1
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Kopp RA, Klapp SHL. Persistent motion of a Brownian particle subject to repulsive feedback with time delay. Phys Rev E 2023; 107:024611. [PMID: 36932532 DOI: 10.1103/physreve.107.024611] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2022] [Accepted: 02/08/2023] [Indexed: 06/18/2023]
Abstract
Based on analytical and numerical calculations we study the dynamics of an overdamped colloidal particle moving in two dimensions under time-delayed, nonlinear feedback control. Specifically, the particle is subject to a force derived from a repulsive Gaussian potential depending on the difference between its instantaneous position, r(t), and its earlier position r(t-τ), where τ is the delay time. Considering first the deterministic case, we provide analytical results for both the case of small displacements and the dynamics at long times. In particular, at appropriate values of the feedback parameters, the particle approaches a steady state with a constant, nonzero velocity whose direction is constant as well. In the presence of noise, the direction of motion becomes randomized at long times, but the (numerically obtained) velocity autocorrelation still reveals some persistence of motion. Moreover, the mean-squared displacement (MSD) reveals a mixed regime at intermediate times with contributions of both ballistic motion and diffusive translational motion, allowing us to extract an estimate for the effective propulsion velocity in presence of noise. We then analyze the data in terms of exact, known results for the MSD of active Brownian particles. The comparison indeed indicates a strong similarity between the dynamics of the particle under repulsive delayed feedback and active motion. This relation carries over to the behavior of the long-time diffusion coefficient D_{eff} which, similarly to active motion, is strongly enhanced compared to the free case. Finally, we show that, for small delays, D_{eff} can be estimated analytically.
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Affiliation(s)
- Robin A Kopp
- Institut für Theoretische Physik, Hardenbergstraße 36, Technische Universität Berlin, D-10623 Berlin, Germany
| | - Sabine H L Klapp
- Institut für Theoretische Physik, Hardenbergstraße 36, Technische Universität Berlin, D-10623 Berlin, Germany
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2
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Holubec V, Geiss D, Loos SAM, Kroy K, Cichos F. Finite-Size Scaling at the Edge of Disorder in a Time-Delay Vicsek Model. PHYSICAL REVIEW LETTERS 2021; 127:258001. [PMID: 35029446 DOI: 10.1103/physrevlett.127.258001] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/2021] [Revised: 09/29/2021] [Accepted: 11/16/2021] [Indexed: 06/14/2023]
Abstract
Living many-body systems often exhibit scale-free collective behavior reminiscent of thermal critical phenomena. But their mutual interactions are inevitably retarded due to information processing and delayed actuation. We numerically investigate the consequences for the finite-size scaling in the Vicsek model of motile active matter. A growing delay time initially facilitates but ultimately impedes collective ordering and turns the dynamical scaling from diffusive to ballistic. It provides an alternative explanation of swarm traits previously attributed to inertia.
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Affiliation(s)
- Viktor Holubec
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
- Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, CZ-180 00 Praha, Czech Republic
| | - Daniel Geiss
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
- Max Planck Institute for Mathematics in the Sciences, D-04103 Leipzig, Germany
| | - Sarah A M Loos
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
- ICTP - International Centre for Theoretical Physics, Strada Costiera 11, 34151, Trieste, Italy
| | - Klaus Kroy
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
| | - Frank Cichos
- Peter Debye Institute for Soft Matter Physics, Universität Leipzig, 04103 Leipzig, Germany
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3
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Cortez MJ, Hong H, Choi B, Kim JK, Josić K. Hierarchical Bayesian models of transcriptional and translational regulation processes with delays. Bioinformatics 2021; 38:187-195. [PMID: 34450624 PMCID: PMC8696106 DOI: 10.1093/bioinformatics/btab618] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2021] [Revised: 08/19/2021] [Accepted: 08/25/2021] [Indexed: 02/05/2023] Open
Abstract
MOTIVATION Simultaneous recordings of gene network dynamics across large populations have revealed that cell characteristics vary considerably even in clonal lines. Inferring the variability of parameters that determine gene dynamics is key to understanding cellular behavior. However, this is complicated by the fact that the outcomes and effects of many reactions are not observable directly. Unobserved reactions can be replaced with time delays to reduce model dimensionality and simplify inference. However, the resulting models are non-Markovian, and require the development of new inference techniques. RESULTS We propose a non-Markovian, hierarchical Bayesian inference framework for quantifying the variability of cellular processes within and across cells in a population. We illustrate our approach using a delayed birth-death process. In general, a distributed delay model, rather than a popular fixed delay model, is needed for inference, even if only mean reaction delays are of interest. Using in silico and experimental data we show that the proposed hierarchical framework is robust and leads to improved estimates compared to its non-hierarchical counterpart. We apply our method to data obtained using time-lapse microscopy and infer the parameters that describe the dynamics of protein production at the single cell and population level. The mean delays in protein production are larger than previously reported, have a coefficient of variation of around 0.2 across the population, and are not strongly correlated with protein production or growth rates. AVAILABILITY AND IMPLEMENTATION Accompanying code in Python is available at https://github.com/mvcortez/Bayesian-Inference. CONTACT kresimir.josic@gmail.com or jaekkim@kaist.ac.kr or cbskust@korea.ac.kr. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Mark Jayson Cortez
- Department of Mathematics, University of Houston, Houston, TX 77204, USA,Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, Laguna 4031, Philippines
| | - Hyukpyo Hong
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea,Biomedical Mathematics Group, Institute for Basic Science, Daejeon 34126, Korea
| | - Boseung Choi
- To whom correspondence should be addressed. or or
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Friston KJ, Da Costa L, Parr T. Some Interesting Observations on the Free Energy Principle. ENTROPY (BASEL, SWITZERLAND) 2021; 23:1076. [PMID: 34441216 PMCID: PMC8391698 DOI: 10.3390/e23081076] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/01/2021] [Revised: 07/26/2021] [Accepted: 08/10/2021] [Indexed: 01/28/2023]
Abstract
Biehl et al. (2021) present some interesting observations on an early formulation of the free energy principle. We use these observations to scaffold a discussion of the technical arguments that underwrite the free energy principle. This discussion focuses on solenoidal coupling between various (subsets of) states in sparsely coupled systems that possess a Markov blanket-and the distinction between exact and approximate Bayesian inference, implied by the ensuing Bayesian mechanics.
