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Knoll G, Lindner B. Information transmission in recurrent networks: Consequences of network noise for synchronous and asynchronous signal encoding. Phys Rev E 2022; 105:044411. [PMID: 35590546 DOI: 10.1103/physreve.105.044411] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2021] [Accepted: 03/04/2022] [Indexed: 06/15/2023]
Abstract
Information about natural time-dependent stimuli encoded by the sensory periphery or communication between cortical networks may span a large frequency range or be localized to a smaller frequency band. Biological systems have been shown to multiplex such disparate broadband and narrow-band signals and then discriminate them in later populations by employing either an integration (low-pass) or coincidence detection (bandpass) encoding strategy. Analytical expressions have been developed for both encoding methods in feedforward populations of uncoupled neurons and confirm that the integration of a population's output low-pass filters the information, whereas synchronous output encodes less information overall and retains signal information in a selected frequency band. The present study extends the theory to recurrent networks and shows that recurrence may sharpen the synchronous bandpass filter. The frequency of the pass band is significantly influenced by the synaptic strengths, especially for inhibition-dominated networks. Synchronous information transfer is also increased when network models take into account heterogeneity that arises from the stochastic distribution of the synaptic weights.
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Affiliation(s)
- Gregory Knoll
- Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany and Physics Department of Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany and Physics Department of Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
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2
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Yan YA, Wang H, Shao J. A unified view of hierarchy approach and formula of differentiation. J Chem Phys 2019; 151:164110. [PMID: 31675858 DOI: 10.1063/1.5126341] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022] Open
Abstract
The stochastic differential equation is a powerful tool for describing the dynamics of a dissipative system in which noise characterizes the influence of the environment. For the Ornstein-Uhlenbeck noise, both the formula of differentiation and the hierarchy approach provide efficient numerical simulations, with the stochastic differential equation transformed into a set of coupled, linear ordinary differential equations. We show that while these two deterministic schemes result in different sets of equations, they can be regarded as two representations of an underlying linear-dynamics. Moreover, by manipulating the involved Ornstein-Uhlenbeck noise, we propose a unified algorithm that may reduce to the hierarchy approach or the formula of differentiation in different limits. We further analyze the numerical performance of this algorithm and find that the hierarchy approach appears to be more efficient for our numerical model studies.
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Affiliation(s)
- Yun-An Yan
- School of Physics and Optoelectronic Engineering, Ludong University, Shandong 264025, China
| | - Haobin Wang
- Department of Chemistry, University of Colorado Denver, Denver, Colorado 80217-3364, USA and Beijing Computational Science Research Center, No. 10 East Xibeiwang Road, Haidian District, Beijing 100193, China
| | - Jiushu Shao
- College of Chemistry and Center for Advanced Quantum Studies, Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, Beijing Normal University, Beijing 100875, China
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Roles of cellular heterogeneity, intrinsic and extrinsic noise in variability of p53 oscillation. Sci Rep 2019; 9:5883. [PMID: 30971810 PMCID: PMC6458166 DOI: 10.1038/s41598-019-41904-9] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2017] [Accepted: 03/15/2019] [Indexed: 01/09/2023] Open
Abstract
The p53 protein is a key mediator of the cellular response to various stress signals. In response to DNA damage, the concentration of p53 can temporally oscillate with fluctuations in both the amplitude and period. The underlying mechanism for p53 variability is not fully understood. Here, we construct a core regulatory network of p53 dynamics comprising the ATM-p53-Wip1 and p53-Mdm2 negative feedback loops. We dissect the contributions of cellular heterogeneity, intrinsic noise, and multiple forms of extrinsic noise to p53 variability in terms of the coefficients of variation of four quantities. Cellular heterogeneity greatly determines the fraction of oscillating cells among a population of isogenic cells. Intrinsic noise—fluctuation in biochemical reactions–has little impact on p53 variability given large amounts of molecules, whereas extrinsic colored noise with proper strength and correlation time contributes much to oscillatory variability in individual cells. With the three sources of noise combined, our results reproduce the experimental observations, suggesting that the long correlation time of colored noise is essential to p53 variability. Compared with previous studies, the current work reveals both the individual and integrated effects of distinct noise sources on p53 variability. This study provides a framework for exploring the variability in oscillations in cellular signaling pathways.
