Physics of Biological Oscillators: New Insights into Non-Equilibrium and Non-Autonomous Systems

Foreword by Alex A. R. Webb
Preface
Acknowledgements
Contents
Contributors
Abbreviations
1 Introduction
1.1 Theory
1.2 Model-Driven and Data-Driven Approaches
1.3 Biological Oscillations
1.4 Applications
1.5 Outlook
1.6 Using the Book
Part ITheory
2 Phase and Amplitude Description of Complex Oscillatory Patterns in Reaction-Diffusion Systems
2.1 Introduction
2.2 Phase-Amplitude Reduction of Limit-Cycle Oscillators
2.3 Phase-Amplitude Reduction of Reaction-Diffusion Systems
2.4 Optimal Entrainment of Oscillatory Patterns with Feedback
2.5 Example: Oscillating Spot in the FitzHugh-Nagumo Model
2.6 Summary
References
3 Reduced Phase Models of Oscillatory Neural Networks
3.1 Introduction
3.2 Networks of Wilson-Cowan Neural Masses
3.3 Phase Reduction of Oscillatory Neural Networks
3.4 Phase Reduction in Face of Strong Coupling
3.5 Phase Reduction in Face of Complex Structural Connectivity
3.6 Conclusion
References
4 Nonautonomous Attractors
4.1 Introduction
4.2 Nonautonomous Difference Equations
4.3 Invariant Sets and Attractors of Processes
4.4 Construction of Forward Attractors
4.4.1 A Counterexample
4.4.2 A Condition Ensuring Forward Convergence
4.5 Forward Attracting Sets
4.5.1 Asymptotic Invariance
4.6 Skew Product Flows
4.6.1 An Example
4.6.2 Attractors of Skew Product Flows
4.6.3 Skew Product Flows as Semi-dynamical Autonomous Systems
4.7 Concluding Remark
References
5 Normal Hyperbolicity for Non-autonomous Oscillators and Oscillator Networks
5.1 Introduction
5.2 Phase-Locking
5.3 Synchronisation of Two Oscillators
5.4 What is Coupling?
5.5 Synchronisation of N Oscillators
5.6 Hierarchical Aggregation
5.7 Normal Hyperbolicity Estimates
5.8 Extension to Class 1 Neurons
5.9 Extension to Chaotic Oscillators
5.10 Conclusion
References
6 Synchronisation and Non-autonomicity
6.1 Introduction
6.2 Non-autonomicity
6.2.1 Thermodynamically Open Versus Isolated
6.2.2 Autonomous Versus Non-autonomous
6.3 Methods
6.3.1 Asymptotic
6.3.2 Finite-Time
6.4 Systems Analysis
6.4.1 Single Oscillator: Periodic Driving
6.4.2 Single Oscillator: Noisy Periodic Driving
6.4.3 Single Oscillator: Time-Varying Frequency Driving
6.4.4 Network: Time-Varying Frequency Driving
6.4.5 Network: Time-Varying Driving Strength
6.5 Summary and Conclusions
References
7 Non-asymptotic-time Dynamics
7.1 Introduction
7.1.1 Classical Dynamical Systems Theory
7.1.2 The Limitations of Long-Time-Asymptotic Analysis
7.1.3 Structure of the Chapter
7.2 The Non-autonomous Adler Equation
7.2.1 Stability and Neutral Stability in the Autonomous Case
7.2.2 Stability and Neutral Stability in the Non-autonomous Case
7.2.3 Quantitative Rate of Synchronisation of Trajectories
7.2.4 Transition from Neutral Stability to Stability
7.2.5 Numerics for the Non-autonomous Adler Equation
7.3 A Toy Model of ``inherently Finite-Time'' Dynamics
7.3.1 Definition and Basic Properties of a Brownian Bridge
7.3.2 Finite-Time Non-autonomous Adler Equation
7.4 The Adler Equation with Sinusoidal Driving
7.4.1 An Initial Long-time-asymptotic Analysis of Dynamics
7.4.2 Analysis Based on Sect. 7.2
7.4.3 Numerics
7.4.4 Comparison of the Above Analyses
7.5 Slow-Fast Finite-Time Dynamical Systems
7.6 Conclusion
References
8 Synchronization of Coupled Oscillators—Phase Transitions and Entropy Production
8.1 Introduction
8.1.1 The Liquid-Gas Phase Transition
8.1.2 Magnetization
8.1.3 Universality in Phase Transitions
8.1.4 Phase Transitions for Coupled Oscillators
8.2 Numerical Simulation and Mathematical Analysis
8.2.1 A Heuristic Approach
8.2.2 The System of ODEs
8.2.3 The Temperature Dependence
8.3 Conclusions and Discussion
8.3.1 The Phase Transition as the Number of States Is Increased
8.3.2 Entropy and Dissipative Structures
8.3.3 Why Mean-Field Works for a 1D System
8.3.4 Implications for Coupled Oscillators in the Wet Lab
