Dissertations / Theses on the topic 'Stochastic dynamics'

We develop a stochastic parametrization, based on a 'simple' deterministic model for the dynamics of steady longshore currents, that produces ensembles that are statistically consistent with field observations of these currents. Unlike deterministic models, stochastic parameterization incorporates randomness and hence can only match the observations in a statistical sense. Unlike statistical emulators, in which the model is tuned to the statistical structure of the observation, stochastic parametrization are not directly tuned to match the statistics of the observations. Rather, stochastic parameterization combines deterministic, i.e physics based models with stochastic models for the "missing physics" to create hybrid models, that are stochastic, but yet can be used for making predictions, especially in the context of data assimilation. We introduce a novel measure of the utility of stochastic models of complex processes, that we call consistency of sensitivity. A model with poor consistency of sensitivity requires a great deal of tuning of parameters and has a very narrow range of realistic parameters leading to outcomes consistent with a reasonable spectrum of physical outcomes. We apply this metric to our stochastic parametrization and show that, the loss of certainty inherent in model due to its stochastic nature is offset by the model's resulting consistency of sensitivity. In particular, the stochastic model still retains the forward sensitivity of the deterministic model and hence respects important structural/physical constraints, yet has a broader range of parameters capable of producing outcomes consistent with the field data used in evaluating the model. This leads to an expanded range of model applicability. We show, in the context of data assimilation, the stochastic parametrization of longshore currents achieves good results in capturing the statistics of observation that were not used in tuning the model.

This doctoral study consists of two parts. In the first part, the quantum statistical effects on the formation process of the heavy ion fusion reactions have been investigated by using the c-number quantum Langevin equation approach. It has been shown that the quantum effects enhance the over-passing probability at low temperatures. In the second part, we have developed a simulation technique for the quantum noises which can be approximated by two-term exponential colored noise.

Gene transcription in individual living cells is inevitably a stochastic and dynamic process. Little is known about how cells and organisms learn to balance the fidelity of transcriptional control and the stochasticity of transcription dynamics. In an effort to elucidate the contribution of environmental signals to this intricate balance, a Three State Model was recently proposed, and the transcription system was assumed to transit among three different functional states randomly. In this work, we employ this model to demonstrate how the stochastic dynamics of gene transcription can be characterized by the three transition parameters. We compute the probability distribution of a zero transcript event and its conjugate, the distribution of the time durations in gene on or gene off periods, the transition frequency between system states, and the transcriptional bursting frequency. We also exemplify the mathematical results by the experimental data on prokaryotic and eukaryotic transcription. The analysis reveals that no promoters will be definitely turned on to transcribe within a finite time period, no matter how strong the induction signals are applied, and how abundant the activators are available. Although stronger extrinsic signals could enhance promoter activation rate, the promoter creates an intrinsic ceiling that no signals could cross over in a finite time. Consequently, among a large population of isogenic cells, only a portion of the cells, but not the whole population, could be induced by environmental signals to express a particular gene within a finite time period. We prove that the gene on duration follows an exponential distribution, and the gene off intervals show a local maximum that is best described by assuming two sequential exponential process. The transition frequencies are determined by a system of stochastic differential equations, or equivalently, an iterative scheme of integral operators. We prove that for each positive integer n , there associates a unique time, called the peak instant, at which the nth transcript synthesis cycle since time zero proceeds most likely. These moments constitute a time series preserving the nature order of n.

This thesis describes the dynamics of both electrons and atoms in periodic potentials. In particular, it explores how such potentials can be used to realise a new type of quantum chaos in which the effective classical Hamiltonian originates from the intrinsically quantum nature of energy bands. Firstly, this study examines electron dynamics in a superlattice with an applied voltage and a tilted magnetic field. This system displays a rare type of chaos known as non-KAM (Kolmogorov-Arnold-Moser) chaos, which switches on abruptly when an applied perturbation reaches certain critical values. The onset of chaos in the system leads to the formation of complex patterns in phase space known as stochastic webs. The electron behaviour under these conditions is analysed both semiclassically and quantum mechanically, and the results compared to experimental studies. We show that the presence of stochastic webs strongly enhances electron transport. We calculate Wigner functions of the electron wavefunction at various times and show that, when compared to the Poincare sections, evidence of stochastic web formation is observed in the quantum mechanical phase space. Two designs of superlattice are studied and we show, in a full quantum mechanical analysis, that the design of the superlattice has a pronounced effect on the probability of inter-miniband tunnelling and hence the calculated and measured transport characteristics. Secondly, we explore the dynamics of an ultra-cold sodium atom falling through an optical lattice whilst confined in a harmonic gutter potential that is tilted at an angle to the lattice axis. We show this system is analogous to the case of an electron in a superlattice, and that the atomic dynamics show similar enhanced transport properties for certain trapping frequencies. We also find that in a full quantum mechanical calculation, the atomic wavepacket tends to fragment as the angle at which the gutter potential is tilted is increased. Finally, we examine the dynamics of a Bose-Einstein condensate falling through an optical lattice whilst confined in a harmonic gutter potential. We vary the strength of the interatomic interaction parameter to investigate the role of interactions in the system and find that, even for small tilt angles, the condensate wavefunction fragments. For large interaction parameters combined with large tilt angles, the wavefunction explodes catastrophically.

The state of a deformed crystal is highly heterogeneous, with plasticity localised into linear and point defects such as dislocations, vacancies and interstitial clusters. The motion of these defects dictate a crystal's mechanical behaviour, but defect dynamics are complicated and correlated by external applied stresses, internal elastic interactions and the fundamentally stochastic influence of thermal vibrations. This thesis is concerned with establishing a rigorous, modern theory of the stochastic and dissipative forces on crystal defects, which remain poorly understood despite their importance in any temperature dependent micro-structural process such as the ductile to brittle transition and irradiation damage. From novel molecular dynamics simulations we parametrise an efficient, stochastic and discrete dislocation model that allows access to experimental time and length scales. Simulated trajectories of thermally activated dislocation motion are in excellent agreement with those measured experimentally. Despite these successes in coarse graining, we find existing theories unable to explain stochastic defect dynamics. To resolve this, we define crystal defects through projection operators, without any recourse to elasticity. By rigorous dimensional reduction we derive explicit analytical forms for the stochastic forces acting on crystal defects, allowing new quantitative insight into the role of thermal fluctuations in crystal plasticity.

This thesis provides a study on stochastic models of financial markets related to problems of asset pricing and hedging, optimal portfolio managing and statistical changepoint detection in trends of asset prices. Chapter 1 develops a general model of a system of interconnected stochastic markets associated with a directed acyclic graph. The main result of the chapter provides sufficient conditions of hedgeability of contracts in the model. These conditions are expressed in terms of consistent price systems, which generalise the notion of equivalent martingale measures. Using the general results obtained, a particular model of an asset market with transaction costs and portfolio constraints is studied. In the second chapter the problem of multi-period utility maximisation in the general market model is considered. The aim of the chapter is to establish the existence of systems of supporting prices, which play the role of Lagrange multipliers and allow to decompose a multi-period constrained utility maximisation problem into a family of single-period and unconstrained problems. Their existence is proved under conditions similar to those of Chapter 1.The last chapter is devoted to applications of statistical sequential methods for detecting trend changes in asset prices. A model where prices are driven by a geometric Gaussian random walk with changing mean and variance is proposed, and the problem of choosing the optimal moment of time to sell an asset is studied. The main theorem of the chapter describes the structure of the optimal selling moments in terms of the Shiryaev–Roberts statistic and the posterior probability process.

This dissertation discusses the construction of some stochastic models for population dynamics with a variety of birth and death rate functions. A general model is constructed considering a fundamental growth rate function of the population while allowing random births and deaths in the population. Four stochastic discrete delay models and two non-delay models using the infinitesimal mean and variance given by birth and death rate functions have been produced and analyzed. In these constructions drift terms are in the form of logistic growth or logistic growth with delay. Logistic growth models are well known to biologists and economists. For each model, the existence and uniqueness of the global solution, non-negativeness of the solution is discussed, and for some models, boundedness of the path is also given. Persistence of the population and the boundary behavior have also been discussed through the hitting times. Here, a new method to analyze the hitting times for a specific class of stochastic delay models is presented. This work is related to and also extends the work of Edward Allen, Linda Allen and Bernt Oksendal in population dynamics.

A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.

