arXiv subjects

William Ott

Publications and source records attributed to William Ott.

At least 19 recordsLinked to original sources

arXiv2025

Distributed delay stabilizes bistable genetic networks

Delay is an inherent feature of genetic regulatory networks. It represents the time required for the assembly of functional regulator proteins. The protein production process is complex, as it includes transcription, translocation, translation, folding, and oligomerization. Because these steps are noisy, the resulting delay associated with protein production is distributed (random). We here consider how distributed delay impacts the dynamics of bistable genetic circuits. We show that for a variety of genetic circuits that exhibit bistability, increasing the noise level in the delay distribution dramatically stabilizes the metastable states. By this we mean that mean residence times in the metastable states dramatically increase. Relevance to Life Sciences. Bistable genetic regulatory networks are ubiquitous in living organisms. Evolutionary processes seem to have tuned such networks so that they switch between metastable states when it is important to do so, but small fluctuations do not cause unwanted switching. Understanding how evolution has tuned the stability of biological switches is an important problem. In particular, such understanding can guide the design of forward-engineered synthetic bistable genetic regulatory networks. Mathematical Content. We use two methods to explain this stabilization phenomenon. First, we introduce and simulate stochastic hybrid models that depend on a switching-rate parameter. These stochastic hybrid models allow us to unfold the distributed-delay models in the sense that, in certain cases, the distributed-delay model can be viewed as a fast-switching limit of the corresponding stochastic hybrid model. Second, we generalize the three-states model, a symbolic model of bistability, and analyze this extension.

q-bio.SC↗

arXiv2025

Attractor learning for spatiotemporally chaotic dynamical systems using echo state networks with transfer learning

In this paper, we explore the predictive capabilities of echo state networks (ESNs) for the generalized Kuramoto-Sivashinsky (gKS) equation, an archetypal nonlinear PDE that exhibits spatiotemporal chaos. Our research focuses on predicting changes in long-term statistical patterns of the gKS model that result from varying the dispersion relation or the length of the spatial domain. We use transfer learning to adapt ESNs to different parameter settings and successfully capture changes in the underlying chaotic attractor. Previous work has shown that transfer learning can be used effectively with ESNs for single-orbit prediction. The novelty of our paper lies in our use of this pairing to predict the long-term statistical properties of spatiotemporally chaotic PDEs. We also show that transfer learning nontrivially improves the length of time that predictions of individual gKS trajectories remain accurate.

math.DS↗

arXiv2023

Correlated Information Reduces Accuracy of Pioneering Decision-Makers

Normative models are often used to describe how humans and animals make decisions. These models treat deliberation as the accumulation of uncertain evidence that terminates with a commitment to a choice. When extended to social groups, such models often assume that individuals make independent observations. However, individuals typically gather evidence from common sources, and their observations are rarely independent. Here we ask: For a group of ideal observers who do not exchange information, what is the impact of correlated evidence on decision accuracy? We show that even when agents are identical, correlated evidence causes decision accuracy to depend on temporal decision order. Surprisingly, the first decider is less accurate than a lone observer. Early deciders are less accurate than late deciders. These phenomena occur despite the fact that the rational observers use the same decision criterion, so they are equally confident in their decisions. We analyze discrete and continuum evidence-gathering models to explain why the first decider is less accurate than a lone observer when evidence is correlated. Pooling the decisions of early deciders using a majority rule does not rescue accuracy in the sense that such pooling results in only modest accuracy gain. Although we analyze an idealized model, we believe that our analysis offers insights that do not depend on exactly how groups integrate evidence and form decisions.

physics.soc-ph↗

arXiv2021

Delay-Induced Uncertainty for a Paradigmatic Glucose-Insulin Model

Medical practice in the intensive care unit is based on the supposition that physiological systems such as the human glucose-insulin system are predictable. We demonstrate that delay within the glucose-insulin system can induce sustained temporal chaos, rendering the system unpredictable. Specifically, we exhibit such chaos for the Ultradian glucose-insulin model. This well-validated, finite-dimensional model represents feedback delay as a three-stage filter. Using the theory of rank one maps from smooth dynamical systems, we precisely explain the nature of the resulting delay-induced uncertainty (DIU). We develop a recipe one may use to diagnose DIU in a general oscillatory dynamical system. For infinite-dimensional delay systems, no analog of the theory of rank one maps exists. Nevertheless, we show that the geometric principles encoded in our DIU recipe apply to such systems by exhibiting sustained temporal chaos for a linear shear flow. Our results are potentially broadly applicable because delay is ubiquitous throughout mathematical physiology.

