这本书提供了一个简短但易于理解的介绍，介绍了一系列相关的数学概念，这些概念在理解大脑和行为方面被证明是有用的。
如果你记录了一群人观看河边场景时的眼动，那么一些人会看河，一些人会看河边的驳船，一些人会看桥上的人，等等，但是如果鸭子起飞，那么所有人都会看它。为什么大脑如此擅长处理这样的生物对象？在这本书中，大脑特别适合利用这些物体的几何特性。
几何方法的核心是流形的概念，它将曲面的概念扩展到许多维度。流形可以通过n维数据点的集合或系统通过状态空间的路径来指定。正如切平面可以用来分析曲面上点的局部线性行为一样，切空间的扩展也可以用来研究流形的局部线性行为。介绍的大多数几何技术都是关于如何使用切线空间。
神经科学的几何方法的例子包括颜色和空间视觉测量的分析，以及眼睛和手臂运动的控制。另外的例子被用来扩展该方法的应用，并表明它导致了研究神经系统的新技术。遵循几何方法的一个优点是，通常可以直观地说明概念，并且示例的所有描述都由综合标题图补充。
这本书是为一位对神经科学感兴趣的读者而写的，他可能在过去被介绍过微积分，但不知道通过几何方法对大脑获得的许多见解。附录简要回顾了神经科学和微积分所需的背景知识。
Mathematical Tools for Neuroscience: A Geometric Approach
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.
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