Stochastic timeseries analysis in electric power systems and paleo-climate data

In this thesis a data science study of elementary stochastic processes is laid, aided with the development of two numerical software programmes, applied to power-grid frequency studies and Dansgaard--Oeschger events in paleo-climate data. Power-grid frequency is a key measure in power grid studies. It comprises the balance of power in a power grid at any instance. In this thesis an elementary Markovian Langevin-like stochastic process is employed, extending from existent literature, to show the basic elements of power-grid frequency dynamics can be modelled in such manner. Through a data science study of power-grid frequency data, it is shown that fluctuations scale in an inverse square-root relation with their size, alike any other stochastic process, confirming previous theoretical results. A simple Ornstein--Uhlenbeck is offered as a surrogate model for power-grid frequency dynamics, with a versatile input of driving deterministic functions, showing not surprisingly that driven stochastic processes with Gaussian noise do not necessarily show a Gaussian distribution. A study of the correlations between recordings of power-grid frequency in the same power-grid system reveals they are correlated, but a theoretical understanding is yet to be developed. A super-diffusive relaxation of amplitude synchronisation is shown to exist in space in coupled power-grid systems, whereas a linear relation is evidenced for the emergence of phase synchronisation. Two Python software packages are designed, offering the possibility to extract conditional moments for Markovian stochastic processes of any dimension, with a particular application for Markovian jump-diffusion processes for one-dimensional timeseries. Lastly, a study of Dansgaard--Oeschger events in recordings of paleoclimate data under the purview of bivariate Markovian jump-diffusion processes is proposed, augmented by a semi-theoretical study of bivariate stochastic processes, offering an explanation for the discontinuous transitions in these events and showing the existence of deterministic couplings between the recordings of the dust concentration and a proxy for the atmospheric temperature

Foundations of Stochastic Thermodynamics

Small systems in a thermodynamic medium --- like colloids in a suspension or the molecular machinery in living cells --- are strongly affected by the thermal fluctuations of their environment. Physicists model such systems by means of stochastic processes. Stochastic Thermodynamics (ST) defines entropy changes and other thermodynamic notions for individual realizations of such processes. It applies to situations far from equilibrium and provides a unified approach to stochastic fluctuation relations. Its predictions have been studied and verified experimentally. This thesis addresses the theoretical foundations of ST. Its focus is on the following two aspects: (i) The stochastic nature of mesoscopic observations has its origin in the molecular chaos on the microscopic level. Can one derive ST from an underlying reversible deterministic dynamics? Can we interpret ST's notions of entropy and entropy changes in a well-defined information-theoretical framework? (ii) Markovian jump processes on finite state spaces are common models for bio-chemical pathways. How does one quantify and calculate fluctuations of physical observables in such models? What role does the topology of the network of states play? How can we apply our abstract results to the design of models for molecular motors? The thesis concludes with an outlook on dissipation as information written to unobserved degrees of freedom --- a perspective that yields a consistency criterion between dynamical models formulated on various levels of description.Comment: Ph.D. Thesis, G\"ottingen 2014, Keywords: Stochastic Thermodynamics, Entropy, Dissipation, Markov processes, Large Deviation Theory, Molecular Motors, Kinesi

Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields

This book covers a broad range of research results in the field of Markov and Semi-Markov chains, processes, systems and related emerging fields. The authors of the included research papers are well-known researchers in their field. The book presents the state-of-the-art and ideas for further research for theorists in the fields. Nonetheless, it also provides straightforwardly applicable results for diverse areas of practitioners

Statistical mechanics of non equilibrium matter: from minimal models to morphogen gradients

Living systems are by definition far from thermodynamic equilibrium, a condition that can be maintained only at the cost of a continuous injection of energy at the microscale, e.g. via cellular metabolic processes, and dissipation into the surrounding environment. The absence of thermodynamic equilibrium, formalised in the breaking of the global detailed balance condition, allows for a wealth of exotic and often counterintuitive phenomena. Our understanding of the capabilities and limitations of living matter has been greatly informed by thermodynamic approaches, which have to be generalised with respect to their traditional counterparts in order to deal with systems subject to strong random fluctuations. The resulting toolkit of stochastic thermodynamics, in particular the concept of entropy production, gives us a quantitative handle on the degree of "non-equilibriumness" of such stochastic processes. Recently, stochastic thermodynamics has benefitted from cross-contamination with the field-theoretic literature and the techniques developed in the latter for the study of collective behaviour have opened the doors to the thermodynamic characterisation of increasingly complex systems. Starting from minimal mathematical models of single active particles and moving up across scales to the level of morphogenetic processes in real organisms (in particular, the formation of morphogen gradients), this thesis contributes to laying the foundations for a bridge between physical understanding and biological insight. While the focus is here on generic mechanisms and on the development of theoretical tools, the applicability to specific experimental scenarios will be pointed out where relevant.Open Acces

Structuring microscopic dynamics with macroscopic feedback: From social insects to artificial intelligence