math.DS↗

arXiv2021

A kinetic theory approach for 2D crowd dynamics with emotional contagion

In this paper, we present a computational modeling approach for the dynamics of human crowds, where the spreading of an emotion (specifically fear) has an influence on the pedestrians' behavior. Our approach is based on the methods of the kinetic theory of active particles. The model allows us to weight between two competing behaviors depending on fear level: the search for less congested areas and the tendency to follow the stream unconsciously (herding). The fear level of each pedestrian influences her walking speed and is influenced by the fear levels of her neighbors. Numerically, we solve our pedestrian model with emotional contagion using an operator splitting scheme. We simulate evacuation scenarios involving two groups of interacting pedestrians to assess how domain geometry and the details of fear propagation impact evacuation dynamics. Further, we reproduce the evacuation dynamics of an experimental study involving distressed ants.

physics.soc-ph↗

arXiv2021

Efficient Evader Detection in Mobile Sensor Networks

Suppose one wants to monitor a domain with sensors, each sensing a small ball-shaped region, but the domain is hazardous enough that one cannot control the placement of the sensors. A prohibitively large number of randomly placed sensors could be required to obtain static coverage. Instead, one can use fewer sensors by providing mobile coverage, a generalization of the static setup wherein every possible evader is detected by the moving sensors in a bounded amount of time. Here, we use topology in order to implement algorithms certifying mobile coverage that use only local data to solve the global problem. Our algorithms do not require knowledge of the sensors' locations, only their connectivity information. We experimentally study the statistics of mobile coverage in two dynamical scenarios. We allow the sensors to move independently (billiard dynamics and Brownian motion), or to locally coordinate their dynamics (collective animal motion models). Our detailed simulations show, for example, that collective motion can enhance performance: The expected time until the mobile sensor network achieves mobile coverage is lower for the D'Orsogna collective motion model than for the billiard motion model. Further, we show that even when the probability of static coverage is low, all possible evaders can nevertheless be detected relatively quickly by the mobile sensor network.

math.DS↗

arXiv2020

Diversity Improves Speed and Accuracy in Social Networks

How does temporally structured private and social information shape collective decisions? To address this question we consider a network of rational agents who independently accumulate private evidence that triggers a decision upon reaching a threshold. When seen by the whole network, the first agent's choice initiates a wave of new decisions; later decisions have less impact. In heterogeneous networks, first decisions are made quickly by impulsive individuals who need little evidence to make a choice, but, even when wrong, can reveal the correct options to nearly everyone else. We conclude that groups comprised of diverse individuals can make more efficient decisions than homogenous ones.

physics.soc-ph↗

arXiv2020

Nonstationary open dynamical systems

This paper studies nonstationary open dynamical systems from the statistical viewpoint. By open, we mean that trajectories may escape through holes in the phase space. By nonstationary, we mean that the dynamical model itself (as well as the holes) may vary with time. Unlike the setting of random dynamical systems, no assumptions are made about the statistics of this variability. We formulate general results that yield conditional memory loss (an analog of decay of correlations) at exponential rates for nonstationary open dynamical systems. We apply our results to a class of nonstationary piecewise-smooth expanding systems in higher dimension with moving holes.

math.DS↗

arXiv2019

Stochastic and Coarse-Grained Two-Dimensional Modeling of Directional Particle Movement

We study the evolution of interacting groups of agents in two-dimensional geometries. We introduce a microscopic stochastic model that includes floor fields modeling the global flow of individual groups as well as local interaction rules. From this microscopic model we derive an analytically-tractable system of conservation laws that governs the evolution of the macroscopic densities. Numerical simulations show good agreement between the system of conservation laws and the microscopic model, though the latter is slightly more diffusive. We conclude by deriving second-order corrections to the system of conservation laws.

