Daniel T. Kaplan -- Research Interests

Assistant Professor
Macalester College
Department of Mathematics
1600 Grand Avenue.
St. Paul, MN 55105 USA
B.A. Swarthmore College, M.S. Stanford University,
M.S. Harvard University, Ph.D. Harvard University

Curriculum Vitae

Preprints

Summer Research Opportunities for Undergraduates

Summer research positions are available to interested students. Computer programming skills are required: CS 20 or 23 or equivalent experience are pre-requisites. Past research students were: Nicholas Weininger, Michael McGeachie, Phil Staffin, Jenny Hunter, Chris Bremer, and Miguel Fidalgo.

Stipend support and housing is available from sources such as the Hughes Program and the Keck Program.

Contact D. Kaplan for more information.

The Analysis of Complex Physiological Variability

Many physiological systems show variability that appears to be irregular and random. Traditionally, such variability is ascribed to changes in the environment of the organism or to changes in activity level. It is now known, however, that nonlinearities in control systems can lead to oscillations that are generated internally. Since physiological systems are invariably nonlinear, physiological variability may contain important information about how the organism is functioning. We are investigating data analysis methods that can help to extract this information from recorded signals such as beat-to-beat heart rate and blood pressure. There are several active research projects along these lines.

Detecting Nonlinearities and Nonstationarities in Heart Rate Data

We are developing simple measures of variability that can be used to probe for interesting structure in heart rate data. These measures include time reversal asymmetries and quantification of repeated patterns or motifs. Using surrogate data, we are able to test whether the detected patterns provide evidence for nonlinear dynamics in heart rate control.

Preprint
data

Interpretation of Heart Rate Variability using Nonlinear Models

Many of the measures of heart rate variability inspired by nonlinear dynamics --- such as entropy and dimension --- are quite abstract, and it is difficult to optimize them for the task of providing useful information to clinicians. We are attempting to develop practical means of using physiologically plausible models of the cardiovascular control system to interpret measured variability. The advantage of using models is that they can present information in a form that is readily assimilated by clinicians. Difficulties arise, however, because the models are not faithful representations of each individual's control system and because the internal variables in the models (e.g., parasympathetic activity) are not directly measurable.

1/f and other long-term Heart Rate Variability

Heart rate, and many other physiological and physical variables, shows long-term correlations that are remarkably consistent in form between individuals. There is currently no universally accepted theory that explains why heart rate has these long-term correlations, and it is unclear whether they can offer any useful information about the cardiovascular control system. One interesting possibility is that the long-term fluctuations reflect the interaction between different elements of the control system that operate on different time scales (e.g., the renal blood-volume pressure-control system and the adaptation of baroreceptors and chemoreceptors), and that the long-term correlations can be used to monitor these subsystems. in the We are developing statistical techniques that will allow us to find an optimal level of description for long-term correlations, and establish appropriate tests for stationarity.

Nonlinear Dynamics in Signal Processing

Detecting Nonlinearity and Nonstationarity

A fundamental question in analyzing aperiodic data is whether the data is generated by a deterministic chaotic process or a stochastic, random process. We have been developing tests to answer this question. To a large extent, these tests for deterministic chaos are based on refuting the Null Hypothesis that the data come from a linear dynamical system with stochastic inputs, and use the surrogate data technique. There are a variety of trivial, non-chaotic nonlinearities than can cause the Null Hypothesis to be rejected, and we are investigating ways to frame alternative Nulls that can allow these situations to be detected.

Using the surrogate data technique, we are also developing practical means for detecting and quantifying process non-stationarity in data.

DT Kaplan (1994), "Exceptional events as evidence for determinism" Physica D 73:38-48 Postscript version

Fixed Points and "Chaos Control"

One of the most exciting developments in nonlinear dynamics has been the idea of chaos control: that the structure of nonlinear dynamical systems can be exploited to allow improved strategies for getting these systems to do things that we want them to do. In biological applications, chaos control seems most often to rely on the existence of unstable fixed points in the system. We are developing statistical techniques for identifying unstable fixed points in biological data.

Nonlinear Filtering

"Filtering" refers to a procedure for separating a desired component --- the signal --- from an undesired component --- the noise. Traditional linear approaches to signal filtering rely on differences in the spectral characteristics of the signal and noise. We are applying nonlinear dynamics techniques to construct useful filtering methods in cases where the spectrum of the signal and noise are not distinct. A particular area of application is the extraction of the fetal component of the electrocardiogram from the combined fetal/maternal ECG measured on the mother's body surface.

Preprint on ECG filtering

Textbook on Nonlinear Dynamics

DT Kaplan and L Glass, Understanding Nonlinear Dynamics Springer-Verlag, 1995

Preface and Table of Contents

Textbook on Resampling and Bootstrapping Statistics

DT Kaplan, Resampling Stats in MATLAB The entire book is on line.

Address:

Daniel Kaplan
Department of Mathematics
Macalester College
1600 Grand Avenue
St. Paul, MN 55105
USA
612-696-6599 (voice)
612-696-6492 (fax)
kaplan@macalester.edu