Mathematical Neuroscience book cover

Mathematical Neuroscience

Mathematical Neuroscience is a book for mathematical biologists seeking to discover the complexities of brain dynamics in an integrative way. It is the first research monograph devoted exclusively to the theory and methods of nonlinear analysis of infinite systems based on functional analysis techniques arising in modern mathematics.

Neural models that describe the spatio-temporal evolution of coarse-grained variables—such as synaptic or firing rate activity in populations of neurons —and often take the form of integro-differential equations would not normally reflect an integrative approach. This book examines the solvability of infinite systems of reaction diffusion type equations in partially ordered abstract spaces. It considers various methods and techniques of nonlinear analysis, including comparison theorems, monotone iterative techniques, a truncation method, and topological fixed point methods. Infinite systems of such equations play a crucial role in the integrative aspects of neuroscience modeling.


neuroscience, theoretical neuroscience, applied mathematics

Hardbound, 208 Pages

Published: August 2013

Imprint: Academic Press

ISBN: 978-0-12-411468-5


  • "Brzychczy,… Kraków and Poznanski…present methods of nonlinear functional analysis and their application to neuroscience. This is the first book, they say, to compile methods of nonlinear analysis to better understand the dynamics associated with solutions of infinite systems of equations. It would be suitable as a textbook for a one-semester graduate course in mathematical neuroscience for neuroscience students seeking tools and mathematics students looking for applications."--Reference & Research Book News, December 2013


  • Part I. Methods of Nonlinear Analysis

    1. Introduction to Part I

    2. Notations, Definitions and Assumptions

    3. Differential Inequalities

    4. Monotone Iterative Methods

    5. Methods of Lower and Upper Solutions

    6. Truncation Method

    7. Fixed Point Method

    8. Stability of Solutions

    PART II. Application of Nonlinear Analysis

    9. Introduction to Part II

    10. Continuous and Discrete Models of Neural Systems

    11. Nonlinear Cable Equations

    12. Reaction-Diffusion Equations


        Further Reading


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