| Preface | p. vii |
| The mathematical analysis of physiological systems: goals and approaches | |
| The goals of mathematical analysis in physiology | p. 2 |
| Outline of dynamic systems | p. 5 |
| Types of dynamic systems - random, deterministic, linear, nonlinear | p. 8 |
| Types of dynamic behaviors - random, fixed point, periodic, quasi-periodic, chaotic | p. 11 |
| Follow the "noise" | p. 13 |
| Chaos and physiology | p. 14 |
| General Bibliography | p. 17 |
| References for Chapter 1 | p. 18 |
| Fundamental signal processing and analysis concepts and measures | |
| Sampled data and continuous distributions | p. 20 |
| Basic statistics | p. 21 |
| Correlation coefficient | p. 24 |
| Linear regression, least-squares, squared-error | p. 25 |
| Random processes, white noise, correlated noise | p. 29 |
| Autocorrelation | p. 30 |
| Concluding remarks | p. 30 |
| References for Chapter 2 | p. 31 |
| Analysis approaches based on linear systems | |
| Definition and properties of linear systems | p. 32 |
| Autocorrelation, cross-correlation, stationarity | p. 33 |
| Fourier transforms and spectral analysis | p. 35 |
| Examples of autocorrelations and frequency spectra | p. 39 |
| Transfer functions of linear systems, Gaussian statistics | p. 43 |
| References for Chapter 3 | p. 44 |
| State-space reconstruction | |
| State variables, state space | p. 45 |
| Time-delay reconstruction | p. 47 |
| A digression on topology | p. 50 |
| How to do the reconstruction correctly | p. 55 |
| Example: detection of fast-phase eye movements | p. 60 |
| Historical notes, examples from the literature | p. 63 |
| Points for further consideration | p. 66 |
| References for Chapter 4 | p. 70 |
| Dimensions | |
| Euclidean dimension and topological dimension | p. 74 |
| Dimension as a scaling process - coastline length, Mandelbrot, fractals, Cantor, Koch | p. 75 |
| Box-counting dimension and correlation dimension | p. 81 |
| Correlation dimension - how to measure it correctly | p. 85 |
| Error bars on dimension estimates | p. 92 |
| Interpretation of the dimension | p. 94 |
| Tracking dimension over time | p. 96 |
| Examples | p. 97 |
| Points for further consideration | p. 99 |
| References for Chapter 5 | p. 102 |
| Surrogate data | |
| The need for surrogates | p. 104 |
| Statistical hypothesis testing | p. 105 |
| Statistical randomization and its implementation | p. 106 |
| Random surrogates | p. 108 |
| Phase-randomization surrogate | p. 109 |
| AAFT surrogate | p. 110 |
| Pseudo-periodic surrogate | p. 113 |
| First differences and surrogates | p. 114 |
| Multivariate surrogates | p. 115 |
| Surrogates tailored to specific physiological hypotheses | p. 117 |
| Examples of different surrogates | p. 118 |
| Physiological examples | p. 121 |
| References for Chapter 6 | p. 122 |
| Nonlinear forecasting | |
| Predictability of prototypical systems | p. 124 |
| Methodology | p. 126 |
| Variations | p. 131 |
| Surrogates, global linear forecasting | p. 132 |
| Time-reversal and amplitude-reversal for detection of nonlinearity | p. 133 |
| Chaos versus colored noise | p. 134 |
| Forecasting of neural spike trains and other discrete events | p. 136 |
| Examples | p. 137 |
| References for Chapter 7 | p. 139 |
| Recurrence analysis | |
| Concept and methodology | p. 141 |
| Recurrence plots of simple systems | p. 143 |
| Recurrence quantification analysis (RQA) | p. 149 |
| Extensions | p. 151 |
| Examples | p. 152 |
| References for Chapter 8 | p. 153 |
| Tests for dynamical interdependence | |
| Concepts | p. 156 |
| Mutual false nearest neighbors | p. 157 |
| Mutual prediction, cross-prediction | p. 160 |
