Rudolf M. Dünki

Indications of nonlinearity, intraindividual specificity and stability of human EEG: the unfolding dimension

Abstract With the help of a parameter which we call the “unfolding dimension”, ww provide a bivariate representation of dynamical EEG analysis. Applied to human EEG, our approach successfully discriminates surrogate data from raw EEG, and similarly, shows human EEG to be both intraindividually specific and stable over time. The heart of our approach is the Grassberger-Procaccia Algorithm for the determination of the correlation dimension D 2 within the context of the well-known “Method of Time Delays”. To guarantee the reproducibility of results, this algorithm as well as an estimate for the K 2 entropy and the determination of f ( α )-spectra have been integrated into a computer program which encompasses an operator/user-independent, automatic and reproducible specification of both an “optimal” time-delay for calculating the correlation integral as well as an “optimal” (scalar invariant) plateau region for the extraction of D 2 and K 2 . Our embedding protocol applied to mathematical systems is shown to be consistent with findings based on other algorithms, in particular, with false nearest neighbor considerations and Sauers minimal embedding criteria. Particular “intelligent” features of the automation and optimization logic plus the inclusion of estimates of the inherent systematic error make the proposed algorithm especially appropriate for applications to biological/psychological data.

Scaling properties in temporal patterns of schizophrenia

Investigations into the patterns of schizophrenia reveal evidence of scaling properties in temporal behaviour. This is shown in the spectral properties of mid-range and long-range (up to two years) daily recordings from a sample of patients drawn at the therapeutic dwelling SOTERIA (Ambuhl et al., in: Springer Series in Synergetics, Vol. 58, eds. Tschacher et al. (Springer, Berlin, 1992) pp. 195–203 and references therein) of the Psychiatric University Hospital in Bern. The therapeutic setting is unique in that it tries to avoid treatment by medication.

Dynamical Systems and the Development of Schizophrenic Symptoms — An Approach to a Formalization

Based on the hypothesis of irregular dynamics in the evolution of schizophrenia (Ciompi et al., 1991), we explored the possibility of describing these processes in terms of dynamical systems (chaos) theory. By analyzing time series of a single schizophrenic patient we found support for the existence of a strange attractor. A short discussion of methods (Grassberger-Procaccia algorithm) that can be applied to dynamical systems completes the theoretical part. This formalization is a basis for further experimental studies as well as for testing and developing models and simulations.

Temporal patterns of human behaviour: are there signs of deterministic 1/f scaling?

Temporal patterns apparently exhibiting scaling properties may originate either from fractal stochastic processes or from causal (i.e., deterministic) dynamics. In general, the distinction between the possible two origins remains a non-trivial task. This holds especially for the interpretation of properties derived from temporal patterns of various types of human behaviour, which were reported repeatedly. We propose here a computational scheme based on a generic intermittency model to test predictability (thus determinism) of a part of a time series with knowledge gathered from another part. The method is applied onto psychodynamic time series related to turns from non-psychosis to psychosis. A nonrandom correlation (ρ=0.76) between prediction and real outcome is found. Our scheme thus provides a particular kind of fractal risk-assessment for this possibly deterministic process. We briefly discuss possible implications of these findings to evaluate the risk to undergo a state transition, in our case a patients risk to enter a next psychotic state. We further point to some problems concerning data sample pecularities and equivalence between data and model setup.

A generic intermittency model and its 1-D meta-map: power laws, invariants and the succession of laminar sequences

Abstract Intermittent behaviour has been found in many systems able to switch between two different dynamic states, e.g. between long laminar phases and short chaotic bursts. Despite the apparently high-dimensional complexity, certain one-dimensional (1-D) maps are known to mimic properties of such dynamics. To these belongs the iterative map x n +1, i = ( x n , i + ( x n , i ) z + ϵ ) mod 1, giving rise to long laminar lengths. The statistics of the laminar lengths are of special interest. Starting from this map, we are interested in the values of x 0, i which arise after passing through the modulo operation. These determine the laminar lengths uniquely. A 1-D meta-map x 0, i = f ( x 0, i −1 ) is derived heuristically. It is used to calculate statistical properties of the laminar phases. Our results show an improvement in the behaviour of short and very long laminar phases as compared to earlier analytical results. Introducing the concept of the generic starting value, we find laminar phases not to be strictly independent of their predecessors.