Geometry and Geodesy on the Primary Visual Cortex as a Surface of Revolution

Lorenzo G Resca,Nicholas A Mecholsky

doi:10.3390/mca25040064

Abstract

Biological mapping of the visual field from the eye retina to the primary visual cortex, also known as occipital area V1, is central to vision and eye movement phenomena and research. That mapping is critically dependent on the existence of cortical magnification factors. Once unfolded, V1 has a convex three-dimensional shape, which can be mathematically modeled as a surface of revolution embedded in three-dimensional Euclidean space. Thus, we solve the problem of differential geometry and geodesy for the mapping of the visual field to V1, involving both isotropic and non-isotropic cortical magnification factors of a most general form. We provide illustrations of our technique and results that apply to V1 surfaces with curve profiles relevant to vision research in general and to visual phenomena such as ‘crowding’ effects and eye movement guidance in particular. From a mathematical perspective, we also find intriguing and unexpected differential geometry properties of V1 surfaces, discovering that geodesic orbits have alternative prograde and retrograde characteristics, depending on the interplay between local curvature and global topology.

Highlights

Since ground-breaking work of Hubel and Wiesel [1], and Daniel and Whitteridge [2], consideration of mapping of the visual field from the eye retina to visual cortex has become increasingly relevant in psychophysics studies of vision in general and eye movements in particular
It turns out that our exact solution of that problem technically circumvents any need of Ricci calculus, let alone any use of sub-Riemannian fiber bundle models designed to account for V1 architecture and connections between cells [37,38,39,40], which we cannot investigate at this time
Visual crowding derives from the breakdown of one’s ability to identify peripheral objects in the presence of other nearby objects. This critically depends on cortical magnification of the visual field in the primary visual cortex, or V1 area [2,3,5,10,27,28,29,30]

Summary

Introduction

Since ground-breaking work of Hubel and Wiesel [1], and Daniel and Whitteridge [2], consideration of mapping of the visual field from the eye retina to visual cortex has become increasingly relevant in psychophysics studies of vision in general and eye movements in particular. Pursuing general relativity studies [6,7,8,9], we developed geodetic procedures that result in first-order differential equations obeyed by geodesic curves over regular two-dimensional (2D) surfaces of revolution, S, embedded in ordinary three-dimensional (3D) Euclidean space, E3. That is a complete formulation and solution of the differential geometry and geodesy problem pertaining to conformal diffeomorphisms between the unit sphere, S2, representing the visual field, and surfaces of revolutions, such as V1 We consider both isotropic and non-isotropic CMF’s. Our geodetic technique applies generally to any regular surface of revolution, derived from a plane profile curve rigidly rotated around a z-axis, providing azimuthal symmetry It produces first-order geodesic-orbit differential equations that can be readily solved by quadrature or numerical integration. Geodetic patterns and their analysis on V1 may critically inform studies of eye movement guidance in visual search [13,14,15,16,17,18,19,20,21,22,23,24,25,26]

Visual Cortical Surfaces

From Ricci Calculus to Geodesic Orbits on Surfaces of Revolution

Geodesy on the Primary Visual Cortex and Diffeomorphic Surfaces of Revolution

Illustrations

Conclusions