arXiv subjects

Tobias Galla

Publications and source records attributed to Tobias Galla.

At least 73 records · Page 4Linked to original sources

arXiv2018

Model reduction methods for classical stochastic systems with fast-switching environments: reduced master equations, stochastic differential equations, and applications

We study classical stochastic systems with discrete states, coupled to switching external environments. For fast environmental processes we derive reduced dynamics for the system itself, focusing on corrections to the adiabatic limit of infinite time scale separation. In some cases, this leads to master equations with negative transition `rates' or bursting events. We devise a simulation algorithm in discrete time to unravel these master equations into sample paths, and provide an interpretation of bursting events. Focusing on stochastic population dynamics coupled to external environments, we discuss a series of approximation schemes combining expansions in the inverse switching rate of the environment, and a Kramers--Moyal expansion in the inverse size of the population. This places the different approximations in relation to existing work on piecewise-deterministic and piecewise-diffusive Markov processes. We apply the model reduction methods to different examples including systems in biology and a model of crack propagation.

cond-mat.stat-mech↗

arXiv2018

Phenotypic switching of populations of cells in a stochastic environment

In biology phenotypic switching is a common bet-hedging strategy in the face of uncertain environmental conditions. Existing mathematical models often focus on periodically changing environments to determine the optimal phenotypic response. We focus on the case in which the environment switches randomly between discrete states. Starting from an individual-based model we derive stochastic differential equations to describe the dynamics, and obtain analytical expressions for the mean instantaneous growth rates based on the theory of piecewise deterministic Markov processes. We show that optimal phenotypic responses are non-trivial for slow and intermediate environmental processes, and systematically compare the cases of periodic and random environments. The best response to random switching is more likely to be heterogeneity than in the case of deterministic periodic environments, net growth rates tend to be higher under stochastic environmental dynamics. The combined system of environment and population of cells can be interpreted as host-pathogen interaction, in which the host tries to choose environmental switching so as to minimise growth of the pathogen, and in which the pathogen employs a phenotypic switching optimised to increase its growth rate. We discuss the existence of Nash-like mutual best-response scenarios for such host-pathogen games.

q-bio.PE↗

arXiv2017

Fast flowing populations are not well mixed

In evolutionary dynamics, well-mixed populations are almost always associated with all-to-all interactions; mathematical models are based on complete graphs. In most cases, these models do not predict fixation probabilities in groups of individuals mixed by flows. We propose an analytical description in the fast-flow limit. This approach is valid for processes with global and local selection, and accurately predicts the suppression of selection as competition becomes more local. It provides a modelling tool for biological or social systems with individuals in motion.

q-bio.PE↗

arXiv2017

Emergence of Cooperative Long-term Market Loyalty in Double Auction Markets

Loyal buyer-seller relationships can arise by design, e.g. when a seller tailors a product to a specific market niche to accomplish the best possible returns, and buyers respond to the dedicated efforts the seller makes to meet their needs. We ask whether it is possible, instead, for loyalty to arise spontaneously, and in particular as a consequence of repeated interaction and co-adaptation among the agents in a market. We devise a stylized model of double auction markets and adaptive traders that incorporates these features. Traders choose where to trade (which market) and how to trade (to buy or to sell) based on their previous experience. We find that when the typical scale of market returns (or, at fixed scale of returns, the intensity of choice) become higher than some threshold, the preferred state of the system is segregated: both buyers and sellers are segmented into subgroups that are persistently loyal to one market over another. We characterize the segregated state analytically in the limit of large markets: it is stabilized by some agents acting cooperatively to enable trade, and provides higher rewards than its unsegregated counterpart both for individual traders and the population as a whole.

econ.GN↗

arXiv2017

The effects of heterogeneity on stochastic cycles in epidemics

Models of biological processes are often subject to different sources of noise. Developing an understanding of the combined effects of different types of uncertainty is an open challenge. In this paper, we study a variant of the susceptible-infective-recovered model of epidemic spread, which combines both agent-to-agent heterogeneity and intrinsic noise. We focus on epidemic cycles, driven by the stochasticity of infection and recovery events, and study in detail how heterogeneity in susceptibilities and propensities to pass on the disease affects these quasi-cycles. While the system can only be described by a large hierarchical set of equations in the transient regime, we derive a reduced closed set of equations for population-level quantities in the stationary regime. We analytically obtain the spectra of quasi-cycles in the linear-noise approximation. We find that the characteristic frequency of these cycles is typically determined by population averages of susceptibilities and infectivities, but that their amplitude depends on higher-order moments of the heterogeneity. We also investigate the synchronisation properties and phase lag between different groups of susceptible and infected individuals.

