G. Bard Ermentrout – Mathematical Foundations of Neuroscience

Автор: G. Bard Ermentrout
Название книги: Mathematical Foundations of Neuroscience
Формат: PDF
Жанр: Психология
Страницы: 432
Качество: Изначально компьютерное, E-book

One can say that the field of computational neuroscience started with the 1952 paper
of Hodgkin and Huxley in which they describe, through nonlinear partial differential
equations, the genesis of the action potential in the giant axon of the squid. These
equations and the methods that arose from this combination of modeling and experiments
have since formed the basis for nearly every subsequent model for active
cells. The Hodgkin–Huxley model and a host of simplified equations that are derived
from it have inspired the development of new and beautiful mathematics. Dynamical
systems and computational methods are now being used to study activity patterns
in a variety of neuronal systems. It is becoming increasingly recognized, by both
experimentalists and theoreticians, that issues raised in neuroscience and the mathematical
analysis of neuronal models provide unique interdisciplinary collaborative
research and educational opportunities.
This book is motivated by a perceived need for an overview of how dynamical
systems and computational analysis have been used in understanding the types of
models that come out of neuroscience. Our hope is that this will help to stimulate
an increasing number of collaborations between mathematicians and other theoreticians,
looking for interesting and relevant problems in applied mathematics and
dynamical systems, and neuroscientists, looking for new ways to think about the
biological mechanisms underlying experimental data.
The book arose out of several courses that the authors have taught. One of these
is a graduate course in computational neuroscience that has students from the disciplines
of psychology, mathematics, computer science, physics, and neuroscience.
Of course, teaching a course to students with such diverse backgrounds presents
many challenges. However, the course provides many opportunities to encourage
students, who may not normally interact with each other, to collaborate on exercises
and projects. Throughout the book are many exercises that involve both computation
and analysis. All of the exercises are motivated by issues that arise from the
biology.
We have attempted to provide a comprehensive introduction to the vocabulary
of neuroscience for mathematicians who are just becoming interested in the field,
but who have struggled with the biological details. Anyone who wants to work in
computational neuroscience should learn these details as this is the only way one
can be sure that the analysis and modeling is actually saying something useful to
biologists. We highly recommend the reader study this material in more detail by
consulting one of the many excellent books devoted primarily to neuroscience. Such
books include those by Kandel et al. [144] and Johnston andWu [139].
We have also tried to provide background material on dynamical systems theory,
including phase plane methods, oscillations, singular perturbations, and bifurcation
analysis. An excellent way to learn this material is by using it, together with computer
simulations, to analyze interesting, concrete examples. The only prerequisites
are a basic knowledge of calculus, knowledge of a little linear algebra (matrices,
eigenvalues), and understanding of some basic theory of ordinary differential equations.
Much of the mathematics is at the level of Strogatz [255].
The book is organized from the bottom up. The first part of the book is concerned
with properties of a single neuron. We start with the biophysics of the cell
membrane, add active ion channels, introduce cable theory, and then derive the
Hodgkin–Huxley model. Chapter 2 is concerned with the basic properties of dendrites.
We then introduce dynamical systems theory, using a simple neuron model to
illustrate the basic concepts. We return to the biology in Chap. 4, where we discuss
the variety of ion channels which have been found in neurons. Chapters 5 and 6
are devoted to bursting oscillations and propagating action potentials, respectively.
Here, we use many of the dynamical systems techniques to describe mechanisms
underlying these behaviors. The second part of the book is concerned with neuronal
networks. In Chap. 7, we describe synaptic channels, which are the primary
way that neurons communicate with each other. Chapters 8 and 9 discuss two different
approaches for studying networks. First, we assume weak coupling and use
phase-response methods. We then demonstrate how one can analyze firing patterns
in neuronal networks using fast/slow analysis. In Chap. 10, we discuss the role of
noise in neuron models. Here, we briefly introduce the reader to the mathematical
theory of stochastic differential equations. Finally, in Chaps. 11 and 12 we discuss
firing rate models and spatially distributed networks.
There is far more material in this book than could be covered in a one-semester
course. Furthermore, some of the material is quite advanced. A course in computational
neuroscience slanted toward mechanisms and dynamics could easily be made
out of the first five chapters along with Chap. 7. These chapters would cover most
of the basics of single-cell modeling as well as introduce students to dynamical systems.
The remainder of such a course could include selections from Chaps. 8–12.
For example, Chap. 11 contains firing rate models, with many applications provided
in Sect. 11.3. Parts of Chap. 12 could comprise the remainder of the course.
For more mathematically inclined students, the elementary dynamics chapter
(Chap. 3) could be skipped and the more technical chapters could be emphasized.
There is lovely nonlinear dynamics in Chaps. 5, 6, 8, and 9, which along with the
earlier chapters could form the core of a mathematical neuroscience course.
There are several recent books that cover some of the same material as in
the present volume. Theoretical Neuroscience by Dayan and Abbott [53] has a
broader range of topics than our book; however, it does not go very deeply into
the mathematical analysis of neurons and networks, nor does it emphasize the
dynamical systems approach. A much more similar book is Dynamical Systems
in Neuroscience by Izhikevich [136]. This book emphasizes the same approach
as we take here; however, the main emphasis of Dynamical Systems in Neuroscience
is on single-neuron behavior. We cover a good deal of single-neuron
biophysics, but include a much larger proportion of theory on systems neuroscience
and applications to networks. There are many specific models and equations
in this text. The forms of these models and their parameters are available at
http://www.math.pitt.edu/bard/bardware/neurobook/allodes.html.
Pittsburgh, PA G. Bard Ermentrout