This book describes fundamental physical principles, together with their mathematical formulations, for modelling the propagation of signals in nerve fibres. Above all, it focuses on the complex electro-mechano-thermal process that produces an ensemble of waves composed of several components, besides the action potential. These components include mechanical waves in the biomembrane and axoplasm, together with the temperature change. Pursuing a step-by-step approach, the content moves from physics and mathematics, to describing the physiological effects, and finally to modelling the coupling effects. The assumptions and hypotheses used for modelling, as well as selected helpful concepts from continuum mechanics, are systematically explained, and the modelling is illustrated using the outcomes of numerical simulation. The book is chiefly intended for researchers and graduate students, providing them with a detailed description of how to model the complex physiological processes in nerve fibres.
Author(s): Jüri Engelbrecht, Kert Tamm, Tanel Peets
Publisher: Springer
Year: 2021
Language: English
Pages: 199
City: Cham
Preface
Acknowledgements
Contents
Chapter 1 Introduction
References
Part I Complexity and Waves
Chapter 2 Complexity
2.1 Complexity of Physical Systems
2.2 Complexity in Biology
2.3 Mathematical Modelling
References
Chapter 3 Waves
3.1 Preliminaries
3.2 Mathematical Physics
3.3 Wave Equations
3.4 Reaction-Diffusion Equations
3.5 Physical Effects
3.6 TheWave Equation with Forcing
3.7 Solitary Waves and solitons
References
Part II Dynamical Processes in Nerve Axons
Chapter 4 Nervous Signals
References
Chapter 5 Dynamical Effects in Nerves
References
Part III Modelling of Dynamical Physiological Processes
Chapter 6 Mathematics of Single Effects
6.1 The Action Potential
6.1.1 The Classical Hodgkin-Huxley Model
6.1.2 The FitzHugh-Nagumo Model
6.1.3 The Evolution Equation
6.2 The LongitudinalWave in a Biomembrane
6.2.1 Model Derivation
6.2.2 Steady Solutions
6.2.3 Solution Types
6.2.4 Solutions Emerging from Arbitrary Initial Conditions
6.2.5 Interaction of Solitons
6.2.6 Discussion
6.3 TheWave Equations with Coupling
References
Chapter 7 Physical Mechanisms
7.1 Basic Elements of an Ensemble of Waves
7.2 Qualitative Observations from Experiments
7.3 Modelling of Coupled Signals and Coupling Forces
7.4 Possible Interactions
7.5 The Concept of Internal Variables
References
Chapter 8 An Ensemble ofWaves
8.1 The Model
8.1.1 Notations, Variables and Parameters in the Model
8.1.2 Coupling Forces
8.2 Energetical Balance
8.3 Simplifications
8.3.1 Action Potential
8.3.2 Pressure Wave in Axoplasm
8.3.3 LongitudinalWave in Biomembrane
8.3.4 Temperature
8.3.5 Neglecting Effects
8.4 Modifications
8.4.1 Properties of the Environment
8.4.2 Properties of Structures
8.4.3 Mechanisms
8.4.4 Systems
8.5 Dimensions
8.5.1 Governing Equations in Physical Units
8.5.2 Coupling Forces in Physical Units
8.5.3 Summary
References
Chapter 9 In Silico Experiments
9.1 Numerical Method
9.2 AP Coupled to Single Mechanical Effects
9.3 The AP Coupled to All Mechanical Effects
9.4 Temperature Changes Accompanying the AP
9.5 Full Coupled Model
9.6 Dimensionalised Example
References
Chapter 10 Final Remarks
References
Appendices
Appendix A The Numerical Scheme
A.1 Initial and Boundary Conditions
A.2 The Derivatives and Integration
A.3 Handling of Mixed Derivatives
A.4 The Time Derivatives VZ•[Z and PZ
A.5 Technical Details and Numerical Accuracy
References
Appendix B The example scripts
B.1 Example: the script for an example
B.2 Example: the script for visualisation
Index
