Elements of Neurogeometry

Transcription

1 Lecture Notes in Morphogenesis Series Editor: Alessandro Sarti Jean Petitot Elements of Neurogeometry Functional Architectures of Vision

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3 Contents 1 Preface The Goal of This Work An Outline of This Work Outline of the First Volume Some Remarks Concerning the Second Volume Limits of This Investigation History, Context, and Acknowledgements References Introduction Origin of Space and Neurogeometry Geometric, Physical, and Sensorimotor Conceptions of Space The Neurogeometric Approach Perceptual Geometry, Neurogeometry, and Gestalt Geometry Geometry s Twofold Way Idealities and Material Processes Mathematical Prerequisites and the Nature of Models Mathematical Structures and Biophysical Data Levels of Investigation: Micro, Meso, and Macro The Context of Cognitive Science Complex Systems and the Physics of the Mental The Philosophical Problem of Cognitive Science Some Examples The Gestalt Concept of Good Continuation Kanizsa s Illusory Contours Entoptic Phenomena The Cut Locus References vii

4 viii Contents 3 Receptive Fields and Profiles, and Wavelet Analysis Structure of the Retino-Geniculo-Cortical Visual Pathways Receptive Fields and Receptive Profiles Structure of the Retina Neurons and Action Potentials Structure of the Photoreceptors Ganglion Cells Retinal Colour Coding Circuitry General Receptive Fields and Neural Coding Visual Neurons as Filters Gabor Wavelets and Derivatives of Gaussians Steerable Filters Linearity Versus Nonlinearity Visual Neurons as Convolution Operators Fine Orientation Discrimination Vision and Wavelets Fourier, Gabor, and Wavelets Wavelets and Group Representation Wavelets and Discontinuities Redundancy of Wavelets Compression and Geometry Matching Pursuit and Rank Coding Feature Detectors Receptive Profiles and Information Theory Signal Decorrelation and Efficient Coding Receptive Profiles and Natural Images Signal Processing and Geometrical Formatting Grid Cells and Place Cells Spatial Navigation Place Cells Grid Cells Head Direction Cells Implementing the Tangent Bundle References Functional Architecture I: The Pinwheels of V The Areas of the Visual Cortex Hypercolumnar Structure of the V1 Area V1 as a Mesoscopic Fibration Bridging Scales : The Mesoscopic Level Fibrations and Engrafted Variables Fibre Bundles V1 as a Geometric Fibre Bundle V1 as a 1-jet Fibre Bundle

5 Contents ix Legendrian Lifts Integrability Condition SEð2Þ Invariance of 1-jets Generalizing the Model Neurophysiology and Its Geometrical Idealization The Pinwheel Structure of V Observation of Pinwheels Limitations of This Analysis Functional Maps as Fields Development of Pinwheels Pinwheels and Evolution End Points and Triple Points Distortions and Defects in the Neighbourhood of the V1=V2 Boundary Topological Universality of Pinwheels Pinwheels as Phase Fields Fields and Coordinates Singularities of a Phase Field Orientation and Iso-orientation Fields Topological Charge and Index Current, Vorticity, and Divergence Helmholtz Equation Illustration Current Conservation Critical Points Mesogeometry and Microphysics Statistics of Pinwheels as Phase Singularities Pinwheels and Gaussian Fields Evolution of Pinwheels as Phase Singularities Pinwheel Singularities Structure in the Vicinity of Singularities The Problem of Resolution Two-Photon Confocal Microscopy Pinwheels and Blow-ups The Geometric Concept of Blow-up Blow-ups and Lines of Dislocations From Blow-up to Fibre Bundle Discrete Versus Continuous Models Different Aspects of Pinwheels Position Orientation Independence and Local Triviality Other Engrafted Variables Spatial Frequency

6 x Contents Generality of Pinwheels Retinotopic Maps and Their Transversality Pinwheels and Ocular Dominance Independent Maps and Transversality Principle Binocularity Blobs and Colour Functionality of Maps Hemispheres and Callosal Connections Homogeneous and Inhomogeneous Qualities Responses to Homogeneous Surfaces Colour Processing References Functional Architectures II: Horizontal Connections and Contact Structure From Pinwheels to Contact Geometry Horizontal Intracortical Connections Semi-local Structures Parallelism and Coaxiality Integration of Contours and Association Field Some Experimental Facts Pop-out, Perceptual Salience, and the Helmholtz Principle Explanation in Terms of Association Fields Confirmation by fmri Relationship with the Horizontal Connections Discretization of the Contact Structure Binding Comparison with Other Data Some Effects of the Horizontal Connections Contextuality of the Receptive Fields Line-Motion Illusion Contact Structure Integrability Condition and Contact Form Contact Structure as a Cartan Connection Non-integrability of the Contact Structure Polarized Heisenberg Group Scale and Characteristic Vectors Jets, Contact Geometry, and Simplexity Illusory Contours as Sub-Riemannian Geodesics Curvature Detectors and 2-jets Data Curvature, 2-Jets, and Engel Structure Good Continuation and the Statistics of Natural Images

7 Contents xi 5.7 Relationship with Wavelets Structure of the V2 Area Colour and Area V Colour Constancy: Semir Zeki and René Thom Objectivism Versus Subjectivism Motion and the MT Area (V5) Models of Direction and Singularities of Functions Projection of Singularity Lines from V2 tomt Swindale s Model Singularities and Universal Unfoldings Swindale Model (Continued) Neural Morphogenesis and Its Genetic Control Guidance of Axon Connections Transcription Factors and Homeoboxes Some Guidance Factors Neurogenesis of the Retina Retinotopy and Neurogenesis of Visual Pathways Dynamical Models of Neural Guidance References Transition to Volume II Introduction Geodesics of the V J Model The V S Model Geodesics of the V S Model Elastica Revisited Sub-Riemannian Diffusion, Heat Kernel, and Non-commutative Harmonic Analysis Confluence Between V J and V S Models Other Themes References Author Index Subject Index