Nonlinear Dynamics in Physiology. A State-Space Approach

Contents
Preface
1. The mathematical analysis of physiological systems: goals and approaches
1.1 The goals of mathematical analysis in physiology
1.2 Outline of dynamic systems
1.3 Types of dynamic systems - random, deterministic, linear, nonlinear
1.4 Types of dynamic behaviors - random, fixed point, periodic, quasi-periodic, chaotic
1.5 Follow the "noise"
1.6 Chaos and physiology
General Bibliography
References for Chapter 1
2. Fundamental signal processing and analysis concepts and measures
2.1 Sampled data and continuous distributions
2.2 Basic statistics
2.3 Correlation coefficient
2.4 Linear regression, least-squares, squared-error
2.5 Random processes, white noise, correlated noise
2.6 Autocorrelation
2.7 Concluding remarks
References for Chapter 2
3. Analysis approaches based on linear systems
3.1 Definition and properties of linear systems
3.2 Autocorrelation, cross-correlation, stationarity
3.3 Fourier transforms and spectral analysis
3.4 Examples of autocorrelations and frequency spectra
3.5 Transfer functions of linear systems, Gaussian statistics
References for Chapter 3
4. State-space reconstruction
4.1 State variables, state space
4.2 Time-delay reconstruction
4.3 A digression on topology
4.4 How to do the reconstruction correctly
4.5 Example: detection of fast-phase eye movements
4.6 Historical notes, examples from the literature
4.7 Points for further consideration
References for Chapter 4
5. Dimensions
5.1 Euclidean dimension and topological dimension
5.2 Dimension as a scaling process - coastline length, Mandelbrot, fractals, Cantor, Koch
5.3 Box-counting dimension and correlation dimension
5.4 Correlation dimension - how to measure it correctly
5.5 Error bars on dimension estimates
5.6 Interpretation of the dimension
5.7 Tracking dimension overtime
5.8 Examples
5.9 Points for further consideration
References for Chapter 5
6. Surrogate data
6.1 The need for surrogates
6.2 Statistical hypothesis testing
6.3 Statistical randomization and its implementation
6.4 Random surrogates
6.5 Phase-randomization surrogate
6.6 AAFT surrogate
6.7 Pseudo-periodic surrogate
6.8 First differences and surrogates
6.9 Multivariate surrogates
6.10 Surrogates tailored to specific physiological hypotheses
6.11 Examples of different surrogates
6.12 Physiological examples
References for Chapter 6
7. Nonlinear forecasting
7.1 Predictability of prototypical systems
7.2 Methodology
7.3 Variations
7.4 Surrogates, global linear forecasting
7.5 Time-reversal and amplitude-reversal for detection of nonlinearity
7.6 Chaos versus colored noise
7.7 Forecasting of neural spike trains and other discrete events
7.8 Examples
References for Chapter 7
8. Recurrence analysis
8.1 Concept and methodology
8.2 Recurrence plots of simple systems
8.3 Recurrence quantification analysis (RQA)
8.4 Extensions
8.5 Examples
References for Chapter 8
9. Tests for dynamical interdependence
9.1 Concepts
9.2 Mutual false nearest neighbors
9.3 Mutual prediction, cross-prediction
9.4 Cross-recurrence, joint recurrence
9.5 Mathematical properties of mappings
9.6 Multivariate surrogates and other test data
9.7 Examples
References for Chapter 9
10. Unstable periodic orbits
10.1 Concepts
10.2 Example
10.3 Physiological examples
References for Chapter 10
11. Other approaches based on the state space
11.1 Properties of mappings
11.2 Parallel flows in state space
11.3 Exceptional events
11.4 Lyapunov exponents
11.5 Deterministic versus stochastic (DVS) analysis
References for Chapter 11
12. Poincaré sections, fixed points, and control of chaotic systems
12.1 Poincaré section
12.2 Fixed points
12.3 Chaos control
12.4 Anticontrol
References for Chapter 12
13. Stochastic measures related to nonlinear dynamical concepts
13.1 Fractal time series, fractional Brownian motion
13.2 fBm, correlation dimension, nonlinear forecasting
13.3 Quantifying fBm: spectrum, autocorrelation, Hurst exponent, detrended fluctuation analysis
13.4 Self-organized criticality
References for Chapter 13
14. From measurements to models
14.1 The nature of the problem
14.2 Approaches to nonlinear system identification
14.3 A reasonable compromise
References for Chapter 14
15. Case study - oculomotor control
15.1 Optokinetic nystagmus - dimension, surrogates, prediction
Recurrence analysis
Correlation dimension
Surrogate data
Filtering
Nonlinear forecasting
Mutual forecasting
Physiological interpretation
15.2 Eye movements and reading ability
References for Chapter 15
16. Case study - motor control
16.1 Postural center of pressure
16.2 Rhythmic movements
References for Chapter 16
17. Case study - neurological tremor
17.1 Physiology background
17.2 Initial studies - evidence for chaos
17.3 Later studies - evidence for randomness
References for Chapter 17
18. Case study - neural dynamics and epilepsy
18.1 Epilepsy background
18.2 Initial dynamical studies
18.3 Dimension as a seizure predictor
18.4 Dynamical similarity as a seizure predictor
18.5 Validation with surrogates, comparison of procedures
References for Chapter 18
19. Case study - cardiac dynamics and fibrillation
19.1 Heart-rate variability
19.2 Noisy clock or chaos?
19.3 Forecasting and chaos
19.4 Detection of imminent fibrillation: point correlation dimension
References for Chapter 19
20. Case study - epidemiology
20.1 Background and early approaches
20.2 Nonlinear forecasting of disease epidemics
References for Chapter 20
21. Case study - psychology
21.1 General concepts
21.2 Psychiatric disorders
21.3 Perception and action
References for Chapter 21
22. Final remarks
References on climatic attractors
Suggested references for further study
Appendix
A.1 State-space reconstruction
A.2 Correlation dimension
A.3 Surrogate data
A.4 Forecasting
A.5 Recurrence plots
A.6 Periodic orbits
A.7 Poincaré sections
A.8 Software packages
A.9 Sources of sample data sets
Index