Complex Stochastic Dynamics Research Articles

Weak Pinning and Long-Range Anticorrelated Motion of Phase Boundaries in Driven Diffusive Systems.

We show that domain walls separating coexisting extremal current phases in driven diffusive systems exhibit complex stochastic dynamics with a subdiffusive temporal growth of position fluctuations due to long-range anticorrelated current fluctuations and a weak pinning at long times. This weak pinning manifests itself in a saturated width of the domain wall position fluctuations that increases sublinearly with the system size. As a function of time t and system size L, the width w(t,L) has a scaling behavior w(t,L)=L^{3/4}f(t/L^{9/4}), with f(u) constant for u≫1 and f(u)∼u^{1/3} for u≪1. An Orstein-Uhlenbeck process with long-range anticorrelated noise is shown to capture this scaling behavior. The exponent 9/4 is a new dynamical exponent for relaxation processes in driven diffusive systems.

The multiple activations in budding yeast S-phase checkpoint are Poisson processes.

Eukaryotic cells activate the S-phase checkpoint signal transduction pathway in response to DNA replication stress. Affected by the noise in biochemical reactions, such activation process demonstrates cell-to-cell variability. Here, through the analysis of microfluidics-integrated time-lapse imaging, we found multiple S-phase checkpoint activations in a certain budding yeast cell cycle. Yeast cells not only varied in their activation moments but also differed in the number of activations within the cell cycle, resulting in a stochastic multiple activation process. By investigating dynamics at the single-cell level, we showed that stochastic waiting times between consecutive activations are exponentially distributed and independent from each other. Finite DNA replication time provides a robust upper time limit to the duration of multiple activations. The mathematical model, together with further experimental evidence from the mutant strain, revealed that the number of activations under different levels of replication stress agreed well with Poisson distribution. Therefore, the activation events of S-phase checkpoint meet the criterion of Poisson process during DNA replication. In sum, the observed Poisson activation process may provide new insights into the complex stochastic dynamics of signal transduction pathways.

Mixing and demixing arising from compression of two semiflexible polymer chains in nanochannels.

We use molecular dynamics simulation to probe the non-equilibrium physics of two nanochannel-confined semiflexible polymers in a homogeneous flow field. We find that for sufficiently stiff chains the internal organization of the two chains takes the form of interwoven folds and circular coils. This organization can lead to mixing or demixing depending on chain stiffness and flow speed. At low and intermediate flow, the two chains adopt a folded configuration, which favours mixing. At high flow, the two chains adopt a predominantly coiled configuration. The coiled configuration results in demixing when the chains are compressed from an initially demixed condition and mixing when the chains are compressed from an initially mixed condition. We find that the mixing/demixing behaviour is governed by the ratio of the number of folded segments of one chain relative to the other at low flow and by the degree of coiling in both chains at high flow. For decreasing stiffness, the chains start to aggregate locally instead of mixing smoothly at low and intermediate flow. In the limit of completely flexible chains, the two chains either completely segregate at low flow, or adopt a locally demixed configuration consisting of large aggregates of one chain relative to the other that undergo complex stochastic dynamics, diffusing, disintegrating, and reforming at intermediate flow. The transition from complete segregation to the aggregate-dominated configuration occurs when the linear intra-chain ordering breaks down.

Additive eigenvectors as optimal reaction coordinates, conditioned trajectories, and time-reversible description of stochastic processes.

A fundamental way to analyze complex multidimensional stochastic dynamics is to describe it as diffusion on a free energy landscape-free energy as a function of reaction coordinates (RCs). For such a description to be quantitatively accurate, the RC should be chosen in an optimal way. The committor function is a primary example of an optimal RC for the description of equilibrium reaction dynamics between two states. Here, additive eigenvectors (addevs) are considered as optimal RCs to address the limitations of the committor. An addev master equation for a Markov chain is derived. A stationary solution of the equation describes a sub-ensemble of trajectories conditioned on having the same optimal RC for the forward and time-reversed dynamics in the sub-ensemble. A collection of such sub-ensembles of trajectories, called stochastic eigenmodes, can be used to describe/approximate the stochastic dynamics. A non-stationary solution describes the evolution of the probability distribution. However, in contrast to the standard master equation, it provides a time-reversible description of stochastic dynamics. It can be integrated forward and backward in time. The developed framework is illustrated on two model systems-unidirectional random walk and diffusion.