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Affiliation(s)
- Karl J. Friston
- The Wellcome Centre for Human Neuroimaging, University College London, London WC1N 3AR, UK; (K.J.F.); (L.D.C.)
| | - Lancelot Da Costa
- The Wellcome Centre for Human Neuroimaging, University College London, London WC1N 3AR, UK; (K.J.F.); (L.D.C.)
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
| | - Thomas Parr
- The Wellcome Centre for Human Neuroimaging, University College London, London WC1N 3AR, UK; (K.J.F.); (L.D.C.)
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5
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Ramstead MJD, Friston KJ, Hipólito I. Is the Free-Energy Principle a Formal Theory of Semantics? From Variational Density Dynamics to Neural and Phenotypic Representations. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E889. [PMID: 33286659 PMCID: PMC7517505 DOI: 10.3390/e22080889] [Citation(s) in RCA: 38] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/17/2020] [Revised: 08/06/2020] [Accepted: 08/07/2020] [Indexed: 12/14/2022]
Abstract
The aim of this paper is twofold: (1) to assess whether the construct of neural representations plays an explanatory role under the variational free-energy principle and its corollary process theory, active inference; and (2) if so, to assess which philosophical stance-in relation to the ontological and epistemological status of representations-is most appropriate. We focus on non-realist (deflationary and fictionalist-instrumentalist) approaches. We consider a deflationary account of mental representation, according to which the explanatorily relevant contents of neural representations are mathematical, rather than cognitive, and a fictionalist or instrumentalist account, according to which representations are scientifically useful fictions that serve explanatory (and other) aims. After reviewing the free-energy principle and active inference, we argue that the model of adaptive phenotypes under the free-energy principle can be used to furnish a formal semantics, enabling us to assign semantic content to specific phenotypic states (the internal states of a Markovian system that exists far from equilibrium). We propose a modified fictionalist account-an organism-centered fictionalism or instrumentalism. We argue that, under the free-energy principle, pursuing even a deflationary account of the content of neural representations licenses the appeal to the kind of semantic content involved in the 'aboutness' or intentionality of cognitive systems; our position is thus coherent with, but rests on distinct assumptions from, the realist position. We argue that the free-energy principle thereby explains the aboutness or intentionality in living systems and hence their capacity to parse their sensory stream using an ontology or set of semantic factors.
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Affiliation(s)
- Maxwell J. D. Ramstead
- Division of Social and Transcultural Psychiatry, Department of Psychiatry, McGill University, Montreal, QC H3A 1A1, Canada
- Culture, Mind, and Brain Program, McGill University, Montreal, QC H3A 1A1, Canada
- Wellcome Centre for Human Neuroimaging, University College London, London, WC1N 3AR, UK;
| | - Karl J. Friston
- Wellcome Centre for Human Neuroimaging, University College London, London, WC1N 3AR, UK;
| | - Inês Hipólito
- Faculty of Arts, Social Sciences, and Humanities, University of Wollongong, Wollongong 2522, Australia;
- Institute of Psychiatry, Psychology and Neuroscience (IoPPN), King’s College London, London SE5 8AF, UK
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Friston KJ, Wiese W, Hobson JA. Sentience and the Origins of Consciousness: From Cartesian Duality to Markovian Monism. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E516. [PMID: 33286288 PMCID: PMC7517007 DOI: 10.3390/e22050516] [Citation(s) in RCA: 57] [Impact Index Per Article: 14.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/04/2020] [Revised: 04/15/2020] [Accepted: 04/16/2020] [Indexed: 12/23/2022]
Abstract
This essay addresses Cartesian duality and how its implicit dialectic might be repaired using physics and information theory. Our agenda is to describe a key distinction in the physical sciences that may provide a foundation for the distinction between mind and matter, and between sentient and intentional systems. From this perspective, it becomes tenable to talk about the physics of sentience and 'forces' that underwrite our beliefs (in the sense of probability distributions represented by our internal states), which may ground our mental states and consciousness. We will refer to this view as Markovian monism, which entails two claims: (1) fundamentally, there is only one type of thing and only one type of irreducible property (hence monism). (2) All systems possessing a Markov blanket have properties that are relevant for understanding the mind and consciousness: if such systems have mental properties, then they have them partly by virtue of possessing a Markov blanket (hence Markovian). Markovian monism rests upon the information geometry of random dynamic systems. In brief, the information geometry induced in any system-whose internal states can be distinguished from external states-must acquire a dual aspect. This dual aspect concerns the (intrinsic) information geometry of the probabilistic evolution of internal states and a separate (extrinsic) information geometry of probabilistic beliefs about external states that are parameterised by internal states. We call these intrinsic (i.e., mechanical, or state-based) and extrinsic (i.e., Markovian, or belief-based) information geometries, respectively. Although these mathematical notions may sound complicated, they are fairly straightforward to handle, and may offer a means through which to frame the origins of consciousness.
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Affiliation(s)
- Karl J. Friston
- The Wellcome Centre for Human Neuroimaging, Institute of Neurology, Queen Square, London WC1N 3AR, UK
| | - Wanja Wiese
- Department of Philosophy, Johannes Gutenberg University Mainz, Jakob-Welder-Weg 18, 55128 Mainz, Germany
| | - J. Allan Hobson
- Division of Sleep Medicine, Harvard Medical School, 74 Fenwood Road, Boston, MA 02115, USA
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7
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Van Vu T, Hasegawa Y. Uncertainty relations for time-delayed Langevin systems. Phys Rev E 2019; 100:012134. [PMID: 31499914 DOI: 10.1103/physreve.100.012134] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2018] [Indexed: 06/10/2023]
Abstract
The thermodynamic uncertainty relation, which establishes a universal trade-off between nonequilibrium current fluctuations and dissipation, has been found for various Markovian systems. However, this relation has not been revealed for non-Markovian systems; therefore, we investigate the thermodynamic uncertainty relation for time-delayed Langevin systems. We prove that the fluctuation of arbitrary dynamical observables is constrained by the Kullback-Leibler divergence between the distributions of the forward path and its reversed counterpart. Specifically, for observables that are antisymmetric under time reversal, the fluctuation is bounded from below by a function of a quantity that can be identified as a generalization of the total entropy production in Markovian systems. We also provide a lower bound for arbitrary observables that are odd under position reversal. The term in this bound reflects the extent to which the position symmetry has been broken in the system and can be positive even in equilibrium. Our results hold for finite observation times and a large class of time-delayed systems because detailed underlying dynamics are not required for the derivation. We numerically verify the derived uncertainty relations using two single time-delay systems and one distributed time-delay system.
