Elements of Neurogeometry

Description

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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“Elements of Neurogeometry” by Jean Petitot delves deep into the realm of neurogeometry, a field that seeks to understand the geometric structures underlying neural processes. Petitot offers a comprehensive account of how the brain perceives and processes geometric information. Petitot’s work is not just a scientific exposition but also a philosophical treatise. He argues that the brain’s geometric processing is not just a passive reception of external stimuli but an active construction. This perspective challenges the traditional Cartesian dualism, suggesting that cognition is deeply intertwined with the very fabric of reality.

Central to Petitot’s theory is the concept of “cortical dynamics.” He posits that the brain’s cortex operates in a manner akin to a dynamic system, where geometric patterns emerge from neural interactions. These patterns, or “neurogeometric structures,” are the brain’s way of representing the external world. One of the book’s significant contributions is the detailed exposition of the mathematical models that underpin neurogeometry. Petitot employs differential geometry and topology to explain how the brain can construct complex geometric representations from simple neural interactions.

Petitot’s work is groundbreaking in its attempt to bridge the gap between abstract mathematical concepts and tangible neural processes. By doing so, he provides a robust framework for understanding geometric cognition, emphasizing that our perception of space and shape is a direct consequence of the brain’s inherent geometric nature. Beyond the theoretical, Petitot touches upon the potential applications of neurogeometry in fields like artificial intelligence and robotics. By understanding how the brain processes geometry, we can potentially design machines that perceive and navigate the world in a manner similar to humans.

While “Elements of Neurogeometry” is undoubtedly a monumental work, it is not without its critics. Some argue that Petitot’s approach is overly mathematical, making it inaccessible to those without a strong background in advanced mathematics. Others feel that while the theory is sound, empirical evidence linking the proposed mathematical models to actual neural processes is lacking.

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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).