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Biele R, Timm C, D'Agosta R. Application of a time-convolutionless stochastic Schrödinger equation to energy transport and thermal relaxation. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2014; 26:395303. [PMID: 25204376 DOI: 10.1088/0953-8984/26/39/395303] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
Quantum stochastic methods based on effective wave functions form a framework for investigating the generally non-Markovian dynamics of a quantum-mechanical system coupled to a bath. They promise to be computationally superior to the master-equation approach, which is numerically expensive for large dimensions of the Hilbert space. Here, we numerically investigate the suitability of a known stochastic Schrödinger equation that is local in time to give a description of thermal relaxation and energy transport. This stochastic Schrödinger equation can be solved with a moderate numerical cost, indeed comparable to that of a Markovian system, and reproduces the dynamics of a system evolving according to a general non-Markovian master equation. After verifying that it describes thermal relaxation correctly, we apply it for the first time to the energy transport in a spin chain. We also discuss a portable algorithm for the generation of the coloured noise associated with the numerical solution of the non-Markovian dynamics.
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Affiliation(s)
- R Biele
- ETSF Scientific Development Center, Departamento de Física de Materiales, Universidad del País Vasco, E-20018 San Sebastián, Spain. Institute of Theoretical Physics, Technische Universität Dresden, D-01062 Dresden, Germany
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5
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Dummer B, Wieland S, Lindner B. Self-consistent determination of the spike-train power spectrum in a neural network with sparse connectivity. Front Comput Neurosci 2014; 8:104. [PMID: 25278869 PMCID: PMC4166962 DOI: 10.3389/fncom.2014.00104] [Citation(s) in RCA: 39] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2014] [Accepted: 08/13/2014] [Indexed: 11/13/2022] Open
Abstract
A major source of random variability in cortical networks is the quasi-random arrival of presynaptic action potentials from many other cells. In network studies as well as in the study of the response properties of single cells embedded in a network, synaptic background input is often approximated by Poissonian spike trains. However, the output statistics of the cells is in most cases far from being Poisson. This is inconsistent with the assumption of similar spike-train statistics for pre- and postsynaptic cells in a recurrent network. Here we tackle this problem for the popular class of integrate-and-fire neurons and study a self-consistent statistics of input and output spectra of neural spike trains. Instead of actually using a large network, we use an iterative scheme, in which we simulate a single neuron over several generations. In each of these generations, the neuron is stimulated with surrogate stochastic input that has a similar statistics as the output of the previous generation. For the surrogate input, we employ two distinct approximations: (i) a superposition of renewal spike trains with the same interspike interval density as observed in the previous generation and (ii) a Gaussian current with a power spectrum proportional to that observed in the previous generation. For input parameters that correspond to balanced input in the network, both the renewal and the Gaussian iteration procedure converge quickly and yield comparable results for the self-consistent spike-train power spectrum. We compare our results to large-scale simulations of a random sparsely connected network of leaky integrate-and-fire neurons (Brunel, 2000) and show that in the asynchronous regime close to a state of balanced synaptic input from the network, our iterative schemes provide an excellent approximations to the autocorrelation of spike trains in the recurrent network.
