Information-theoretic approach for the characterization of interactions in nonlinear dynamical systems

Symbolic time series analysis provides us a solid and broadly used toolkit for the characterization of interactions between nonlinear dynamical systems. In this thesis, information-theoretic measures are evaluated with respect to their capability to characterize interactions between dynamical systems. We investigate several important limitations of these measures which may appear when experimental data exhibit strong correlations. It is demonstrated that a high degree of static and/or long-term temporal correlations can, in general, lead to the incorrect inference of directionality of interactions between underlying dynamical systems. In this thesis, we propose two complementary information-theoretic measures which can provide a better characterization of the directionality of interactions in cases where the influence of such correlations in data cannot be neglected. First, the proposed information-theoretic measures are applied to characterize interactions between dynamical model systems with known equations of motion. Finally, they are applied to characterize interactions between multi-channel electroencephalographic recordings from epilepsy patients undergoing the presurgical diagnostics.Informationstheoretischer Ansatz zur Charakterisierung von Interaktionen in nichtlinearen dynamischen Systemen Mit Hilfe der Zeitreihenanalyse können Interaktionen zwischen natürlichen dynamischen Systemen anhand experimenteller Daten charakterisiert werden. In den letzten Jahren wurde eine Reihe von Maßen vorgestellt, die darauf abzielen, neben der Interaktionsrichtung auch die Interaktionsstärke zu bestimmen. Die zur Charakterisierung von Interaktionsrichtungen konzipierte Transferentropie zeichnet sich gerade durch eine besonders hohe Rauschtoleranz gegenüber anderen Maßen aus. Ziel der vorliegenden Arbeit ist es, zwei Limitationen, die die Interpretierbarkeit der Charakterisierungen mit der bisher vorgeschlagenen Transferentropie einschränken, zu untersuchen und auszuräumen. Zum einen wird ein Verfahren entwickelt und implementiert, mit dem langreichweitige Korrelationen besser beobachtet werden können, zum anderen werden Korrekturen vorgeschlagen, die den Einfluss so genannter statischer Korrelationen berücksichtigen. Bei Charakterisierungen von Interaktionsrichtungen mit Hilfe der Transferentropie konnten langreichweitige Korrelationen nur durch die Abschätzung von hochdimensionalen Wahrscheinlichkeitsräumen berücksichtigt werden. Für diese Abschätzung sind sehr viele Datenpunkte innerhalb des Beobachtungsintervalls notwendig, was bei Felddaten, gemessen an unbekannten Systemen, mit der Annahme der Stationarität in einem Beobachtungsintervall konkurriert. Um diese Beschränkung zu umgehen, wird in dieser Dissertation eine Verallgemeinerung des Konzepts der Entropie im Sinne von Lempel-Ziv auf das Maß der Transferentropie übertragen. Hierdurch können langreichweitige Korrelationen ohne die Abschätzung eines hochdimensionalen Wahrscheinlichkeitsraums bestimmt werden. Zeitgleiche Korrelationen der zugrunde liegenden Signale - so genannte statische Korrelationen - können die Interpretierbarkeit der Charakterisierung einschränken. Zur Berücksichtigung statistischer Korrelationen mit den bisher vorgestellten Maßen war ebenfalls eine mit einem großen Rechenaufwand verbundene Abschätzung hochdimensionaler Wahrscheinlichkeiten notwendig. In der vorliegenden Dissertation wird eine Korrektur der Transferentropie zur Abschätzung der statischen Korrelationen vorgeschlagen, ohne höherdimensionale Terme berechnen zu müssen. Durch die in dieser Arbeit vorgestellten Maße und Korrekturen kann die Charakterisierung der Interaktionsrichtung verbessert werden. Dabei wird anhand prototypischer Modellsysteme mit chaotischen Dynamiken demonstriert, dass die Charakterisierungen mit Hilfe der vorgeschlagenen Maße und Korrekturen gerade bei Systemen, die ohne Zeitversatz interagieren, besser interpretierbar sind. Weiterhin wurden Interaktionsstärke und Interaktionsrichtung an Zeitreihen hirnelektrischer Aktivität von Epilepsiepatienten bestimmt und mit Charakterisierungen der Transferentropie verglichen. Hierbei lässt sich zusammenfassen, dass sich mit den in dieser Arbeit vorgestellten Maßen Kontraste unterschiedlicher Interaktionsrichtungen besser auflösen lassen

