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Elements of Neurogeometry Functional Architectures of Vision Lecture Notes in Morphogenesis Lecture Notes in Morphogenesis Series Editor: Alessandro Sarti Jean Petitot Elements of Neurogeometry Functional Architectures of Vision Lecture Notes in Morphogenesis Series editor Alessandro Sarti, CAMS Center for Mathematics, CNRS-EHESS, Paris, France e-mail: [email protected] More information about this series at http://www.springer.com/series/11247 Jean Petitot Elements of Neurogeometry Functional Architectures of Vision 123 Jean Petitot CAMS, EHESS Paris France Translated by Stephen Lyle ISSN 2195-1934 ISSN 2195-1942 (electronic) Lecture Notes in Morphogenesis ISBN 978-3-319-65589-5 ISBN 978-3-319-65591-8 (eBook) DOI 10.1007/978-3-319-65591-8 Library of Congress Control Number: 2017950247 Translation from the French language edition: Neurogéométrie de la vision by Jean Petitot, © Les Éditions de l’École Polytechnique 2008. All Rights Reserved © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Of Interest for Neurogeometry by Jean Petitot o Les Catastrophes de la Parole. De Roman Jakobson à René Thom, Maloine, Paris, 1985. o Morphogenesis of Meaning, (trans. F. Manjali), Peter Lang, Bern, 2003. o Neurogéométrie de la vision. Modèles mathématiques et physiques des archi- tectures fonctionnelles, Les Éditions de l’École Polytechnique, Distribution Ellipses, Paris, 2008. o Cognitive Morphodynamics. Dynamical Morphological Models of Constituency in Perception and Syntax, (with R. Doursat), Peter Lang, Bern, 2011. ****** o (ed.) Logos et Théorie des Catastrophes, Colloque de Cerisy à partir de l’œuvre de René Thom, Éditions Patiño, Genève, 1988. o (ed., with F. Varela, J.-M. Roy and B. Pachoud) Naturalizing Phenomenology. Issues in Contemporary Phenomenology and Cognitive Science, Stanford University Press, 1999. o (ed. with J. Lorenceau), “Neurogeometry and Visual Perception”, Special issue of the Journal of Physiology-Paris, 97, 2003. o (ed., with M. Bitbol and P. Kerszberg) Constituting Objectivity. Transcendental Perspectives on Modern Physics, The Western Ontario Series in Philosophy of Science, vol. 74, Springer: Berlin, New York, 2009. o (ed. with A. Sarti and G. Citti), “Neuromathematics of vision”, Special Issue of the Journal of Physiology-Paris, 103, 1–2, 2009. v Contents 1 Preface .................................................. 1 1.1 The Goal of This Work ................................ 1 1.2 An Outline of This Work ............................... 3 1.2.1 Outline of the First Volume ...................... 3 1.2.2 Some Remarks Concerning the Second Volume ....... 7 1.2.3 Limits of This Investigation ...................... 9 1.3 History, Context, and Acknowledgements .................. 10 References................................................ 19 2 Introduction.............................................. 21 2.1 Origin of Space and Neurogeometry....................... 21 2.1.1 Geometric, Physical, and Sensorimotor Conceptions of Space ..................................... 21 2.1.2 The Neurogeometric Approach .................... 24 2.2 Perceptual Geometry, Neurogeometry, and Gestalt Geometry.... 25 2.3 Geometry’s ‘Twofold Way’ ............................. 27 2.4 Idealities and Material Processes ......................... 28 2.5 Mathematical Prerequisites and the Nature of Models ......... 30 2.6 Mathematical Structures and Biophysical Data ............... 32 2.7 Levels of Investigation: Micro, Meso, and Macro ............ 33 2.8 The Context of Cognitive Science ........................ 34 2.9 Complex Systems and the Physics of the Mental ............. 36 2.10 The Philosophical Problem of Cognitive Science ............. 37 2.11 Some Examples ...................................... 38 2.11.1 The Gestalt Concept of Good Continuation........... 38 2.11.2 Kanizsa’s Illusory Contours....................... 39 2.11.3 Entoptic Phenomena ............................ 40 2.11.4 The Cut Locus................................. 41 References................................................ 42 vii viii Contents 3 Receptive Fields and Profiles, and Wavelet Analysis ............. 