Overview
- Illustrates the fascinating interactions between mathematics and neuroscience
- Describes geometrical models of the functional architecture of the visual cortex
- Presents a variety of examples
- Includes supplementary material: sn.pub/extras
Part of the book series: Lecture Notes in Morphogenesis (LECTMORPH)
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About this book
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.
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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
Table of contents (6 chapters)
Authors and Affiliations
About the author
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
Book Title: Elements of Neurogeometry
Book Subtitle: Functional Architectures of Vision
Authors: Jean Petitot
Series Title: Lecture Notes in Morphogenesis
DOI: https://doi.org/10.1007/978-3-319-65591-8
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2017
Hardcover ISBN: 978-3-319-65589-5Published: 24 November 2017
Softcover ISBN: 978-3-319-88063-1Published: 31 August 2018
eBook ISBN: 978-3-319-65591-8Published: 08 November 2017
Series ISSN: 2195-1934
Series E-ISSN: 2195-1942
Edition Number: 1
Number of Pages: XV, 379
Number of Illustrations: 71 b/w illustrations, 186 illustrations in colour
Additional Information: Original French edition published by Les Éditions de l’École Polytechnique, Palaiseau, 2008
Topics: Mathematical and Computational Biology, Mathematical Models of Cognitive Processes and Neural Networks, Geometry