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Affiliation(s)
- Tan Van Vu
- Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
| | - Yoshihiko Hasegawa
- Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
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8
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Loos SAM, Klapp SHL. Force-linearization closure for non-Markovian Langevin systems with time delay. Phys Rev E 2017; 96:012106. [PMID: 29347056 DOI: 10.1103/physreve.96.012106] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2017] [Indexed: 06/07/2023]
Abstract
This paper is concerned with the Fokker-Planck (FP) description of classical stochastic systems with discrete time delay. The non-Markovian character of the corresponding Langevin dynamics naturally leads to a coupled infinite hierarchy of FP equations for the various n-time joint distribution functions. Here, we present an approach to close the hierarchy at the one-time level based on a linearization of the deterministic forces in all members of the hierarchy starting from the second one. This leads to a closed equation for the one-time probability density in the steady state. Considering two generic nonlinear systems, a colloidal particle in a sinusoidal or bistable potential supplemented by a linear delay force, we demonstrate that our approach yields a very accurate representation of the density as compared to quasiexact numerical results from direct solution of the Langevin equation. Moreover, the results are significantly improved against those from a small-delay approximation and a perturbation-theoretical approach. We also discuss the possibility of accessing transport-related quantities, such as escape times, based on an additional Kramers approximation. Our approach applies to a wide class of models with nonlinear deterministic forces.
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Affiliation(s)
- Sarah A M Loos
- Institut für Theoretische Physik, Hardenbergstr. 36, Technische Universität Berlin, D-10623 Berlin, Germany
| | - Sabine H L Klapp
- Institut für Theoretische Physik, Hardenbergstr. 36, Technische Universität Berlin, D-10623 Berlin, Germany
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9
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Xiao T. Heat dissipation and information flow for delayed bistable Langevin systems near coherence resonance. Phys Rev E 2016; 94:052109. [PMID: 27967182 DOI: 10.1103/physreve.94.052109] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2016] [Indexed: 11/07/2022]
Abstract
In this paper, stochastic thermodynamics of delayed bistable Langevin systems near coherence resonance is discussed. We calculate the heat dissipation rate and the information flow of a delayed bistable Langevin system under various noise intensities. Both the heat dissipation rate and the information flow are found to be bell-shaped functions of the noise intensity, which implies that coherence resonance manifests itself in the thermodynamic properties.
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Affiliation(s)
- Tiejun Xiao
- Guizhou Provincial Key Laboratory of Computational Nano-Material Science, Guizhou Synergetic Innovation Center of Scientific Big Data for Advanced Manufacturing Technology, Guizhou Education University, Guiyang 550018, People's Republic of China
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10
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11
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Segmentation of uterine fibroid ultrasound images using a dynamic statistical shape model in HIFU therapy. Comput Med Imaging Graph 2015; 46 Pt 3:302-14. [PMID: 26459767 DOI: 10.1016/j.compmedimag.2015.07.004] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2014] [Revised: 06/24/2015] [Accepted: 07/13/2015] [Indexed: 11/20/2022]
Abstract
Segmenting the lesion areas from ultrasound (US) images is an important step in the intra-operative planning of high-intensity focused ultrasound (HIFU). However, accurate segmentation remains a challenge due to intensity inhomogeneity, blurry boundaries in HIFU US images and the deformation of uterine fibroids caused by patient's breathing or external force. This paper presents a novel dynamic statistical shape model (SSM)-based segmentation method to accurately and efficiently segment the target region in HIFU US images of uterine fibroids. For accurately learning the prior shape information of lesion boundary fluctuations in the training set, the dynamic properties of stochastic differential equation and Fokker-Planck equation are incorporated into SSM (referred to as SF-SSM). Then, a new observation model of lesion areas (named to RPFM) in HIFU US images is developed to describe the features of the lesion areas and provide a likelihood probability to the prior shape given by SF-SSM. SF-SSM and RPFM are integrated into active contour model to improve the accuracy and robustness of segmentation in HIFU US images. We compare the proposed method with four well-known US segmentation methods to demonstrate its superiority. The experimental results in clinical HIFU US images validate the high accuracy and robustness of our approach, even when the quality of the images is unsatisfactory, indicating its potential for practical application in HIFU therapy.
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12
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Rosinberg ML, Munakata T, Tarjus G. Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042114. [PMID: 25974446 DOI: 10.1103/physreve.91.042114] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2015] [Indexed: 06/04/2023]
Abstract
Response lags are generic to almost any physical system and often play a crucial role in the feedback loops present in artificial nanodevices and biological molecular machines. In this paper, we perform a comprehensive study of small stochastic systems governed by an underdamped Langevin equation and driven out of equilibrium by a time-delayed continuous feedback control. In their normal operating regime, these systems settle in a nonequilibrium steady state in which work is permanently extracted from the surrounding heat bath. By using the Fokker-Planck representation of the dynamics, we derive a set of second-law-like inequalities that provide bounds to the rate of extracted work. These inequalities involve additional contributions characterizing the reduction of entropy production due to the continuous measurement process. We also show that the non-Markovian nature of the dynamics requires a modification of the basic relation linking dissipation to the breaking of time-reversal symmetry at the level of trajectories. The modified relation includes a contribution arising from the acausal character of the reverse process. This, in turn, leads to another second-law-like inequality. We illustrate the general formalism with a detailed analytical and numerical study of a harmonic oscillator driven by a linear feedback, which describes actual experimental setups.
