Tackling the Inverse Problem for Non-Autonomous Systems: Application to the Life Sciences.

The common assumption that a dynamical system found in nature can be considered as isolated and autonomous is frequently a poor approximation. In reality, there are always external influences, and these are often too strong to ignore. In the case of an interacting oscillatory systems, they may e.g. modify their natural frequencies or coupling amplitudes. The main objective of this thesis is to study, detect and understand in greater detail the effect of external dynamical influences on interacting self-sustained oscillators. Theoretical framework for the analysis of synchronization between non-autonomous oscillating systems is discussed. Multiple-scale analysis is applied on a phase oscillators model with slowly varying frequency. This analysis revealed the analytic form of the synchronization state with respect to slow and fast time-variations. Limit-cycle oscillators are used to study amplitude dynamics and to investigate synchronization transitions, which occur in the bifurcation points where the equilibrium solution for the phase difference and amplitudes changes their stability. Bifurcation diagrams as functions of coupling parameters are also constructed. In a case of non-autonomous interacting oscillators, the phase difference varies dynamically, the external influences can be the cause for synchronization transitions between different synchronization orders, and lag synchronization is hardly achievable. It is also demonstrated that the time-variations of the form of the coupling function alone can be the cause for synchronization transitions. A method is introduced for analysis of interactions between time-dependent coupled oscillators, based on the signals they generate. It distinguishes unsynchronized dynamics from noise-induced phase slips, and enables the evolution of the coupling functions and other parameters to be followed. The technique is based on Bayesian inference of the time-evolving parameters, achieved by shaping the prior densities to incorporate knowledge of previous samples. The dynamics can be inferred from phase variables, in which case a finite number of Fourier base functions are used, or from state variables exploiting the model state base functions. The latter is used for detection of generalized synchronization. The method is tested numerically and applied to reveal and quantify the time-varying nature of synchronization, directionality and coupling functions from cardiorespiratory and analogue signals. It is found that, in contrast to many systems with time-invariant coupling functions, the functional relations for the interactions of an open (biological) system can in itself be a time-varying process. The cardiorespiratory analysis demonstrated that not only the parameters, but also the functional relationships, can be time-varying, and the new technique can effectively follow their evolution. The proposed theory and methods are applied for the analysis of biological oscillatory systems affected by external dynamical influences. The main investigation is performed on physiological measurements under conditions where the breathing frequency is varied linearly in a deterministic way, which introduces non-autonomous time-variability into the oscillating system. Methods able to track time-varying characteristics are applied to signals from the cardiovascular, and the sympathetic neural systems. The time-varying breathing process significantly affected the functioning and regulation of several physiological mechanisms, demonstrating a clear imprint of the particular form of externally induced time-variation. Specifically, the low breathing frequencies provoked more information flow, interfering the coordination and increasing the coupling strength between the oscillatory processes. Statistical analyses are performed to identify significant relationships. The proposed inferential method is applied to cardiorespiratory signals of this kind. The technique successfully identified that the cardiorespiratory coordination depends on, and is regulated to a great extent by, the respiration dynamics. The time-varying respiration acted as a cause for synchronization transitions between different orders. Additional complexity is encountered by the coupling functions which are also identified as time-varying processes. A technique based on wavelet synchrosqueezed transform shows how the instantaneous phase can be extracted from complex mixed-mode signals with time-varying characteristics. The latter is demonstrated on several physiological signals of this kind. The dynamical characterization for the reproducibility of blood flow is shown to be more appropriate than the time-averaged analysis. This also implies that care must be taken when external perturbations are made consecutively. Finally, the study focuses on analysis of analogue simulation of two non-autonomous van der Pol oscillators. The oscillators are unidirectionally coupled, and the frequency of the first oscillator is externally and periodically perturbed. The analogue simulation presents another model which encounters real experimental noise. The intermittent synchronization and the corresponding transitions are detected both through phase, and generalized synchronization, based on a common inferential basis