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Affiliation(s)
- Benjamin Dummer
- Theory of Complex Systems and Neurophysics, Bernstein Center for Computational Neuroscience Berlin, Germany ; Department of Physics, Humboldt Universität zu Berlin Berlin, Germany
| | - Stefan Wieland
- Theory of Complex Systems and Neurophysics, Bernstein Center for Computational Neuroscience Berlin, Germany ; Department of Physics, Humboldt Universität zu Berlin Berlin, Germany
| | - Benjamin Lindner
- Theory of Complex Systems and Neurophysics, Bernstein Center for Computational Neuroscience Berlin, Germany ; Department of Physics, Humboldt Universität zu Berlin Berlin, Germany
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6
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Kliese R, Taimre T, Bakar AAA, Lim YL, Bertling K, Nikolić M, Perchoux J, Bosch T, Rakić AD. Solving self-mixing equations for arbitrary feedback levels: a concise algorithm. APPLIED OPTICS 2014; 53:3723-3736. [PMID: 24921138 DOI: 10.1364/ao.53.003723] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/26/2013] [Accepted: 04/28/2014] [Indexed: 06/03/2023]
Abstract
Self-mixing laser sensors show promise for a wide range of sensing applications, including displacement, velocimetry, and fluid flow measurements. Several techniques have been developed to simulate self-mixing signals; however, a complete and succinct process for synthesizing self-mixing signals has so far been absent in the open literature. This article provides a systematic numerical approach for the analysis of self-mixing sensors using the steady-state solution to the Lang and Kobayashi model. Examples are given to show how this method can be used to synthesize self-mixing signals for arbitrary feedback levels and for displacement, distance, and velocity measurement. We examine these applications with a deterministic stimulus and discuss the velocity measurement of a rough surface, which necessitates the inclusion of a random stimulus.
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7
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Moix JM, Cao J. A hybrid stochastic hierarchy equations of motion approach to treat the low temperature dynamics of non-Markovian open quantum systems. J Chem Phys 2013; 139:134106. [DOI: 10.1063/1.4822043] [Citation(s) in RCA: 95] [Impact Index Per Article: 8.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023] Open
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8
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Zhong X, Zhao Y. Non-Markovian stochastic Schrödinger equation at finite temperatures for charge carrier dynamics in organic crystals. J Chem Phys 2013; 138:014111. [DOI: 10.1063/1.4773319] [Citation(s) in RCA: 49] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
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9
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Biele R, D'Agosta R. A stochastic approach to open quantum systems. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2012; 24:273201. [PMID: 22713734 DOI: 10.1088/0953-8984/24/27/273201] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.
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Affiliation(s)
- R Biele
- Institut für Theoretische Physik, Technische Universität Dresden, D-01062 Dresden, Germany.
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10
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Zhang W, Zhong X, Zhao Y. Electron mobilities of n-type organic semiconductors from time-dependent wavepacket diffusion method: pentacenequinone derivatives. J Phys Chem A 2012; 116:11075-82. [PMID: 22483224 DOI: 10.1021/jp3023474] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
Abstract
The electron mobilities of two n-type pentacenequinone derivative organic semiconductors, 5,7,12,14-tetraaza-6,13-pentacenequinone (TAPQ5) and 1,4,8,11-tetraaza-6,13-pentacenequinone (TAPQ7), are investigated with use of the methods of electronic structure and quantum dynamics. The electronic structure calculations reveal that the two key parameters for the control of electron transfer, reorganization energy and electronic coupling, are similar for these two isomerization systems, and the charge carriers essentially display one-dimensional transport properties. The mobilities are then calculated by using the time-dependent wavepacket diffusion approach in which the dynamic fluctuations of the electronic couplings are incorporated via their correlation functions obtained from molecular dynamics simulations. The predicted mobility of TAPQ7 crystal is about six times larger than that of TAPQ5 crystal. Most interestingly, Fermi's golden rule predicts the mobilities very close to those from the time-dependent wavepacket diffusion method, even though the electronic couplings are explicitly large enough to make the perturbation theory invalid. The possible reason is analyzed from the dynamic fluctuations.
