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.

• The first focused introduction to the use of nonlinear analysis with an infinite dimensional approach to theoretical neuroscience
• Combines functional analysis techniques with nonlinear dynamical systems applied to the study of the brain
• Introduces powerful mathematical techniques to manage the dynamics and challenges of infinite systems of equations applied to neuroscience modeling

neuroscience, theoretical neuroscience, applied mathematics

About the Authors

Foreword

Preface

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 Systems

3.4 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

5: Methods of Lower and Upper Solutions

5.1 Some Remarks in Connection with Applications of Numerical Methods

5.2 On Constructions of Upper and Lower Solutions

5.3 Positive Solutions

5.4 Some Remarks on Strongly Coupled Systems

5.5 Estimation of Convergence Speed for Different Iterative Methods

6: Truncation Method

6.1 Introduction

6.2 Truncation Method for Infinite Countable Systems

6.3 Truncation Method for Infinite Uncountable Systems

6.4 Relation Between Continuous and Discrete Infinite-Dimensional Models

6.5 Conclusion

7: Fixed Point Method

7.1 Introduction

7.2 Theorems on Fixed Point Mapping

7.3 Banach Theorem for Contraction Mappings

7.4 Schauder Fixed Point Theorem for Compact Mappings

7.5 Leray-Schauder Theorem for Compact Mappings

8: Stability of Solutions

8.1 Introduction

8.2 Existence of Solutions for Infinite Systems

8.3 Stability of Solutions of Infinite Systems

II Application of Nonlinear Analysis

9: Introduction to Part II

10: Continuous and Discrete Models of Neural Systems

10.1 Introduction

10.2 Mathematical Motivations

10.3 The Formulation of Problems

10.4 Observations

10.5 Applications

10.6 Conclusions

11: Nonlinear Cable Equations

11.1 Introduction

11.2 Nonlinear Cable Equations

11.3 Comparison of Solutions for Continuous and Discrete Cable Equations

11.4 Application of Comparison Theorem

11.5 Conclusions

12: Reaction-Diffusion Equations

12.1 Introduction

12.2 Ellipticity and Parabolicity

12.3 Transformation of Reaction-Diffusion Equations

12.4 Reaction-Diffusion Equations in Diffusion Processes

12.5 Monotone Iterative Methods for Finite Systems

12.6 Extension of Monotone Iterative Methods to Infinite Systems

12.7 Conclusion

Appendix

A.1 List of Symbols Used for Approximation Sequences

A.2 Existence and Uniqueness Theorems

A.3 Integral Representations of Solutions

A.4 Weak C-Solution

A.5 Integral Transformation

Further Reading

References

Index