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.
Hardbound,
Published: June 2013
Imprint: Academic Press
ISBN: 978-0-12-411468-5
Contents
1. Introduction
1.1. Dynamic brain information processing
1.2. Different levels of neuronal organization
1.3. Nonlinear Cable Equations
1.4. Tapering cable model of a Neuron
1.5. Infinite systems of tapering neuron models
1.6. Dynamic continuity and integrative modelling2. Mathematical Preliminaries
2.1 Notations and assumptions
2.2. Sets and domains
2.3. Vector spaces of continuous functions
2.4. Cones and order
2.5. Ellipticity and parabolicity
2.6. Compactness, convexity, and connectedness
2.7. Initial and boundary conditions E
2.8. Examples of nonlinear boundary value problems3. Monotone Iterative Methods
4. Methods of Lower and Upper Solutions
3.1. Method of direct iterations
3.2. Chaplygin method
3.3. Some examples of Chaplygin method
3.4. Some examples of monotone iterative method
3.5. Method of direct iterations in unbounded domains
3.6. Notes and comments
4.1. On construction of upper and lower solutions
4.2. Positive solutions
4.3. Strongly coupled systems of equations
4.4. Connection with applications of numerical methods
4.5. Estimation of convergence speed for different iterative methods
4.6. Notes and comments5. Truncation Method
6. Fixed Point Method
5.1. Truncation method for discrete systems
5.2. Truncation method for continuous systems
5.3. Relation between continuous and discrete infinite-dimensional models
5.4. Some problems from neurobiology
5.5. Notes and comments
6.1. Introduction to continuous linear transformations
6.2. The Banach theorem for contraction mappings
6.3. The Schauder fixed point theorem for compact mappings
6.4. The Leray-Schauder theorem for compact mappings
6.5. Some applications of fixed point theorems
6.6. Notes and comments7. Stability of Solutions
8. Methods of Differential Inequalities
7.1. Existence and stability properties of solutions
7.2. Weak solutions
7.3. Existence of solutions of infinite systems
7.4. Stability of solutions of infinite systems
7.5. Notes and comments
8.1. Comparison theorems for existence and uniqueness of solutions
8.2. Application of comparison theorems
8.3. Maximum principles
8.4. Cauchy problem from neurobiology
8.5. Notes and Comments9. Neuronal Models in Infinite Dimensional Spaces
10. Neuronal Models and their Finite-Dimensional Projections
9.1. What is infinite dimensional analysis?
9.2. Infinite systems of partial differential equations
9.3. The formulation of problems from neuroscience
9.4. Application to some neuronal models
9.5. Notes and comments
10.1. The lumped parameter assumption
10.2. Application of the truncation method to cable problems
10.3. Relation between continuous and discrete infinite-dimensional cable models
10.4. Some further problems and conclusions
10.5. Notes and comments11. Finite and Infinite Systems of Quasilinear Equations
12. Infinite Systems of Nonlinear Equations
11.1. Some properties of quasilinear equations
11.2. Some examples of quasilinear equations
11.3. Monotone iterative methods for finite systems
11.4. Extension of monotone iterative methods to infinite systems
11.5. Notes and comments
12.1. Some properties of nonlinear equations
12.2. Comparison theorems for infinite systems
12.3. Weak inequalities for infinite systems
12.4. Strong inequalities for infinite systems
12.5. Notes and commentsList of symbols used for approximation sequences
Appendix