physics.soc-ph↗

arXiv2018

Large deviations for Gaussian diffusions with delay

Dynamical systems driven by nonlinear delay SDEs with small noise can exhibit important rare events on long timescales. When there is no delay, classical large deviations theory quantifies rare events such as escapes from metastable fixed points. Near such fixed points, one can approximate nonlinear delay SDEs by linear delay SDEs. Here, we develop a fully explicit large deviations framework for (necessarily Gaussian) processes $X_t$ driven by linear delay SDEs with small diffusion coefficients. Our approach enables fast numerical computation of the action functional controlling rare events for $X_t$ and of the most likely paths transiting from $X_0 = p$ to $X_T=q$. Via linear noise local approximations, we can then compute most likely routes of escape from metastable states for nonlinear delay SDEs. We apply our methodology to the detailed dynamics of a genetic regulatory circuit, namely the co-repressive toggle switch, which may be described by a nonlinear chemical Langevin SDE with delay.

math.PR↗

arXiv2017

Modeling mechanical interactions in growing populations of rod-shaped bacteria

Advances in synthetic biology allow us to engineer bacterial collectives with pre-specified characteristics. However, the behavior of these collectives is difficult to understand, as cellular growth and division as well as extra-cellular fluid flow lead to complex, changing arrangements of cells within the population. To rationally engineer and control the behavior of cell collectives we need theoretical and computational tools to understand their emergent spatiotemporal dynamics. Here, we present an agent-based model that allows growing cells to detect and respond to mechanical interactions. Crucially, our model couples the dynamics of cell growth to the cell's environment: Mechanical constraints can affect cellular growth rate and a cell may alter its behavior in response to these constraints. This coupling links the mechanical forces that influence cell growth and emergent behaviors in cell assemblies. We illustrate our approach by showing how mechanical interactions can impact the dynamics of bacterial collectives growing in microfluidic traps.

q-bio.CB↗

arXiv2016

Effects of cell cycle noise on excitable gene circuits

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a 3-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

q-bio.MN↗

arXiv2015

A scale-free method for testing the proportionality of branch lengths between two phylogenetic trees

We introduce a scale-free method for testing the proportionality of branch lengths between two phylogenetic trees that have the same topology and contain the same set of taxa. This method scales both trees to a total length of 1 and sums up the differences for each branch. Compared to previous methods, ours yields a fully symmetrical score that measures proportionality without being affected by scale. We call this score the normalized tree distance (NTD). Based on real data, we demonstrate that NTD scores are distributed unimodally, in a manner similar to a lognormal distribution. The NTD score can be used to, for example, detect co-evolutionary processes and measure the accuracy of branch length estimates.

q-bio.QM↗

arXiv2014

Modeling delay in genetic networks: From delay birth-death processes to delay stochastic differential equations

Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove that the distance between these two descriptions, as measured by expectations of functionals of the processes, converges to zero with increasing system size. Further, we prove that the delay birth-death process converges to the thermodynamic limit as system size tends to infinity. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the spatial and temporal distributions of transition pathways in metastable systems, oscillatory behavior in negative feedback circuits, and cross-correlations between nodes in a network. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.

q-bio.MN↗

arXiv2013

One-sided shift spaces over infinite alphabets

We define a notion of (one-sided) shift spaces over infinite alphabets. Unlike many previous approaches to shift spaces over countable alphabets, our shift spaces are compact Hausdorff spaces. We examine shift morphisms between these shift spaces, and identify three distinct classes that generalize the shifts of finite type. We show that when our shift spaces satisfy a property that we call "row-finite", then shift morphisms on them may be identified with sliding block codes. As applications, we show that if two (possibly infinite) directed graphs have edge shifts that are conjugate, then the groupoids of the graphs are isomorphic, and the C*-algebras of the graphs are isomorphic.

math.OA↗

arXiv2013

Transcriptional delay stabilizes bistable gene networks

Transcriptional delay can significantly impact the dynamics of gene networks. Here we examine how such delay affects bistable systems. We investigate several stochastic models of bistable gene networks and find that increasing delay dramatically increases the mean residence times near stable states. To explain this, we introduce a non-Markovian, analytically tractable reduced model. The model shows that stabilization is the consequence of an increased number of failed transitions between stable states. Each of the bistable systems that we simulate behaves in this manner.

q-bio.MN↗

arXiv2012

Memory loss for time-dependent dynamical systems

This paper discusses the evolution of probability distributions for certain time-dependent dynamical systems. Exponential loss of memory is proved for expanding maps and for one-dimensional piecewise expanding maps with slowly varying parameters.

math.DS↗

arXiv2010

From limit cycles to strange attractors

We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.

math.DS↗