Thesis investigates the dynamics of financial markets. Nowadays, this is one of the emergent fields in physics and requires a multidisciplinary approach. The thesis studies the first work made by the financial mathematicians and presents those in a more comprehensible form for a physicist.

Option pricing is perhaps most complete problem. Until very recently, stochastic differential equations theory was solely applied to finance by mathematicians. The thesis reviews the theory of Black-Scholes and pays attention to questions that had not interested too much to the mathematicians but that are of importance from a physicist point of view. Among other things, thesis derives the so-called Black-Scholes option price following the rules used by physicists (Stratonovich). Mathematicians have been using Itô convention for deriving this price and thesis founds that both approaches are equivalent. Thesis also focus on the martingale option pricing which directly relates the stock probability density to the option price. The thesis optimizes the martingale method to implement it in cases where only the characteristic function is known.

The study of the correlations observed in markets conform the second block of the thesis. Good knowledge of correlations is essential to perform predictions. In this sense, two diffusive models are presented. First model proposes a market described by a singular two-dimensional process driven by an Ornstein-Uhlenbeck process where noise source is Gaussian and white. The model correctly describes the volatility as a function of time by considering the memory effects in the stock price changes. This model gives reason of the market inefficiencies due to the absence of liquidity or any other type of market interties. These correlations appear to have a a long range persistence in the option price and entails a remarkable influence in the risk due to holding an option. The second model is a stochastic volatility model. In this case, prices are described by a two-dimensional process with two Gaussian white noise sources and where volatility follows an Ornstein-Uhlenbeck process. Their statistical properties are studied and these describe most of the empirical market properties such as the leverage effect.

Thesis investigates the dynamics of financial markets. Nowadays, this is one of the emergent fields in physics and requires a multidisciplinary approach. The thesis studies the first work made by the financial mathematicians and presents those in a more comprehensible form for a physicist. Option pricing is perhaps most complete problem. Until very recently, stochastic differential equations theory was solely applied to finance by mathematicians. The thesis reviews the theory of Black-Scholes and pays attention to questions that had not interested too much to the mathematicians but that are of importance from a physicist point of view. Among other things, thesis derives the so-called Black-Scholes option price following the rules used by physicists (Stratonovich). Mathematicians have been using Itô convention for deriving this price and thesis founds that both approaches are equivalent. Thesis also focus on the martingale option pricing which directly relates the stock probability density to the option price. The thesis optimizes the martingale method to implement it in cases where only the characteristic function is known. The study of the correlations observed in markets conform the second block of the thesis. Good knowledge of correlations is essential to perform predictions. In this sense, two diffusive models are presented. First model proposes a market described by a singular two-dimensional process driven by an Ornstein-Uhlenbeck process where noise source is Gaussian and white. The model correctly describes the volatility as a function of time by considering the memory effects in the stock price changes. This model gives reason of the market inefficiencies due to the absence of liquidity or any other type of market interties. These correlations appear to have a a long range persistence in the option price and entails a remarkable influence in the risk due to holding an option. The second model is a stochastic volatility model. In this case, prices are described by a two-dimensional process with two Gaussian white noise sources and where volatility follows an Ornstein-Uhlenbeck process. Their statistical properties are studied and these describe most of the empirical market properties such as the leverage effect.

Condensation is a special class of phase transition which has been observed throughout the natural and social sciences. The understanding of dynamics towards condensation on a mathematically rigorous level is currently a major research topic. Starting the system from homogeneous initial conditions, the time evolution of the condensed phase often exhibits an interesting coarsening phenomenon of mass transport between cluster sites. In this thesis, we study the coarsening dynamics in several condensing stochastic particle systems. First, we consider the single site dynamics in general stochastic particle systems of misanthrope type with bounded rates on a complete graph. In the limit of diverging system size, we establish convergence to a Markovian non-linear birth death chain, described by a mean-field equation also known from exchange-driven growth processes. Conservation of mass in the particle system leads to conservation of the first moment for the limiting dynamics, and to non-uniqueness of stationary measures. The proof is based on a coupling to branching processes via the graphical construction and establishing uniqueness of the solution for the limit dynamics. As particularly interesting examples we discuss the dynamics of two models that exhibit a condensation transition and their connection to exchange-driven growth processes. The first model is the zero-range process with bounded jump rates. It is well known that zero-range processes with decreasing jump rates exhibit a condensation transition under certain conditions. The mean-field limit of the single site dynamics leads to a non-linear birth death chain describing the coarsening behaviour. We introduce a size-biased version of the single site process, which provides an effective tool to analyse the dynamics of the condensed phase without finite size effects. The second model is the inclusion process, which has unbounded jump rates and also exhibits the condensation phenomenon. However, in this case, the mean-field equation is derived differently, and the single site process is in the form of a standard birth death chain. In addition to the site and size-biased processes, we derive some exact results on the system through duality. We compute the time dependent covariance using the self-duality of inclusion processes and a two-particle dual process. Our results are based on exact computations and are corroborated by detailed simulation data, which contribute to a rigorous understanding of the approach to stationarity in the thermodynamic limit of diverging system size and particle number.

There is a growing body of evidence to suggest that stem cell populations from both the embryo and the adult are heterogeneous in their gene expression patterns. However, the underlying mechanisms are not well understood. This thesis explores cell-to-cell variability in both multipotent and pluripotent stem cell populations using mathematical models to provide a theoretical framework to understand the collective dynamics of stem cell populations. In the first part of the thesis we investigate the possibility that fluctuations in the transcription factor Nanog { which is central to the embryonic stem cell transcriptional regulatory network (ESCTRN) { regulate population variability by controlling important feedback mechanisms. Our analyses reveal the ESC TRN is rich in feedback, with global feedback structure critically dependent on Nanog, Oct4 and Sox2, which collectively participate in over two thirds of all feedback loops. Using a general measure of feedback centrality we show that removal of Nanog severely compromises the global feedback structure of the ESC TRN. These analyses indicate that Nanog fluctuations regulate population heterogeneity by transiently activating different regulatory subnetworks, driving transitions between a Nanog-expressing, feedback-rich, robust and self-perpetuating pluripotent state and a Nanog-diminished, feedback-sparse and differentiation-sensitive state. The majority of studies characterising heterogeneity in Nanog expression have used live-cell fluorescent reporter strategies. However, recent evidence suggests that these reporters may not give a faithful reflection of endogenous Nanog expression because the introduction of the reporter construct can perturb the kinetics of the underlying regulatory network. To investigate the role of Nanog further we therefore sought to model in detail the dynamics of Nanog expression in heterozygous fluorescent knock-in reporter cell lines. We develop chemical master equation, chemical Langevin equation and reaction rate equation models of the reporter system to determine how this might disturb normal Nanog transcriptional control. Our analyses indicate that the reporter construct can weaken the strength of autoactivatory feedback loops that are central to Nanog regulation, and thereby qualitatively perturbs endogenous Nanog dynamics. These results question the efficacy of commonly used reporter strategies and therefore have important implications for the design and use of synthetic reporters in general, not just for Nanog. In the second part of this thesis we consider the dynamics of populations of multipotent adult hematopoietic stem cells (HSCs). It is known that fluctuations within individual HSCs allow them to transit stochastically between functionally distinct metastable states, while the overall population distribution of expression remains stable. To investigate the relationship between single cell and population-level dynamics we propose a theoretical framework that views cellular multipotency as an instance of maximum entropy statistical inference, in which an underlying ergodic stochastic process gives rise to robust variability within the cell population. We illustrate this view by analysing expression fluctuations of the stem cell surface marker Sca1 in mouse HSCs and find that the observed dynamics naturally lie close to a critical state, thereby producing a diverse population that is able to respond rapidly to environmental changes. Although we focus on Sca1 dynamics, comparable expression fluctuations are known to generate functional diversity in other mammalian stem cell systems, including in pluripotent stem cells. Thus, the generation of ergodic expression fluctuations may be a generic way in which cell populations maintain robust multilineage differentiation potential under environmental uncertainty.