q-bio.PE↗

arXiv2017

Effects of population growth on the success of invading mutants

Understanding if and how mutants reach fixation in populations is an important question in evolutionary biology. We study the impact of population growth has on the success of mutants. To systematically understand the effects of growth we decouple competition from reproduction; competition follows a birth--death process and is governed by an evolutionary game, while growth is determined by an externally controlled branching rate. In stochastic simulations we find non-monotonic behaviour of the fixation probability of mutants as the speed of growth is varied; the right amount of growth can lead to a higher success rate. These results are observed in both coordination and coexistence game scenarios, and we find that the 'one-third law' for coordination games can break down in the presence of growth. We also propose a simplified description in terms of stochastic differential equations to approximate the individual-based model.

q-bio.PE↗

arXiv2016

The prevalence of chaotic dynamics in games with many players

We study adaptive learning in a typical p-player game. The payoffs of the games are randomly generated and then held fixed. The strategies of the players evolve through time as the players learn. The trajectories in the strategy space display a range of qualitatively different behaviors, with attractors that include unique fixed points, multiple fixed points, limit cycles and chaos. In the limit where the game is complicated, in the sense that the players can take many possible actions, we use a generating-functional approach to establish the parameter range in which learning dynamics converge to a stable fixed point. The size of this region goes to zero as the number of players goes to infinity, suggesting that complex non-equilibrium behavior, exemplified by chaos, may be the norm for complicated games with many players.

econ.GN↗

arXiv2016

Sojourn times and fixation dynamics in multi-player games with fluctuating environments

We study evolutionary multi-player games in finite populations, subject to fluctuating environments. The population undergoes a birth-death process with absorbing states, and the environment follows a Markovian process, resulting in a fluctuating payoff matrix for the evolutionary game. Our focus is on the fixation or extinction of a single mutant in a population of wildtypes. We show that the nonlinear nature of fitnesses in multi-player games gives rise to an intricate interplay of selection, genetic drift and environmental fluctuations. This generates effects not seen in simpler two-player games. To analyse trajectories towards fixation we analytically calculate sojourn times for general birth-death processes in populations of two types of individuals and in fluctuating environments.

q-bio.PE↗

arXiv2016

Mechanisms of stochastic onset and termination of atrial fibrillation episodes: Insights using a cellular automaton model

Mathematical models of cardiac electrical excitation are increasingly complex, with multiscale models seeking to represent and bridge physiological behaviours across temporal and spatial scales. The increasing complexity of these models makes it computationally expensive to both evaluate long term (>60 seconds) behaviour and determine sensitivity of model outputs to inputs. This is particularly relevant in models of atrial fibrillation (AF), where individual episodes last from seconds to days, and inter-episode waiting times can be minutes to months. Potential mechanisms of transition between sinus rhythm and AF have been identified but are not well understood, and it is difficult to simulate AF for long periods of time using state-of-the-art models. In this study, we implemented a Moe-type cellular automaton on a novel, topologically correct surface geometry of the left atrium. We used the model to simulate stochastic initiation and spontaneous termination of AF, arising from bursts of spontaneous activation near pulmonary veins. The simplified representation of atrial electrical activity reduced computational cost, and so permitted us to investigate AF mechanisms in a probabilistic setting. We computed large numbers (~10^5) of sample paths of the model, to infer stochastic initiation and termination rates of AF episodes using different model parameters. By generating statistical distributions of model outputs, we demonstrated how to propagate uncertainties of inputs within our microscopic level model up to a macroscopic level. Lastly, we investigated spontaneous termination in the model and found a complex dependence on its past AF trajectory, the mechanism of which merits future investigation.

q-bio.TO↗

arXiv2016

Stochastic evolution in populations of ideas

It is known that learning of players who interact in a repeated game can be interpreted as an evolutionary process in a population of ideas. These analogies have so far mostly been established in deterministic models, and memory loss in learning has been seen to act similarly to mutation in evolution. We here propose a representation of reinforcement learning as a stochastic process in finite "populations of ideas". The resulting birth-death dynamics has absorbing states and allows for the extinction or fixation of ideas, marking a key difference to mutation-selection processes in finite populations. We characterize the outcome of evolution in populations of ideas for several classes of symmetric and asymmetric games.

q-bio.PE↗

arXiv2016

Stochastic transport in the presence of spatial disorder: fluctuation-induced corrections to homogenization

Motivated by uncertainty quantification in natural transport systems, we investigate an individual-based transport process involving particles undergoing a random walk along a line of point sinks whose strengths are themselves independent random variables. We assume particles are removed from the system via first-order kinetics. We analyse the system using a hierarchy of approaches when the sinks are sparsely distributed, including a stochastic homogenization approximation that yields explicit predictions for the extrinsic disorder in the stationary state due to sink strength fluctuations. The extrinsic noise induces long-range spatial correlations in the particle concentration, unlike fluctuations due to the intrinsic noise alone. Additionally, the mean concentration profile, averaged over both intrinsic and extrinsic noise, is elevated compared with the corresponding profile from a uniform sink distribution, showing that the classical homogenization approximation can be a biased estimator of the true mean.

