Many chaotic and complicated systems cannot be analysed by traditional methods. In 1987 P.Bak, C.Tang, and K.A.Wiesenfeld developed a new concept called Self-Organised Criticality (SOC) to explain the behaviour of composite systems containing a large number of elements that interact over a short range. In general this theory applies to complex systems that naturally evolve to a critical state in which a minor event starts a chain reaction that can affect any number of elements in the system. It was later shown that many complex phenomena such as flux pinning in superconductors, dynamics of granular systems, earthquakes, droplet formation and biological evolution show signs of SOC. The dynamics of complex systems in nature often occurs in terms of punctuation, or avalanches rather than following a smooth, gradual path. Extremal dynamics is used to model the temporal evolution of many different complex systems. Specifically the Bak-Sneppen evolution model, the Sneppen interface depinning model, the Zaitsev flux creep model, invasion percolation, and several other depinning models. This thesis considers extremal dynamics at constant flux where M>1 smallest barriers are simultaneously updated as opposed to models in the limit of zero flux where only the smallest barrier is updated. For concreteness, we study the Bak-Sneppen (BS) evolution model [Phys. Rev. Lett. 71, 4083 (1993)]. M=1 corresponds to the original BS model. The aim of the present work is to understand analytically through mean field theory the random neighbour version of the generalised BS model and verify the results against the computer simulations. This is done in order to scrutinise the trustworthiness of our numerical simulations. The computer simulations are found to be identical with results obtained from the analytical approach. Due to this agreement, we know that our simulations will produce reliable results for the nearest neighbour version of the generalised BS model. Since the nearest neighbour version of the generalised BS model cannot be solved analytically, we have to rely on simulations. We investigate the critical behaviour of both versions of the model using the scaling theory. We look at various distributions and their scaling properties, and also measure the critical exponents accurately verifying whether the scaling relations holds. The effect of increasing from M=1 to M>1 is surprising with dramatic decrease in size of the scaling regime.

The brain is a complex system par excellence. Its intricate structure has become clearer recently, and it has been reported that it shares some properties common to complex networks, such as the small-world property, the presence of hubs, and assortative mixing, among others. These properties provide the brain with a robust architecture appropriate for efficient information transmission across different brain regions. Nevertheless, how these topological properties emerge in neural networks is still an open question. Moreover, in the last decade the observation of neuronal avalanches in neocortical circuits suggested the presence of self-organised criticality in neural systems. The occurrence of this kind of dynamics implies several benefits to neural computation. However, the mechanisms that give rise to critical behaviour in these systems, and how they interact with other neuronal processes such as synaptic plasticity are not fully understood. In this thesis, we study self-organised criticality and neural systems in the context of complex networks. Our work differs from other similar approaches by stressing the importance of analysing the influence of hubs, high clustering coefficients, and synaptic plasticity into the collective dynamics of the system. Additionally, we introduce a metric that we call node success to assess the effectiveness of a spike in terms of its capacity to trigger cascading behaviour. We present a synaptic plasticity rule based on this metric, which enables the system to reach the critical state of its collective dynamics without the need to fine-tune any control parameter. Our results suggest that retro-synaptic signals could be responsible for the emergence of self-organised criticality in brain networks. Furthermore, based on the measure of node success, we find what kind of topology allows nodes to be more successful at triggering cascades of activity. Our study comprises four different scenarios: i) static synapses, ii) dynamic synapses under spike-timing-dependent plasticity (STDP), iii) dynamic synapses under node-success-driven plasticity (NSDP), and iv) dynamic synapses under both NSDP and STDP mechanisms. We observe that small-world structures emerge when critical dynamics are combined with STDP mechanisms in a particular type of topology. Moreover, we go beyond simple spike pairs of STDP, and implement spike triplets to assess their influence on the dynamics of the system. To the best of our knowledge this is the first study that implements this version of STDP in the context of critical dynamics in complex networks.