Les neurones interagissent à travers leur potentiel de membrane qui a en général une évolution temporelle complexe due aux nombreuses entrées synaptiques irrégulières reçues. Cette évolution est mieux décrite en termes probabilistes, en raison de ces entrées irrégulières ou «bruit synaptique». L'évolution temporelle du potentiel de membrane est stochastique mais aussi déterministe: stochastique, car conduite par des entrées synaptiques qui arrivent de façon aléatoire dans le temps, et déterministe, car un neurone biologique a une évolution temporelle très similaire quand soumis à une même séquence d'entrées synaptiques. Nous étudions les propriétés statistiques d'un modèle simplifié de neurone soumis à des entrées à taux variable d'où en résulte l'évolution non-stationnaire du potentiel de membrane. Nous considérons un modèle passif de membrane neuronale, sans mécanisme de décharge neuronale, soumis à des entrées à courant ou à conductance sous la forme d'un processus de «shot noise». Les fluctuations du potentiel de membrane sont aussi modélisées par un processus stochastique similaire, de «shot noise» filtré. Nous avons analysé les propriétés statistiques de ces processus dans le cadre des transformations de processus ponctuels de Poisson. Des propriétés de ces transformations sont dérivées les statistiques non-stationnaires du processus. Nous obtenons ainsi des expressions analytiques exactes pour les moments et cumulants du processus filtré dans le cas général des taux d'entrée variables. Ce travail ouvre de nombreuses perspectives pour l'analyse de neurones dans les conditions in vivo, en présence d'entrées synaptiques intenses et bruitées
Neurons interact through their membrane potential that generally has a complex time evolution due to numerous irregular synaptic inputs received. This complex time evolution is best described in probabilistic terms due to this irregular or "noisy" activity. The time evolution of the membrane potential is therefore both stochastic and deterministic: it is stochastic since it is driven by random input arrival times, but also deterministic, since subjecting a biological neuron to the same sequence of input arrival times often results in very similar membrane potential traces. In this thesis, we investigated key statistical properties of a simplified neuron model under nonstationary input from other neurons that results in nonstationary evolution of membrane potential statistics. We considered a passive neuron model without spiking mechanism that is driven by input currents or conductances in the form of shot noise processes. Under such input, membrane potential fluctuations can be modeled as filtered shot noise currents or conductances. We analyzed the statistical properties of these filtered processes in the framework of Poisson Point Processes transformations. The key idea is to express filtered shot noise as a transformation of random input arrival times and to apply the properties of these transformations to derive its nonstationary statistics. Using this formalism we derive exact analytical expressions, and useful approximations, for the mean and joint cumulants of the filtered process in the general case of variable input rate. This work opens many perspectives for analyzing neurons under in vivo conditions, in the presence of intense and noisy synaptic inputs

All real-world populations are composed of a finite number of individuals. Due to the intrinsically random nature of interactions between individuals, the dynamics of finite-sized populations are stochastic processes. Additionally, for many types of interaction not all effects occur instantaneously. Instead there are delays before effects are felt. The centrepiece of this thesis is a method of analytically studying stochastic population dynamics with delay reactions. Dynamics with delay reactions are non-Markovian, meaning many of the widely used techniques to study stochastic processes break down. It is not always possible to formulate the master equation, which is a common starting point for analysis of stochastic effects in population dynamics. We follow an alternative method, and derive an exact functional integral approach which is capable of capturing the effects of both stochasticity and delay in the same modelling framework. Our work builds on previous techniques developed in statistical physics, in particular the Martin-Siggia-Rose-Janssen-de Dominicis functional integral. The functional integral approach does not rely on an particular constraints on the population dynamics, for example the choice of delay distribution. Functional integrals can not in general be solved exactly. We show how the functional integral can be used to derive the deterministic, chemical Langevin, and linear-noise approximations for stochastic dynamics with delay. In the later chapters we extend Gillespie’s approximate method of studying stochastic dynamics with delay reactions, which can be used to derive the chemical Langevin equation, by-pass the functional integral. We also derive an extension to the functional integral approach so that it also covers systems with interruptible delay reactions. To demonstrate the applicability of our results we consider various models of population dynamics, arising from ecology, epidemiology, developmental biology, and chemistry. Our analytical calculations are found to provide excellent agreement with exact numerical simulations.

The topic of this thesis is the stochastic dynamics of biochemical reaction systems. The importance of the intrinsic fluctuations in these systems has become more widely appreciated in recent years, and should be accounted for when modelling such systems mathematically. These models are described as continuous time Markov processes and their dynamics defined by a master equation. Analytical progress is made possible by the use of the van Kampen system-size expansion, which splits the dynamics into a macroscopic component plus stochastic corrections, statistics for which can then be obtained. In the first part of this thesis, the terms obtained from the expansion are written down for an arbitrary model, enabling the expansion procedure to be automated and implemented in the software package COPASI. This means that the fluctuation analysis may be used in tandem with other tools in COPASI, in particular parameter scanning and optimisation. This scheme is then extended so that models involving multiple compartments (e.g. cells) may be studied. This increases the range of models that can be evaluated in this fashion. The second part of this thesis also concerns these multi-compartment models, and examines how oscillations can synchronise across a population of cells. This has been observed in many biochemical processes, such as yeast glycolysis. However, the vast majority of modelling of such systems has used the deterministic framework, which ignores the effect of fluctuations. It is now widely known that the type of models studied here can exhibit sustained temporal oscillations when formulated stochastically, despite no such oscillations being found in the deterministic version of the model. Using the van Kampen expansion as a starting point, multi-cell models are studied, to see how stochastic oscillations in one cell may influence, and be influenced by, oscillations in neighbouring cells. Analytical expressions are found, indicating whether or not the oscillations will synchronise across multiple cells and, if synchronisation does occur, whether the oscillations synchronise in phase, or with a phase lag.

In this thesis, I present two methods of fast variable elimination in stochastic systems. Their application to models of population dynamics from ecology, epidemiology and population genetics, is explored. In each application, care is taken to develop the models at the microscale, in terms of interactions between individuals. Such an approach leads to well-defined stochastic systems for finite population sizes. These systems are then approximated at the mesoscale, and expressed as stochastic differential equations. It is in this setting the elimination techniques are developed. In each model a deterministically stable state is assumed to exist, about which the system is linearised. The eigenvalues of the system's Jacobian are used to identify the existence of a separation of timescales. The fast and slow directions are then given locally by the associated eigenvectors. These are used as approximations for the fast and slow directions in the full non-linear system. The general aim is then to remove these fast degrees of freedom and thus arrive at an approximate, reduced-variable description of the dynamics on a slow subspace of the full system. In the first of the methods introduced, the conditioning method, the noise of the system is constrained so that it cannot leave the slow subspace. The technique is applied to an ecological model and a susceptible-exposed-infectious-recovered epidemiological model, in both instances providing a reduced system which preserves the behaviour of the full model to high precision. The second method is referred to as the projection matrix method. It isolates the components of the noise on the slow subspace to provide its reduced description. The method is applied to a generalised Moran model of population genetics on islands, between which there is migration. The model is successfully reduced from a system in as many variables as there are islands, to an effective description in a single variable. The same methodology is later applied to the Lotka-Volterra competition model, which is found under certain conditions to behave as a Moran model. In both cases the agreement between the reduced system and stochastic simulations of the full model is excellent. It is stressed that the ideas behind both the conditioning and projection matrix methods are simple, their application systematic, and the results in very good agreement with simulations for a range of parameter values. When the methods are compared however, the projection matrix method is found in general to provide better results.

This thesis is concerned with quantifying the dynamical role of stochasticity in models of recurrent epidemics. Although the simulation of stochastic models can accurately capture the qualitative epidemic patterns of childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns. The novel aspect of this thesis is the use of analytic methods to quantify the results from simulations. All the models are formulated as continuous time Markov processes, the temporal evolutions of which is described by a master equation. This is expanded in the inverse system size, which decomposes the full stochastic dynamics into a macroscopic part, described by deterministic equations, plus a stochastic fluctuating part. The first part examines the inclusion of non-exponential latent and infectious periods into the the standard susceptible-infectious-recovered model. The method of stages is used to formulate the problem as a Markov process and thus derive a power spectrum for the stochastic oscillations. This model is used to understand the dynamics of whooping cough, which we show to be the mixture of an annual limit cycle plus resonant stochastic oscillations. This limit cycle is generated by the time-dependent external forcing, but we show that the spectrum is close to that predicted by the unforced model. It is demonstrated that adding distributed infectious periods only changes the frequency and amplitude of the stochastic oscillations---the basic mechanisms remain the same. In the final part of this thesis, the effect of seasonal forcing is studied with an analysis of the full time-dependent master equation. The comprehensive nature of this approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in measles epidemics. In the pre-vaccination regime the dynamics are dominated by a period doubling bifurcation, which leads to large biennial oscillations in the deterministic dynamics. Vaccination is shown to move the system away from the biennial limit cycle and into a region where there is an annual limit cycle and stochastic oscillations, similar to whooping cough. Finite size effects are investigated and found to be of considerable importance for measles dynamics, especially in the biennial regime.