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Affiliation(s)
- WeiWei Zhang
- State Key Laboratory for Physical Chemistry of Solid Surfaces, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, 361005, People's Republic of China
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11
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Zhong X, Zhao Y. Charge carrier dynamics in phonon-induced fluctuation systems from time-dependent wavepacket diffusion approach. J Chem Phys 2011; 135:134110. [DOI: 10.1063/1.3644965] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022] Open
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12
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Si YB, Zhong XX, Zhang WW, Zhao Y. Theoretical Investigation on Triplet Excitation Energy Transfer in Fluorene Dimer. CHINESE J CHEM PHYS 2011. [DOI: 10.1088/1674-0068/24/05/538-546] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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13
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Leahy C, Dainty C. A non-stationary model for simulating the dynamics of ocular aberrations. OPTICS EXPRESS 2010; 18:21386-96. [PMID: 20941035 DOI: 10.1364/oe.18.021386] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
Abstract
The time-evolution of ocular aberrations has been the subject of many studies, but so far there has been little discussion involving the modelling of the underlying temporal statistics. This paper presents a non-stationary modelling approach based on a coloured-noise generator, which can be applied to ocular aberration dynamics. The model parameters are computed from measured ocular aberration data. A custom-built aberrometer based on a Shack-Hartmann sensor was used for measurement. We present simulations based on our modelling approach, and validate them through comparison to real data. This work could be useful in areas such as the testing of ophthalmic devices and the development of improved algorithms for laser refractive surgery.
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Affiliation(s)
- C Leahy
- Applied Optics Group, School of Physics, National University of Ireland, Galway.
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14
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Tian L. Correcting low-frequency noise with continuous measurement. PHYSICAL REVIEW LETTERS 2007; 98:153602. [PMID: 17501347 DOI: 10.1103/physrevlett.98.153602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2006] [Indexed: 05/15/2023]
Abstract
Low-frequency noise presents a serious source of decoherence in solid-state qubits. When combined with a continuous weak measurement of the eigenstates, low-frequency noise induces a second-order relaxation between the qubit states. Here, we show that the relaxation provides a unique approach to calibrate the low-frequency noise in the time domain. By encoding one qubit with two physical qubits that are alternatively calibrated, quantum-logic gates with high fidelity can be performed.
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Affiliation(s)
- L Tian
- National Institute of Standards and Technology, 100 Bureau Drive, Stop 8423, Gaithersburg, Maryland 20878, USA
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15
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Abstract
The detection of seizure in the newborn is a critical aspect of neurological research. Current automatic detection techniques are difficult to assess due to the problems associated with acquiring and labelling newborn electroencephalogram (EEG) data. A realistic model for newborn EEG would allow confident development, assessment and comparison of these detection techniques. This paper presents a model for newborn EEG that accounts for its self-similar and nonstationary nature. The model consists of background and seizure submodels. The newborn EEG background model is based on the short-time power spectrum with a time-varying power law. The relationship between the fractal dimension and the power law of a power spectrum is utilized for accurate estimation of the short-time power law exponent. The newborn EEG seizure model is based on a well-known time-frequency signal model. This model addresses all significant time-frequency characteristics of newborn EEG seizure which include; multiple components or harmonics, piecewise linear instantaneous frequency laws and harmonic amplitude modulation. Estimates of the parameters of both models are shown to be random and are modelled using the data from a total of 500 background epochs and 204 seizure epochs. The newborn EEG background and seizure models are validated against real newborn EEG data using the correlation coefficient. The results show that the output of the proposed models have a higher correlation with real newborn EEG than currently accepted models (a 10% and 38% improvement for background and seizure models, respectively).
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Affiliation(s)
- Luke Rankine
- Perinatal Research Centre, University of Queensland, Royal Brisbane and Women's Hospital, Brisbane QLD 4029, Australia.
