Course Director:  Daniel Kaplan  Macalester College 
Lecturers:  Leon Glass  McGill University 
Thomas Schreiber  MPIPKS 
Lab Software: a tar file of Matlab Mfiles
NTSA  H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press 
UND  D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, SpringerVerlag 
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 online. 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 

9:3010:20  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:

10:2011:10  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:

11:1011:30  Break  
11:3012:20  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. 

12:30 2:00  Lunch: UT Conf. Center Rm 404  
2:00  2:50  Foundations of NLTS Analysis:
D. Kaplan 
Much of nonlinear timeseries 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 lagembedding of time series; analysis of structure in system trajectories based on fractals and estimation of dimension.  Readings:

2:50 3:40  Fractal Dimensions: Promises and Problems:
T. Schreiber 

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  
8:4010:20  Nonlinear Function Estimation:
D. Kaplan 
Many techniques in nonlinear timeseries 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:

10:2010:40  Break  
10:4012:20  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 

2:50 3:40  1/f Noise:
D. Kaplan 
Many biological and physiological time series show longterm correlations of a sort called "1/f noise" or, more generally, "powerlaw noise." This lecture will discuss the interpretation of powerlaw signals in terms of selfsimilarity, introduce methods for characterizing the signals, and discuss the possible interpretation of 1/f noise in terms of nonstationarity.  Readings:

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  Contradance 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. 
9:3010:20  Hypothesis testing on time series:
D. Kaplan 
This lecture is about how to pose time series questions in terms of the statistical hypothesistesting 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., phaserandomized, amplitudeadjusted, polished, impulsive, annealed) for generating surrogate data and computing pvalues.  Readings

10:2010:40  Break  
10:4012:20  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 surrogatedata 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 lowdimensional chaos:
T. Schreiber 

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.  
9:3010:20  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 
10:2010:40  Break  
10:4011:30  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.  
11:3012:00  Wrapup session and adjournment
D. Kaplan and T. Schreiber 

12:00  Lunch UT Conf Center Rm 404 