Numerical Stochastic Perturbation theory is a powerful tool for estimating high-order perturbative expansions in lattice quantum field theory. The standard algorithm based on the Langevin equation, however, suffers from several limitations which in practice restrict the potential of this technique: first of all it is not exact, a sequence of simulations with finer and finer discretization of the relevant equations have to be performed in order to extrapolate away the systematic errors in the results; and, secondly, the numerical simulations suffer from critical slowing down as the continuum limit of the theory is approached. In this thesis I investigate some alternative methods which improve upon the standard approach. In particular, I present a formulation of Numerical Stochastic Perturbation theory based on the Generalised Hybrid Molecular Dynamics algorithm and a study of the recently proposed Instantaneous Stochastic Perturbation Theory. The viability of these methods is investigated in φ4 theory.

This Thesis studies different cellular and cell population processes driven by non-linear and stochastic dynamics. The problems addressed here gravitate around the concepts of transient dynamics and relaxation from a perturbed to a steady state. In this regard, in all processes studied, stochastic fluctuations, either intrinsically present in or externally applied to these systems play an important and constructive role, by either driving the systems out of equilibrium, interfering with the underlying deterministic laws, or establishing suitable levels of heterogeneity. The first part of the Thesis is committed the analysis of genetically regulated transient cellular processes. Here, we analyse, from a theoretical standpoint, three genetic circuits with pulsed excitable dynamics. We show that all circuits can work in two different excitable regimes, in contrast to what was previously speculated. We also study how, in the presence of molecular noise, these excitable circuits can generate periodic polymodal pulses due to the combination of two noise induced phenomena: stabilisation of an unstable spiral point and coherence resonance. We also studied an excitable genetic mechanism for the regulation of the transcriptional fluctuations observed in some pluripotency factors in Embryonic Stem cells. In the embryo, pluripotency is a transient cellular state and the exit of cells from it seems to be associated with transcriptional fluctuations. In regard to pluripotency control, we also propose a novel mechanism based on the post-translational regulation of a small set of four pluripotency factors. We have validated the theoretical model, based on the formation of binary complexes among these factors, with quantitative experimental data at the single-cell level. The model suggests that the pluripotency state does not depend on the cellular levels of a single factor, but rather on the equilibrium of correlations between the different proteins. In addition, the model is able to anticipate the phenotype of several mutant cell types and suggests that the regulatory function of the protein interactions is to buffer the transcriptional activity of Oc4, a key pluripotency factor. In the second part of the Thesis we studied the behaviour of a computational cell signalling network of the human fibroblast in the presence of external fluctuations and signals. The results obtained here indicate that the network responds in a nontrivial manner to background chatter, both intrinsically and in the presence of external periodic signals. We show that these responses are consequence of the rerouting of the signal to different network information-transmission paths that emerge as noise is modulated. Finally, we also study the cell population dynamics during the formation of microbial biofilms, wrinkled pellicles of bacteria glued by an extracellular matrix that are one of the simplest cases of self-organised multicellular structures. In this Thesis we develop a spatiotemporal model of cellular growth and death that accounts for the experimentally observed patterns of massive bacterial death that precede wrinkle formation in biofilms. These localised patterns focus mechanical forces during biofilm expansion and trigger the formation of the characteristic ridges. In this sense, the proposed model suggests that the death patterns emerge from the mobility changes in bacteria due to the production of extracellular matrix and the spatially inhomogeneous cellular growth. An important prediction of the model is that matrix productions is crucial for the appearance of the patterns and, therefore for winkle formation. We have also experimentally validated validated this prediction with matrix deficient bacterial strains, which show neither death patterns nor wrinkles.
En aquesta Tesi s’estudien diferents processos intracel·lulars i de poblacions cel·lulars regits per dinàmica estocàstica i no lineal. El problemes biològics tractats graviten al voltant el concepte de dinàmica transitòria i de relaxació d’un estat dinàmic pertorbat a l’estat estacionari. En aquest sentit, en tots els processos estudiats, les fluctuacions estocàstiques, presents intrínsecament o aplicades de forma externa, hi tenen un paper constructiu, ja sigui empenyent els sistemes fora de l’equilibri, interferint amb les lleis deterministes subjacents, o establint els nivells d’heterogeneïtat necessaris. La primera part de la Tesi es dedica a l’estudi de processos cel·lulars transitoris regulats genèticament. En ella analitzem des d’un punt de vista teòric tres circuits genètics de control de polsos excitables i, contràriament al que s’havia especulat anteriorment, establim que tots ells poden treballar en dos tipus de règim excitable. Analitzem també com, en presència de soroll molecular, aquests circuits excitables poden generar polsos periòdics i multimodals degut a la combinació de dos fenòmens induïts per soroll: l’estabilització estocàstica d’estats inestables i la ressonància de coherència. D’altra banda, estudiem com un mecanisme genètic excitable pot ser el responsable de regular a nivell transcripcional les fluctuacions que s’observen experimentalment en alguns factors de pluripotència en cèl·lules mare embrionàries. En l’embrió, la pluripotència és un estat cel·lular transitori i la sortida de les cèl·lules d’aquest sembla que està associada a fluctuacions transcripcionals. En relació al control de la pluripotència, presentem també un nou mecanisme basat en la regulació post-traduccional d’un petit conjunt de 4 factors de pluripotència. El model teòric proposat, basat en la formació de complexos entre els diferents factors de pluripotència, l’hem validat mitjançant experiments quantitatius en cèl·lules individuals. El model postula que l’estat de pluripotència no depèn dels nivells cel·lulars d’un únic factor, sinó d’un equilibri de correlacions entre diverses proteïnes. A més, prediu el fenotip de cèl·lules mutants i suggereix que la funció reguladora de les interaccions entre les quatre proteïnes és la d’esmorteir l’activitat transcripcional d’Oct4, un dels principals factors de pluripotència. En el segon apartat de la Tesi estudiem el comportament d’una xarxa computacional de senyalització cel·lular de fibroblast humà en presència de senyals externs fluctuants i cíclics. Els resultats obtinguts mostren que la xarxa respon de forma no trivial a les fluctuacions ambientals, fins i tot en presència d’una senyal externa. Diferents nivells de soroll permeten modular la resposta de la xarxa, mitjançant la selecció de rutes alternatives de transmissió de la informació. Finalment, estudiem la dinàmica de poblacions cel·lulars durant la formació de biofilms, pel·lícules arrugades d’aglomerats de bacteris que conformen un dels exemples més simples d’estructures multicel·lulars autoorganitzades. En aquesta Tesi presentem un model espai-temporal de creixement i mort cel·lular motivat per l’evidència experimental sobre l’aparició de patrons de mort massiva de bacteris previs a la formació de les arrugues dels biofilms. Aquests patrons localitzats concentren les forces mecàniques durant l’expansió del biofilm i inicien la formació de les arrugues característiques. En aquest sentit, el model proposat explica com es formen els patrons de mort a partir dels canvis de mobilitat dels bacteris deguts a la producció de matriu extracel·lular combinats amb un creixement espacialment heterogeni. Una important predicció del model és que la producció de matriu és un procés clau per a l’aparició dels patrons i, per tant de les arrugues. En aquest aspecte, els nostres resultats experimentals en bacteris mutants que no produeixen components essencials de la matriu, confirmen les prediccions.

Models computacionals basats en xarxes a gran escala d'inspiració neurobiològica permeten descriure els correlats neurals de la decisió observats en certes àrees corticals com una transició entre atractors de la xarxa cortical. L'estimulació provoca un canvi en el paisatge d'atractors que afavoreix la transició entre l'atractor neutre inicial a un dels atractors associats a les eleccions categòriques. El soroll present en el sistema introdueix indeterminació en les transicions. En aquest treball mostrem l'existència de dos mecanismes de decisió qualitativament diferents, cadascun amb signatures psicofísiques diferenciades. El mecanisme que apareix a baixes intensitats, induït exclusivament pel soroll, dóna lloc a temps de decisió distribuïts asimètricament, amb una mitjana dictada per l'amplitud del soroll.A més, tant els temps de decisió com el rendiment psicofísic són funcions decreixents de l'estimulació externa. També proposem dos mètodes, un basat en l'aproximació macroscòpica i un altre en la teoria de la varietat central, que simplifiquen la descripció de sistemes estocàstics multistables.
Computational models based on large-scale, neurobiologically-inspired networks describe the decision-related activity observed in some cortical areas as a transition between attractors of the cortical network. Stimulation induces a change in the attractor configuration and drives the system out from its initial resting attractor to one of the existing attractors associated with the categorical choices. The noise present in the system renders transitions random. We show that there exist two qualitatively different mechanisms for decision, each with distinctive psychophysical signatures. The decision mechanism arising at low inputs, entirely driven by noise, leads to skewed distributions of decision times, with a mean governed by the amplitude of the noise. Moreover, both decision times and performances are monotonically decreasing functions of the overall external stimulation. We also propose two methods, one based on the macroscopic approximation and one based on center manifold theory, to simplify the description of multistable stochastic neural systems.