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

Connectivity Analysis of Electroencephalograms in Epilepsy

This dissertation introduces a novel approach at gauging patterns of informa- tion flow using brain connectivity analysis and partial directed coherence (PDC) in epilepsy. The main objective of this dissertation is to assess the key characteristics that delineate neural activities obtained from patients with epilepsy, considering both focal and generalized seizures. The use of PDC analysis is noteworthy as it es- timates the intensity and direction of propagation from neural activities generated in the cerebral cortex, and it ascertains the coefficients as weighted measures in formulating the multivariate autoregressive model (MVAR). The PDC is used here as a feature extraction method for recorded scalp electroencephalograms (EEG) as means to examine the interictal epileptiform discharges (IEDs) and reflect the phys- iological changes of brain activity during interictal periods. Two experiments were set up to investigate the epileptic data by using the PDC concept. For the investigation of IEDs data (interictal spike (IS), spike and slow wave com- plex (SSC), and repetitive spikes and slow wave complex (RSS)), the PDC analysis estimates the intensity and direction of propagation from neural activities gener- ated in the cerebral cortex, and analyzes the coefficients obtained from employing MVAR. Features extracted by using PDC were transformed into adjacency matrices using surrogate data analysis and were classified by using the multilayer Perceptron (MLP) neural network. The classification results yielded a high accuracy and pre- cision number. The second experiment introduces the investigation of intensity (or strength) of information flow. The inflow activity deemed significant and flowing from other regions into a specific region together with the outflow activity emanating from one region and spreading into other regions were calculated based on the PDC results and were quantified by the defined regions of interest. Three groups were considered for this study, the control population, patients with focal epilepsy, and patients with generalized epilepsy. A significant difference in inflow and outflow validated by the nonparametric Kruskal-Wallis test was observed for these groups. By taking advantage of directionality of brain connectivity and by extracting the intensity of information flow, specific patterns in different brain regions of interest between each data group can be revealed. This is rather important as researchers could then associate such patterns in context to the 3D source localization where seizures are thought to emanate in focal epilepsy. This research endeavor, given its generalized construct, can extend for the study of other neurological and neurode- generative disorders such as Parkinson, depression, Alzheimers disease, and mental illness

Cerebral Synchrony Assessment Tutorial: A General Review on Cerebral Signals' Synchronization Estimation Concepts and Methods

The human brain is ultimately responsible for all thoughts and movements that the body produces. This allows humans to successfully interact with their environment. If the brain is not functioning properly many abilities of human can be damaged. The goal of cerebral signal analysis is to learn about brain function. The idea that distinct areas of the brain are responsible for specific tasks, the functional segregation, is a key aspect of brain function. Functional integration is an important feature of brain function, it is the concordance of multiple segregated brain areas to produce a unified response. There is an amplified feedback mechanism in the brain called reentry which requires specific timing relations. This specific timing requires neurons within an assembly to synchronize their firing rates. This has led to increased interest and use of phase variables, particularly their synchronization, to measure connectivity in cerebral signals. Herein, we propose a comprehensive review on concepts and methods previously presented for assessing cerebral synchrony, with focus on phase synchronization, as a tool for brain connectivity evaluation

Evaluation of Granger causality measures for constructing networks from multivariate time series

Granger causality and variants of this concept allow the study of complex dynamical systems as networks constructed from multivariate time series. In this work, a large number of Granger causality measures used to form causality networks from multivariate time series are assessed. These measures are in the time domain, such as model-based and information measures, the frequency domain and the phase domain. The study aims also to compare bivariate and multivariate measures, linear and nonlinear measures, as well as the use of dimension reduction in linear model-based measures and information measures. The latter is particular relevant in the study of high-dimensional time series. For the performance of the multivariate causality measures, low and high dimensional coupled dynamical systems are considered in discrete and continuous time, as well as deterministic and stochastic. The measures are evaluated and ranked according to their ability to provide causality networks that match the original coupling structure. The simulation study concludes that the Granger causality measures using dimension reduction are superior and should be preferred particularly in studies involving many observed variables, such as multi-channel electroencephalograms and financial markets.Comment: 24 pages, 5 figures, to be published in Entrop

Brain Connectivity Networks for the Study of Nonlinear Dynamics and Phase Synchrony in Epilepsy

Assessing complex brain activity as a function of the type of epilepsy and in the context of the 3D source of seizure onset remains a critical and challenging endeavor. In this dissertation, we tried to extract the attributes of the epileptic brain by looking at the modular interactions from scalp electroencephalography (EEG). A classification algorithm is proposed for the connectivity-based separation of interictal epileptic EEG from normal. Connectivity patterns of interictal epileptic discharges were investigated in different types of epilepsy, and the relation between patterns and the epileptogenic zone are also explored in focal epilepsy. A nonlinear recurrence-based method is applied to scalp EEG recordings to obtain connectivity maps using phase synchronization attributes. The pairwise connectivity measure is obtained from time domain data without any conversion to the frequency domain. The phase coupling value, which indicates the broadband interdependence of input data, is utilized for the graph theory interpretation of local and global assessment of connectivity activities. The method is applied to the population of pediatric individuals to delineate the epileptic cases from normal controls. A probabilistic approach proved a significant difference between the two groups by successfully separating the individuals with an accuracy of 92.8%. The investigation of connectivity patterns of the interictal epileptic discharges (IED), which were originated from focal and generalized seizures, was resulted in a significant difference ( ) in connectivity matrices. It was observed that the functional connectivity maps of focal IED showed local activities while generalized cases showed global activated areas. The investigation of connectivity maps that resulted from temporal lobe epilepsy individuals has shown the temporal and frontal areas as the most affected regions. In general, functional connectivity measures are considered higher order attributes that helped the delineation of epileptic individuals in the classification process. The functional connectivity patterns of interictal activities can hence serve as indicators of the seizure type and also specify the irritated regions in focal epilepsy. These findings can indeed enhance the diagnosis process in context to the type of epilepsy and effects of relative location of the 3D source of seizure onset on other brain areas