| Course Director: | Daniel Kaplan | Macalester College |
| Lecturers: | Leon Glass | McGill University |
| Thomas Schreiber | MPIPKS |
Lab Software: a tar file of Matlab M-files
| NTSA | H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press |
| UND | D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, Springer-Verlag |
| REVIEW | T. Schreiber, Interdisciplinary Application of Nonlinear Time Series Methods This reprint, provided in PDF format here by permission of the author, offers a consise summary of many of the topics in NTSA. |
| Clocks | L. Glass and M. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton Univ. Press |
| NR | W.H. Press et al. Numerical Recipes in C: The Art of Scientific
Computing, Cambridge Univ. Press
NOTE: This book is available on-line. We will not be using the computer programs from the book, but the text explanations are often excellent. |
| Reprints | Various reprints, in PDF format, are available through links below. They are provided by permission of the authors. |
| 8:30- 9:20 | Registration and Continental Breakfast | ||
| 9:20- 9:30 | Introduction and Objectives:
D. Kaplan |
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| 9:30-10:20 | Nonlinear Dynamics in Physiology:
L. Glass |
Physiological systems are extraordinarily complex at all levels of organization. Whether looking at a single cell, an organ, an organism, or an organism interacting with its environment, there is a remarkable amount of detail concerning interatctions and structure. Even the most accurate mathematical models of biological systems are gross simplifications of the real system. Nonlinear dynamics offers a way to think about qualitative features of dynamics in physiological systems. Nonlinear dynamics focuses on the conditions that lead to steady states, stable oscillations, and chaos in dynamical systems, and clarifies the way changes in parameters lead to different dynamic regimes. I discuss applications to medicine with illustration of the way different disease conditions may be associated with changes in qualitative features of physiological dynamics. Some diseases may be considered "dynamical disease" and are associated with changes in values of parameters characterizing physiological control systems. | Readings:
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| 10:20-11:10 | Linear Dynamics: Phenomena and Analysis:
D. Kaplan |
Although our ultimate interest in this course is the analysis of data stemming from nonlinear dynamics, many of the most powerful tools in time series analysis are based on linear models and many of the ideas in nonlinear time series analysis make reference to linear tools and concepts, e.g. surrogate data or Lyapunov exponents. This lecture will introduce (briefly) basic concepts of linear dynamics (stability, modes, input/output, superposition) and analysis tools such as the autocorrelation function, power spectrum, and transfer function. | Readings:
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| 11:10-11:30 | Break | ||
| 11:30-12:20 | Resetting and Entrainment of Oscillators:
L. Glass |
Single stimuli delivered to a biological oscillator lead to a resetting of the oscillation. The magnitude of the resetting depends on the amplitude and the phase of the stimulus. Qualitative aspects of resetting reflect nonlinear dynamical properties of biological oscillators. Periodic stimulation of a biological oscillator leads to a variety of different rhythms including entrainment with specific ratios between the stimulus and the oscillation or perhaps chaotic dynamics. Nonlinear dynamics offers a precise way to think about these phenomena and to make predictions concerning the effects of single and periodic stimulation on oscillating systems. |
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| 12:30- 2:00 | Lunch: UT Conf. Center Rm 404 | ||
| 2:00 - 2:50 | Foundations of NLTS Analysis:
D. Kaplan |
Much of nonlinear time-series analysis is based on rather simple (but perhaps subtle) tools: identification of nonlinear relationships between past and present using Poincare plots; reconstructive modeling of the system's state space using lag-embedding of time series; analysis of structure in system trajectories based on fractals and estimation of dimension. | Readings:
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| 2:50- 3:40 | Fractal Dimensions: Promises and Problems:
T. Schreiber |
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| 3:40- 4:00 | Break | ||
| 4:00- 4:45 | Lab Introduction: Matlab:
D. Kaplan |
For those participants with no previous experience using the MATLAB software, an introductory session will be held. Those who have used MATLAB (e.g. in Courses 1 or 2) will want to proceed directly to the Lab session. | |
| 4:00- 5:30 | Lab: Poincare Oscillators | ||
| 6:00 | Dinner: UT Conf. Center Rm 404 | ||
| 8:00- 8:40 | Breakfast | ||
| 8:40-10:20 | Nonlinear Function Estimation:
D. Kaplan |
Many techniques in nonlinear time-series analysis involve constructing a model, based on the observed data, of the system under study. Such models are useful for a variety of purposes: making predictions, detecting determinism or nonlinearity in the data, comparing two or more systems under different conditions; reducing noise and estimating system characteristics such as Lyapunov exponents. This lecture will cover the fundamentals of fitting linear and nonlinear functions to data, major architectures of models, and the statistics of comparing models. | Readings:
