Elements of Neurogeometry
Elements of Neurogeometry

Author: Jean Petitot

Publisher: Springer

Published: 2017-11-08

Total Pages: 388

ISBN-13: 3319655914

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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.

Geometric Science of Information
Geometric Science of Information

Author: Frank Nielsen

Publisher: Springer Nature

Published: 2021-07-14

Total Pages: 929

ISBN-13: 3030802094

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This book constitutes the proceedings of the 5th International Conference on Geometric Science of Information, GSI 2021, held in Paris, France, in July 2021. The 98 papers presented in this volume were carefully reviewed and selected from 125 submissions. They cover all the main topics and highlights in the domain of geometric science of information, including information geometry manifolds of structured data/information and their advanced applications. The papers are organized in the following topics: Probability and statistics on Riemannian Manifolds; sub-Riemannian geometry and neuromathematics; shapes spaces; geometry of quantum states; geometric and structure preserving discretizations; information geometry in physics; Lie group machine learning; geometric and symplectic methods for hydrodynamical models; harmonic analysis on Lie groups; statistical manifold and Hessian information geometry; geometric mechanics; deformed entropy, cross-entropy, and relative entropy; transformation information geometry; statistics, information and topology; geometric deep learning; topological and geometrical structures in neurosciences; computational information geometry; manifold and optimization; divergence statistics; optimal transport and learning; and geometric structures in thermodynamics and statistical physics.

Neuromathematics of Vision
Neuromathematics of Vision

Author: Giovanna Citti

Publisher: Springer Science & Business Media

Published: 2014-02-08

Total Pages: 378

ISBN-13: 3642344445

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This book is devoted to the study of the functional architecture of the visual cortex. Its geometrical structure is the differential geometry of the connectivity between neural cells. This connectivity is building and shaping the hidden brain structures underlying visual perception. The story of the problem runs over the last 30 years, since the discovery of Hubel and Wiesel of the modular structure of the primary visual cortex, and slowly cams towards a theoretical understanding of the experimental data on what we now know as functional architecture of the primary visual cortex. Experimental data comes from several domains: neurophysiology, phenomenology of perception and neurocognitive imaging. Imaging techniques like functional MRI and diffusion tensor MRI allow to deepen the study of cortical structures. Due to this variety of experimental data, neuromathematematics deals with modelling both cortical structures and perceptual spaces. From the mathematical point of view, neuromathematical call for new instruments of pure mathematics: sub-Riemannian geometry models horizontal connectivity, harmonic analysis in non commutative groups allows to understand pinwheels structure, as well as non-linear dimensionality reduction is at the base of many neural morphologies and possibly of the emergence of perceptual units. But at the center of the neurogeometry is the problem of harmonizing contemporary mathematical instruments with neurophysiological findings and phenomenological experiments in an unitary science of vision. The contributions to this book come from the very founders of the discipline.

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging
Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

Author: Ke Chen

Publisher: Springer Nature

Published: 2023-02-24

Total Pages: 1981

ISBN-13: 3030986616

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This handbook gathers together the state of the art on mathematical models and algorithms for imaging and vision. Its emphasis lies on rigorous mathematical methods, which represent the optimal solutions to a class of imaging and vision problems, and on effective algorithms, which are necessary for the methods to be translated to practical use in various applications. Viewing discrete images as data sampled from functional surfaces enables the use of advanced tools from calculus, functions and calculus of variations, and nonlinear optimization, and provides the basis of high-resolution imaging through geometry and variational models. Besides, optimization naturally connects traditional model-driven approaches to the emerging data-driven approaches of machine and deep learning. No other framework can provide comparable accuracy and precision to imaging and vision. Written by leading researchers in imaging and vision, the chapters in this handbook all start with gentle introductions, which make this work accessible to graduate students. For newcomers to the field, the book provides a comprehensive and fast-track introduction to the content, to save time and get on with tackling new and emerging challenges. For researchers, exposure to the state of the art of research works leads to an overall view of the entire field so as to guide new research directions and avoid pitfalls in moving the field forward and looking into the next decades of imaging and information services. This work can greatly benefit graduate students, researchers, and practitioners in imaging and vision; applied mathematicians; medical imagers; engineers; and computer scientists.

Perceptual Organization in Computer and Biological Vision
Perceptual Organization in Computer and Biological Vision

Author: James Elder

Publisher: Frontiers Media SA

Published: 2024-08-22

Total Pages: 220

ISBN-13: 2832553451

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A principal challenge for both biological and machine vision systems is to integrate and organize the diversity of cues received from the environment into the coherent global representations we experience and require to make good decisions and take effective actions. Early psychological investigations date back more than 100 years to the seminal work of the Gestalt school. Yet in the last 50 years, neuroscientific and computational approaches to understanding perceptual organization have become equally important, and a full understanding requires integration of all three approaches. This highly interdisciplinary Research Topic welcomes contributions spanning Computer Science, Psychology, and Neuroscience, with the aim of presenting a single, unified collection that will encourage integration and cross-fertilization across disciplines.

A Critical Reflection on Automated Science
A Critical Reflection on Automated Science

Author: Marta Bertolaso

Publisher: Springer Nature

Published: 2020-02-05

Total Pages: 302

ISBN-13: 3030250016

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This book provides a critical reflection on automated science and addresses the question whether the computational tools we developed in last decades are changing the way we humans do science. More concretely: Can machines replace scientists in crucial aspects of scientific practice? The contributors to this book re-think and refine some of the main concepts by which science is understood, drawing a fascinating picture of the developments we expect over the next decades of human-machine co-evolution. The volume covers examples from various fields and areas, such as molecular biology, climate modeling, clinical medicine, and artificial intelligence. The explosion of technological tools and drivers for scientific research calls for a renewed understanding of the human character of science. This book aims precisely to contribute to such a renewed understanding of science.

Neurodynamics
Neurodynamics

Author: Stephen Coombes

Publisher: Springer Nature

Published: 2023-05-09

Total Pages: 513

ISBN-13: 3031219163

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This book is about the dynamics of neural systems and should be suitable for those with a background in mathematics, physics, or engineering who want to see how their knowledge and skill sets can be applied in a neurobiological context. No prior knowledge of neuroscience is assumed, nor is advanced understanding of all aspects of applied mathematics! Rather, models and methods are introduced in the context of a typical neural phenomenon and a narrative developed that will allow the reader to test their understanding by tackling a set of mathematical problems at the end of each chapter. The emphasis is on mathematical- as opposed to computational-neuroscience, though stresses calculation above theorem and proof. The book presents necessary mathematical material in a digestible and compact form when required for specific topics. The book has nine chapters, progressing from the cell to the tissue, and an extensive set of references. It includes Markov chain models for ions, differential equations for single neuron models, idealised phenomenological models, phase oscillator networks, spiking networks, and integro-differential equations for large scale brain activity, with delays and stochasticity thrown in for good measure. One common methodological element that arises throughout the book is the use of techniques from nonsmooth dynamical systems to form tractable models and make explicit progress in calculating solutions for rhythmic neural behaviour, synchrony, waves, patterns, and their stability. This book was written for those with an interest in applied mathematics seeking to expand their horizons to cover the dynamics of neural systems. It is suitable for a Masters level course or for postgraduate researchers starting in the field of mathematical neuroscience.