cond-mat.stat-mech↗

arXiv2016

Time scales and species coexistence in chaotic flows

Empirical observations in marine ecosystems have suggested a balance of biological and advection time scales as a possible explanation of species coexistence. To characterise this scenario, we measure the time to fixation in neutrally evolving populations in chaotic flows. Contrary to intuition the variation of time scales does not interpolate straightforwardly between the no-flow and well-mixed limits; instead we find that fixation is the slowest at intermediate Damköhler numbers, indicating long-lasting coexistence of species. Our analysis shows that this slowdown is due to spatial organisation on an increasingly modularised network. We also find that diffusion can either slow down or speed up fixation, depending on the relative time scales of flow and evolution.

q-bio.PE↗

arXiv2016

Birth-death processes with quenched uncertainty and intrinsic noise

The dynamics of populations is frequently subject to intrinsic noise. At the same time unknown interaction networks or rate constants can present quenched uncertainty. Existing approaches often involve repeated sampling of the quenched disorder and then running the stochastic birth-death dynamics on these samples. In this paper we take a different view, and formulate an effective jump process, representative of the ensemble of quenched interactions as a whole. Using evolutionary games with random payoff matrices as an example, we develop an algorithm to simulate this process, and we discuss diffusion approximations in the limit of weak intrinsic noise.

q-bio.PE↗

arXiv2016

Intrinsic noise in systems with switching environments

We study individual-based dynamics in finite populations, subject to randomly switching environmental conditions. These are inspired by models in which genes transition between on and off states, regulating underlying protein dynamics. Similarly switches between environmental states are relevant in bacterial populations and in models of epidemic spread. Existing piecewise-deterministic Markov process (PDMP) approaches focus on the deterministic limit of the population dynamics while retaining the randomness of the switching. Here we go beyond this approximation and explicitly include effects of intrinsic stochasticity at the level of the linear-noise approximation. Specifically we derive the stationary distributions of a number of model systems, in good agreement with simulations. This improves existing approaches which are limited to the regimes of fast and slow switching.

cond-mat.stat-mech↗

arXiv2015

When the mean is not enough: Calculating fixation time distributions in birth-death processes

Studies of fixation dynamics in Markov processes predominantly focus on the mean time to absorption. This may be inadequate if the distribution is broad and skewed. We compute the distribution of fixation times in one-step birth-death processes with two absorbing states. These are expressed in terms of the spectrum of the process, and we provide different representations as forward-only processes in eigenspace. These allow efficient sampling of fixation time distributions. As an application we study evolutionary game dynamics, where invading mutants can reach fixation or go extinct. We also highlight the median fixation time as a possible analog of mixing times in systems with small mutation rates and no absorbing states, whereas the mean fixation time has no such interpretation.

cond-mat.stat-mech↗

arXiv2015

Assessing Measures of Atrial Fibrillation Clustering via Stochastic Models of Episode Recurrence and Disease Progression

Atrial fibrillation (AF) is a leading cause of morbidity and mortality. AF prevalence increases with age, which is attributed to pathophysiological changes that aid AF initiation and perpetuation. Current state-of-the-art models are only capable of simulating short periods of atrial activity at high spatial resolution, whilst the majority of clinical recordings are based on infrequent temporal datasets of limited spatial resolution. Being able to estimate disease progression informed by both modelling and clinical data would be of significant interest. In addition an analysis of the temporal distribution of recorded fibrillation episodes AF density can provide insights into recurrence patterns. We present an initial analysis of the AF density measure using a simplified idealised stochastic model of a binary time series representing AF episodes. The future aim of this work is to develop robust clinical measures of progression which will be tested on models that generate long-term synthetic data. These measures would then be of clinical interest in deciding treatment strategies.

q-bio.TO↗

arXiv2015

Bursting noise in gene expression dynamics: Linking microscopic and mesoscopic models

The dynamics of short-lived mRNA results in bursts of protein production in gene regulatory networks. We investigate the propagation of bursting noise between different levels of mathematical modelling, and demonstrate that conventional approaches based on diffusion approximations can fail to capture bursting noise. An alternative coarse-grained model, the so-called piecewise deterministic Markov process (PDMP), is seen to outperform the diffusion approximation in biologically relevant parameter regimes. We provide a systematic embedding of the PDMP model into the landscape of existing approaches, and we present analytical methods to calculate its stationary distribution and switching frequencies.

q-bio.MN↗

arXiv2015

Modelling the progression of atrial fibrillation: A stochastic individual-based approach

We propose a stochastic individual-based model of the progression of atrial fibrillation (AF). The model operates at patient level over a lifetime and is based on elements of the physiology and biophysics of AF, making contact with existing mechanistic models. The outputs of the model are times when the patient is in normal rhythm and AF, and we carry out a population-level analysis of the statistics of disease progression. While the model is stylised at present and not directly predictive, future improvements are proposed to tighten the gap between existing mechanistic models of AF, and epidemiological data, with a view towards model-based personalised medicine.

q-bio.PE↗