Deterministic modeling of the diffusive memristor.

Recently developed diffusive memristors have gathered a large amount of research attention due to their unique property to exhibit a variety of spiking regimes reminiscent to that found in biological cells, which creates a great potential for their application in neuromorphic systems of artificial intelligence and unconventional computing. These devices are known to produce a huge range of interesting phenomena through the interplay of regular, chaotic, and stochastic behavior. However, the character of these interplays as well as the instabilities responsible for different dynamical regimes are still poorly studied because of the difficulties in analyzing the complex stochastic dynamics of the memristive devices. In this paper, we introduce a new deterministic model justified from the Fokker-Planck description to capture the noise-driven dynamics that noise has been known to produce in the diffusive memristor. This allows us to apply bifurcation theory to reveal the instabilities and the description of the transition between the dynamical regimes.

Estimation of distribution algorithms for the computation of innovation estimators of diffusion processes

Innovation Method is a recognized method for the estimation of parameters in diffusion processes. It is well known that the quality of the Innovation Estimator strongly depends on an adequate selection of the initial value for the parameters when a local optimization algorithm is used in its computation. Alternatively, in this paper, we use a strategy based on a modern method for solving global optimization problems, Estimation of Distribution Algorithms (EDAs). We study the feasibility of a specific EDA - a continuous version of the Univariate Marginal Distribution Algorithm (UMDAc) - for the computation of the Innovation Estimators. Through numerical simulations, we show that the considered global optimization algorithms substantially improves the effectiveness of the Innovation Estimators for different types of diffusion processes with complex nonlinear and stochastic dynamics.

Hopf Bifurcations in a Delayed Microscopic Model of Credit Risk Contagion

This paper is concerned with the proof of the existence of Hopf bifurcations in a mathematical model recently proposed in (T. Chen, X. Li, and J. He, Abstract and Applied Analysis 2014, 456764 (2014)) for understanding the complex stochastic dynamics phenomena of credit risk contagion in the financial market. S pecifically the model consists in an ordinary differential e quation with time-delay. Moreover, by using the normal form theory and center manifold argument, the stability, direction, and peri od of bifurcating periodic solutions are gained.

Level-crossing statistics of a passive scalar dispersed in a neutral boundary layer

Abstract The concentration of a passive scalar dispersed in a turbulent flow exhibits a complex stochastic dynamics. In this paper, we present a minimalist stochastic model that resembles the concentration statistics of a passive scalar emitted from a localized source in a neutral boundary layer. The model provides closed forms for the crossing rates and times — the mean frequency of exceeding a certain concentration level and the mean time above it. Three concentration statistics are needed as model inputs: the mean, the standard deviation, and the integral scale. By giving analytical relationships also for these statistics, we provide a completely closed methodology that may serve as a rapid and practical tool to estimate the dynamics of a pollutant dispersed in the atmosphere. Results are validated against wind-tunnel measurements.

A robust data cleaning procedure for eddy covariance flux measurements

Abstract. The sources of systematic error responsible for introducing significant biases in the eddy covariance (EC) flux computation are manifold, and their correct identification is made difficult by the lack of reference values, by the complex stochastic dynamics, and by the high level of noise characterizing raw data. This work contributes to overcoming such challenges by introducing an innovative strategy for EC data cleaning. The proposed strategy includes a set of tests aimed at detecting the presence of specific sources of systematic error, as well as an outlier detection procedure aimed at identifying aberrant flux values. Results from tests and outlier detection are integrated in such a way as to leave a large degree of flexibility in the choice of tests and of test threshold values, ensuring scalability of the whole process. The selection of best performing tests was carried out by means of Monte Carlo experiments, whereas the impact on real data was evaluated on data distributed by the Integrated Carbon Observation System (ICOS) research infrastructure. Results evidenced that the proposed procedure leads to an effective cleaning of EC flux data, avoiding the use of subjective criteria in the decision rule that specifies whether to retain or reject flux data of dubious quality. We expect that the proposed data cleaning procedure can serve as a basis towards a unified quality control strategy for EC datasets, in particular in centralized data processing pipelines where the use of robust and automated routines ensuring results reproducibility constitutes an essential prerequisite.

