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Oxidative stress is a fundamental stimulus to which eukaryotic cells respond via many channels. Among these channels are both protein systems that process oxidative stress, such as the 2-Cys peroxiredoxin-thioredoxin system (PTRS), as well as changes in transcriptional activity that target outcomes such as growth, damage control and repair, or cell death. Recent work has revealed connections between the PTRS and temporal phases of transcriptional activity involving famous transcription factors like p53 and FOXO1. To examine potential mechanisms for these connections, we implement an existing dynamical systems mathematical model for the PTRS. We hypothesize that dose-dependent hyperoxidation events enact ultrasensitive switches by which the PTRS can categorize stress severity and activate appropriate transcriptional responses. Using numerical simulations of the PTRS in human cells, we provide a proof of principle for staggered, switchlike hyperoxidation of peroxiredoxins (Prx) as well as an underlying mechanism requiring simultaneous signaling by Prx I and II. Then, we use our model to make testable predictions about individual Prx knockouts as well as the affinity for hydrogen peroxide of Prx across oxidation states. This study provides direction for future experimental work and sheds light into the mechanisms underlying oxidative stress response in human cells., Comment: 21 pages, 8 figures, 1 table. Bachelor's Honors Thesis more...

Constraining the many biological parameters that govern cortical dynamics is computationally and conceptually difficult because of the curse of dimensionality. This paper addresses these challenges by proposing (1) a novel data-informed mean-field (MF) approach to efficiently map the parameter space of network models; and (2) an organizing principle for studying parameter space that enables the extraction biologically meaningful relations from this high-dimensional data. We illustrate these ideas using a large-scale network model of the Macaque primary visual cortex. Of the 10-20 model parameters, we identify 7 that are especially poorly constrained, and use the MF algorithm in (1) to discover the firing rate contours in this 7D parameter cube. Defining a "biologically plausible" region to consist of parameters that exhibit spontaneous Excitatory and Inhibitory firing rates compatible with experimental values, we find that this region is a slightly thickened codimension-1 submanifold. An implication of this finding is that while plausible regimes depend sensitively on parameters, they are also robust and flexible provided one compensates appropriately when parameters are varied. Our organizing principle for conceptualizing parameter dependence is to focus on certain 2D parameter planes that govern lateral inhibition: Intersecting these planes with the biologically plausible region leads to very simple geometric structures which, when suitably scaled, have a universal character independent of where the intersections are taken. In addition to elucidating the geometry of the plausible region, this invariance suggests useful approximate scaling relations. Our study offers, for the first time, a complete characterization of the set of all biologically plausible parameters for a detailed cortical model, which has been out of reach due to the high dimensionality of parameter space. more...

Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key dynamical mechanisms. In the absence of special dynamical features such as scale separation or symmetries, the time evolution of these variables typically exhibits memory effects. Recent work has found a variety of data-driven model reduction methods to be effective for representing such non-Markovian dynamics, but their scope and dynamical underpinning remain incompletely understood. Here, we study data-driven model reduction from a dynamical systems perspective. For both chaotic and randomly-forced systems, we show the problem can be naturally formulated within the framework of Koopman operators and the Mori-Zwanzig projection operator formalism. We give a heuristic derivation of a NARMAX (Nonlinear Auto-Regressive Moving Average with eXogenous input) model from an underlying dynamical model. The derivation is based on a simple construction we call Wiener projection, which links Mori-Zwanzig theory to both NARMAX and to classical Wiener filtering. We apply these ideas to the Kuramoto-Sivashinsky model of spatiotemporal chaos and a viscous Burgers equation with stochastic forcing., Comment: Substantial revisions, including additional references. To appear in Journal of Computational Physics more...