In this thesis, several stochastic models are investigated, which are subjected to irreversible dynamics. Motivation for the presented work stems, on the one hand, from particular physical systems under consideration, which are modeled by the studied stochastic processes. Besides that, the models discussed in this thesis are, on the other hand, generally interesting from the point of view of statistical physics, since they describe systems far from thermodynamic equilibrium. Interesting properties to be encountered are, e.g., dynamical scaling behavior or continuous phase transitions. The first issue to be addressed, is the investigation of irreversibly aggregating systems, where the main emphasis is laid on aggregation of monopolarly charged clusters suspended in a fluid. For this purpose, rate equations are analyzed and Brownian dynamics simulations are performed. It is shown that the system crosses over from power-law cluster growth to sub-logarithmic cluster growth. Asymptotically, the cluster size distribution evolves towards a universal scaling form, which implies a 'self-focussing' of the size distribution. Another emphasis of this thesis is the investigation of nonequilibrium critical phenomena, in particular, the study of phase transitions into absorbing states (states that may be reached irreversibly). To this end, the continuous nonequilibrium phase transition of directed percolation, which serves as a paradigm for absorbing-state phase transitions, is analyzed by a novel approach. Despite the lack of a partition function for directed percolation, this novel approach follows the ideas of Yang-Lee theory of equilibrium statistical mechanics, by investigating the complex roots of the survival probability. Stochastic models such as directed percolation mimic spreading processes, e.g., the spreading of an infectious disease. The effect of long-time memory, which is not included in directed percolation and which corresponds to immunization in epidemic spreading, is investigated through an appropriate model. This model includes dynamical percolation (perfect immunization) as a special case, as well as directed percolation (no immunization). The critical behavior of this model is studied by means of Monte Carlo simulations, in particular for weak immunization. A further generalization is investigated, which allows spontaneous mutations and different species of spreading agents (pathogens). Restricting the analysis to perfect immunization and two spatial dimensions, it is shown by Monte Carlo simulations, that immunization leads to a crossover from dynamical to directed percolation. Other properties of this model are discussed in detail.

In this thesis, we consider a class of stochastic dynamics running backwards, so called backward stochastic differential equations (BSDEs) in the literature. We demonstrate BSDEs can be reformulated as functional differential equations defined on path spaces, and therefore solving BSDEs is equivalent to solving the associated functional differential equations. With such observation we can solve BSDEs on general filtered probability space satisfying the usual conditions, and in particular without the requirement of the martingale representation. We further solve the above functional differential equations numerically, and propose a numerical scheme based on the time discretization and the Picard iteration. This in turn also helps us solve the associated BSDEs numerically. In the second part of the thesis, we consider a class of BSDEs with quadratic growth (QBSDEs). By using the functional differential equation approach introduced in this thesis and the idea of the Cole-Hopf transformation, we first solve the scalar case of such QBSDEs on general filtered probability space satisfying the usual conditions. For a special class of QBSDE systems (not necessarily scalar) in Brownian setting, we do not use such Cole-Hopf transformation at all, and instead introduce the weak solution method, which is to use the strong solutions of forward backward stochastic differential equations (FBSDEs) to construct the weak solutions of such QBSDE systems. Finally we apply the weak solution method to a specific financial problem in the credit risk setting, where we modify the Merton's structural model for credit risk by using the idea of indifference pricing. The valuation and the hedging strategy are characterized by a class of QBSDEs, which we solve by the weak solution method.

The motor control of anthropomorphic robotic systems is a challenging computational task mainly because of the high levels of redundancies such systems exhibit. Optimality principles provide a general strategy to resolve such redundancies in a task driven fashion. In particular closed loop optimisation, i.e., optimal feedback control (OFC), has served as a successful motor control model as it unifies important concepts such as costs, noise, sensory feedback and internal models into a coherent mathematical framework. Realising OFC on realistic anthropomorphic systems however is non-trivial: Firstly, such systems have typically large dimensionality and nonlinear dynamics, in which case the optimisation problem becomes computationally intractable. Approximative methods, like the iterative linear quadratic gaussian (ILQG), have been proposed to avoid this, however the transfer of solutions from idealised simulations to real hardware systems has proved to be challenging. Secondly, OFC relies on an accurate description of the system dynamics, which for many realistic control systems may be unknown, difficult to estimate, or subject to frequent systematic changes. Thirdly, many (especially biologically inspired) systems suffer from significant state or control dependent sources of noise, which are difficult to model in a generally valid fashion. This thesis addresses these issues with the aim to realise efficient OFC for anthropomorphic manipulators. First we investigate the implementation of OFC laws on anthropomorphic hardware. Using ILQG we optimally control a high-dimensional anthropomorphic manipulator without having to specify an explicit inverse kinematics, inverse dynamics or feedback control law. We achieve this by introducing a novel cost function that accounts for the physical constraints of the robot and a dynamics formulation that resolves discontinuities in the dynamics. The experimental hardware results reveal the benefits of OFC over traditional (open loop) optimal controllers in terms of energy efficiency and compliance, properties that are crucial for the control of modern anthropomorphic manipulators. We then propose a new framework of OFC with learned dynamics (OFC-LD) that, unlike classic approaches, does not rely on analytic dynamics functions but rather updates the internal dynamics model continuously from sensorimotor plant feedback. We demonstrate how this approach can compensate for unknown dynamics and for complex dynamic perturbations in an online fashion. A specific advantage of a learned dynamics model is that it contains the stochastic information (i.e., noise) from the plant data, which corresponds to the uncertainty in the system. Consequently one can exploit this information within OFC-LD in order to produce control laws that minimise the uncertainty in the system. In the domain of antagonistically actuated systems this approach leads to improved motor performance, which is achieved by co-contracting antagonistic actuators in order to reduce the negative effects of the noise. Most importantly the shape and source of the noise is unknown a priory and is solely learned from plant data. The model is successfully tested on an antagonistic series elastic actuator (SEA) that we have built for this purpose. The proposed OFC-LD model is not only applicable to robotic systems but also proves to be very useful in the modelling of biological motor control phenomena and we show how our model can be used to predict a wide range of human impedance control patterns during both, stationary and adaptation tasks.

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
As proteínas e outros peptídeos são cadeias de aminoácidos que desempenham funções biológicas específicas dentro de um organismo. A funcionalidade dessas estruturas depende da sua organização tridimensional, portanto é importante determinar quais são os fatores que controlam o bom enovelamento. Se a sequência é conhecida em principio poder-se-ia predizer sua estrutura 3D mediante uma dinâmica molecular de todos os átomos da sequência e das moléculas de água circundantes, mas é claro que esse tipo de simulação é inviável com os recursos computacionais atuais. Alternativamente, consideramos modelos simplificados que levem em conta somente as características principais de cada monômero e das partículas do meio. Efetuamos simulações de dinâmica molecular, considerando interações do tipo Lennard Jones entre monômeros (distinguindo entre monômeros polares e hidrofóbicos) e adicionalmente incorporando uma força estocástica (Langevin) para complementar a influência do meio aquoso. Consideramos diversas sequências lineares, simétricas e de comprimento fixo, evoluindo no espaço bi ou tridimensional. Como resultado destas simulações, podemos descrever a evolução temporal no espaço de conformações mediante variáveis efetivas ou coordenadas de reação, tais como o raio de giro, a distância entre as extremidades ou o número de contatos entre monômeros não ligados. Da análise das séries temporais dessas variáveis efetivas, extraímos os coeficientes que permitem construir seja a equação diferencial estocástica do movimento das variáveis efetivas ou a equação de Fokker-Planck associada. Estas equações para um número reduzido de graus de liberdade permitem, em princípio, obter informações sobre mudanças conformacionais, difíceis de acessar na descrição completa no espaço de fases original, de alta dimensionalidade. Discutimos as vantagens e limitações desta abordagem.
Proteins and other peptides are aminoacid chains that perform specific biological functions within an organism. The functionality of these structures depends on their three-dimensional organization, so it is important to determine what are the factors that control the proper folding. If the sequence is known, in principle it would be possible to predict its 3D structure by means of molecular dynamics of all atoms of the sequence and the surrounding water molecules, but it is clear that this type of simulation is not feasible with the current computational resources. Alternatively, we consider simplified models that take into account only the main characteristics of each monomer and the particles of the medium. We have performed molecular dynamics simulations, considering the LennardJones-like interactions between monomers (distinguishing between polar and hydrophobic monomers) and additionally incorporating a stochastic (Langevin) force to complement the influence of the aqueous medium. We considered several linear sequences, symmetric, with fixed-length, evolving in the tri or bi-dimensional space. As a result of these simulations, we can describe the temporal evolution in the space of conformations through effective variables or reaction coordinates, such as gyration radius, distance between ends or number of contacts between unbound monomers. From the analysis of the time series of the effective variables, we extract the coefficients that allow to build the stochastic differential equation of motion of the effective variables or its associated Fokker-Planck equation. These equations for a limited number of degrees of freedom provide, in principle, information on conformational changes, which are difficult to access in the description of the original, high dimensional, phase. We discuss the advantages and limitations of this approach.

