horizontal flow of any number of discrete visible points. The results provide insight into the convergence of visual and vestibular sensory signals for self-motion and indicate perceptual algorithms by which primitive visual and vestibular signals may be integrated for self-motion perception.