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Affiliation(s)
- M L Rosinberg
- Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie, CNRS UMR 7600, 4 place Jussieu, 75252 Paris Cedex 05, France
| | - T Munakata
- Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
| | - G Tarjus
- Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie, CNRS UMR 7600, 4 place Jussieu, 75252 Paris Cedex 05, France
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13
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McKetterick TJ, Giuggioli L. Exact dynamics of stochastic linear delayed systems: application to spatiotemporal coordination of comoving agents. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042135. [PMID: 25375466 DOI: 10.1103/physreve.90.042135] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/14/2014] [Indexed: 06/04/2023]
Abstract
Delayed dynamics result from finite transmission speeds of a signal in the form of energy, mass, or information. In stochastic systems the resulting lagged dynamics challenge our understanding due to the rich behavioral repertoire encompassing monotonic, oscillatory, and unstable evolution. Despite the vast literature, quantifying this rich behavior is limited by a lack of explicit analytic studies of high-dimensional stochastic delay systems. Here we fill this gap for systems governed by a linear Langevin equation of any number of delays and spatial dimensions with additive Gaussian noise. By exploiting Laplace transforms we are able to derive an exact time-dependent analytic solution of the Langevin equation. By using characteristic functionals we are able to construct the full time dependence of the multivariate probability distribution of the stochastic process as a function of the delayed and nondelayed random variables. As an application we consider interactions in animal collective movement that go beyond the traditional assumption of instantaneous alignment. We propose models for coordinated maneuvers of comoving agents applicable to recent empirical findings in pigeons and bats whereby individuals copy the heading of their neighbors with some delay. We highlight possible strategies that individual pairs may adopt to reduce the variance in their velocity difference and/or in their spatial separation. We also show that a minimum in the variance of the spatial separation at long times can be achieved with certain ratios of measurement to reaction delay.
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Affiliation(s)
- Thomas John McKetterick
- Bristol Centre for Complexity Sciences, University of Bristol, Bristol BS8 1UG, United Kingdom and Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UG, Kingdom
| | - Luca Giuggioli
- Bristol Centre for Complexity Sciences, University of Bristol, Bristol BS8 1UG, United Kingdom and Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UG, Kingdom and School of Biological Sciences, University of Bristol, Bristol BS8 1UG, United Kingdom
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14
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Loos SAM, Gernert R, Klapp SHL. Delay-induced transport in a rocking ratchet under feedback control. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052136. [PMID: 25353768 DOI: 10.1103/physreve.89.052136] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2014] [Indexed: 06/04/2023]
Abstract
Based on the Fokker-Planck equation we investigate the transport of an overdamped colloidal particle in a static, asymmetric periodic potential supplemented by a time-dependent, delayed feedback force, F(fc). For a given time t, F(fc) depends on the status of the system at a previous time t-τ(D), with τ(D) being a delay time, specifically on the delayed mean particle displacement (relative to some "switching position"). For nonzero delay times F(fc)(t) develops nearly regular oscillations, generating a net current in the system. Depending on the switching position, this current is nearly as large or even larger than that in a conventional open-loop rocking ratchet. We also investigate thermodynamic properties of the delayed nonequilibrium system and we suggest an underlying Langevin equation which reproduces the Fokker-Planck results.
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Affiliation(s)
- Sarah A M Loos
- Institut für Theoretische Physik, Hardenbergstr. 36, Technische Universität Berlin, D-10623 Berlin, Germany
| | - Robert Gernert
- Institut für Theoretische Physik, Hardenbergstr. 36, Technische Universität Berlin, D-10623 Berlin, Germany
| | - Sabine H L Klapp
- Institut für Theoretische Physik, Hardenbergstr. 36, Technische Universität Berlin, D-10623 Berlin, Germany
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15
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Lafuerza LF, Toral R. Stochastic description of delayed systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2013; 371:20120458. [PMID: 23960216 DOI: 10.1098/rsta.2012.0458] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
We study general stochastic birth and death processes including delay. We develop several approaches for the analytical treatment of these non-Markovian systems, valid, not only for constant delays, but also for stochastic delays with arbitrary probability distributions. The interplay between stochasticity and delay and, in particular, the effects of delay in the fluctuations and time correlations are discussed.
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Affiliation(s)
- L F Lafuerza
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos, CSIC-UIB, Campus UIB, 07122 Palma de Mallorca, Spain
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16
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Brett T, Galla T. Stochastic processes with distributed delays: chemical Langevin equation and linear-noise approximation. PHYSICAL REVIEW LETTERS 2013; 110:250601. [PMID: 23829723 DOI: 10.1103/physrevlett.110.250601] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2013] [Indexed: 06/02/2023]
Abstract
We develop a systematic approach to the linear-noise approximation for stochastic reaction systems with distributed delays. Unlike most existing work our formalism does not rely on a master equation; instead it is based upon a dynamical generating functional describing the probability measure over all possible paths of the dynamics. We derive general expressions for the chemical Langevin equation for a broad class of non-Markovian systems with distributed delay. Exemplars of a model of gene regulation with delayed autoinhibition and a model of epidemic spread with delayed recovery provide evidence of the applicability of our results.
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Affiliation(s)
- Tobias Brett
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom.
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17
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D’Odorico P, Ridolfi L, Laio F. Precursors of state transitions in stochastic systems with delay. THEOR ECOL-NETH 2013. [DOI: 10.1007/s12080-013-0188-2] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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18
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Shepherd T, Prince SJD, Alexander DC. Interactive lesion segmentation with shape priors from offline and online learning. IEEE TRANSACTIONS ON MEDICAL IMAGING 2012; 31:1698-712. [PMID: 22547455 DOI: 10.1109/tmi.2012.2196285] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/14/2023]
Abstract
In medical image segmentation, tumors and other lesions demand the highest levels of accuracy but still call for the highest levels of manual delineation. One factor holding back automatic segmentation is the exemption of pathological regions from shape modelling techniques that rely on high-level shape information not offered by lesions. This paper introduces two new statistical shape models (SSMs) that combine radial shape parameterization with machine learning techniques from the field of nonlinear time series analysis. We then develop two dynamic contour models (DCMs) using the new SSMs as shape priors for tumor and lesion segmentation. From training data, the SSMs learn the lower level shape information of boundary fluctuations, which we prove to be nevertheless highly discriminant. One of the new DCMs also uses online learning to refine the shape prior for the lesion of interest based on user interactions. Classification experiments reveal superior sensitivity and specificity of the new shape priors over those previously used to constrain DCMs. User trials with the new interactive algorithms show that the shape priors are directly responsible for improvements in accuracy and reductions in user demand.