We study a class of importance sampling methods for stochastic differential equations (SDEs). A small-noise analysis is performed, and the results suggest that a simple symmetrization procedure can significantly improve the performance of our importance sampling schemes when the noise is not too large. We demonstrate that this is indeed the case for a number of linear and nonlinear examples. Potential applications, e.g., data assimilation, are discussed., Comment: Added brief discussion of Hamilton-Jacobi equation. Also made various minor corrections. To appear in Communciations in Applied Mathematics and Computational Science more...

We compare two approaches to the predictive modeling of dynamical systems from partial observations at discrete times. The first is continuous in time, where one uses data to infer a model in the form of stochastic differential equations, which are then discretized for numerical solution. The second is discrete in time, where one directly infers a discrete-time model in the form of a nonlinear autoregression moving average model. The comparison is performed in a special case where the observations are known to have been obtained from a hypoelliptic stochastic differential equation. We show that the discrete-time approach has better predictive skills, especially when the data are relatively sparse in time. We discuss open questions as well as the broader significance of the results., Comment: 25 pages more...

Highly connected recurrent neural networks often produce chaotic dynamics, meaning their precise activity is sensitive to small perturbations. What are the consequences for how such networks encode streams of temporal stimuli? On the one hand, chaos is a strong source of randomness, suggesting that small changes in stimuli will be obscured by intrinsically generated variability. On the other hand, recent work shows that the type of chaos that occurs in spiking networks can have a surprisingly low-dimensional structure, suggesting that there may be "room" for fine stimulus features to be precisely resolved. Here we show that strongly chaotic networks produce patterned spikes that reliably encode time-dependent stimuli: using a decoder sensitive to spike times on timescales of 10's of ms, one can easily distinguish responses to very similar inputs. Moreover, recurrence serves to distribute signals throughout chaotic networks so that small groups of cells can encode substantial information about signals arriving elsewhere. A conclusion is that the presence of strong chaos in recurrent networks does not prohibit precise stimulus encoding., Comment: 8 figures more...

Implicit samplers are algorithms for producing independent, weighted samples from multi-variate probability distributions. These are often applied in Bayesian data assimilation algorithms. We use Laplace asymptotic expansions to analyze two implicit samplers in the small noise regime. Our analysis suggests a symmetrization of the algo- rithms that leads to improved (implicit) sampling schemes at a rel- atively small additional cost. Computational experiments confirm the theory and show that symmetrization is effective for small noise sampling problems. more...

We propose a procedure to generate dynamical networks with bursty, possibly repetitive and correlated temporal behaviors. Regarding any weighted directed graph as being composed of the accumulation of paths between its nodes, our construction uses random walks of variable length to produce time-extended structures with adjustable features. The procedure is first described in a general framework. It is then illustrated in a case study inspired by a transportation system for which the resulting synthetic network is shown to accurately mimic the empirical phenomenology. more...

Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strong-attraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based Morris-Lecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phase-amplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phase-reduction technique. As an explicit application of this phase-amplitude framework, we consider in some detail the response of a generic planar model where the strong-attraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response., Comment: 26 pages, 11 figures more...

This paper considers the problem of designing time-dependent, real-time control policies for controllable nonlinear diffusion processes, with the goal of obtaining maximally-informative observations about parameters of interest. More precisely, we maximize the expected Fisher information for the parameter obtained over the duration of the experiment, conditional on observations made up to that time. We propose to accomplish this with a two-step strategy: when the full state vector of the diffusion process is observable continuously, we formulate this as an optimal control problem and apply numerical techniques from stochastic optimal control to solve it. When observations are incomplete, infrequent, or noisy, we propose using standard filtering techniques to first estimate the state of the system, then apply the optimal control policy using the posterior expectation of the state. We assess the effectiveness of these methods in 3 situations: a paradigmatic bistable model from statistical physics, a model of action potential generation in neurons, and a model of a simple ecological system. more...