Low-dimensional projection of stochastic cell-signalling dynamics via a variational approach.

Noise and fluctuations play vital roles in signal transduction in cells. Various numerical techniques for its simulation have been proposed, most of which are not efficient in cellular networks with a wide spectrum of timescales. In this paper, based on a recently developed variational technique, low-dimensional structures embedded in complex stochastic reaction dynamics are unfolded which sheds light on new design principles of efficient simulation algorithm for treating noise in the mesoscopic world. This idea is effectively demonstrated in several popular regulation models with an empirical selection of test functions according to their reaction geometry, which not only captures complex distribution profiles of different molecular species but also considerably speeds up the computation.

Efficient estimators for likelihood ratio sensitivity indices of complex stochastic dynamics.

We demonstrate that centered likelihood ratio estimators for the sensitivity indices of complex stochastic dynamics are highly efficient with low, constant in time variance and consequently they are suitable for sensitivity analysis in long-time and steady-state regimes. These estimators rely on a new covariance formulation of the likelihood ratio that includes as a submatrix a Fisher information matrix for stochastic dynamics and can also be used for fast screening of insensitive parameters and parameter combinations. The proposed methods are applicable to broad classes of stochastic dynamics such as chemical reaction networks, Langevin-type equations and stochastic models in finance, including systems with a high dimensional parameter space and/or disparate decorrelation times between different observables. Furthermore, they are simple to implement as a standard observable in any existing simulation algorithm without additional modifications.

Path-Space Information Bounds for Uncertainty Quantification and Sensitivity Analysis of Stochastic Dynamics

Uncertainty quantification is a primary challenge for reliable modeling and simulation of complex stochastic dynamics. Such problems are typically plagued by incomplete information that may enter as uncertainty in the model parameters, or even in the model itself. Furthermore, due to their dynamic nature, we need to assess the impact of these uncertainties on both finite and long-time behavior of the stochastic models and derive corresponding uncertainty bounds for observables of interest. A special class of such challenges is parametric uncertainties in the model, in particular sensitivity analysis along with the corresponding sensitivity bounds for stochastic dynamics. Moreover, sensitivity analysis can be further complicated in models with a high number of parameters that render straightforward approaches, such as gradient methods, impractical. In this paper, we derive uncertainty and sensitivity bounds for path-space observables of stochastic dynamics in terms of new goal-oriented divergences; the lat...

Mechanistic stochastic model of histone modification pattern formation.

BackgroundThe activity of a single gene is influenced by the composition of the chromatin in which it is embedded. Nucleosome turnover, conformational dynamics, and covalent histone modifications each induce changes in the structure of chromatin and its affinity for regulatory proteins. The dynamics of histone modifications and the persistence of modification patterns for long periods are still largely unknown.ResultsIn this study, we present a stochastic mathematical model that describes the molecular mechanisms of histone modification pattern formation along a single gene, with non-phenomenological, physical parameters. We find that diffusion and recruitment properties of histone modifying enzymes together with chromatin connectivity allow for a rich repertoire of stochastic histone modification dynamics and pattern formation. We demonstrate that histone modification patterns at a single gene can be established or removed within a few minutes through diffusion and weak recruitment mechanisms of histone modification spreading. Moreover, we show that strong synergism between diffusion and weak recruitment mechanisms leads to nearly irreversible transitions in histone modification patterns providing stable patterns. In the absence of chromatin connectivity spontaneous and dynamic histone modification boundaries can be formed that are highly unstable, and spontaneous fluctuations cause them to diffuse randomly. Chromatin connectivity destabilizes this synergistic system and introduces bistability, illustrating state switching between opposing modification states of the model gene. The observed bistable long-range and localized pattern formation are critical effectors of gene expression regulation.ConclusionThis study illustrates how the cooperative interactions between regulatory proteins and the chromatin state generate complex stochastic dynamics of gene expression regulation.Electronic supplementary materialThe online version of this article (doi:10.1186/1756-8935-7-30) contains supplementary material, which is available to authorized users.