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Affiliation(s)
- Tony Shepherd
- Turku PET Centre and Department of Oncology and Radiotherapy, Turku University Hospital, Turku, Finland.
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19
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D'Odorico P, Laio F, Ridolfi L. Noise-sustained fluctuations in stochastic dynamics with a delay. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:041106. [PMID: 22680419 DOI: 10.1103/physreve.85.041106] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2011] [Indexed: 06/01/2023]
Abstract
Delayed responses to external drivers are ubiquitous in environmental, social, and biological processes. Delays may induce oscillations, Hopf bifurcations, and instabilities in deterministic systems even in the absence of nonlinearities. Despite recent advances in the study of delayed stochastic differential equations, the interaction of random drivers with delays remains poorly understood. In particular, it is unclear whether noise-induced behaviors may emerge from these interactions. Here we show that noise may enhance and sustain transient periodic oscillations inherent to deterministic delayed systems. We investigate the conditions conducive to the emergence and disappearance of these dynamics in a linear system in the presence of both additive and multiplicative noise.
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Affiliation(s)
- Paolo D'Odorico
- Department of Environmental Sciences, University of Virginia, 291 McCormick Road, Charlottesville, Virginia 22904-4123, USA
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20
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FRANK TD, FRIEDRICH R, BEEK PJ. TIME SERIES ANALYSIS OF MULTIVARIATE TIME-DELAYED SYSTEMS WITH NOISE: APPLICATIONS TO LASER PHYSICS AND HUMAN MOVEMENT. STOCH DYNAM 2011. [DOI: 10.1142/s0219493705001456] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
A data analysis method is proposed to reconstruct evolution equations of stochastic systems with time-delayed feedback from experimental time series. Multivariate systems involving noise sources with arbitrary correlation times are considered.
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Affiliation(s)
- T. D. FRANK
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
| | - R. FRIEDRICH
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
| | - P. J. BEEK
- Faculty of Human Movement Sciences, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands
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21
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Lafuerza LF, Toral R. Exact solution of a stochastic protein dynamics model with delayed degradation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:051121. [PMID: 22181383 DOI: 10.1103/physreve.84.051121] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2011] [Indexed: 05/31/2023]
Abstract
We study a stochastic model of protein dynamics that explicitly includes delay in the degradation. We rigorously derive the master equation for the processes and solve it exactly. We show that the equations for the mean values obtained differ from others intuitively proposed and that oscillatory behavior is not possible in this system. We discuss the calculation of correlation functions in stochastic systems with delay, stressing the differences with Markovian processes. The exact results allow us to clarify the interplay between stochasticity and delay.
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Affiliation(s)
- L F Lafuerza
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), Palma de Mallorca, Spain
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22
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Lafuerza LF, Toral R. Role of delay in the stochastic creation process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:021128. [PMID: 21928970 DOI: 10.1103/physreve.84.021128] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2011] [Indexed: 05/31/2023]
Abstract
We develop an approximate theoretical method to study discrete stochastic birth and death models that include a delay time. We analyze the effect of the delay in the fluctuations of the system and obtain that it can qualitatively alter them. We also study the effect of distributed delay. We apply the method to a protein-dynamics model that explicitly includes transcription and translation delays. The theoretical model allows us to understand in a general way the interplay between stochasticity and delay.
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Affiliation(s)
- L F Lafuerza
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos, CSIC-UIB, Campus UIB, E-07122 Palma de Mallorca, Spain
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23
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Jiang H, Xiao T, Hou Z. Stochastic thermodynamics for delayed Langevin systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061144. [PMID: 21797339 DOI: 10.1103/physreve.83.061144] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2011] [Revised: 04/25/2011] [Indexed: 05/31/2023]
Abstract
We discuss stochastic thermodynamics (ST) for delayed Langevin systems in this paper. By using the general principles of ST, the first-law-like energy balance and trajectory-dependent entropy s(t) can be well defined in a way that is similar to that in a system without delay. Because the presence of time delay brings an additional entropy flux into the system, the conventional second law (Δs(tot))≥0 no longer holds true, where Δs(tot) denotes the total entropy change along a stochastic path and (·) stands for the average over the path ensemble. With the help of a Fokker-Planck description, we introduce a delay-averaged trajectory-dependent dissipation functional η[χ(t)] which involves the work done by a delay-averaged force F(x,t) along the path χ(t) and equals the medium entropy change Δs(m)[x(t)] in the absence of delay. We show that the total dissipation functional R=Δs+η, where Δs denotes the system entropy change along a path, obeys (R)≥0, which could be viewed as the second law in the delayed system. In addition, the integral fluctuation theorem (e(-R))=1 also holds true. We apply these concepts to a linear Langevin system with time delay and periodic external force. Numerical results demonstrate that the total entropy change (Δs(tot)) could indeed be negative when the delay feedback is positive. By using an inversing-mapping approach, we are able to obtain the delay-averaged force F(x,t) from the stationary distribution and then calculate the functional R as well as its distribution. The second law (R)≥0 and the fluctuation theorem are successfully validated.
