Hybrid stochastic simplifications for multiscale gene networks

<p>Abstract</p> <p>Background</p> <p>Stochastic simulation of gene networks by Markov processes has important applications in molecular biology. The complexity of exact simulation algorithms scales with the number of discrete jumps to be performed. Approximate schemes reduce the computational time by reducing the number of simulated discrete events. Also, answering important questions about the relation between network topology and intrinsic noise generation and propagation should be based on general mathematical results. These general results are difficult to obtain for exact models.</p> <p>Results</p> <p>We propose a unified framework for hybrid simplifications of Markov models of multiscale stochastic gene networks dynamics. We discuss several possible hybrid simplifications, and provide algorithms to obtain them from pure jump processes. In hybrid simplifications, some components are discrete and evolve by jumps, while other components are continuous. Hybrid simplifications are obtained by partial Kramers-Moyal expansion <abbrgrp><abbr bid="B1">1</abbr><abbr bid="B2">2</abbr><abbr bid="B3">3</abbr></abbrgrp> which is equivalent to the application of the central limit theorem to a sub-model. By averaging and variable aggregation we drastically reduce simulation time and eliminate non-critical reactions. Hybrid and averaged simplifications can be used for more effective simulation algorithms and for obtaining general design principles relating noise to topology and time scales. The simplified models reproduce with good accuracy the stochastic properties of the gene networks, including waiting times in intermittence phenomena, fluctuation amplitudes and stationary distributions. The methods are illustrated on several gene network examples.</p> <p>Conclusion</p> <p>Hybrid simplifications can be used for onion-like (multi-layered) approaches to multi-scale biochemical systems, in which various descriptions are used at various scales. Sets of discrete and continuous variables are treated with different methods and are coupled together in a physically justified approach.</p

Reaction Networks and Population Dynamics

Reaction systems and population dynamics constitute two highly developed areas of research that build on well-defined model classes, both in terms of dynamical systems and stochastic processes. Despite a significant core of common structures, the two fields have largely led separate lives. The workshop brought the communities together and emphasised concepts, methods and results that have, so far, appeared in one area but are potentially useful in the other as well

A Bayesian approach to simultaneously characterize the stochastic and deterministic components of a system

The present work provides a Bayesian approach to learn plausible models capable of characterizing complex time series in which deterministic and stochastic phenomena concur. Two main approaches are actually developed. The first approach, is a simple superposition model grounded on the hypothesis that the interactions between the stochastic and deterministic phenomena are negligible. To enable this model to capture complex dynamics, the stochastic part is assumed to be a fractal signal. Under the assumptions of this model, an analysis method is proposed, enabling the characterization of the fractal stochastic component and the estimation the deterministic part. The second main approach relies on Stochastic Differential Equations (SDEs) to model systems where the stochastic and deterministic part interact. First, a non-parametric estimation method for SDEs is developed, using recent advances from Gaussian processes. Finally, the thesis studies how to overcome the main constraint that the use of SDEs imposes: the Markovianity assumption. To that end, a new structured variational autoencoder with latent SDE dynamics is proposed. All the methods are tested on both synthetic and real signals, demonstrating its ability to capture the behavior of complex systems

Comportement asymptotique de processus avec sauts et applications pour des modèles avec branchement

