Modelling and analysing neuronal and epidemiological dynamics on structured static and dynamic networks

Many of the technological, social and biological systems we observe and partake in in our everyday lives can be described as networks of interacting elements. Network- based research can then be performed to improve our understanding about the structural features of such complex networks, the behaviour of processes occurring within such complex networks, and the interaction between the two. During my PhD I have considered neuronal and epidemiological dynamics occurring on complex networks, with the main aim of improving model realism by incorporating spatial or local structure whilst maintaining model tractability. In total I have considered three network-based research projects which are included in this thesis in chronological order. This thesis begins with an introduction to the study of complex networks and processes occurring on complex networks. Comparisons are drawn between the approaches of neuroscience and epidemiology-based network studies, including consideration of the difficulties regarding modelling local spatial structure. Chapter 2 considers an existing model describing the activity-dependent growth and development of a network of excitatory and inhibitory neurons embedded in space. A systematic investigation of the effects of various spatial arrangements of neurons on the resultant electrical dynamics finds that increased spatial proximity between inhibitory neurons leads to oscillatory dynamics. Chapter 3 utilises the edge-based compartmental modelling approach. Existing research is extended to derive and validate equations describing the evolution of a susceptible-infected-recovered (SIR) epidemic process occurring on a dual-layer multiplex network incorporating heterogeneity in the structure, type and duration of connections between individuals. Chapter 4 considers pairwise models describing the SIR epidemic process and derives and validates analytic expressions for the epidemic threshold, an improvement on existing results. This thesis concludes with a discussion of the research contained in Chapters 2-4, including suggestions for improvements and future research ideas.