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Affiliation(s)
- Huijun Jiang
- Hefei National Laboratory for Physical Science at Microscale and Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui, People's Republic of China
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24
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Ceux T, Montagne G, Buekers MJ. The integration of temporally shifted visual feedback in a synchronization task: The role of perceptual stability in a visuo-proprioceptive conflict situation. Hum Mov Sci 2011; 29:893-909. [PMID: 20663579 DOI: 10.1016/j.humov.2010.04.008] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2008] [Revised: 04/13/2010] [Accepted: 04/25/2010] [Indexed: 10/19/2022]
Abstract
The present study examined whether the beneficial role of coherently grouped visual motion structures for performing complex (interlimb) coordination patterns can be generalized to synchronization behavior in a visuo-proprioceptive conflict situation. To achieve this goal, 17 participants had to synchronize a self-moved circle, representing the arm movement, with a visual target signal corresponding to five temporally shifted visual feedback conditions (0%, 25%, 50%, 75%, and 100% of the target cycle duration) in three synchronization modes (in-phase, anti-phase, and intermediate). The results showed that the perception of a newly generated perceptual Gestalt between the visual feedback of the arm and the target signal facilitated the synchronization performance in the preferred in-phase synchronization mode in contrast to the less stable anti-phase and intermediate mode. Our findings suggest that the complexity of the synchronization mode defines to what extent the visual and/or proprioceptive information source affects the synchronization performance in the present unimanual synchronization task.
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Affiliation(s)
- Tanja Ceux
- Department of Biomedical Kinesiology, Katholieke Universiteit Leuven, Belgium.
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25
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Liu B, Yan S, Wang Q. Intrinsic noise and Hill dynamics in the p53 system. J Theor Biol 2010; 269:104-8. [PMID: 20974153 DOI: 10.1016/j.jtbi.2010.10.009] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2010] [Revised: 07/29/2010] [Accepted: 10/07/2010] [Indexed: 10/18/2022]
Abstract
The p53 feedback loop can induce cellular senescence, cell cycle arrest and apoptosis in response to various stresses, including DNA damage, hypoxia and nutrient deprivation. Using a stochastic model of the negative feedback circuit involving p53 and its inhibitor Mdm2, we present the different oscillatory dynamics at the single-cell and population-cell levels as described in the experiments, and the resonant nature of the oscillations is captured. The stationary distributions of protein populations are characterized by non-Gaussian nature which is due to the interplay between time delay and nonlinearity of reactions.
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Affiliation(s)
- Bo Liu
- College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China
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26
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Kimizuka M, Munakata T, Rosinberg ML. Stochastic dynamics of N bistable elements with global time-delayed interactions: towards an exact solution of the master equations for finite N. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:041129. [PMID: 21230260 DOI: 10.1103/physreve.82.041129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2010] [Indexed: 05/30/2023]
Abstract
We consider a network of N noisy bistable elements with global time-delayed couplings. In a two-state description, where elements are represented by Ising spins, the collective dynamics is described by an infinite hierarchy of coupled master equations which was solved at the mean-field level in the thermodynamic limit. When the number of elements is finite, as is the case in actual laser networks, an analytical description was deemed so far intractable and numerical studies seemed to be necessary. In this paper we consider the case of two interacting elements and show that a partial analytical description of the stationary state is possible if the stochastic process is time symmetric. This requires some relationship between the transition rates to be satisfied.
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Affiliation(s)
- M Kimizuka
- Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
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27
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Kimizuka M, Munakata T. Stochastic dynamics in systems with unidirectional delay coupling: two-state description. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:021139. [PMID: 19792109 DOI: 10.1103/physreve.80.021139] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2009] [Indexed: 05/28/2023]
Abstract
We study stochastic dynamics of two-state particles coupled unidirectionally with delay. We give exact results for the stationary distribution function p(st) and the time correlation function (TCF) when the system consists of two (N=2) and three (N=3) particles. Based on these results, effects of delay are discussed and compared with the N=1 case, studied by Tsimring and Pikovsky [Phys. Rev. Lett. 87, 250602 (2001)]. Next, we consider the general N -particle system, for which we give exact expressions for p(st) and the TCF, which are inferred based on the N=2 and N=3 solutions and then confirmed via detailed arguments. It is pointed out that the stationary state is mapped to Ising spin model with ferro(antiferro)magnetic interaction when delay feedback is positive (negative).
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Affiliation(s)
- Makoto Kimizuka
- Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
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28
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Galla T. Intrinsic fluctuations in stochastic delay systems: theoretical description and application to a simple model of gene regulation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:021909. [PMID: 19792153 DOI: 10.1103/physreve.80.021909] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2009] [Indexed: 05/28/2023]
Abstract
The effects of intrinsic noise on stochastic delay systems is studied within an expansion in the inverse system size. We show that the stochastic nature of the underlying dynamics may induce oscillatory behavior in parameter ranges where the deterministic system does not sustain cycles, and compute the power spectra of these stochastic oscillations analytically, in good agreement with simulations. The theory is developed in the context of a simple one-dimensional toy model, but is applicable more generally. Gene regulatory systems in particular often contain only a small number of molecules, leading to significant fluctuations in messenger RNA (mRNA) and protein concentrations. As an application we therefore study a minimalistic model of the expression levels of hes1 mRNA and Hes1 protein, representing the simple motif of an autoinhibitory feedback loop and motivated by its relevance to somite segmentation.
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Affiliation(s)
- Tobias Galla
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom.
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29
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Munakata T, Iwama S, Kimizuka M. Linear stochastic system with delay: energy balance and entropy production. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:031104. [PMID: 19391899 DOI: 10.1103/physreve.79.031104] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2008] [Revised: 12/17/2008] [Indexed: 05/27/2023]
Abstract
We study the energy balance in a linear stochastic dynamics with delay under the impact of an external periodic force. The linearity of the model, in combination with a response function method, enables us to perform detailed analytic calculations of each term in the energy balance equation. From this, we discuss thermodynamics and entropy production rate sigma . With use of the delay time tau and strength of the external force A0 , sigma is simply expressed as sigma=sigma_{D,1}(tau)+A_{0};{2}eta(tau) , with both sigma_{D,1}(tau) and eta(tau) positive definite. We thus conclude that even when there is no external force (A_{0}=0) , the entropy production rate sigma=sigma_{D,1}(tau) is positive, meaning that the delay force produces work, which is dissipated into a reservoir. Numerical experiments are performed to confirm theoretical results.