Aging as a process of deficit accumulation: its utility and origin.

Individuals of the same age differ greatly with respect to their health status and life span. We have suggested that the health status of individuals can be represented by the number of health deficits that they accumulate during their life. We have suggested that this can be measured by a fitness-frailty index (or just a frailty index), which is the ratio of the deficits present in a person to the total number of deficits considered (e.g. available in a given database or experimental procedure). Further, we have proposed that the frailty index represents the biological age of the individual, and suggested an algorithm for its estimation. In investigations by many groups, the frailty index has shown reproducible properties such as: age-specific, nonlinear increase, higher values in women, strong association with mortality and other adverse outcomes, and universal limit to its increase. At the level of individual, the frailty index shows complex stochastic dynamics, reflecting both stochasticity of the environment and the ability to recover from various illnesses. Most recently, we have proposed that the origin of deficit accumulation lies in the interaction between the environment, the organism and its ability to recover. We apply a stochastic dynamics framework to illustrate that the average recovery time increases with age, mimicking the age-associated increase in deficit accumulation.

Consentaneous agent-based and stochastic model of the financial markets.

We are looking for the agent-based treatment of the financial markets considering necessity to build bridges between microscopic, agent based, and macroscopic, phenomenological modeling. The acknowledgment that agent-based modeling framework, which may provide qualitative and quantitative understanding of the financial markets, is very ambiguous emphasizes the exceptional value of well defined analytically tractable agent systems. Herding as one of the behavior peculiarities considered in the behavioral finance is the main property of the agent interactions we deal with in this contribution. Looking for the consentaneous agent-based and macroscopic approach we combine two origins of the noise: exogenous one, related to the information flow, and endogenous one, arising form the complex stochastic dynamics of agents. As a result we propose a three state agent-based herding model of the financial markets. From this agent-based model we derive a set of stochastic differential equations, which describes underlying macroscopic dynamics of agent population and log price in the financial markets. The obtained solution is then subjected to the exogenous noise, which shapes instantaneous return fluctuations. We test both Gaussian and q-Gaussian noise as a source of the short term fluctuations. The resulting model of the return in the financial markets with the same set of parameters reproduces empirical probability and spectral densities of absolute return observed in New York, Warsaw and NASDAQ OMX Vilnius Stock Exchanges. Our result confirms the prevalent idea in behavioral finance that herding interactions may be dominant over agent rationality and contribute towards bubble formation.

A stochastic approach to modeling the dynamics of natural ventilation systems

Heating ventilation and air conditioning (HVAC) systems in residential and commercial buildings make up 16% of the United States energy consumption. Utilizing natural ventilation strategies is a low energy solution to reduce the energy used by building environmental control systems. Design of effective natural ventilation strategies is challenging because of inherent stochasticity in interior (machine loads, number of people) and exterior conditions (wind load, outside temperature). However, by exploiting the natural dynamics of building systems, efficient design and control seems possible. We explore this idea by introducing a stochastic approach to analyze the natural dynamics of building systems (under natural ventilation) by explicitly incorporating the effects of stochastic wind speeds and stochastic internal loads. We show that complex dynamics in the form of bi-stable behavior emerges when considering a single zone building with stochastic inputs. We show that neglecting these complex stochastic dynamics leads to inaccurate predictions in the thermal response, especially for natural ventilation. We compute the sensitivity of the system with respect to various system parameters which provide insight into developing robust design guidelines. The techniques presented aid in the design process, and are a step toward adaptive, efficient, robust control of natural ventilation systems.