| Cross-recurrence, joint recurrence | p. 165 |
| Mathematical properties of mappings | p. 169 |
| Multivariate surrogates and other test data | p. 170 |
| Examples | p. 171 |
| References for Chapter 9 | p. 172 |
| Unstable periodic orbits | |
| Concepts | p. 175 |
| Example | p. 176 |
| Physiological examples | p. 178 |
| References for Chapter 10 | p. 179 |
| Other approaches based on the state space | |
| Properties of mappings | p. 181 |
| Parallel flows in state space | p. 183 |
| Exceptional events | p. 185 |
| Lyapunov exponents | p. 186 |
| Deterministic versus stochastic (DVS) analysis | p. 187 |
| References for Chapter 11 | p. 188 |
| Poincare sections, fixed points, and control of chaotic systems | |
| Poincare section | p. 190 |
| Fixed points | p. 193 |
| Chaos control | p. 203 |
| Anticontrol | p. 208 |
| References for Chapter 12 | p. 209 |
| Stochastic measures related to nonlinear dynamical concepts | |
| Fractal time series, fractional Brownian motion | p. 211 |
| fBm, correlation dimension, nonlinear forecasting | p. 214 |
| Quantifying fBm: spectrum, autocorrelation, Hurst exponent, detrended fluctuation analysis | p. 216 |
| Self-organized criticality | p. 217 |
| References for Chapter 13 | p. 218 |
| From measurements to models | |
| The nature of the problem | p. 220 |
| Approaches to nonlinear system identification | p. 221 |
| A reasonable compromise | p. 222 |
| References for Chapter 14 | p. 223 |
| Case study - oculomotor control | |
| Optokinetic nystagmus - dimension, surrogates, prediction | p. 225 |
| Recurrence analysis | p. 227 |
| Correlation dimension | p. 229 |
| Surrogate data | p. 230 |
| Filtering | p. 234 |
| Nonlinear forecasting | p. 235 |
| Mutual forecasting | p. 237 |
| Physiological interpretation | p. 239 |
| Eye movements and reading ability | p. 239 |
| References for Chapter 15 | p. 240 |
| Case study - motor control | |
| Postural center of pressure | p. 242 |
| Rhythmic movements | p. 246 |
| References for Chapter 16 | p. 250 |
| Case study - neurological tremor | |
| Physiology background | p. 252 |
| Initial studies - evidence for chaos | p. 253 |
| Later studies - evidence for randomness | p. 256 |
| References for Chapter 17 | p. 260 |
| Case study - neural dynamics and epilepsy | |
| Epilepsy background | p. 262 |
| Initial dynamical studies | p. 263 |
| Dimension as a seizure predictor | p. 265 |
| Dynamical similarity as a seizure predictor | p. 268 |
| Validation with surrogates, comparison of procedures | p. 271 |
| References for Chapter 18 | p. 273 |
| Case study - cardiac dynamics and fibrillation | |
| Heart-rate variability | p. 276 |
| Noisy clock or chaos? | p. 278 |
| Forecasting and chaos | p. 280 |
| Detection of imminent fibrillation: point correlation dimension | p. 283 |
| References for Chapter 19 | p. 289 |
| Case study - epidemiology | |
| Background and early approaches | p. 292 |
| Nonlinear forecasting of disease epidemics | p. 294 |
| References for Chapter 20 | p. 299 |
| Case study - psychology | |
| General concepts | p. 302 |
| Psychiatric disorders | p. 303 |
| Perception and action | p. 306 |
| References for Chapter 21 | p. 309 |
| Final remarks | |
| References on climatic attractors | p. 313 |
| Suggested references for further study | p. 313 |
| Appendix | |
| State-space reconstruction | p. 316 |
| Correlation dimension | p. 320 |
| Surrogate data | p. 322 |
| Forecasting | p. 323 |
| Recurrence plots | p. 326 |
| Periodic orbits | p. 327 |
| Poincare sections | p. 328 |
| Software packages | p. 330 |
| Sources of sample data sets | p. 333 |
| Index | p. 335 |
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