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Affiliation(s)
- Toyonori Munakata
- Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
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30
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Siefert M. Practical criterion for delay estimation using random perturbations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:026215. [PMID: 17930129 DOI: 10.1103/physreve.76.026215] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2007] [Indexed: 05/25/2023]
Abstract
Many systems contain an internal time delay, which significantly influences their dynamical properties. Methods to estimate this delay from times series in the presence of dynamical noise are not systematically studied. Addressing this problem, we demonstrate that it is sufficient to analyze the system's response to short-correlated external disturbances or internal noise. Following this idea, it is shown for linear and nonlinear systems, as well as for periodic dynamics, that the delay can be estimated by analyzing the correlation function. This method covers the case of strong noise and multiple delays.
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Affiliation(s)
- M Siefert
- Institute of Physics, University of Potsdam, Potsdam, Germany.
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31
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Richardson MJ, Schmidt RC, Kay BA. Distinguishing the noise and attractor strength of coordinated limb movements using recurrence analysis. BIOLOGICAL CYBERNETICS 2007; 96:59-78. [PMID: 16953458 DOI: 10.1007/s00422-006-0104-6] [Citation(s) in RCA: 44] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2005] [Accepted: 07/24/2006] [Indexed: 05/11/2023]
Abstract
The variability of coupled rhythmic limb movements is assumed to be a consequence of the strength of a movement's attractor dynamic and a constant stochastic noise process that continuously perturbs the movement system away from this dynamic. Recently, it has been suggested that the nonlinear technique of recurrence analysis can be used to index the effects of noise and attractor strength on movement variability. To test this, three experiments were conducted in which the attractor strength of bimanual wrist-pendulum movements (using coordination mode, movement frequency and detuning), as well as the magnitude of stochastic perturbations affecting the variability of these movements (using a temporally fluctuating visual metronome) was manipulated. The results of these experiments demonstrate that recurrence analysis can index parametric changes in the attractor strength of coupled rhythmic limb movements and the magnitude of metronome induced stochastic perturbations independently. The results of Experiments 1 and 2 also support the claim that differences between the variability of inphase and antiphase coordination, and between slow and fast movement frequencies are due to differences in attractor strength. In contrast to the standard assumption that the noise that characterizes interlimb coordination remains constant for different magnitudes of detuning (Delta omega) the results of Experiment 3 suggest that the magnitude of noise increases with increases in |Delta omega|.
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Affiliation(s)
- Michael J Richardson
- Department of Psychology, Colby College, 5550 Mayflower Hill, Waterville, ME 04901, USA.
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32
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Frank TD, Friedrich R, Beek PJ. Stochastic order parameter equation of isometric force production revealed by drift-diffusion estimates. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:051905. [PMID: 17279937 DOI: 10.1103/physreve.74.051905] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2006] [Revised: 07/27/2006] [Indexed: 05/13/2023]
Abstract
We address two questions that are central to understanding human motor control variability: what kind of dynamical components contribute to motor control variability (i.e., deterministic and/or random ones), and how are those components structured? To this end, we derive a stochastic order parameter equation for isometric force production from experimental data using drift-diffusion estimates. We show that the force variability increases with the required force output because of a decrease of deterministic stability and an accompanying increase of noise intensity. A structural analysis reveals that the deterministic component consists of a linear control loop, while the random component involves a noise source that scales with force output. In addition, we present evidence for the existence of a subject-independent overall noise level of human isometric force production.
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Affiliation(s)
- T D Frank
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 48149 Münster, Germany
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33
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Wu D, Zhu S. Brownian motor with time-delayed feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:051107. [PMID: 16802918 DOI: 10.1103/physreve.73.051107] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2005] [Revised: 02/27/2006] [Indexed: 05/10/2023]
Abstract
An inertial Brownian motor with time-delayed feedback driven by an unbiased time-periodic force is investigated. It is found that the mean velocity and the rectification efficiency are decreased when the noise intensity is increased. While the shape of the mean velocity and the rectification efficiency can be changed from one peak to two peaks when the time delay is increased, the symmetry in the velocity probability distribution function is broken when the delay time is increased.
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Affiliation(s)
- Dan Wu
- School of Physical Science and Technology, Suzhou University, Suzhou, Jiangsu 215006, People's Republic of China
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34
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Patanarapeelert K, Frank TD, Friedrich R, Beek PJ, Tang IM. Theoretical analysis of destabilization resonances in time-delayed stochastic second-order dynamical systems and some implications for human motor control. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:021901. [PMID: 16605356 DOI: 10.1103/physreve.73.021901] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2005] [Revised: 10/12/2005] [Indexed: 05/08/2023]
Abstract
A linear stochastic delay differential equation of second order is studied that can be regarded as a Kramers model with time delay. An analytical expression for the stationary probability density is derived in terms of a Gaussian distribution. In particular, the variance as a function of the time delay is computed analytically for several parameter regimes. Strikingly, in the parameter regime close to the parameter regime in which the deterministic system exhibits Hopf bifurcations, we find that the variance as a function of the time delay exhibits a sequence of pronounced peaks. These peaks are interpreted as delay-induced destabilization resonances arising from oscillatory ghost instabilities. On the basis of the obtained theoretical findings, reinterpretations of previous human motor control studies and predictions for future human motor control studies are provided.
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Affiliation(s)
- K Patanarapeelert
- Faculty of Science, Department of Mathematics, Mahidol University, Rama VI Road, Bangkok 10400, Thailand
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35
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Frank TD. Delay Fokker-Planck equations, Novikov's theorem, and Boltzmann distributions as small delay approximations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:011112. [PMID: 16089942 DOI: 10.1103/physreve.72.011112] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2005] [Revised: 03/10/2005] [Indexed: 05/03/2023]
Abstract
We study time-delayed stochastic systems that can be described by means of so-called delay Fokker-Planck equations. Using Novikov's theorem, we first show that the theory of delay Fokker-Planck equations is on an equal footing with the theory of ordinary Fokker-Planck equations. Subsequently, we derive stationary distributions in the case of small time delays. In the case of additive noise systems, these distributions can be cast into the form of Boltzmann distributions involving effective potential functions.
