Many natural and artificial systems can be modelled by ensembles of
coupled oscillators. These types of systems can exhibit various
synchronisation phenomena, where the interaction between the oscillators
leads them to some kind of coherent behaviour, despite heterogeneities in
the system. Moreover, many such systems are subject to a timevariable
environment which effectively drives them. Many examples can be found in
living systems, e.g., the dynamics of a cell is strongly dependent on the
ever-changing intra- and extra-cellular ionic concentrations.
Motivated by these considerations, this thesis investigates the effect of
time-varying parameters on synchronisation and stability in ensembles of
coupled oscillators. Timevariability is a crucial ingredient of the
dynamics of many real-life systems, and interest in it is only recently
starting to grow. Such systems are in general described by nonautonomous
equations, which are hard to treat in general. This present work aims at
answering questions such as: Can time-variability be
detrimental/beneficial to synchronisation? If so, under which conditions?
Can time-variability seed new dynamical phenomena? How can one best treat
The systems studied can be divided into two categories. First, the effect
of a driving oscillator with a time-varying frequency is investigated. It
is shown that increasing the amplitude of the frequency modulation can
increase the size of the stability region in parameter space, under
general assumptions. Short-term dynamics and stability properties are also
investigated, and their dynamics is shown to be of importance. Second, the
effect of time-varying couplings between the oscillators is considered.
This is shown to be able to make the synchronous state unstable and yield
Overall, the thesis illustrates that time-variability can be either
beneficial or detrimental to synchronous dynamics, and investigates in
detail and gives insight about cases of both. It argues towards the
general fact that short-term dynamics is often crucial to a physically
relevant understanding of nonautonomous systems.
The superfluid state, often obtained in the laboratory using liquid helium at very low temperatures, provides the basis for a wide range of interesting experiments. One large field of research relates to the study of turbulence in a superfluid, referred to as quantum turbulence due to the importance of quantum mechanical behaviour in the description of this phenomenon. An important experimental tool in the study of quantum turbulence is the mechanical resonator, with many different types of oscillator seeing use. Variations in the resonance can be related to the properties of the fluid through an understanding of the drag applied to the object. The many seemingly disparate measurements reported here were performed in the hope of providing background for future development of improved experimental techniques. In an effort to develop an improved method for determining oscillator properties, measurements have been made using a multifrequency lock-in amplifier in superfluid 4He. The results obtained show that the multifrequency lock-in can be used to obtain results equivalent to the traditional method while reducing the time required. Due to the possibility of vastly increased sensitivity to changes in effective mass, tests were performed in 4He using a new form of oscillator with a 100 nm by 100 nm square cross-section, significantly smaller than other available devices. The resonant frequency of these resonators, referred to as nanobeams, was varied from 0.6 MHz to 8.5 MHz by using beams of different length. Measurements of the resonator response as a function of temperature show that the beams can successfully probe the fluid, though the current theory is found to be insufficient to exactly quantify the dependence seen. A possible observation of turbulence generated by a nanobeam is also reported. Despite an observed critical velocity significantly different to theoretical predictions, all other measurements are consistent with a turbulent transition. As the eventual goal is to use nanobeams for measurements in 3He-B, the drag on high frequencyoscillators in 3He-B was also studied. Measurements on 4 devices of different frequencies found that the current model of damping remains adequate beyond the expected frequency limit for this model. Observations of anomalous increases in the damping for a single resonator in 3He-B are also discussed. As this unexpected damping is only seen for small, sensitive resonators there is concern that similar effects could hinder interpretation of future nanobeam measurements in 3He-B. Efforts were made to understand the source of this damping, and hence explain why it is seen for only one of three nominally identical oscillators, though no conclusive explanation could be found.
Synchronous, coherent interaction is key for the functioning of our brain.
The coordinated interplay between neurons and neural circuits allows to
perceive, process and transmit information in the brain. As such,
synchronization phenomena occur across all scales. The coordination of
oscillatory activity between cortical regions is hypothesized to underlie
the concept of phase synchronization. Accordingly, phase models have found
their way into neuroscience.
The concepts of neural synchrony and oscillations are introduced in
Chapter 1 and linked to phase synchronization phenomena in oscillatory
Chapter 2 provides the necessary mathematical theory upon which a sound
phase description builds. I outline phase reduction techniques to distill
the phase dynamics from complex oscillatory networks. In Chapter 3 I apply
them to networks of weakly coupled Brusselators and of Wilson-Cowan neural
masses. Numerical and analytical approaches are compared against each
other and their sensitivity to parameter regions and nonlinear coupling
schemes is analysed.
In Chapters 4 and 5 I investigate synchronization phenomena of complex
phase oscillator networks. First, I study the effects of network-network
interactions on the macroscopic dynamics when coupling two symmetric
populations of phase oscillators. This setup is compared against a single
network of oscillators whose frequencies are distributed according to a
symmetric bimodal Lorentzian. Subsequently, I extend the applicability of
the Ott-Antonsen ansatz to parameterdependent oscillatory systems. This
allows for capturing the collective dynamics of coupled oscillators when
additional parameters influence the individual dynamics.
