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This book provides a brief but accessible introduction to a set of related, mathematical ideas that have proved useful in understanding the brain and behaviour.
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.
Acknowledgements
Mind and Brain
Objects in Mind
Objects on the Brain
Using Your Grey Matter
About This Book
Biological Objects
Configuration Spaces
Tangent Spaces
Moving Around
Measurements
Linear Transformations
Principal Components Analysis
Hebbian Learning
Anti-Hebbian Learning
From Local to Global
Critical Points
Receptive and Association Fields
Bump Functions
Gradient Space
Actions
Slow-Fast Systems
Velocity Commands
Deciding What to interact With
Unexpected Consequences of Nonlinear Behaviour
Brain and Body
Mechanics of Biological Tissue
Linear Behaviour
Nonlinear Behaviour
Production of Voice Sounds
Heteroclinic Cycles
Analysis of Experimental Measurements
Delay Embedding
An Unstable Neural Pathway
Topological Data Analysis
Where Are We Going With All This?
Review of the Geometric Approach
Alternative Approaches
Mathematics for Neurons
Further Reading
Brain
Geometry
Appendix: Background Material
Brains
Calculus
Glossary of Terms with Hidden Assumptions