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Affiliation(s)
- T D Frank
- Institute for Theoretical Physics, University of Münster, Germany
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36
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Frank TD. Delay Fokker-Planck equations, perturbation theory, and data analysis for nonlinear stochastic systems with time delays. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:031106. [PMID: 15903405 DOI: 10.1103/physreve.71.031106] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2004] [Indexed: 05/02/2023]
Abstract
We study nonlinear stochastic systems with time-delayed feedback using the concept of delay Fokker-Planck equations introduced by Guillouzic, L'Heureux, and Longtin. We derive an analytical expression for stationary distributions using first-order perturbation theory. We demonstrate how to determine drift functions and noise amplitudes of this kind of systems from experimental data. In addition, we show that the Fokker-Planck perspective for stochastic systems with time delays is consistent with the so-called extended phase-space approach to time-delayed systems.
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Affiliation(s)
- T D Frank
- Institute for Theoretical Physics, University of Münster, Germany
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37
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Budini AA, Cáceres MO. Functional characterization of linear delay Langevin equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:046104. [PMID: 15600457 DOI: 10.1103/physreve.70.046104] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2004] [Indexed: 05/24/2023]
Abstract
We present an exact functional characterization of linear delay Langevin equations driven by any noise structure defined through its characteristic functional. This method relies on the possibility of finding an explicitly analytical expression for each realization of the delayed stochastic process in terms of those of the driving noise. General properties of the transient dissipative dynamics are analyzed. The corresponding interplay with a color Gaussian noise is presented. As a full application of our functional method we study a model for population growth with non-Gaussian fluctuations: the Gompertz model driven by multiplicative white shot noise.
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Affiliation(s)
- Adrián A Budini
- Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
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38
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Hasegawa H. Augmented moment method for stochastic ensembles with delayed couplings. I. Langevin model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:021911. [PMID: 15447519 DOI: 10.1103/physreve.70.021911] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2003] [Revised: 04/02/2004] [Indexed: 05/24/2023]
Abstract
By employing a semianalytical dynamical mean-field approximation theory previously proposed by the author [H. Hasegawa, Phys. Rev. E 67, 041903 (2003)], we have developed an augmented moment method (AMM) in order to discuss dynamics of an N -unit ensemble described by Langevin equations with delays. In an AMM, original N -dimensional stochastic delay differential equations (SDDEs) are transformed to infinite-dimensional deterministic DEs for means and correlations of local as well as global variables. Infinite-order DEs arising from the non-Markovian property of SDDE, are terminated at the finite level m in the level-m AMM (AMMm), which yields (3+m)-dimensional deterministic DEs. Model calculations have been made for linear and nonlinear Langevin models. The stationary solution of AMM for the linear Langevin model with N=1 is nicely compared to the exact result. In the nonlinear Langevin ensemble, the synchronization is shown to be enhanced near the transition point between the oscillating and nonoscillating states. Results calculated by AMM6 are in good agreement with those obtained by direct simulations.
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Affiliation(s)
- Hideo Hasegawa
- Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan.
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39
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Hasegawa H. Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:021912. [PMID: 15447520 DOI: 10.1103/physreve.70.021912] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2003] [Revised: 04/02/2004] [Indexed: 05/24/2023]
Abstract
Dynamics of FitzHugh-Nagumo (FN) neuron ensembles with time-delayed couplings subject to white noises, has been studied by using both direct simulations and a semianalytical augmented moment method (AMM) which has been proposed in a preceding paper [H. Hasegawa, Phys. Rev. E 70, 021911 (2004)]. For N-unit FN neuron ensembles, AMM transforms original 2N-dimensional stochastic delay differential equations (SDDEs) to infinite-dimensional deterministic DEs for means and correlation functions of local and global variables. Infinite-order recursive DEs are terminated at the finite level m in the level-m AMM (AMMm), yielding 8(m+1)-dimensional deterministic DEs. When a single spike is applied, the oscillation may be induced if parameters of coupling strength, delay, noise intensity and/or ensemble size are appropriate. Effects of these parameters on the emergence of the oscillation and on the synchronization in FN neuron ensembles have been studied. The synchronization shows the fluctuation-induced enhancement at the transition between nonoscillating and oscillating states. Results calculated by AMM5 are in fairly good agreement with those obtained by direct simulations.
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Affiliation(s)
- Hideo Hasegawa
- Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan.
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Frank TD. Analytical results for fundamental time-delayed feedback systems subjected to multiplicative noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:061104. [PMID: 15244537 DOI: 10.1103/physreve.69.061104] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2004] [Indexed: 05/24/2023]
Abstract
We study the stochastic behavior of fundamental time-delayed feedback systems subjected to multiplicative noise. We derive exact results for the first and second moments and the autocorrelation function. For a particular class of systems we show how the variance depends on the amplitude of the multiplicative noise. Furthermore, we identify parameter regions of stationary solutions with finite and infinite variances. Finally, we suggest that delay-induced Lévy flights may occur in time-delayed feedback systems involving multiplicative noise.
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Affiliation(s)
- T D Frank
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
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Houlihan J, Goulding D, Busch T, Masoller C, Huyet G. Experimental investigation of a bistable system in the presence of noise and delay. PHYSICAL REVIEW LETTERS 2004; 92:050601. [PMID: 14995292 DOI: 10.1103/physrevlett.92.050601] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2003] [Indexed: 05/24/2023]
Abstract
We experimentally analyze the behavior of a non-Markovian bistable system with noise, using a vertical cavity surface emitting laser with time-delayed optoelectronic feedback. The effects of the delayed feedback are observed in the probability distribution of the residence times of the two orthogonal polarization states, and in the polarization-resolved power spectrum. They agree well with recent theoretical predictions based on a two-state model with transition rates depending on an earlier state of the system. We also observe experimentally and explain theoretically that the residence time probability distribution deviates from exponential decay for residence times close to (and smaller than) the delay time.
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Affiliation(s)
- J Houlihan
- Physics Department, National University of Ireland, University College Cork, Cork, Ireland
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