The V1-V2-V3 complex: quasiconformal dipole maps in primate striate and extra-striate cortex

The mapping function w = k log(z + a) is a widely accepted approximation to the topographic structure of primate V1 foveal and parafoveal regions. A better model, at the cost of an additional parameter, captures the full field topographic map in terms of the dipole map function w = k log[(z + a)/(z + b)]. However, neither model describes topographic shear since they are both explicitly complex-analytic or conformal. In this paper, we adopt a simple ansatz for topographic shear in V1, V2, and V3 that assumes that cortical topographic shear is rotational, i.e. a compression along iso-eccentricity contours. We model the constant rotational shear with a quasiconformal mapping, the wedge mapping. Composing this wedge mapping with the dipole mapping provides an approximation to V1, V2, and V3 topographic structure, effectively unifying all three areas into a single V1-V2-V3 complex using five independent parameters. This work represents the first full-field, multi-area, quasiconformal model of striate and extra-striate topographic map structure.

[1]  D. Hubel,et al.  The pattern of ocular dominance columns in macaque visual cortex revealed by a reduced silver stain , 1975, The Journal of comparative neurology.

[2]  L. Ahlfors.  Complex analysis : an introduction to the theory of analytic functions of one complex variable / Lars V. Ahlfors , 1984 .

[3]  H. Hochstadt.  Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable; 3rd ed. (Lars V. Ahlfors) , 1980 .

[4]  J. Kaas.  Theories of Visual Cortex Organization in Primates , 1997 .

[5]  L. Spillmann,et al.  Visual Perception: The Neurophysiological Foundations , 1989 .

[6]  E. DeYoe,et al.  Mapping striate and extrastriate visual areas in human cerebral cortex. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[7]  G. Glover,et al.  Retinotopic organization in human visual cortex and the spatial precision of functional MRI. , 1997, Cerebral cortex.

[8]  R Gattass,et al.  Representation of the visual field in the second visual area in the Cebus monkey , 1988, The Journal of comparative neurology.

[9]  J. Allman,et al.  The dorsomedial cortical visual area: A third tier area in the occipital lobe of the owl monkey (aotus trivirgatus) , 1975, Brain Research.

[10]  G. Blasdel,et al.  Functional Retinotopy of Monkey Visual Cortex , 2001, The Journal of Neuroscience.

[11]  Eric L. Schwartz,et al.  Design considerations for a space-variant visual sensor with complex-logarithmic geometry , 1990, [1990] Proceedings. 10th International Conference on Pattern Recognition.

[12]  George T. Symm,et al.  An integral equation method in conformal mapping , 1966 .

[13]  J. Kaas,et al.  Evidence for a Modified V3 with Dorsal and Ventral Halves in Macaque Monkeys , 2002, Neuron.

[14]  E. Schwartz,et al.  On the mathematical structure of the visuotopic mapping of macaque striate cortex. , 1985, Science.

[15]  Eric L. Schwartz,et al.  Anatomical and physiological correlates of visual computation from striate to infero-temporal cortex , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[16]  Anna W. Roe,et al.  The Functional Architecture of Area V2 in the Macaque Monkey , 1997 .

[17]  B M Dow,et al.  The mapping of visual space onto foveal striate cortex in the macaque monkey , 1985, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[18]  J W Belliveau,et al.  Borders of multiple visual areas in humans revealed by functional magnetic resonance imaging. , 1995, Science.

[19]  E. Switkes,et al.  Deoxyglucose analysis of retinotopic organization in primate striate cortex. , 1982, Science.

[20]  H. Helmholtz.  Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. , 1858 .

[21]  L. Ahlfors,et al.  Lectures on quasiconformal mappings , 1966 .

[22]  D. Whitteridge,et al.  The representation of the visual field on the cerebral cortex in monkeys , 1961, The Journal of physiology.

[23]  J. Horton,et al.  Quadrantic visual field defects. A hallmark of lesions in extrastriate (V2/V3) cortex. , 1991, Brain : a journal of neurology.

[24]  Alex T. Pang.  Line-drawing algorithms for parallel machines , 1990, IEEE Computer Graphics and Applications.

[25]  Lars V. Ahlfors.  An introduction to the theory of analytic functions of one complex variable , 1978 .

[26]  Eric L. Schwartz,et al.  Computing Minimal Distances on Polyhedral Surfaces , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[27]  T. ATION.  COMPUTING MINIMAL DISTANCES ON ARBITRARY POLYHEDRAL SURFACES , .

[28]  Tristan Needham,et al.  Visual Complex Analysis , 1997 .

[29]  Kathleen S. Rockland,et al.  Primary Visual Cortex in Primates , 1994, Cerebral Cortex.

[30]  A. M. Dale,et al.  BORDERS OF MULTIPLE VISUAL AREAS IN HUMANS REVEALED BY FUNCTIONAL MRI , 1995 .

[31]  John H. R. Maunsell,et al.  The visual field representation in striate cortex of the macaque monkey: Asymmetries, anisotropies, and individual variability , 1984, Vision Research.

[32]  J. Kaas,et al.  Extrastriate Cortex in Primates , 1997, Cerebral Cortex.

[33]  L. Ahlfors,et al.  RIEMANN'S MAPPING THEOREM FOR VARIABLE METRICS* , 1960 .

[34]  Lars Valerian Ahlfors.  Conformality with Respect to Riemannian Metrics , 1982 .

[35]  Hugh R. Wilson,et al.  10 – THE PERCEPTION OF FORM: Retina to Striate Cortex , 1989 .

[36]  J. Kaas,et al.  Connectional and Architectonic Evidence for Dorsal and Ventral V3, and Dorsomedial Area in Marmoset Monkeys , 2001, The Journal of Neuroscience.

[37]  Eric L. Schwartz,et al.  Computational Studies of the Spatial Architecture of Primate Visual Cortex , 1994 .

[38]  Eric L. Schwartz,et al.  Computational anatomy and functional architecture of striate cortex: A spatial mapping approach to perceptual coding , 1980, Vision Research.

[39]  Eric L. Schwartz,et al.  Conformal image warping , 2018, IEEE Computer Graphics and Applications.