Piecewise-smooth stochastic systems have attracted a lot of interest in the last decades in engineering science and mathematics. Many investigations have focused only on one-dimensional problems. This thesis deals with simple two-dimensional piecewise-smooth stochastic systems in the absence of detailed balance. We investigate the simplest example of such a system, which is a pure dry friction model subjected to coloured Gaussian noise. The nite correlation time of the noise establishes an additional dimension in the phase space and gives rise to a non-vanishing probability current. Our investigation focuses on stick-slip transitions, which can be related to a critical value of the noise correlation time. Analytical insight is provided by applying the uni ed coloured noise approximation. Afterwards, we extend our previous model by adding viscous friction and a constant force. Then we perform a similar analysis as for the pure dry friction case. With parameter values close to the deterministic stick-slip transition, we observe a non-monotonic behaviour of the probability of sticking by increasing the correlation time of the noise. As the eigenvalue spectrum is not accessible for the systems with coloured noise, we consider the eigenvalue problem of a dry friction model with displacement, velocity and Gaussian white noise. By imposing periodic boundary conditions on the displacement and using a Fourier ansatz, we can derive an eigenvalue equation, which has a similar form in comparison to the known one-dimensional problem for the velocity only. The eigenvalue analysis is done for the case without a constant force and with a constant force separately. Finally, we conclude our ndings and provide an outlook on related open problems.

Condensation is a special class of phase transition which has been observed throughout the natural and social sciences. The understanding of the critical behaviour of such systems is a very active area of current research, in particular a mathematical description of the formation and time evolution of the condensate. In this thesis we study these phenomena in several models. In particular we focus on the recently introduced inclusion process, and we compare it with related classical mass transport models such as zero range processes. We first give a brief review of relevant definitions and properties of interacting particle systems, in particular recent literatures on the condensation and stationary behaviour of a large class of interacting particle systems with stationary product measures, which forms the theoretical basis of this thesis. The second part of this thesis is on the dynamics of condensation in the inclusion process on a one-dimensional periodic lattice in the thermodynamic limit. This generalises recent results which were limited to finite lattices and symmetric dynamics. Our main focus is firstly on totally asymmetric dynamics which have not been studied before, which we compare to exact solutions for symmetric systems. We identify all the relevant dynamical regimes and corresponding time scales as a function of the system size, including a coarsening regime where clusters move on the lattice and exchange particles, leading to a growing average cluster size. After establishing the general approach to study dynamics of condensation in totally asymmetric processes, we extend the results to more general partially asymmetric cases as well as higher dimensional cases. In the third part of this thesis we derive some preliminary exact results on symmetric systems through duality, which recovers heuristic results in previous chapter and allows us to treat coarsening in the infinite lattice directly.

Biological brains exhibit many interesting and complex behaviours. Understanding of the mechanisms behind brain behaviours is critical for continuing advancement in fields of research such as artificial intelligence and medicine. In particular, synchronisation of neuronal firing is associated with both improvements to and degeneration of the brain's performance; increased synchronisation can lead to enhanced information-processing or neurological disorders such as epilepsy and Parkinson's disease. As a result, it is desirable to research under which conditions synchronisation arises in neural networks and the possibility of controlling its prevalence. Stochastic ensembles of FitzHugh-Nagumo elements are used to model neural networks for numerical simulations and bifurcation analysis. The FitzHugh-Nagumo model is employed because of its realistic representation of the flow of sodium and potassium ions in addition to its advantageous property of allowing phase plane dynamics to be observed. Network characteristics such as connectivity, configuration and size are explored to determine their influences on global synchronisation generation in their respective systems. Oscillations in the mean-field are used to detect the presence of synchronisation over a range of coupling strength values. To ensure simulation efficiency, coupling strengths between neurons that are identical and fixed with time are investigated initially. Such networks where the interaction strengths are fixed are referred to as homogeneously coupled. The capacity of controlling and altering behaviours produced by homogeneously coupled networks is assessed through the application of weak and strong delayed feedback independently with various time delays. To imitate learning, the coupling strengths later deviate from one another and evolve with time in networks that are referred to as heterogeneously coupled. The intensity of coupling strength fluctuations and the rate at which coupling strengths converge to a desired mean value are studied to determine their impact upon synchronisation performance. The stochastic delay differential equations governing the numerically simulated networks are then converted into a finite set of deterministic cumulant equations by virtue of the Gaussian approximation method. Cumulant equations for maximal and sub-maximal connectivity are used to generate two-parameter bifurcation diagrams on the noise intensity and coupling strength plane, which provides qualitative agreement with numerical simulations. Analysis of artificial brain networks, in respect to biological brain networks, are discussed in light of recent research in sleep theory.

The immune system protects the body against invading pathogens. For an immune response to occur, a T cell must encounter a rare antigen-presenting-cell (APC) presenting its cognate antigen. The time it takes for this encounter depends upon how quickly the T cell is moving, as well as how many APCs carrying the T cells cognate antigen are present. First passage processes are used to derive an equation for the encounter time of a T cell with one of N APCs. Using this time, a rate of encounter is established, and used throughout this thesis. The encounter rate is dependent upon the radius of the lymph node, the effective radius of the APC, and the diffusivity of the T cell. However, the diffusivity of T cells has not been clearly established. In vivo imaging data is used to develop a systematic method for determining the diffusivity of a population of T cells. Due to in vivo imaging experiments having a limited sized imaging volume, a confinement effect is observed. The expected squared displacement of imaged cells is calculated, and the level at which a confinement plateau should be observed is determined. T cell activation, in lymph nodes, relies upon encounters with APCs, but the number of APCs required to initiate a T cell response is currently unknown. Using mathematical models, in combination with experimental work, the probability of T cell-APC encounters can be quantified. The probability of a T cell, residing in the lymph node for twenty four hours, to interact with APCs is calculated. Extrapolating the developed models to later times and lower cell numbers than can be achieved experimentally, a minimum number of APCs required to initiate a T cell response, for typical human T cell precursor frequencies, is estimated. It has been proposed that regulatory T cells suppress effector T cells via a three way interaction with APCs, as a method of preventing autoimmunity. A stochastic model of these interactions is developed and explored. The steady state of the system is found to depend upon the rate of encounter of T cells and APCs, as well as the number of APCs. Stochastic effects are observed in the model, which affect the state of the system, and are not observed in a deterministic approach. Interactions between T cells and APCs, in lymph nodes, are crucial for initiating cell-mediated adaptive immune responses. However, how these interactions cause activation of the T cells is not yet fully understood. Three hypotheses have been proposed for the method of T cell activation. These hypotheses are investigated, and models developed, in an attempt to quantify the observed stages of the activation process. It is found that experimental results can, in part, be explained by a probabilistic approach.