Chapter 6 draws the line to experimental data. The phase time series of
resting state MEG data display large-scale brain activity at the edge of
criticality. After reducing neurophysiological phase models from the
underlying dynamics of Wilson-Cowan and Freeman neural masses, they are
analyzed with respect to two complementary notions of critical dynamics.
A general discussion and an outlook of future work are provided in the
final Chapter 7.
Hodgkin and Huxley won a Nobel Prize for their passive model of the squid axon. Their model describes the voltage across a cell plasma membrane, based on measurements in the squid axon. The model explains the generation of action potential. It was published over 60 years ago, however, their model still represents the paradigm in neurobiology . Hodgkin and Huxley used the voltage-clamping method to do their measurements of currents across the membrane of the axon. All measured currents are caused by the diffusion of ions due to their electrochemical gradients. Due to the voltage-clamping method that they used, there was no need to include active transport in their model, therefore, metabolism was ignored in their study. In reality, metabolism is required to produce ATP, which is required to operate the ATPases that regenerates the electrochemical gradient of the cations Na+ and K+ and results in maintenance of the plasma membrane potential . Therefore, it is still unknown how the energy state of a cell is involved in the generation of the plasma membrane potential, and what is the origin of the fluctuations in the voltage across the membrane of a cell. Here we discuss results of free-running whole-cell patch-clamp recordings of the resting membrane potential of jurkat T cells . Since the voltage was not clamped in these experiments, it is plausible to assume that a metabolism is required to pump the cations against their electrochemical gradients. These pumps have been shown to be crucial in maintenance of the plasma membrane potential . To study the interactions between the plasma membrane potential and metabolism, we analysed data recorded in yeast cells in suspension [4, 5]. The measurements include the energy state of the cell evaluated from the intracellular level of ATP in the yeast population, and the mitochondrial membrane potential obtained by a fluorescent recording . In addition, nicotinamide adenine dinucleotide NAD and hydrogen H substance (NADH), plays a role in the chemical process that generates energy for the cell, as well as the intracellular pH were measured. All measured parameters were oscillating over time under aerobic/anaerobic shift. The results were analysed using time series analysis methods that allow for time-localised analyses of the underlying dynamics [7, 8, 9]. We will present results of analysis of interaction between cellular functional processes and argue that the metabolism is driving them. The results suggest that the mitochondrial F0F1-ATPase might be involved in the mechanism by which glycolytic oscillations are driving the oscillations in the mitochondrial membrane potential and the cytosolic pH. The results were modelled as phase oscillators of glycolysis, cytosolic pH and the mitochondrial membrane potential. This model regenerates the signals measured from yeast cells and show approximately the same main mode frequency as the original data.
This PhD thesis explores the hydrodynamics and the performance of a moored array of Wave Energy Converters (WECs). A comparison is made between the performance of an array with bottom mooring lines and the performance of an array that uses shared mooring lines (i.e. inter-body mooring connections). The fundamental equations for arrays with shared moorings are developed in the frequency-domain where simple spherical devices moving in the three translational modes are considered. The numerical model uses linear hydrodynamic and hydrostatic forces, while the mooring connections are linearised using perturbation theory. This is further elaborated upon by considering cylindrical devices, so that the three rotational modes may be considered, as well as their hydrodynamic coupling with the three translational modes. This moored array of cylindrical devices moving in all six Degrees-of-Freedom (DoF) is used as a foundation to explore the hydrodynamics and the performance of an array of floating Oscillating Water Column (OWC) devices known as spar-buoy OWC. Real fluid viscous effects are accounted for by using a linear approximation. The frequency-domain numerical model of the moored array of spar-buoy OWCs is used as a basis for a stochastic model, where the array's performance for a Portuguese wave climate is assessed. The performance of arrays with shared mooring connections VS arrays with bottom mooring connections is explored in both the frequency and stochastic domain models for the spar-buoy OWC array. Both models have confirmed that there is minimal difference in performance between the two mooring systems, which is desirable because the one has been suggested as a more economically viable solution than the other due to drastic reductions in the amount of mooring cables and anchors.
A challenging goal in neuroscience is that of identifying specific brain
patterns characterising autistic spectrum disorder (ASD). Genetic studies,
together with investigations based on magnetic resonance imaging (MRI) and
functional MRI, support the idea that distinctive structural features
could exist in the ASD brain. In the developing brains of babies and small
children, structural differences could provide the basis for different
brain connectivity, giving rise to macroscopic effects detectable by e.g.
electroencephalography (EEG). A significant body of research has already
been conducted in this direction, mainly computing spectral power and
coherence. Perhaps due to methodological limitations, together with high
variability within and between the cohorts investigated, results have not
been in complete agreement, and it is therefore still the case that the
diagnosis of ASD is based on behavioural tests and interviews.
