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.