References
Part IIModel-Driven and Data-Driven Approaches
9 On Localised Modes in Bio-inspired Hierarchically Organised Oscillatory Chains
9.1 Introduction
9.2 Model with First-Order Branches
9.3 Models with Second-Order Branches
9.4 Model with a Chain of Arbitrary Order of Hierarchy
9.5 Conclusions
References
10 Useful Transformations from Non-autonomous to Autonomous Systems
10.1 Introduction
10.2 Appearance of Oscillations in Mathematical Modelling of the Cardio-Respiratory System
10.2.1 Intrinsic Oscillations
10.2.2 Forced Oscillations
10.3 Alternative Transformations
10.3.1 Transforming a System of Ordinary Differential Equations with Sinusoidal Inputs
10.3.2 Transformation of a Non-autonomous Boolean Network
10.4 Conclusions
References
11 Coupling Functions in Neuroscience
11.1 Introduction
11.1.1 Coupling Function Basics
11.2 Suitability of Coupling Functions for Neuroscience
11.2.1 Coupling Functions on Neuronal Level
11.2.2 Coupling Functions on Brainwave Level
11.3 Theory and Methods for Coupling Functions in Neuroscience
11.4 Conclusions and Discussions
References
12 Phase Reconstruction with Iterated Hilbert Transforms
12.1 Introduction and Overview
12.2 Nonlinear Oscillators and Phase Reduction
12.3 Phase Reconstruction and Iterative Hilbert Transform Embeddings
12.3.1 Waveform, Phase and Demodulation
12.3.2 Embeddings, Hilbert Transform, and Phase-Amplitude Mixing
12.3.3 Iterated HT Embeddings
12.4 Numerical Experiments
12.4.1 Deterministic Oscillations
12.4.2 Reconstruction of the Phase Response Curve from Observation
12.4.3 Noisy Oscillations
12.5 Conclusion and Open Problems
References
Part IIIBiological Oscillators
13 Oscillations in Yeast Glycolysis
13.1 Introduction
13.2 Variables Measured in Oscillating Glycolysis
13.3 The Dynamics of Intracellular Water Modulate Glycolytic Oscillations
13.3.1 Macromolecular Crowding and the Dynamics of Intracellular Water
13.3.2 Coupling of Dynamics of Intracellular Water to Glycolysis
13.4 A Potential Role of Glycolytic Oscillations
13.5 Concluding Remarks
References
14 Oscillations, Rhythms and Synchronized Time Bases: The Key Signatures of Life
14.1 Introduction: Life as a Complex System
14.1.1 Thermodynamics of the Living State
14.1.2 Metabolic Fluxes, Flows and Turnover of Constituents and Component Organelles
14.1.3 Time Domains of Living Systems
14.1.4 Biological Oscillations
14.1.5 Rhythms and Clock-Like Timekeepers
14.1.6 Ultradian (Circahoralian) Clocks
14.1.7 The Cell-Division Cycle
14.1.8 Oscillations in Glycolysis
14.1.9 Cellular Respiratory Oscillations
14.1.10 Time Structure Discovered in Self-synchronous Yeast Continuous Cultures
14.1.11 Non-linear Dynamics of the Self-synchronous Culture: Chaos and Fractals
14.2 General Conclusions
References
15 Glycolytic Oscillations in Cancer Cells
15.1 Introduction
15.2 Mechanism of Glycolytic Oscillations in Cancer Cells
15.3 Crabtree Effect and Warburg Effect
15.4 Dynamical Quorum Sensing and Kuramoto Desynchronization
15.5 Glycolytic Oscillations in HeLa Cervical Cancer Cells
15.6 Glycolytic Oscillations in Prostate Cancer Cells (DU145)
15.7 Mathematical Model for Glycolytic Oscillations in Cancer Cells
15.8 Conditions for Glycolytic Oscillations in Cancer Cells
15.9 Malignancy of Cancer Cells and Glycolytic Oscillations
15.10 Summary
References
16 Mechanism and Consequence of Vasomotion
16.1 Background Information on Small Arteries and Their Function
16.2 The Concept of Vasomotion
16.3 How Does Vasomotion Develop?
16.4 Microvascular Networks
16.5 Vasomotion Depends on the Hemodynamic Status and May Have Consequences for Tissue Oxygenation
References
17 Biological Oscillations of Vascular Origin and Their Meaning: In Vivo Studies of Arteriolar Vasomotion
17.1 Historical Background
17.2 Vasomotion in Hamster Dorsal Skin Fold Window Preparation
17.3 Vasomotion in Rat Closed Cranial Window
17.4 Conclusions
References
