My current research interests are located in the field of analysis and modeling of multivariate (possibly spatiotemporal) time series data obtained from complex nonlinear systems. My focus lies on actual applications and interdisciplinary cooperation. I am involved in research projects in nonlinear dynamics, brain science, finance and hydrodynamics.
My most extensive research experience lies in the analysis of neurophysiological data from clinical sources, mainly given as electroencephalographic recordings (EEG). In a previous project (conducted partly in Kiel, Germany, partly at ISM Tokyo and partly in Davis, California) I have worked on hypersynchroneous brain dynamics in children suffering from a specific kind of epilepsy, known as Rolandic epilepsy, which so far is not yet understood in terms of the underlying brain dynamics. During this project I have collected a data base comprising of more than 1000 carefully selected data sets from 102 patients (all from the hospital of neuropediatrics at Kiel) mostly between 5 and 15 years of age. This data base is quite unique and presents the necessary foundation for closer investigation of this disease in particular, and of human brain dynamics in childhood in general. Recently I have also begun collecting a data base of EEG recordings displaying epileptic absence seizures.
The human brain is commonly regarded as the most complex natural structure known to science, and in order to achieve progress in understanding its dynamics new statistical techniques for analysis and modelling of spatiotemporal time series data will have to be developed. Until recently mainly phenomenological approaches to modelling given EEG time series data have been prevailing, but it would be desirable to gradually replace them by models directly refering to the electrical generators of the EEG signal. These will contain a much larger number of degrees of freedom than available in the surface EEG, even if recordings with a high number of electrodes are performed. So far the problem of estimating the spatiotemporally extended dynamical state of a human brain in the space of electrical generators, given only the small amount of information contained in surface EEG recordings (posing an "inverse problem"), has been addressed, and approximately solved, only for the static case, i.e. for a fixed point of time; the result is known as "low resolution brain electromagnetic tomography" (LORETA).
A highly challenging and promising approach to EEG modeling consists of combining the estimation of a spatial solution to the inverse problem with temporal (i.e. dynamical) modelling. Only through this combination we could hope to reveal the details of actual brain dynamics, most of which are inaccessible to direct observation. I have already been able to demonstrate that in principle this goal can be achieved by reconstructing the EEG generators through Kalman filtering and estimating model parameters by Maximum Likelihood. This work is conducted in a collaboration with the ISM Tokyo, the Cuban Neuroscience Center and the Neuropediatric hospital of Kiel University, Germany.
As a result of these efforts it is hoped that much more accurate inverse solutions are obtained, both in terms of spatial resolution and residual errors. The availability of a reliable estimation procedure for inverse solutions would open up a vast field of applications to brain science and clinical research. Moreover, the applicability of Maximum Likelihood estimation to spatiotemporal data will itself provide essential information on modelling brain dynamics, since it provides a measure of success for iterative refinement of models; a wide variety of linear and nonlinear spatiotemporal autoregressive models is waiting for being explored. In this context it would be particularly interesting whether nonlinear models will actually be able to outperform linear models in terms of improving the likelihood, since so far this has only rarely been demonstrated for real-world EEG data. Also the generalisation of Kalman filtering to the spatiotemporal case constitutes a promising field for research and applications to a variety of data analysis problems, e.g. in meteorology, geophysics or hydrodynamics. Once sufficient experience has been gained with the application of this new filtering approach to EEG data, I intend to explore also its application within these other fields; with my former research group at Kiel University, Germany, I am still involved in collaborative research on hydrodynamic model systems (mainly Taylor-Couette flow) using high-precision experiments. Such experiments provide the unique occasion to analyse experimental data obtained from a well-controlled high-dimensional system.
More direct access to spatiotemporal brain dynamics than from EEG time series data can be obtained from functional magnetic resonance imaging (fMRI), but the temporal resolution is much lower than in the case of EEG. Nevertheless the new results on spatiotemporal autogressive model estimation should also be applied to MRI time series data. In this case, as with EEG, the idea of extending the concept of whitening from the temporal to the spatial domain will be a crucial point.
A comprehensive dynamical model for filtering real-world EEG time series will have to model not only brain dynamics, but also various artifacts of technical or physiological origin, which are known to severely contaminate the signal. So far no sufficiently reliable algorithm for automatic artifact detection and filtering has been developed. It is hoped that through estimation of spatiotemporal models by Maximum Likelihood it will become possible to clearly isolate and remove various of the more frequent artifacts. For roughly periodic artifacts originating from the power supply system, a variant of seasonal adjustment should be employed. This could provide highly useful tools for practical work with EEG data; due to the lack of such tools, it has until now not been possible to introduce statistical techniques for filtering and modelling into practical clinical work with EEG data, e.g. for the purpose of establishing diagnoses. Until now clinical analysis of EEG data is still largely limited to visual inspection, without any involvement of quantitative methods. By close cooperation with clinical collaborators I hope to be able to establish the use of quantitative statistical techniques in practical clinical data analysis.
An important aim of present research on EEG time series is the identification of characteristic features of specific diagnoses. As an example, in the case of Rolandic epilepsy one would like to predict hypersynchroneous brain dynamics during sleep by using only awake-state data, although in many cases the latter renders a perfectly normal, healthy impression. In order to approach this task, techniques for classification or clustering of time series are needed, but so far only few studies on this topic have been published. In previous work (so far unpublished) I have applied a technique for clustering of time series by use of multivariate linear correlation information to the aforementioned EEG data base; this work was done in cooperation with R. Shumway from UC Davis, California, who had proposed this technique for the analysis of seismic signals. I intend to continue this line of work by using improved measures of correlation in given data, exploring in particular nonlinear measures borrowed from information theory like mutual information. It is also to be expected that refined spatiotemporal models of brain dynamics, as they will be obtained from spatiotemporal Kalman filtering, will provide improved features for the purposes of clustering or classification of EEG time series. Another important question to be investigated by using inverse solutions of brain dynamics is the issue of causality relationships between different parts of brain.
Furthermore, I am currently participating in a cooperation with M. Mueller and G. Baier from UAEM, Cuernavaca, Mexico, on the analysis of correlations in multivariate EEG time series from patients suffering from absence epilepsy. In this project it is our aim to identify precursors of imminent epileptic seizures from the correlation structure of the time series. This approach to the description of multivariate correlations is based on Random Matrix Theory, as developed for the analysis of energy spectra in nuclear physics.
Spatially extended high-dimensional systems may display strong nonlinearities, which sometimes can be detected by the estimation of fractal dimension (in particular "correlation dimension"). In previous work I was able to obtain finite estimates of fractal dimension for certain dynamical states in Taylor-Couette flow and also for certain financial time series, namely daily returns of exchange rates. The implications of these results for appropriate modeling are not yet clear and deserve further investigation. Both of these two cases will be suitable candidates for the development and application of advanced approaches to multivariate modeling. Fractal dimension itself, having been very popular in the nonlinear dynamics community during the previous 15 years, remains an interesting tool for the analysis of times series data; it may offer an attractive link between the fields of autoregressive modelling and of general nonlinear correlation measures, which so far has been hardly explored. Currently I am employing fractal dimension for investigating the interaction between spatial and temporal correlations in reconstructed state spaces of nonlinear deterministic time series (multiple Lorenz attractors). I hope that these investigations may contribute to bridging the gap between statistical modelling of time series on the one side and nonlinear dynamical systems theory on the other side.