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 inte.

12.5 Monotone Iterative Methods for Finite Systems.

Half Title; Title Page; Copyright; Contents; About the Authors; Foreword; Preface; Part I Methods of Nonlinear Analysis; 1 Introduction to Part I; 2 Preliminary Considerations; 2.1 Sets and Domains; 2.2 Banach and Hölder Spaces; 2.3 Cones and Ordered Spaces; 2.4 Ellipticity and Parabolicity; 2.5 Notations of Functional Dependence; 2.6 Initial and Boundary Conditions; 2.7 Fundamental Assumptions and Conditions; 2.8 Lower and Upper Solutions; 2.9 Stability of Solutions of Infinite Systems; 3 Differential Inequalities; 3.1 Introduction; 3.2 Comparison Theorems for Finite Systems.

3.3 Maximum Principles for Finite Systems3.4 Comparison Theorems for Infinite Systems; 3.5 Infinite Systems of Nonlinear Differential Inequalities; 3.6 Ellipticity and Parabolicity of Nonlinear Inequalities; 3.7 Weak Differential Inequalities for Infinite Systems; 3.8 Strong Differential Inequalities for Infinite Systems; 4 Monotone Iterative Methods; 4.1 Method of Direct Iterations; 4.2 Chaplygin Method; 4.3 Certain Variants of the Chaplygin Method; 4.4 Certain Variants of Monotone Iterative Methods; 4.5 Another Variant of the Monotone Iterative Method.

4.6 Method of Direct Iterations in Unbounded Domains…