Biological information processing is often carried out by complex networks of interconnected dynamical units. A basic question about such networks is that of reliability: if the same signal is presented many times with the network in different initial states, will the system entrain to the signal in a repeatable way? Reliability is of particular interest in neuroscience, where large, complex networks of excitatory and inhibitory cells are ubiquitous. These networks are known to autonomously produce strongly chaotic dynamics - an obvious threat to reliability. Here, we show that such chaos persists in the presence of weak and strong stimuli, but that even in the presence of chaos, intermittent periods of highly reliable spiking often coexist with unreliable activity. We elucidate the local dynamical mechanisms involved in this intermittent reliability, and investigate the relationship between this phenomenon and certain time-dependent attractors arising from the dynamics. A conclusion is that chaotic dynamics do not have to be an obstacle to precise spike responses, a fact with implications for signal coding in large networks., Comment: 15 pages, 6 figures more...

Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are "sufficiently weak", an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of "sticky" phase-space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience. more...

We review some recent results surrounding a general mechanism for producing chaotic behavior in periodically-kicked oscillators. The key geometric ideas are illustrated via a simple linear shear model. more...

We report the results of a numerical study of nonequilibrium steady states for a class of Hamiltonian models. In these models of coupled matter-energy transport, particles exchange energy through collisions with pinned-down rotating disks. In [Commun. Math. Phys. 262 (2006)], Eckmann and Young studied 1D chains and showed that certain simple formulas give excellent approximations of energy and particle density profiles. Keeping the basic mode of interaction in [Eckmann-Young], we extend their prediction scheme to a number of new settings: 2D systems on different lattices, driven by a variety of boundary (heat bath) conditions including the use of thermostats. Particle-conserving models of the same type are shown to behave similarly. The second half of this paper examines memory and finite-size effects, which appear to impact only minimally the profiles of the models tested in [Eckmann-Young]. We demonstrate that these effects can be significant or insignificant depending on the local geometry. Dynamical mechanisms are proposed, and in the case of directional bias in particle trajectories due to memory, correction schemes are derived and shown to give accurate predictions. more...

We show that Markov couplings can be used to improve the accuracy of Markov chain Monte Carlo calculations in some situations where the steady-state probability distribution is not explicitly known. The technique generalizes the notion of control variates from classical Monte Carlo integration. We illustrate it using two models of nonequilibrium transport. more...

We study the reliability of large networks of coupled neural oscillators in response to fluctuating stimuli. Reliability means that a stimulus elicits essentially identical responses upon repeated presentations. We view the problem on two scales: neuronal reliability, which concerns the repeatability of spike times of individual neurons embedded within a network, and pooled-response reliability, which addresses the repeatability of the total synaptic output from the network. We find that individual embedded neurons can be reliable or unreliable depending on network conditions, whereas pooled responses of sufficiently large networks are mostly reliable. We study also the effects of noise, and find that some types affect reliability more seriously than others. more...

We study the reliability of phase oscillator networks in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations, regardless of the network's initial condition. In this paper, we extend previous results on two-cell networks to larger systems. The first issue that arises is chaos in the absence of inputs, which we demonstrate and interpret in terms of reliability. We give a mathematical analysis of networks that can be decomposed into modules connected by an acyclic graph. For this class of networks, we show how to localize the source of unreliability, and address questions concerning downstream propagation of unreliability once it is produced., Comment: Part 2 of 2. Part 1 can be found at 0708.3061 more...

This paper concerns the reliability of a pair of coupled oscillators in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations regardless of the network's initial condition. Our main result is that both reliable and unreliable behaviors occur in this network for broad ranges of coupling strengths, even though individual oscillators are always reliable when uncoupled. A new finding is that at low input amplitudes, the system is highly susceptible to unreliable responses when the feedforward and feedback couplings are roughly comparable. A geometric explanation based on shear-induced chaos at the onset of phase-locking is proposed., Comment: Part 1 of 2. Part 2 can be found at 0708.3063 more...

