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© 2017

Elements of Neurogeometry

Functional Architectures of Vision

Benefits

  • Illustrates the fascinating interactions between mathematics and neuroscience

  • Describes geometrical models of the functional architecture of the visual cortex

  • Presents a variety of examples

Book

Part of the Lecture Notes in Morphogenesis book series (LECTMORPH)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Jean Petitot
    Pages 1-20
  3. Jean Petitot
    Pages 21-43
  4. Jean Petitot
    Pages 347-366
  5. Back Matter
    Pages 367-379

About this book

Introduction

This book describes several mathematical models of the primary visual cortex, referring them to a vast ensemble of experimental data and putting forward an original geometrical model for its functional architecture, that is, the highly specific organization of its neural connections. The book spells out the geometrical algorithms implemented by this functional architecture, or put another way, the “neurogeometry” immanent in visual perception. Focusing on the neural origins of our spatial representations, it demonstrates three things: firstly, the way the visual neurons filter the optical signal is closely related to a wavelet analysis; secondly, the contact structure of the 1-jets of the curves in the plane (the retinal plane here) is implemented by the cortical functional architecture; and lastly, the visual algorithms for integrating contours from what may be rather incomplete sensory data can be modelled by the sub-Riemannian geometry associated with this contact structure.

As such, it provides readers with the first systematic interpretation of a number of important neurophysiological observations in a well-defined mathematical framework. The book’s neuromathematical exploration appeals to graduate students and researchers in integrative-functional-cognitive neuroscience with a good mathematical background, as well as those in applied mathematics with an interest in neurophysiology.

Keywords

Functional architecture Association field Helmholtz equation Contact structure Fibre bundle Gaussian field Geodesic Gestalt Heisenberg group Frobenius integrability Legendrian lift Orientation (hyper) column Parallel transport Pinwheel Receptive profile Sub-Riemannian geometry Transversality Universal unfolding of singularity Wavelet

Authors and affiliations

  1. 1.CAMS, EHESSParisFrance

About the authors

Jean Petitot specializes in mathematical modeling in  the cognitive sciences. Former student and teacher at the École Polytechnique, he is currently Professor at the Mathematics Center of the École des Hautes Études en Sciences Sociales in Paris. He is a member of the International Academy of Philosophy of Science.  Having worked for several years on the theory of singularities in differential geometry, he was one of the first to become interested in René Thom's morphodynamic models of visual perception  and phonetics in the 1970s.

He is the author of several books, such as Neurogeomtrie de la vision (École Polytechnique, Ellipses, 2008), ‘Physique du Sen’ (Centre Nationale de la Recherche Scientifique, 1992),  ‘Morphogenèse du Sens’ (Presses Universitaires de France, 1985;  English transl. Peter Lang, 2004), five other books, and more than 300 papers. He is also co-editor of ‘Constituting Objectivity’ (Springer, 2009), ‘Neurogeometry and Visual Perception’ (J. of Physiology-Paris, 2003), ‘Au Nom du Sens’, a tribute to Umberto Eco (Grasset, 2000), ‘Naturalizing Phenomenology’ (Stanford University Press, 1999), ‘Logos et Théorie des Catastrophes’, a tribute to René Thom, (Patiño, 1988). 

Bibliographic information