Overview
- Intuitive explanation of mathematical ideas
- Example applications from enduring areas of neuroscience research
- Self-contained, with diagrams of all key insights
Part of the book series: Lecture Notes in Morphogenesis (LECTMORPH)
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Table of contents (9 chapters)
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About this book
If you record the eye movements of a group of people watching a riverside scene then some will look at the river, some will look at the barge by the side of the river, some will look at the people on the bridge, and so on, but if a duck takes off then everybody will look at it. How come the brain is so adept at processing such biological objects? In this book it is shown that brains are especially suited to exploiting the geometric properties of such objects.
Central to the geometric approach is the concept of a manifold, which extends the idea of a surface to many dimensions. The manifold can be specified by collections of n-dimensional data points or by the paths of a system through state space. Just as tangent planes can be used to analyse the local linear behaviour of points on a surface, so the extension to tangent spaces can be used to investigate the local linear behaviour of manifolds. The majority of the geometric techniques introduced are all about how to do things with tangent spaces.
Examples of the geometric approach to neuroscience include the analysis of colour and spatial vision measurements and the control of eye and arm movements. Additional examples are used to extend the applications of the approach and to show that it leads to new techniques for investigating neural systems. An advantage of following a geometric approach is that it is often possible to illustrate the concepts visually and all the descriptions of the examples are complemented by comprehensively captioned diagrams.
The book is intended for a reader with an interest in neuroscience who may have been introduced to calculus in the past but is not aware of the many insights obtained by a geometric approach to the brain. Appendices contain brief reviews of the required background knowledge in neuroscience and calculus.
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Bibliographic Information
Book Title: Mathematical Tools for Neuroscience
Book Subtitle: A Geometric Approach
Authors: Richard A. Clement
Series Title: Lecture Notes in Morphogenesis
DOI: https://doi.org/10.1007/978-3-030-98495-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
Hardcover ISBN: 978-3-030-98494-6Published: 22 April 2022
Softcover ISBN: 978-3-030-98497-7Published: 22 April 2023
eBook ISBN: 978-3-030-98495-3Published: 21 April 2022
Series ISSN: 2195-1934
Series E-ISSN: 2195-1942
Edition Number: 1
Number of Pages: X, 162
Number of Illustrations: 15 b/w illustrations, 106 illustrations in colour
Topics: Mathematical and Computational Biology, Neurosciences, Biostatistics