In a stochastic Lotka-Volterra model on a two-dimensional square lattice with periodic boundary conditions and subject to occupation restrictions, there exists an extinction threshold for the predator population that separates a stable active two-species coexistence phase from an inactive state wherein only prey survive. When investigating the non-equilibrium relaxation of the predator density in the vicinity of the phase transition point, we observe critical slowing-down and algebraic decay of the predator density at the extinction critical point. The numerically determined critical exponents are in accord with the established values of the directed percolation universality class. Following a sudden predation rate change to its critical value, one finds critical aging for the predator density autocorrelation function that is also governed by universal scaling exponents. This aging scaling signature of the active-to-absorbing state phase transition emerges at significantly earlier times than the stationary critical power laws, and could thus serve as an advanced indicator of the (predator) population's proximity to its extinction threshold. In order to study boundary effects, we split the system into two patches: Upon setting the predation rates at two distinct values, one half of the system resides in an absorbing state where only the prey survives, while the other half attains a stable coexistence state wherein both species remain active. At the domain boundary, we observe a marked enhancement of the predator population density, the minimum value of the correlation length, and the maximum attenuation rate. Boundary effects become less prominent as the system is successively divided into subdomains in a checkerboard pattern, with two different reaction rates assigned to neighboring patches. We furthermore add another predator species into the system with the purpose of studying possible origins of biodiversity. Predators are characterized with individual predation efficiencies and death rates, to which "Darwinian" evolutionary adaptation is introduced. We find that direct competition between predator species and character displacement together play an important role in yielding stable communities. We develop another variant of the lattice predator-prey model to help understand the killer- prey relationship of two different types of E. coli in a biological experiment, wherein the prey colonies disperse all over the plate while the killer cell population resides at the center, and a "kill zone" of prey forms immediately surrounding the killer, beyond which the prey population gradually increases outward.
Ph. D.

The aim of this thesis is to deepen into the understanding of the mechanisms responsible for the generation of complex and stochastic dynamics, as well as emerging phenomena, in the human brain. We study typical features from the mesoscopic scale, i.e., the scale in which the dynamics is given by the activity of thousands or even millions of neurons. At this scale the synchronous activity of large neuronal populations gives rise to collective oscillations of the average voltage potential. These oscillations can easily be recorded using electroencephalography devices (EEG) or measuring the Local Field Potentials (LFPs). In Chapter 5 we show how the communication between two cortical columns (mesoscopic structures) can be mediated efficiently by a microscopic neural network. We use the synchronization of both cortical columns as a probe to ensure that an effective communication is established between the three neural structures. Our results indicate that there are certain dynamical regimes from the microscopic neural network that favor the correct communication between the cortical columns: therefore, if the LFP frequency of the neural network is of around 40Hz, the synchronization between the cortical columns is more robust compared to the situation in which the neural network oscillates at a lower frequency (10Hz). However, microscopic topological characteristics of the network also influence communication, being a small-world structure the one that best promotes the synchronization of the cortical columns. Finally, this Chapter shows how the mediation exerted by the neural network cannot be substituted by the average of its activity, that is, the dynamic properties of the microscopic neural network are essential for the proper transmission of information between all neural structures. The oscillatory brain electrical activity is largely dependent on the interplay between excitation and inhibition. In Chapter 6 we study how groups of cortical columns show complex patterns of cortical excitation and inhibition taking into account their topological features and the strength of their couplings. These cortical columns segregate between those dominated by excitation and those dominated by inhibition, affecting the synchronization properties of networks of cortical columns. In Chapter 7 we study a dynamic regime by which complex patterns of synchronization between chaotic oscillators appear spontaneously in a network. We show what conditions must a set of coupled dynamical systems fulfill in order to display heterogeneity in synchronization. Therefore, our results are related to the complex phenomenon of synchronization in the brain, which is a focus of study nowadays. Finally, in Chapter 8 we study the ability of the brain to compute and process information. The novelty here is our use of complex synchronization in the brain in order to implement basic elements of Boolean computation. In this way, we show that the partial synchronization of the oscillations in the brain establishes a code in terms of synchronization / non-synchronization (1/0, respectively), and thus all simple Boolean functions can be implemented (AND, OR, XOR, etc.). We also show that complex Boolean functions, such as a flip-flop memory, can be constructed in terms of states of dynamic synchronization of brain oscillations.
L'objectiu d'aquesta Tesi és aprofundir en la comprensió dels mecanismes responsables de la generació de dinàmica complexa i estocàstica, així com de fenòmens emergents, en el cervell humà. Estudiem la fenomenologia característica de l'escala mesoscòpica, és a dir, aquella en la que la dinàmica característica ve donada per l'activitat de milers de neurones. En aquesta escala l'activitat síncrona de grans poblacions neuronals dóna lloc a un fenomen col·lectiu pel qual es produeixen oscil·lacions del seu potencial mitjà. Aquestes oscil·lacions poden ser fàcilment enregistrades mitjançant aparells d'electroencefalograma (EEG) o enregistradors de Potencials de Camp Local (LFP). En el Capítol 5 mostrem com la comunicació entre dos columnes corticals (estructures mesoscòpiques) pot ser conduïda de forma eficient per una xarxa neuronal microscòpica. De fet, emprem la sincronització de les dues columnes corticals per comprovar que s'ha establert una comunicació efectiva entre les tres estructures neuronals. Els resultats indiquen que hi ha règims dinàmics de la xarxa neuronal microscòpica que afavoreixen la correcta comunicació entre les columnes corticals: si la freqüència típica de LFP a la xarxa neuronal està al voltant dels 40Hz la sincronització entre les columnes corticals és més robusta que a una menor freqüència (10Hz). La topologia de la xarxa microscòpica també influeix en la comunicació, essent una estructura de tipus món petit (small-world) la que més afavoreix la sincronització. Finalment, la mediació de xarxa neuronal no pot ser substituïda per la mitjana de la seva activitat, és a dir, les propietats dinàmiques microscòpiques són imprescindibles per a la correcta transmissió d'informació entre totes les escales cerebrals. L'activitat elèctrica oscil·latòria cerebral ve donada en gran mesura per la interacció entre excitació i inhibició neuronal. En el Capítol 6 estudiem com grups de columnes corticals mostren patrons complexos d'excitació i inhibició segons quina sigui la seva topologia i d'acoblament. D'aquesta manera les columnes corticals se segreguen entre aquelles dominades per l'excitació i aquelles dominades per la inhibició, influint en les capacitats de sincronització de xarxes de columnes corticals. En el Capítol 7 estudiem un règim dinàmic segons el qual patrons complexos de sincronització apareixen espontàniament en xarxes d'oscil·ladors caòtics. Mostrem quines condicions s'han de donar en un conjunt de sistemes dinàmics acoblats per tal de mostrar heterogeneïtat en la sincronització, és a dir, coexistència de sincronitzacions. D'aquesta manera relacionem els nostres resultats amb el fenomen de sincronització complexa en el cervell. Finalment, en el Capítol 8 estudiem com el cervell computa i processa informació. La novetat aquí és l'ús que fem de la sincronització complexa de columnes corticals per tal d'implementar elements bàsics de computació Booleana. Mostrem com la sincronització parcial de les oscil·lacions cerebrals estableix un codi neuronal en termes de sincronització/no sincronització (1/0, respectivament) amb el qual totes les funcions Booleanes simples poden ésser implementades (AND, OR, XOR, etc). Mostrem, també, com emprant xarxes mesoscòpiques extenses les capacitats de computació creixen proporcionalment. Així funcions Booleanes complexes, com una memòria del tipus flip-flop, pot ésser construïda en termes d'estats de sincronització dinàmica d'oscil·lacions cerebrals.