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16
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Lindner B, Chacron MJ, Longtin A. Integrate-and-fire neurons with threshold noise: a tractable model of how interspike interval correlations affect neuronal signal transmission. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:021911. [PMID: 16196608 PMCID: PMC5283900 DOI: 10.1103/physreve.72.021911] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2005] [Indexed: 05/04/2023]
Abstract
Many neurons exhibit interval correlations in the absence of input signals. We study the influence of these intrinsic interval correlations of model neurons on their signal transmission properties. For this purpose, we employ two simple firing models, one of which generates a renewal process, while the other leads to a nonrenewal process with negative interval correlations. Different methods to solve for spectral statistics in the presence of a weak stimulus (spike train power spectra, cross spectra, and coherence functions) are presented, and their range of validity is discussed. Using these analytical results, we explore a lower bound on the mutual information rate between output spike train and input stimulus as a function of the system's parameters. We demonstrate that negative correlations in the baseline activity can lead to enhanced information transfer of a weak signal by means of noise shaping of the background noise spectrum. We also show that an enhancement is not compulsory--for a stimulus with power exclusively at high frequencies, the renewal model can transfer more information than the nonrenewal model does. We discuss the application of our analytical results to other problems in neuroscience. Our results are also relevant to the general problem of how a signal affects the power spectrum of a nonlinear stochastic system.
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Affiliation(s)
- Benjamin Lindner
- Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Canada K1N-6N5
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17
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Kirchner JW. Aliasing in 1/f(alpha) noise spectra: origins, consequences, and remedies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:066110. [PMID: 16089823 DOI: 10.1103/physreve.71.066110] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2003] [Revised: 11/22/2004] [Indexed: 05/03/2023]
Abstract
The scaling exponent of a 1/f(alpha) noise time series is commonly estimated from the power-law slope of its Fourier power spectrum. Here I show that because 1/f(alpha) noises typically have significant power above the Nyquist frequency, measurements of their power spectra will often be severely distorted by aliasing, not only near the Nyquist frequency, but also far below it. I show that spectral aliasing typically leads to large systematic biases in the scaling exponents, and thus the fractal dimensions, that are estimated from the power-law slopes of 1/f(alpha) noise spectra. I describe a simple spectral filtering method that corrects the distortions introduced by spectral aliasing, and recovers the broadband spectrum of 1/f(alpha) noises. Like a Wiener filter, this filtering method does not require that the correct spectrum is known in advance. I illustrate this filtering technique using two environmental noise spectra that are distorted by aliasing.
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Affiliation(s)
- James W Kirchner
- Department of Earth and Planetary Science, University of California, Berkeley, California 94720-4767, USA.
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18
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Wu MM, Billah KY, Shinozuka M. Systematic adiabatic analysis of a nonlinear oscillator with inertia driven by colored noise. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:3377-3380. [PMID: 9963813 DOI: 10.1103/physreve.52.3377] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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19
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Wu M, Billah KY, Shinozuka M. Numerical solution of the Kardar-Parisi-Zhang equation with a long-range spatially correlated noise. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:995-998. [PMID: 9962738 DOI: 10.1103/physreve.51.995] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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20
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Yordanov OI, Nickolaev NI. Self-affinity of time series with finite domain power-law power spectrum. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:R2517-R2520. [PMID: 9961639 DOI: 10.1103/physreve.49.r2517] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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21
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Bittner ER, Light JC. Quantum stochastic approach for molecule/surface scattering. I. Atom–phonon interactions. J Chem Phys 1993. [DOI: 10.1063/1.465648] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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22
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Bulsara AR, Schieve WC, Jacobs EW. Reply to "Melnikov function and homoclinic chaos induced by weak perturbations". PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:3193-3195. [PMID: 9960962 DOI: 10.1103/physreve.48.3193] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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23
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Billah KY, Shinozuka M. Reply to the "Comment on 'Numerical method for colored-noise generation and its application to a bistable system' ". PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:8031-8033. [PMID: 9908160 DOI: 10.1103/physreva.46.8031] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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24
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Mannella R, Palleschi V. Comment on "Numerical method for colored-noise generation and its application to a bistable system". PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:8028-8030. [PMID: 9908159 DOI: 10.1103/physreva.46.8028] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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25
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Billah KY, Shinozuka M. Stabilization of a nonlinear system by multiplicative noise. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:R4779-R4781. [PMID: 9906589 DOI: 10.1103/physreva.44.r4779] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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