L'objectif de ce travail est d'étudier le comportement en temps long d'un modèle de particules avec une interaction de type branchement. Plus précisément, les particules se déplacent indépendamment suivant une dynamique markovienne jusqu'au temps de branchement, où elles donnent naissance à de nouvelles particules dont la position dépend de celle de leur mère et de son nombre d'enfants. Dans la première partie de ce mémoire nous omettons le branchement et nous étudions le comportement d'une seule lignée. Celle-ci est modélisée via un processus de Markov qui peut admettre des sauts, des parties diffusives ou déterministes par morceaux. Nous quantifions la convergence de ce processus hybride à l'aide de la courbure de Wasserstein, aussi nommée courbure grossière de Ricci. Cette notion de courbure, introduite récemment par Joulin, Ollivier, et Sammer correspond mieux à l'étude des processus avec sauts. Nous établissons une expression du gradient du semigroupe des processus de Markov stochastiquement monotone, qui nous permet d'expliciter facilement leur courbure. D'autres bornes fines de convergence en distance de Wasserstein et en variation totale sont aussi établies. Dans le même contexte, nous démontrons qu'un processus de Markov, qui change de dynamique suivant un processus discret, converge rapidement vers un équilibre, lorsque la moyenne des courbures des dynamiques sous-jacentes est strictement positive. Dans la deuxième partie de ce mémoire, nous étudions le comportement de toute la population de particules. Celui-ci se déduit du comportement d'une seule lignée grâce à une formule many-to-one, c'est-à-dire un changement de mesure de type Girsanov. Via cette transformation, nous démontrons une loi des grands nombres et établissons une limite macroscopique, pour comparer nos résultats aux résultats déjà connus en théorie des équations aux dérivées partielles. Nos résultats sont appliqués sur divers modèles ayant des applications en biologie et en informatique. Parmi ces modèles, nous étudierons le comportement en temps long de la plus grande particule dans un modèle simple de population structurée en tailleThe aim of this work is to study the long time behavior of a branching particle model. More precisely, the particles move independently from each other following a Markov dynamics until the branching event. When one of these events occurs, the particle produces some random number of individuals whose position depends on the position of its mother and her number of offspring. In the first part of this thesis, we only study one particle line and we ignore the branching mechanism. So we are interested by the study of a Markov process which can jump, diffuse or be piecewise deterministic. The long time behavior of these hybrid processes is described with the notion of Wasserstein or coarse Ricci curvature. This notion of curvature, introduced by Joulin, Ollivier and Sammer, is more appropriate for the study of processes with jumps. We establish an expression of the gradient of the Markov semigroup of stochastically monotone processes which gives the curvature of these processes. Others sharp bounds of convergence, in Wasserstein distance and total variation distance, are also established. In the same way, we prove that if a Markov process evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of a second Markov process, then it is exponentially ergodic, under the assumption that the mean of the curvature of the underlying dynamics is positive. In the second part of the work, we study all the population. Its behaviour can be deduced to the study of the first part using a Girsavov-type transform which is called a many-to-one formula. Using this relation, we establish a law of large numbers and a macroscopic limit, in order to compare our results to the well know results on deterministic setting. Several examples, based on biology and computer science problems, illustrate our results, including the study of the largest individual in a size-structured population modelPARIS-EST-Université (770839901) / SudocSudocFranceF

Taming neuronal noise with large networks

How does reliable computation emerge from networks of noisy neurons? While individual neurons are intrinsically noisy, the collective dynamics of populations of neurons taken as a whole can be almost deterministic, supporting the hypothesis that, in the brain, computation takes place at the level of neuronal populations. Mathematical models of networks of noisy spiking neurons allow us to study the effects of neuronal noise on the dynamics of large networks. Classical mean-field models, i.e., models where all neurons are identical and where each neuron receives the average spike activity of the other neurons, offer toy examples where neuronal noise is absorbed in large networks, that is, large networks behave like deterministic systems. In particular, the dynamics of these large networks can be described by deterministic neuronal population equations. In this thesis, I first generalize classical mean-field limit proofs to a broad class of spiking neuron models that can exhibit spike-frequency adaptation and short-term synaptic plasticity, in addition to refractoriness. The mean-field limit can be exactly described by a multidimensional partial differential equation; the long time behavior of which can be rigorously studied using deterministic methods. Then, we show that there is a conceptual link between mean-field models for networks of spiking neurons and latent variable models used for the analysis of multi-neuronal recordings. More specifically, we use a recently proposed finite-size neuronal population equation, which we first mathematically clarify, to design a tractable Expectation-Maximization-type algorithm capable of inferring the latent population activities of multi-population spiking neural networks from the spike activity of a few visible neurons only, illustrating the idea that latent variable models can be seen as partially observed mean-field models. In classical mean-field models, neurons in large networks behave like independent, identically distributed processes driven by the average population activity -- a deterministic quantity, by the law of large numbers. The fact the neurons are identically distributed processes implies a form of redundancy that has not been observed in the cortex and which seems biologically implausible. To show, numerically, that the redundancy present in classical mean-field models is unnecessary for neuronal noise absorption in large networks, I construct a disordered network model where networks of spiking neurons behave like deterministic rate networks, despite the absence of redundancy. This last result suggests that the concentration of measure phenomenon, which generalizes the ``law of large numbers'' of classical mean-field models, might be an instrumental principle for understanding the emergence of noise-robust population dynamics in large networks of noisy neurons

Spatial and stochastic epidemics : theory, simulation and control

It is now widely acknowledged that spatial structure and hence the spatial position of host populations plays a vital role in the spread of infection. In this work I investigate an ensemble of techniques for understanding the stochastic dynamics of spatial and discrete epidemic processes, with especial consideration given to SIR disease dynamics for the Levins-type metapopulation. I present a toolbox of techniques for the modeller of spatial epidemics. The highlight results are a novel form of moment closure derived directly from a stochastic differential representation of the epidemic, a stochastic simulation algorithm that asymptotically in system size greatly out-performs existing simulation methods for the spatial epidemic and finally a method for tackling optimal vaccination scheduling problems for controlling the spread of an invasive pathogen

