Mathematical tools for neuroscience. A geometric approach (Q2122510)

In this book, the author provides us with a relatively short but accessible introduction to a set of related mathematical ideas that have proven useful for understanding the brain and behaviour. It is shown that brains are especially suited to exploiting the geometric properties of biological objects. This manuscript spreads over nine chapters. Chapter 1 (Mind and Brain) discusses the basic terminology of dynamical systems theory and linear algebra. If the behaviour of a dynamical system is described geometrically by the trajectory of a point corresponding to its state, then the changes in the firing rates of a population of neurons can be used emulate the changes in the coordinates of the state of the system as its behaviour changes. The manifold defined by the collection of possible trajectories of a dynamical system is referred to as a state space and an equilibrium state is referred to as a fixed point of the state space. Chapter 2 (Biological Objects) is devoted to detailed consideration of trajectory introduced in the context of an animal navigating a path towards a required destination. The concepts are also introduced: fibre bundle, symmetry operations, orthogonal group, tangent space, Veronese surface, local navigation, path integration and grid cells. In Chapter 3 measurements are used to illustrate that data can be simplified by applying principal components analysis, resulting in a partition of the data according to how the measurements vary with each other. The neural implementation of this procedure corresponds to Hebbian learning and the complementary procedure of anti-Hebbian learning is also introduced using the cerebellum as an example. In Chapter 4 (From Local to Global) the task of making use of a retinal image is used as an example of how measurements from the outline of a manifold can be used to recover a global description by applying Morse theory. A useful tool for this task is the bump function which is a smooth function that is non-zero for all points in the neighbourhood of a given point on the surface, and zero everywhere else. The simplest form of interaction with a biological object that an animal can make is a discrete behavioural event or an action. In Chapter 5 (Actions) a geometric description of an action as a slow-fast system is introduced and applied to eye and arm movements. For convenience, in the context of slow-fast modelling, the velocity command neurons will be referred to as burst neurons and the neurons specifying the slow manifold will be referred to as pause neurons. In Chapter 6 (Brain and Body) the linear modelling of biological tissue is introduced and applied to the orbital plant and the oculomotor neural integrator. The linear approach is extended to simple nonlinear behaviour, using the larynx as an example. Nonlinearities lead to a greater range of behaviours of which heteroclinic cycles form one example. Chapter 7 is devoted to the analysis of experimental measurements. A review of the geometric approach is given in Chapter 8. Background Material discussed in this book are considered in Appendix. The monograph is exclusively professionally written and the materials are presented in an attractive way. Obviously, research in this important scientific direction can be successfully expanded with the inclusion of specialized software tools implemented in computer algebraic systems designed for scientific calculations.