This thesis describes a step-by-step characterisation and comparison of
brain dynamics from ASD and neurotypical subjects, based on the analysis
of multi-probe EEG time-series from male children aged 3-5 years. The
methods applied are all ones that take explicit account of the
intrinsically non-linear, open, and time-variable nature of the system.
Time-frequency representations were first computed from the time-series to
evaluate the spectral power and to categorise the ranges encompassing
different activities as low-frequency (LF, 0.8-3.5 Hz),
mid-range-frequency (MF, 3.5-12 Hz) or high-frequency (HF, 12-48 Hz). The
spatial pathways for the propagation of neuronal activity were then
investigated by calculation of wavelet phase coherence. Finally, deeper
insight into brain connectivity was achieved by computation of the
dynamical cross-frequency coupling between triplets of spatially
distributed phases. In doing so, dynamical Bayesian inference was used to
find the coupling parameters between the oscillators in the
spatially-distributed network. The sets of parameters extracted by this
means allowed evaluation of the strength of particular coupling components
of the triplet LF, MF→HF, and enabled reconstruction of the coupling
functions. By investigation of the form of the coupling functions, the
thesis goes beyond conventional measures like the directionality and
strength of an interaction, and reveals subtler features of the underlying
The measured power distributions highlight differences between ASD and
typically developing children in the preferential frequency range for
local synchronisation of neuronal activity: the relative power is
generally higher at LF and HF, and lower at MF, in the ASD case. The phase
coherence maps from ASD subjects also exhibited differences, with lower
connectivity at LF and MF in the frontal and fronto-occipital pairs, and
higher coherence at high frequencies for central links. There was higher
inter-subject variability in a comparison of the forms of coupling
functions in the ASD group; and a weaker coupling in their theta-gamma
range, which can be linked with the cognitive features of the disorder.
In conclusion, the approach developed in this thesis gave promising
preliminary results, suggesting that a biomarker for ASD could be defined
in terms of the described patterns of functional and effective
connectivity computed from EEG measurements.
Contributors:Krishna Kumar, Roshan
The past decade has seen a new paradigm in solid state physics, where a
new class of layered crystals can be thinned down to a monolayer and
exhibit drastic changes in their electronic and optical properties in
comparison to their bulk counterpart. Graphene was the first, and
certainly most outstanding, of this set of so called two-dimensional (2D)
materials. Aside from its obvious appeal which earnt its discovery the
2010 Nobel Prize, the electronic properties of graphene are truly unique.
Perhaps the most familiar is its linear electron dispersion which hosts
quasi-particles that obey the Dirac equation. This has enabled the study
of a plethora of transport phenomena, as well as the realisation of novel
device architectures that will be used in the next generation electronics.
In general, experimental signatures of electron transport are most
prominent at liquid helium temperatures when lattice vibrations are weak,
for example in quantum hall physics. In this Thesis, we explore the regime
of intermediate temperatures where the physics of interest is strongest
between 100 and 300 K. Equipped with the state of the art high quality
graphene samples, we demonstrate novel electron transport unique to
The experimental work consists of two themes. In the first work, we study
hydrodynamic electron flow in graphene encapsulated with hexagonal boron
nitride devices. At elevated temperatures, electron-electron collisions
become significant, and the electron viscosity starts to influence the
steady state current distribution in a variety of surprising ways. In the
first work, we perform transport experiments on standard graphene hall
bars in a unique measurement geometry which allows the detection of
negative non-local voltages intrinsic to viscous flow. In another
experiment, we study viscous electron flow through graphene
nano-constrictions/classical point contacts. Here, we observed anomalous
temperature dependence in the conductance measured across the
constriction. Specifically, the conductance increases with increasing
temperature and even exceeded the semi-classical limit which is expected
for single-particle ballistic transport. The underlying mechanism
originates from electron-electron collisions, which, counter-intuitively,
act to enhance current flow.
In the second work, we slightly change our experimental system by studying
magneto transport in a graphene/hexagonal boron nitride superlattice. Owed
to the large periodicity of the superlattice unit cell, these devices have
allowed experimental observation of the long sought Hofstadter butterfly,
which addresses the electronic dispersion of electrons in a periodic
potential and magnetic field.
Here, we again go to elevated temperatures, where all the spectral gaps
related to Hofstadter butterflies are completely smeared, and instead find
a new type of quantum oscillation. These new oscillations are periodic in
1/B with a frequency corresponding to one flux quantum piercing the
superlattice unit cell. Whilst these oscillations are related to
Hofstadter physics, they are in fact more primal in origin. The most
fascinating feature is their robustness with respect to increasing
temperature. The oscillations are easily observable at room temperature in
fields as low as 3 T and still remained prominent at 373 K, the boiling
point of water