Environmental management and restoration under unified risk and uncertainty using robustified dynamic Orlicz risk

Environmental management and restoration should be designed such that the risk and uncertainty owing to nonlinear stochastic systems can be successfully addressed. We apply the robustified dynamic Orlicz risk to the modeling and analysis of environmental management and restoration to consider both the risk and uncertainty within a unified theory. We focus on the control of a jump-driven hybrid stochastic system that represents macrophyte dynamics. The dynamic programming equation based on the Orlicz risk is first obtained heuristically, from which the associated Hamilton-Jacobi-Bellman (HJB) equation is derived. In the proposed Orlicz risk, the risk aversion of the decision-maker is represented by a power coefficient that resembles a certainty equivalence, whereas the uncertainty aversion is represented by the Kullback-Leibler divergence, in which the risk and uncertainty are handled consistently and separately. The HJB equation includes a new state-dependent discount factor that arises from the uncertainty aversion, which leads to a unique, nonlinear, and nonlocal term. The link between the proposed and classical stochastic control problems is discussed with a focus on control-dependent discount rates. We propose a finite difference method for computing the HJB equation. Finally, the proposed model is applied to an optimal harvesting problem for macrophytes in a brackish lake that contains both growing and drifting populations

Properties and advances of probabilistic and statistical algorithms with applications in finance

This thesis is concerned with the construction and enhancement of algorithms involving probability and statistics. The main motivation for these are problems that appear in finance and more generally in applied science. We consider three distinct areas, namely, credit risk modelling, numerics for McKean Vlasov stochastic differential equations and stochastic representations of Partial Differential Equations (PDEs), therefore the thesis is split into three parts. Firstly, we consider the problem of estimating a continuous time Markov chain (CTMC) generator from discrete time observations, which is essentially a missing data problem in statistics. These generators give rise to transition probabilities (in particular probabilities of default) over any time horizon, hence the estimation of such generators is a key problem in the world of banking, where the regulator requires banks to calculate risk over different time horizons. For this particular problem several algorithms have been proposed, however, through a combination of theoretical and numerical results we show the Expectation Maximisation (EM) algorithm to be the superior choice. Furthermore we derive closed form expressions for the associated Wald confidence intervals (error) estimated by the EM algorithm. Previous attempts to calculate such intervals relied on numerical schemes which were slower and less stable. We further provide a closed form expression (via the Delta method) to transfer these errors to the level of the transition probabilities, which are more intuitive. Although one can establish more precise mathematical results with the Markov assumption, there is empirical evidence suggesting this assumption is not valid. We finish this part by carrying out empirical research on non-Markov phenomena and propose a model to capture the so-called rating momentum. This model has many appealing features and is a natural extension to the Markov set up. The second part is based on McKean Vlasov Stochastic Differential Equations (MV-SDEs), these Stochastic Differential Equations (SDEs) arise from looking at the limit, as the number of weakly interacting particles (e.g. gas particles) tends to infinity. The resulting SDE has coefficients which can depend on its own law, making them theoretically more involved. Although MV-SDEs arise from statistical physics, there has been an explosion in interest recently to use MV-SDEs in models for economics. We firstly derive an explicit approximation scheme for MV-SDEs with one-sided Lipschitz growth in the drift. Such a condition was observed to be an issue for standard SDEs and required more sophisticated schemes. There are implicit and explicit schemes one can use and we develop both types in the setting of MV-SDEs. Another main issue for MVSDEs is, due to the dependency on their own law they are extremely expensive to simulate compared to standard SDEs, hence techniques to improve computational cost are in demand. The final result in this part is to develop an importance sampling algorithm for MV-SDEs, where our measure change is obtained through the theory of large deviation principles. Although importance sampling results for standard SDEs are reasonably well understood, there are several difficulties one must overcome to apply a good importance sampling change of measure in this setting. The importance sampling is used here as a variance reduction technique although our results hint that one may be able to use it to reduce propagation of chaos error as well. Finally we consider stochastic algorithms to solve PDEs. It is known one can achieve numerical advantages by using probabilistic methods to solve PDEs, through the so-called probabilistic domain decomposition method. The main result of this part is to present an unbiased stochastic representation for a first order PDE, based on the theory of branching diffusions and regime switching. This is a very interesting result since previously (Itô based) stochastic representations only applied to second order PDEs. There are multiple issues one must overcome in order to obtain an algorithm that is numerically stable and solves such a PDE. We conclude by showing the algorithm’s potential on a more general first order PDE