This doctoral thesis outlines several methodological advances in network science aimed
towards uncovering rapid, complex interdependencies of electromagnetic brain activity
recorded from the Electroencephalogram (EEG). This entails both new analyses and
modelling of EEG brain network topologies and a novel approach to analyse rapid dynamics
of connectivity. Importantly, we implement these advances to provide novel insights into
pathological brain function in Alzheimer’s disease.
We introduce the concept of hierarchical complexity of network topology, providing both an
index to measure it and a model to simulate it. We then show that the topology of functional
connectivity estimated from EEG recordings is hierarchically complex, existing in a scale
between random and star-like topologies, this is a paradigm shift from the established
understanding that complexity arises between random and regular topologies. We go
on to consider the density appropriate for binarisation of EEG functional connectivity, a
methodological step recommended to produce compact and unbiased networks, in light of its
new-found hierarchical complexity. Through simulations and real EEG data, we show the
benefit of going beyond often recommended sparse representations to account for a broader
range of hierarchy level interactions.
After this, we turn our attention to assessing dynamic changes in connectivity. By constructing
a unified framework for multivariate signals and graphs, inspired by network science and graph
signal processing, we introduce graph-variate signal analysis which allows us to capture rapid
fluctuations in connectivity robust to spurious short-term correlations. We define this for
three pertinent brain connectivity estimates- Pearson’s correlation coefficient, coherence and
phase-lag index- and show its benefit over standard dynamic connectivity measures in a range
of simulations and real data.
Applying these novel methods to EEG datasets of the performance of visual short-term memory
binding tasks by familial and sporadic Alzheimer’s disease patients, we uncover disorganisation
of the topological hierarchy of EEG brain function and abnormalities of transient phase-based
activity which paves the way for new interpretations of the disease’s affect on brain function.
Hierarchical complexity and graph-variate dynamic connectivity are entirely new methods for
analysing EEG brain networks. The former provides new interpretations of complexity in static
connectivity patterns while the latter enables robust analysis of transient temporal connectivity
patterns, both at the frontiers of analysis. Although designed with EEG functional connectivity
in mind, we hope these techniques will be picked up in the broader field, having consequences
for research into complex networks in general