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| 10:20-10:40 | Break | ||
| 10:40-12:20 | Entrainment and Control:
L. Glass |
Since stimulatation delivered to biological oscillators affects the oscillation, there is the possibility to harness our knowledge of nonlinear dynamics to practical effects. Single stimuli can lead to an annhilation of oscillation (which would be a good thing if the oscillation was a potentially fatal cardiac arrhythmia). Stimulation can also be delivered according to algorithms based on our knowledge of the nonlinear dynamical foundation for the oscillations in order to control dangerous rhythms. | Readings: |
| 12:30- 2:00 | Lunch: UT Conf. Center Rm 404 | ||
| 2:00- 2:50 | Predictability, noise, and the Lyapunov exponent:
T. Schreiber |
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| 2:50- 3:40 | 1/f Noise:
D. Kaplan |
Many biological and physiological time series show long-term correlations of a sort called "1/f noise" or, more generally, "power-law noise." This lecture will discuss the interpretation of power-law signals in terms of self-similarity, introduce methods for characterizing the signals, and discuss the possible interpretation of 1/f noise in terms of non-stationarity. | Readings:
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| 3:40- 4:00 | Break | ||
| 4:00- 5:40 | Lab: Function estimation, dimensions | ||
| 6:00 | Dinner: UT Conf. Center Rm 404 | ||
| 7:00 | Walk (or drive) to Laurel Theatre | ||
| 7:30 | Laurel Theatre | Basic steps of contra dancing | |
| 8:00 | Laurel Theatre | Contra-dance with Music by Contraindicated from Brasstown, NC and calling by Keith Cornett of Lexington, KY | |
| 8:00- 8:40 | Breakfast | ||
| 8:40- 9:30 | Detecting chaos experimentally and practical applications of nonlinear
dynamics:
L. Glass |
In real systems, dynamics arises from a combination of deterministic and stochastic processes. Since by definition, chaos is a property of some deterministic systems, it is difficult (or perhaps does not make sense) to identify chaotic dynaics in real (as opposed to model) systems. However, there is a large literature that proposes to identify chaos in real systems. A "strict constructionist" attitude questions many of the algorithms that have been proposed. A more important question is: what is the mechanism of complex rhythms in physiological systems, and how could one profitably put our knowledge of nonlinear dynamics to use. | J. Cardio.
Electrophys. |
| 9:30-10:20 | Hypothesis testing on time series:
D. Kaplan |
This lecture is about how to pose time series questions in terms of the statistical hypothesis-testing framework. "Surrogate data" is a set of techniques for generating sample time series from a specific null hypothesis. We will motivate and introduce a variety of methods (e.g., phase-randomized, amplitude-adjusted, polished, impulsive, annealed) for generating surrogate data and computing p-values. | Readings
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| 10:20-10:40 | Break | ||
| 10:40-12:20 | Test statistics and artifacts:
T. Schreiber and D. Kaplan |
Surrogate data works in conjunction with a "test statistic" such as nonlinear predictability or the correlation dimension. We will discuss which test statistics are appropriate in various settings. There will be an emphasis on the interpretation of surrogate-data results: what precisely is the null hypothesis for the different types of surrogate data; what alternative hypotheses exist; what artifacts can be introduced in the generation of surrogate data and how can they be avoided. | Readings
test statistics |
| 12:30- 2:00 | Lunch: UT Conf. Center Rm 404 | ||
| 2:00- 2:50 | Beyond low-dimensional chaos:
T. Schreiber |
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| 2:50- 3:40 | Lab: power spectra | ||
| 3:40- 4:00 | Break | ||
| 4:00- 5:40 | Lab: surrogate data | ||
| 6:00 | Dinner: UT Conf. Center Rm 404 | ||
| 8:00- 8:40 | Breakfast | ||
| 8:40- 9:30 | Correlation and coupling: Basics
D. Kaplan |
Although early interest in nonlinear dynamics has focussed on isolated systems, there is a growing set of techniques for characterizing the interacting between systems. This lecture will introduce simple linear and nonlinear techniques for characterizing interactions, correlations, and coupling between systems. | |
| 9:30-10:20 | Generalized measures of coupling
T. Schreiber |
The holy grail of coupling: how to detect coupling of any sort from data. This talk will describe one recently developed technique that illustrates the possibilities and difficulties involved in the general detection and characterization of coupling. | Transinformation |
| 10:20-10:40 | Break | ||
| 10:40-11:30 | Phase synchronization
D. Kaplan |
This lecture will review some recent work by the Potsdam nonlinear dynamics group to examine the synchronization of oscillators in terms of phase (but not necessarily amplitude). This work appears to have particular relevance to the interaction of physiological oscillators such as respiration and the heart. | |
| 11:30-12:00 | Wrap-up session and adjournment
D. Kaplan and T. Schreiber |
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| 12:00 | Lunch UT Conf Center Rm 404 | ||