Above true strain rates of $10^4$ s$^{-1}$ FCC metals exhibit a rapid increase in strength. Understanding of the physical mechanisms behind this strength transition is hindered by the number and interdependence of candidate mechanisms. Broadly, contributions to strength can be split into `instantaneous' effects and the more permanent `structural' ones. In this thesis a series of experiments are presented which are designed to separate the two types of contribution. Chapter 2 outlines the basics of dislocation plasticity, based on the seminal works of Taylor and Orowan. It then progresses on to discuss recent experimental and theoretical work on the understanding of slip as avalanche behaviour. Chapter 3 summarises traditional modelling approaches for instantaneous strength contributions which are routinely applied below $10^4$ s$^{-1}$. It then continues on to outline a number of different approaches which have been adopted to attempt to explain and model the strength transition. Chapter 4 outlines the methods used in the earliest stages of the study: Instron and split Hopkinson pressure bar methods. Both methods are well established, and cover the majority of the range of rates under study. Emphasis is made on minimising experimental sources of error, and subsequently accounting for those which are unavoidable. Finally, the specimen material is introduced and is shown to be fit for purpose. Chapter 5 presents a set of mechanical tests of specimens at strain rates between $10^4-10^5$~s$^{-1}$. The softening of the specimens with increased temperature is observed to increase with strain rate, both in absolute terms and when normalised to the 300 K measurement for each strain rate. The observations are most easily explained if the strength transition is due to an increase in early stage work hardening, however, some anomalous behaviours remain. Chapter 6 introduces a new experimental technique; direct impact Hopkinson pressure bars, required to perform experiments shown to be necessary by the results of Chapter 5. Photon Doppler velocimetry is applied to the projectiles used in experiments, removing one of the most significant flaws of the technique, and creating a more confident basis with which to perform further experimental work. Chapter 7 presents a series of `jump tests' at ambient temperatures. Specimens are deformed at strain rates ranging from $10^{-2}$ to $10^5$~s$^{-1}$ to a fixed strain of 0.1, then reloaded to yield at a strain rate of $10^{-1}$. The yield point at reload is shown to have the same rapid upturn as seen when the specimens were deforming at high rates, providing strong evidence that the increase in strength is due to changes in the underlying dislocation structure, rather than a dynamic effect, as it remains even when the high strain rate is removed. Chapter 8 continues on from the conclusions of Chapter 7. Jump tests are expanded to a variety of temperatures and strains, to provide a more complete characterisation of metal behaviour. No dramatic change in the saturation of work hardening is observed to coincide with the increase in early stage work hardening. Chapter 9 discusses discrepancies between contemporary high rate models and recent developments in the understanding of plasticity being an avalanche process. Potential consequences of incorporating avalanche plasticity into high rate models are explored. Particular attention is paid to Brown's observation that based on quasi static observations of avalanche behaviour, the formation of dislocation avalanches will begin to fail at strain rates of approximately $10^4$ s$^{-1}$. Consequences of the progressive breakdown of avalanche behaviour are discussed with respect to the experimental observations presented in earlier chapters. In Chapter 10, we will discuss the key conclusions of the work. Finally, a number of avenues are proposed for building upon the current work both theoretically and experimentally.

In seismology, an important goal is to attain a better understanding of the earthquake process. In this study of the physics of aftershock generation, I couple statistical analysis with modelling of physical processes in the postseismic period. I present a theoretical formulation for the distribution of interevent times for aftershock sequences obeying the empirically well established Omori law. As opposed to claims by other authors, this work demonstrates that the duration of the time interval between two successive earthquakes cannot be used to identify whether or not they belong to the same aftershock sequence or occur as a result of the same underlying process. This implies that a proper understanding of earthquake interevent time distributions is necessary before conclusions regarding the physics of the earthquake process are drawn. In a discussion of self-organised criticality (SOC) in relation to empirical laws in seismology, I find that Omori's law for aftershocks cannot be used as evidence for the theory of SOC. Instead, I consider that the occurrence of aftershocks in accordance with Omori's law is a result of a physical process that can be modelled and understood. I analyse characteristic features in the spatiotemporal distribution of aftershocks in the south Iceland seismic zone, following the two M6.5 June 2000 earthquakes and a M4.5 earthquake in September, 1999. These features include an initially constant aftershock rate, whose duration is larger following a larger main shock, and a subsequent power law decay that is interrupted by distinct and temporary deviations in terms of rate increases and decreases. Based on pore pressure diffusion modelling, I interpret these features in terms of main shock initiated diffusion processes. I conclude that thorough data analysis and physics-based modelling are essential components in attempts to improve our understanding of processes governing the occurrence of earthquakes.