Simulation of stochastic network dynamics via entropic matching

The simulation of complex stochastic network dynamics arising, for instance, from models of coupled biomolecular processes remains computationally challenging. Often, the necessity to scan a model's dynamics over a large parameter space renders full-fledged stochastic simulations impractical, motivating approximation schemes. Here we propose an approximation scheme which improves upon the standard linear noise approximation while retaining similar computational complexity. The underlying idea is to minimize, at each time step, the Kullback-Leibler divergence between the true time evolved probability distribution and a Gaussian approximation (entropic matching). This condition leads to ordinary differential equations for the mean and the covariance matrix of the Gaussian. For cases of weak nonlinearity, the method is more accurate than the linear method when both are compared to stochastic simulations.

A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics

We propose a new sensitivity analysis methodology for complex stochastic dynamics based on the relative entropy rate. The method becomes computationally feasible at the stationary regime of the process and involves the calculation of suitable observables in path space for the relative entropy rate and the corresponding Fisher information matrix. The stationary regime is crucial for stochastic dynamics and here allows us to address the sensitivity analysis of complex systems, including examples of processes with complex landscapes that exhibit metastability, non-reversible systems from a statistical mechanics perspective, and high-dimensional, spatially distributed models. All these systems exhibit, typically non-Gaussian stationary probability distributions, while in the case of high-dimensionality, histograms are impossible to construct directly. Our proposed methods bypass these challenges relying on the direct Monte Carlo simulation of rigorously derived observables for the relative entropy rate and Fisher information in path space rather than on the stationary probability distribution itself. We demonstrate the capabilities of the proposed methodology by focusing here on two classes of problems: (a) Langevin particle systems with either reversible (gradient) or non-reversible (non-gradient) forcing, highlighting the ability of the method to carry out sensitivity analysis in non-equilibrium systems; and, (b) spatially extended kinetic Monte Carlo models, showing that the method can handle high-dimensional problems.

Transfer Entropy as a Log-Likelihood Ratio

Transfer entropy, an information-theoretic measure of time-directed information transfer between joint processes, has steadily gained popularity in the analysis of complex stochastic dynamics in diverse fields, including the neurosciences, ecology, climatology, and econometrics. We show that for a broad class of predictive models, the log-likelihood ratio test statistic for the null hypothesis of zero transfer entropy is a consistent estimator for the transfer entropy itself. For finite Markov chains, furthermore, no explicit model is required. In the general case, an asymptotic χ2 distribution is established for the transfer entropy estimator. The result generalizes the equivalence in the Gaussian case of transfer entropy and Granger causality, a statistical notion of causal influence based on prediction via vector autoregression, and establishes a fundamental connection between directed information transfer and causality in the Wiener-Granger sense.

A discrete dynamical system for the greedy strategy at collective Parrondo games

We consider a collective version of Parrondo's games with probabilities parameterized by ρ ∈ (0, 1) in which a fraction φ ∈ (0, 1] of an infinite number of players collectively choose and individually play at each turn the game that yields the maximum average profit at that turn. Dinís and Parrondo [L. Dinís and J.M.R. Parrondo, Optimal strategies in collective Parrondo games, Europhys. Lett. 63 (2003), pp. 319–325] and Van den Broeck and Cleuren [C. Van den Broeck and B. Cleuren, Parrondo games with strategy, in Noise in Complex Systems and Stochastic Dynamics II, Z. Gingl, J.M. Sancho, L. Schimansky-Geier, and J. Kertesz, eds., SPIE, Bellingham, WA, 2004, pp. 109–118.] studied the asymptotic behaviour of this greedy strategy, which corresponds to a piecewise-linear discrete dynamical system in a subset of the plane, for ρ = 1/3 and three choices of φ. We study its asymptotic behaviour for all (ρ, φ) ∈ (0, 1) × (0, 1], finding that there is a globally asymptotically stable equilibrium if φ ≤ 2/3 and, typically, a unique (asymptotically stable) limit cycle if φ > 2/3 (‘typically’ because there are rare cases with two limit cycles). Asymptotic stability results for φ > 2/3 are partly conjectural.