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Abstract

Complex systems, composed of numerous interacting subsystems, exhibit intricate dynamics
that are challenging to model and understand. This dissertation focuses on advancing
methodologies for detecting the direction and quantifying the strength of couplings in such
systems and presents their application to real data. The basics of dynamical systems and
chaos are first presented, followed by an introduction to couplings and synchronization.
Then, the methodology for detecting couplings from measured time series is presented.
This includes the Granger causality, the phase dynamics methods, the information methods,
and the state space methods, with an emphasis on the latter two. An alternative
approach called perturbation experiments, applicable to systems that can be directly influenced,
is also presented. The subsequent chapters present original research and practical
applications.
The first part of the research addresses the solvability of the inverse problem, specifically
whether the coupling strength parameter can be inferred from time series by computing
information flow. Findings reveal this is possible for weak couplings but not when the
response system is highly sensitive to perturbations. Additionally, for strong diffusive
couplings, information flow decreases monotonically with increasing coupling strength,
making the inverse problem solvable, given that coupling is strong enough.
The research’s second part improves the state space methods for directional coupling
detection. By introducing cross-distance vectors and analyzing their response to coupling,
the initial tails of the cross-distance vectors are recognized as containing information about
the coupling. A new measure is defined that quantifies the prominence of this initial tail,
enhancing the accuracy of the state space measures and reducing the risk of false positives.
This advancement increases the reliability of such methods, broadening their applicability
across various scientific fields. Furthermore, a rank-based state space measure is introduced
and combined with surrogate data. Application to an EEG database demonstrates
the measure’s enhanced ability to differentiate between focal and nonfocal EEG signals,
highlighting its potential in clinical applications.
The final part presents a perturbation experiment and an analysis of the resulting
time series. The experiment is conducted on patients with severe aortic valve stenosis
and a healthy control group, with the goal of quantifying neurovascular coupling strength.
Results show that computing the visually evoked cerebral blood flow velocity response to
quantify the coupling strength can effectively distinguish between the aortic stenosis group
and the healthy control group, as found by detecting a statistically significant difference.
Overall, this dissertation provides contributions to the field of detecting interactions
within complex systems, offering improved methods for detecting and quantifying directional
couplings and demonstrating their practical applications in medical research.