Nonlinear Dynamics in Physiology. A State-Space Approach

Description

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