The purpose of the senses of animals (and humans) is to translate information available in the external environment into internal information that can be processed by the brain. In the case of the olfactory sense -- the sense of smell -- this is information about the type and concentration of odourants. In the last 15 years major progress has been made in the experimental understanding of the first two stages of the olfactory sense: the signal transduction inside the cilia of the olfactory receptor neurons and the first 'relay station' in the brain, the olfactory bulb, as well as the connection between these two. Theoretical studies that classify the experimentally achieved knowledge or help in testing different biological hypotheses are only starting to be developed. The present work aims to contribute to the theoretical understanding of the first two stages of the olfactory sense. The first processing of the olfactory information, the olfactory signal transduction, is accomplished by a complex chemical network in the sensory cells with the task of coding the available information reliably over a wide range of stimulus strength. In the present work, methods from nonlinear dynamics combined with network theory (namely stoichiometric network analysis) are used to identify a specific negative feedback mechanism that accounts for a number of recently measured experimental results, e.g. oscillations in calcium concentration or the adaptation of the cell towards strong stimuli. This feedback is an experimentally well-established inhibition of cationic channels by the calcium-loaded form of the protein calmodulin. The results of the set of coupled nonlinear deterministic differential equations describing these dynamics agree quantitatively with experimental data. A bifurcation analysis of the system considered shows the robustness of the oscillatory solution against changes in parameters used. It also gives predictions that could serve as an experimental test of the proposed mechanism. Further abstraction and simplification of this specific signal transduction unit leads to a stochastic two-level system with negative feedback, that can not only be found in signalling systems but also in other branches of cell biology, e.g. regulated enzyme activity or in transcription dynamics. Whereas the description outlined above is fully deterministic, here the model system is intrinsically noisy. The influence of the feedback on the intrinsic noise as well as on the signalling properties of the module are analysed in detail by computing mean values, correlation and response functions of the two dynamical system variables using different analytical approaches. Common to all of them is that the intrinsic noise of the system is calculated from its dynamics rather than being introduced by hand. A master equation is used to get generally valid expressions for the mean values. Correlation and response functions for weak feedback are calculated within a path-integral description, and an easier self-consistent method with restricted validity is developed for future extensions of the module such as, e.g., the inclusion of diffusion. The results of the analytical methods are compared to each other and to the results of extended numerical simulations. The considered quantities allow for statements regarding the quality of the signal transduction properties of this module and the positive and negative effects of feedback on it. Going one step up in the information processing in the olfactory sense, another system is found that shows interesting dynamics during development and is influenced by stochastic effects: the formation of the neural map on the surface of the olfactory bulb -- stage two in the olfactory system. The dynamics of this very complex biological pattern formation process is studied mostly numerically focusing on three different aspects of axonal growth. Possible chemical guidance cues and the reaction of axonal growth cones to them are described using different levels of detail. There is strong experimental evidence for interactions among growing axons which is implemented in different ways into models. Finally, axon turnover is considered and used in the most promising simulation approach, where many axons grow as interacting directed random walkers. For each of these aspects, qualitative features of respective experiments are reproduced
Die Sinne der Tiere (und Menschen) dienen dazu, Informationen über die Aussenwelt in neuronale, ' interne' Information zu 'übersetzen'. Im Falle des Geruchssinns sind dies Informationen über die Art und Konzentration von Geruchsstoffen. In den letzten 15 Jahren wurden grosse Fortschritte im experimentellen Verständnis der ersten beiden Stufen des Geruchssinns gemacht, sowohl was die Signaltransduktion in den Zilien der Geruchszellen betrifft, als auch bezüglich der ersten 'Schaltstelle' im Gehirn, dem olfaktorischen Bulbus (sowie in der Verbindung dieser beiden Stufen). Die Entwicklung theoretischer Studien, die die experimentell gewonnenen Daten klassifizieren können, befindet sich dagegen erst am Anfang. Ziel der vorliegenden Arbeit ist es, zum theoretischen Verständnis dieser ersten beiden Stufen beizutragen. Die erste Verarbeitung der olfaktorischen Information, die olfaktorische Signaltransduktion, wird durch ein komplexes chemisches Netzwerk in den Sinneszellen bewerkstelligt. In dieser Dissertation werden Methoden der nichtlinearen Dynamik, kombiniert mit Netzwerktheorie (stöchiometrische Netzwerkanalyse) benutzt, um einen negativen Rückkopplungsmechanismus zu identifizieren, der einige in neuerer Zeit gewonnene experimentelle Ergebnisse erklären kann, u.a. Oszillationen der Kalziumkonzentration oder die Anpassung der Zelle an starke Reize. Bei dieser Rückkopplung handelt es sich um eine experimentell gut bestätigte Hemmung eines Kationenkanals durch den Kalziumkomplex des Proteins Calmodulin. Das Ergebnis der vier gekoppelten nichtlinearen deterministischen Differenzialgleichungen, die das dynamische Verhalten des Systems beschreiben, stimmt quantitativ mit experimentellen Daten überein. Eine Bifurkationsanalyse zeigt die Robustheit der oszillierenden Lösung gegenüber Veränderungen der verwendeten Parameter und macht Vorhersagen möglich, die als experimentelle Tests des vorgeschlagenen Mechanismus dienen können. Eine weitere Abstrahierung der oben beschriebenen Signaltransduktionseinheit führt zu einem stochastischen Zweiniveausystem mit negativer Rückkopplung, das nicht nur in Signalsystemen gefunden werden kann, sondern auch in anderen Bereichen der Zellbiologie. Im Gegensatz zu der oben beschriebenen, komplett deterministischen Beschreibung zeigt das hier betrachtete Modellsystem intrinsisches Rauschen. Der Einfluss der Rückkopplung auf das Rauschen sowie auf die Signalübertragungseigenschaften des Moduls werden detailliert analysiert, indem mit Hilfe verschiedener analytischer Methoden Mittelwerte, Korrelations- und Antwortfunktionen des Systems ausgerechnet werden. Diese Methoden habe alle gemein, dass das intrinsische Rauschen des Systems aus der Dynamik selbst berechnet wird und nicht ' von Hand' eingefügt wird. Um allgemeingültige Ausdrücke für die Mittelwerte zu bekommen, wird eine Mastergleichung aufgestellt und gelöst. Die Korrelations- und Antwortfunktionen werden für schwache Rückkopplung mit Hilfe einer Pfadintegralmethode ausgerechnet, und eine einfachere, selbstkonsistente Methode begrenzter Gültigkeit wird für mögliche Erweiterungen des Systems, z.B. die Berücksichtigung von Diffusion, entwickelt. Die Ergebnisse der verschiedenen analytischen Methoden werden miteinander und mit den Ergebnissen ausführlicher numerischer Simulationen verglichen. Die betrachteten Grössen ermöglichen Aussagen über die Qualität der Signaltransduktion dieses Moduls sowie über die positiven und negativen Effekte der Rückkopplung auf diese. Ein weiteres Beispiel für interessante und von stochastischen Effekten beeinflusste Dynamik findet man einen Schritt weiter in der olfaktorischen Signalverarbeitung: Die während der Entwicklung stattfindende Ausbildung der neuronalen Karte auf der Oberfläche des olfaktorischen Bulbus, der zweiten Stufe des olfaktorischen Systems. Die Dynamik dieser sehr komplexen biologischen Musterbildung wird mittels numerischer Simulationen untersucht, wobei der Schwerpunkt auf drei verschiedene Aspekte axonalen Wachstums gesetzt wird. Die Reaktion axonaler Wachstumskegel auf mögliche chemische Signalstoffe wird verschieden detailliert beschrieben. Es gibt deutliche experimentelle Hinweise auf Wechselwirkung zwischen Axonen, was in den Modellen auf verschiedene Arten implementiert wird. Schliesslich wird die Erneuerung der Axone betrachtet und im vielversprechendsten Modell, in dem viele Axone als wechselwirkende gerichtete random walkers simuliert werden, berücksichtigt und analysiert. Für jeden dieser drei Aspekte können entsprechende experimentelle Ergebnisse qualitativ reproduziert werden

Despite numerous advances in experimental methodologies capable of addressing the various phenomenon occurring on metal surfaces, atomic scale resolution of the microscopic dynamics remains elusive for most systems. Computational models of the processes may serve as an alternative tool to fill this void. To this end, parallel molecular dynamics simulations of self-diffusion on metal surfaces have been developed and employed to address microscopic details of the system. However these simulations are not without their limitations and prove to be computationally impractical for a variety of chemically relevant systems, particularly for diffusive events occurring in the low temperature regime. To circumvent this difficulty, a corresponding coarse-grained representation of the surface is also developed resulting in a reduction of the required computational effort by several orders of magnitude, and this description becomes all the more advantageous with increasing system size and complexity. This representation provides a convenient framework to address fundamental aspects of diffusion in nonequilibrium environments and an interesting mechanism for directing diffusive motion along the surface is explored. In the ensuing discussion, additional topics including transition state theory in noisy systems and the construction of a checking function for protein structure validation are outlined. For decades the former has served as a cornerstone for estimates of chemical reaction rates. However, in complex environments transition state theory most always provides only an upper bound for the true rate. An alternative approach is described that may alleviate some of the difficulties associated with this problem. Finally, one of the grand challenges facing the computational sciences is to develop methods capable of reconstructing protein structure based solely on readily-available sequence information. Herein a checking function is developed that may prove useful for addressing whether a particular proposed structure is a viable possibility.