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.

Different animals have different visual systems and so presumably have different ways of seeing. How does the way in which we see depend on the optical, neural and motor components of our visual systems? Originally published in 1993, the mathematical tools needed to answer this question are introduced in this book. Elementary linear algebra is used to describe the transformations of the stimulus that occur in the formation of the optical, neural and motor images in the human visual system. The distinctive feature of the approach is that transformations are specified with enough rigour for readers to be able to set up their own models and generate predictions from them. Underlying the approach of this book is the goal of providing a self-contained source for the derivation of the basic equations of vision science. An introductory section on vector and matrix algebra covers the mathematical techniques which are applied to both sensory and motor aspects of the visual system, and the intervening steps in the mathematical arguments are given in full, in order to make the derivation of the equations easier to follow. A subsidiary goal of this book is to demonstrate the utility of current desktop computer packages which make the application of mathematics very easy. All the numerical results were produced using only a spreadsheet or mathematics package, and example calculations are included in the text.

Different animals have different visual systems and so presumably have different ways of seeing. How does the way in which we see depend on the optical, neural and motor components of our visual systems? Originally published in 1993, the mathematical tools needed to answer this question are introduced in this book. Elementary linear algebra is used to describe the transformations of the stimulus that occur in the formation of the optical, neural and motor images in the human visual system. The distinctive feature of the approach is that transformations are specified with enough rigour for readers to be able to set up their own models and generate predictions from them. Underlying the approach of this book is the goal of providing a self-contained source for the derivation of the basic equations of vision science. An introductory section on vector and matrix algebra covers the mathematical techniques which are applied to both sensory and motor aspects of the visual system, and the intervening steps in the mathematical arguments are given in full, in order to make the derivation of the equations easier to follow. A subsidiary goal of this book is to demonstrate the utility of current desktop computer packages which make the application of mathematics very easy. All the numerical results were produced using only a spreadsheet or mathematics package, and example calculations are included in the text.

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.