45 3.1 Structure of the Retino-Geniculo-Cortical Visual Pathways ..... 45 3.2 Receptive Fields and Receptive Profiles .................... 50 3.2.1 Structure of the Retina .......................... 50 3.2.2 Neurons and Action Potentials .................... 51 3.2.3 Structure of the Photoreceptors .................... 52 3.2.4 Ganglion Cells................................. 58 3.2.5 Retinal Colour Coding Circuitry ................... 59 3.2.6 General Receptive Fields and Neural Coding ......... 62 3.3 Visual Neurons as Filters ............................... 73 3.3.1 Gabor Wavelets and Derivatives of Gaussians......... 73 3.3.2 Steerable Filters................................ 76 3.3.3 Linearity Versus Nonlinearity ..................... 77 3.3.4 Visual Neurons as Convolution Operators............ 79 3.3.5 Fine Orientation Discrimination.................... 84 3.4 Vision and Wavelets................................... 85 3.4.1 Fourier, Gabor, and Wavelets ..................... 85 3.4.2 Wavelets and Group Representation ................ 88 3.4.3 Wavelets and Discontinuities...................... 89 3.4.4 Redundancy of Wavelets ......................... 89 3.4.5 Compression and Geometry....................... 92 3.4.6 Matching Pursuit and Rank Coding................. 92 3.5 Feature Detectors ..................................... 94 3.6 Receptive Profiles and Information Theory.................. 95 3.6.1 Signal Decorrelation and Efficient Coding............ 95 3.6.2 Receptive Profiles and Natural Images .............. 97 3.7 Signal Processing and Geometrical Formatting ............... 101 3.8 Grid Cells and Place Cells .............................. 102 3.8.1 Spatial Navigation .............................. 102 3.8.2 Place Cells.................................... 103 3.8.3 Grid Cells .................................... 103 3.8.4 Head Direction Cells ............................ 107 3.8.5 Implementing the Tangent Bundle.................. 108 References................................................ 108 4 Functional Architecture I: The Pinwheels of V1 ................. 113 4.1 The Areas of the Visual Cortex .......................... 114 4.2 Hypercolumnar Structure of the V1 Area ................... 120 4.3 V1 as a Mesoscopic Fibration............................ 127 4.3.1 ‘Bridging Scales’: The Mesoscopic Level ............ 127 4.3.2 Fibrations and Engrafted Variables ................. 127 4.3.3 Fibre Bundles ................................. 129 4.3.4 V1 as a Geometric Fibre Bundle ................... 130 4.3.5 V1 as a 1-jet Fibre Bundle ....................... 132 Contents ix 4.3.6 Legendrian Lifts ............................... 135 4.3.7 Integrability Condition........................... 135 4.3.8 SEð2Þ Invariance of 1-jets ........................ 136 4.3.9 Generalizing the Model .......................... 137 4.3.10 Neurophysiology and Its Geometrical Idealization...... 138 4.4 The Pinwheel Structure of V1 ........................... 138 4.4.1 Observation of Pinwheels ........................ 138 4.4.2 Limitations of This Analysis ...................... 147 4.4.3 Functional Maps as Fields........................ 147 4.4.4 Development of Pinwheels ....................... 151 4.4.5 Pinwheels and Evolution ......................... 156 4.4.6 End Points and Triple Points...................... 158 4.4.7 Distortions and Defects in the Neighbourhood of the V1=V2 Boundary .............................. 162 4.5 Topological Universality of Pinwheels ..................... 165 4.6 Pinwheels as Phase Fields .............................. 170 4.6.1 Fields and Coordinates .......................... 170 4.6.2 Singularities of a Phase Field ..................... 171 4.6.3 Orientation and Iso-orientation Fields ............... 173 4.6.4 Topological Charge and Index..................... 174 4.6.5 Current, Vorticity, and Divergence ................. 174 4.6.6 Helmholtz Equation............................. 179 4.6.7 Illustration .................................... 180 4.6.8 Current Conservation............................ 187 4.6.9 Critical Points
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