Physical processes rely on the transmission of energy and information across scales. In the last century, theoretical tools have been developed in the field of statistical physics to infer macroscopic properties starting from a microscopic description of the system. However, less attention has been devoted to the remodelling of microscopic degrees of freedom by macroscopic feedback. In recent years, ideas from non-equilibrium physics have been applied to characterise biological and artificial intelligence systems. These systems share in common their structure in discrete scales of organisation that perform specialised functions. To correctly regulate these functions, the accurate transmission of information across scales is crucial. In this thesis we study the role of macroscopic feedback in the remodelling of microscopic degrees of freedom in two paradigmatic examples, one taken from the field of biology, the self-organisation of specialisation and plasticity in a social wasp, and one from artificial intelligence, the remodelling of deep neural networks in a stochastic many-particle system. In the first part of this thesis we study how the primitively social wasp Polistes canadensis simultaneously achieves robust specialization and rapid plasticity. Combining a unique experimental strategy correlating time-resolved measurements across vastly different scales with a theoretical approach, we characterise the re-establishment of the social steady state after queen removal. We show that Polistes integrates antagonistic processes on multiple scales to distinguish between extrinsic and intrinsic perturbations and thereby achieve both robust specialisation and rapid plasticity. Furthermore, we show that the long-term stability of the social structure relies on the regulation of transcriptional noise by dynamic DNA methylation. In the second part of this thesis, we ask whether emergent collective interactions can be used to remodel deep neural networks. To this end, we study a paradigmatic stochastic manyparticle model where the dynamics are defined by the reaction rates of single particles, given by the output of distinct deep neural networks. The neural networks are in turn dynamically remodelled using deep reinforcement learning depending on the previous history of the system. In particular, we implement this model as a one dimensional stochastic lattice gas. Our results show the formation of two groups of particles that move in opposite directions, diffusively at early times and ballistically over longer time-scales, with the transition between these regimes corresponding to the time-scale of left/right symmetry breaking at the level of individual particles. Over a hierarchy of characteristic time-scales these particles develop emergent, increasingly complex interactions characterised by short-range repulsion and long-range attraction. As a result, the system asymptotically converges to a regime characterised by the presence of anti-ferromagnetic particle clusters. To conclude, we characterise the impact of memory effects and demographic disorder on the dynamics. Together, our results shed light on how non-equilibrium systems can employ macroscopic feedback to regulate the propagation of fluctuations across scales

Markov field models of molecular kinetics

Computer simulations such as molecular dynamics (MD) provide a possible means to understand protein dynamics and mechanisms on an atomistic scale. The resulting simulation data can be analyzed with Markov state models (MSMs), yielding a quantitative kinetic model that, e.g., encodes state populations and transition rates. However, the larger an investigated system, the more data is required to estimate a valid kinetic model. In this work, we show that this scaling problem can be escaped when decomposing a system into smaller ones, leveraging weak couplings between local domains. Our approach, termed independent Markov decomposition (IMD), is a first-order approximation neglecting couplings, i.e., it represents a decomposition of the underlying global dynamics into a set of independent local ones. We demonstrate that for truly independent systems, IMD can reduce the sampling by three orders of magnitude. IMD is applied to two biomolecular systems. First, synaptotagmin-1 is analyzed, a rapid calcium switch from the neurotransmitter release machinery. Within its C2A domain, local conformational switches are identified and modeled with independent MSMs, shedding light on the mechanism of its calcium-mediated activation. Second, the catalytic site of the serine protease TMPRSS2 is analyzed with a local drug-binding model. Equilibrium populations of different drug-binding modes are derived for three inhibitors, mirroring experimentally determined drug efficiencies. IMD is subsequently extended to an end-to-end deep learning framework called iVAMPnets, which learns a domain decomposition from simulation data and simultaneously models the kinetics in the local domains. We finally classify IMD and iVAMPnets as Markov field models (MFM), which we define as a class of models that describe dynamics by decomposing systems into local domains. Overall, this thesis introduces a local approach to Markov modeling that enables to quantitatively assess the kinetics of large macromolecular complexes, opening up possibilities to tackle current and future computational molecular biology questions

Multi-Scale Mathematical Modelling of Brain Networks in Alzheimer's Disease

Perturbations to brain network dynamics on a range of spatial and temporal scales are believed to underpin neurological disorders such as Alzheimer’s disease (AD). This thesis combines quantitative data analysis with tools such as dynamical systems and graph theory to understand how the network dynamics of the brain are altered in AD and experimental models of related pathologies. Firstly, we use a biophysical neuron model to elucidate ionic mechanisms underpinning alterations to the dynamics of principal neurons in the brain’s spatial navigation systems in an animal model of tauopathy. To uncover how synaptic deficits result in alterations to brain dynamics, we subsequently study an animal model featuring local and long-range synaptic degeneration. Synchronous activity (functional connectivity; FC) between neurons within a region of the cortex is analysed using two-photon calcium imaging data. Long-range FC between regions of the brain is analysed using EEG data. Furthermore, a computational model is used to study relationships between networks on these different spatial scales. The latter half of this thesis studies EEG to characterize alterations to macro-scale brain dynamics in clinical AD. Spectral and FC measures are correlated with cognitive test scores to study the hypothesis that impaired integration of the brain’s processing systems underpin cognitive impairment in AD. Whole brain computational modelling is used to gain insight into the role of spectral slowing on FC, and elucidate potential synaptic mechanisms of FC differences in AD. On a finer temporal scale, microstate analyses are used to identify changes to the rapid transitioning behaviour of the brain’s resting state in AD. Finally, the electrophysiological signatures of AD identified throughout the thesis are combined into a predictive model which can accurately separate people with AD and healthy controls based on their EEG, results which are validated on an independent patient cohort. Furthermore, we demonstrate in a small preliminary cohort that this model is a promising tool for predicting future conversion to AD in patients with mild cognitive impairment