18 Phase Coherence of Finger Skin Blood Flow Oscillations Induced by Controlled Breathing in Humans
18.1 Introduction
18.2 Methodology
18.2.1 Participants
18.2.2 Measurements
18.2.3 Data Analysis
18.3 Results
18.4 Discussion
18.5 Conclusion
References
19 Complexity-Based Analysis of Microvascular Blood Flow in Human Skin
19.1 Introduction
19.2 Assessment of Microvascular Blood Flow
19.3 Analysis of Microvascular Blood Flow Signals in the Spectral Domain
19.3.1 Fast Fourier Transform
19.3.2 Wavelet Transform
19.3.3 Application of Spectral Domain Analysis
19.4 Information and Complexity-Based Analysis of Microvascular Blood Flow Signals
19.4.1 Lempel-Ziv Complexity (LZC)-Based Analysis
19.4.2 Entropy-Based and Effort to Compress Complexity Analysis
19.4.3 Application of Complexity-Based Analysis of Laser Doppler BF Signals
19.5 Multiscale Frequency, Complexity and Scale
19.6 Other Descriptors of Time- and Frequency-Domain Characteristics of the Microvascular Blood Flux Signal
19.6.1 Time Localised Phase Coherence
19.6.2 Attractor Reconstruction
19.7 Concluding Remarks
References
20 Sleep-Related Modulations of Heart Rate Variability, ECG, and Cardio-Respiratory Coupling
20.1 Introduction
20.2 Nocturnal Electrocardiography (ECG) Recordings to Monitor Changes in Autonomous Nervous System Activity During Sleep
20.3 Non-linear Analysis of Heart Rate Variability
20.4 Cyclical Variation of Heart Rate with Sleep Apnea
20.5 Detection of Sleep Apnea Through Changes in Heart Rate and ECG Morphology
20.6 Cardiopulmonary Coupling and Cardiorespiratory Phase Synchronization
20.7 Discussion
20.8 Summary
References
21 Brain Morphological and Functional Networks: Implications for Neurodegeneration
21.1 Introduction
21.2 Graph Theory and the Brain
21.2.1 Brain Network Node
21.2.2 Brain Network Edge
21.2.3 Brain Network Modules
21.2.4 Dynamical Functional Networks
21.3 Morphology and Function
21.4 Implications for Neurodegeneration
21.5 Conclusion
21.6 Glossary
References
Part IVApplications
22 Predicting Epileptic Seizures—An Update
22.1 Introduction
22.2 Seizure Prediction and Time Series Analysis
22.3 Performance Evaluation of Seizure Prediction Algorithms
22.4 Devices for Seizure Prediction
22.5 Open Questions and Outlook
References
23 General Anaesthesia and Oscillations in Human Physiology: The BRACCIA Project
23.1 Definition and Mechanisms
23.2 Monitoring of General Anaesthesia
23.3 The Braccia Concept: Anaesthesia Monitoring—Bringing it All Together?
23.4 The Braccia Results, so far
References
24 Processed EEG as a Measure of Brain Activity During Anaesthesia
24.1 Introduction. The Need of Anaesthesia Monitoring
24.2 Main EEG Patterns During Anaesthesia
24.3 Algorithms and Monitoring Design Approaches
24.4 Depth of Anaesthesia Monitoring Validation
24.5 Advantages and Challenges in Depth of Anaesthesia Monitoring
24.6 Conclusions
References
25 Medical Products Inspired by Biological Oscillators: Intermittent Pneumatic Compression and the Microcirculation
25.1 The Microcirculation
25.1.1 Overview
25.1.2 Structure and Function of the Microcirculation
25.2 Endogenous Vascular Oscillators
25.3 Medical Products Inducing Oscillations Through Intermittent Impulse Compression
References
26 Phase Coherence Between Cardiovascular Oscillations in Malaria: The Basis for a Possible Diagnostic Test
26.1 Introduction
26.2 Materials and Methods
26.2.1 Subjects and Plan of the Study
26.2.2 Data Acquisition
26.3 Analysis of Cardiovascular Time Series
26.3.1 Spectral Analysis
26.3.2 Extracting the Instantaneous Heart Frequency
26.3.3 Wavelet Phase Coherence
26.3.4 Statistical analysis
26.4 Results
26.4.1 Effect of Malaria on Blood Pressure, Respiration Frequency, and Skin Temperature
26.4.2 Detecting Oscillations Using Time-Frequency Analysis
26.5 Non-invasive Diagnosis of Malaria
26.6 Discussion
References
Part VOutlook
27 Physics of Biological Oscillators: Outlook
References
Index