Guided by a geometric understanding developed in earlier works of Wang and Young, we carry out some numerical studies of shear-induced chaos. The settings considered include periodic kicking of limit cycles, random kicks at Poisson times, and continuous-time driving by white noise. The forcing of a quasi-periodic model describing two coupled oscillators is also investigated. In all cases, positive Lyapunov exponents are found in suitable parameter ranges when the forcing is suitably directed. more...

We present the results of a detailed study of energy correlations at steady state for a 1-D model of coupled energy and matter transport. Our aim is to discover -- via theoretical arguments, conjectures, and numerical simulations -- how spatial covariances scale with system size, their relations to local thermodynamic quantities, and the randomizing effects of heat baths. Among our findings are that short-range covariances respond quadratically to local temperature gradients, and long-range covariances decay linearly with macroscopic distance. These findings are consistent with exact results for the simple exclusion and KMP models., Comment: 30 pages, 20 figures more...

This letter concerns the reliability of coupled oscillator networks in response to fluctuating inputs. Reliability means that (following a transient) an input elicits identical responses upon repeated presentations, regardless of the system's initial condition. Here, we analyze this property for two coupled oscillators, demonstrating that oscillator networks exhibit both reliable and unreliable dynamics for broad ranges of coupling strengths. We further argue that unreliable dynamics are characterized by strange attractors with random SRB measures, implying that though unreliable, the responses lie on low-dimensional sets. Finally, we show that 1:1 phase locking in the zero-input system corresponds to high susceptibility for unreliable responses. A geometric explanation is proposed. more...

The Hodgkin-Huxley model describes action potential generation in certain types of neurons and is a standard model for conductance-based, excitable cells. Following the early work of Winfree and Best, this paper explores the response of a spontaneously spiking Hodgkin-Huxley neuron model to a periodic pulsatile drive. The response as a function of drive period and amplitude is systematically characterized. A wide range of qualitatively distinct responses are found, including entrainment to the input pulse train and persistent chaos. These observations are consistent with a theory of kicked oscillators developed by Qiudong Wang and Lai-Sang Young. In addition to general features predicted by Wang-Young theory, it is found that most combinations of drive period and amplitude lead to entrainment instead of chaos. This preference for entrainment over chaos is explained by the structure of the Hodgkin-Huxley phase resetting curve., Comment: Minor revisions; modified Fig. 3; added references more...

This paper presents a systematic numerical study of the effects of noise on the invariant probability densities of dynamical systems with varying degrees of hyperbolicity. It is found that the rate of convergence of invariant densities in the small-noise limit is frequently governed by power laws. In addition, a simple heuristic is proposed and found to correctly predict the power law exponent in exponentially mixing systems. In systems which are not exponentially mixing, the heuristic provides only an upper bound on the power law exponent. As this numerical study requires the computation of invariant densities across more than 2 decades of noise amplitudes, it also provides an opportunity to discuss and compare standard numerical methods for computing invariant probability densities., Comment: 27 pages, 19 figures, revised with minor corrections more...

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on $ {\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $ \exp (C |s|^\delta) $ where $ \delta $ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound $ \exp (C |s|^{n+1}) $, and it gives new bounds on the number of resonances (scattering poles) of $ \Gamma \backslash {\mathbb H}^{n+1} $. The proof of this result is based on the application of holomorphic $ L^2$-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider $ \Gamma \backslash {\mathbb H}^{n+1} $ as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic $L^2$-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. more...

This paper presents numerical evidence that for quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy $E$ scales like $\hbar^{-\frac{D(K_E)+1}{2}}$ as $\hbar\to{0}$. Here, $K_E$ denotes the subset of the classical energy surface $\{H=E\}$ which stays bounded for all time under the flow generated by the Hamiltonian $H$ and $D(K_E)$ denotes its fractal dimension. Since the number of bound states in a quantum system with $n$ degrees of freedom scales like $\hbar^{-n}$, this suggests that the quantity $\frac{D(K_E)+1}{2}$ represents the effective number of degrees of freedom in scattering problems., Comment: 24 pages, including 44 figures more...