KurSL: a model of coupled oscillators based on Kuramoto's coupling and Sturm-Liouville theory

Methods commonly used to analyse oscillatory systems, such as short-time Fourier or wavelet transforms, require predefined oscillatory structures or ne-tuning of method's parameters. These limitations may be detrimental for an adequate component description and can introduce bias to the interpretation. This thesis addresses the challenge of identifying interacting components in a signal by introducing a model of coupled oscillators. The proposed model consists of two parts: Sturm-Liouville self-adjoint ordinary differential equation (ODE) and Kuramoto's coupling model. The resulting model, KurSL, is described by a set of coupled ODEs producing general amplitude- and frequency-modulated mutually interacting oscillations. The complexity of these equations depends on the definition of the coupling function, the number of oscillators and the initial state of each oscillator. Thus, the performance of the KurSL decomposition can be characterised in terms of the model parameters optimisation. After introducing the model, the thesis provides analysis and discussion of the KurSL with examples of its usage. The method is firstly tested on various synthetic data that were generated from simulated stationary and dynamical processes. Such testing allows capturing various characteristics that are desirable in coupled oscillatory components such as phase and amplitude dynamics. Subsequently, experiments were performed on empirical EEG signals recorded from patients with epilepsy. Validation of these experiments is through comparisons to different orders of the KurSL and to other time-frequency methods. Overall results indicate that the KurSL method provides a more detailed description of oscillatory processes than the Huang-Hilbert transform and it provides insights comparable to manually tuned short-time Fourier transform and Morlet-based wavelet time-frequency representations. However, the advantage of the KurSL is that the similar results can be achieved with a finite number of components. Moreover, in contrast to the mentioned representations which, due to finite resolution, are unable to localise time-frequency events precisely, the KurSL provides an instantaneous description. This exactness allows to identify any modulations in both time and frequency domains and thus better describe the behaviour of the analysed system

On the Phase Coupling of Two Components Mixing in Empirical Mode Decomposition

This paper investigates frequency mixing effect of empirical mode decomposition (EMD) and explores whether it can be explained by simple phase coupling between components of the input signal. The input is assumed to be a linear combination of harmonic oscillators. The hypothesis was tested assuming that phases of input signals’ components would couple according to Kuramoto’s model. Using a Kuramoto’s model with as many oscillators as the number of intrinsic mode functions (result of EMD), the model’s parameters were adjusted by a particle swarm optimization (PSO) method. The results show that our hypothesis is plausible, however, a different coupling mechanism than the simple sine-coupling Kuramoto’s model are likely to give better results

Objective Empirical Mode Decomposition metric

Empirical Mode Decomposition (EMD) is a data driven technique for extraction of oscillatory components from data. Although it has been introduced over 15 years ago, its mathematical foundations are still missing which also implies lack of objective metrics for decomposed set evaluation. Most common technique for assessing results of EMD is their visual inspection, which is very subjective. This article provides objective measures for assessing EMD results based on the original definition of oscillatory components