On the equivalence and properties of noisy neural and probabilistic RAM nets

A formal identity is demonstrated between the equations describing the dynamical evolution of a net of noisy RAMs and those for a net of noisy neurons, under certain assumptions on the latter. The dynamical evolution of such noisy nets is investigated both analytically and by computer, and found, in the generic case, to evolve to a unique stable fixed point for the RAM/neuron output probabilities.     

基于稀疏贝叶斯学习的复杂网络拓扑估计

提出了一种噪声环境下复杂网络拓扑估计方法, 仅利用含噪时间序列估计未知结构混沌系统的动力学方程和参数, 以及由混沌系统组成的复杂网络的拓扑结构、节点动力学方程、所有参数、 节点间耦合方向和耦合强度.通过采用动力学方程的统一形式, 将动力系统方程结构和参数估计看成线性回归问题的系数估计, 该估计问题利用贝叶斯压缩传感的信号重建算法求解, 含噪信号的模型重建使用相关向量机方法,即通过稀疏贝叶斯学习求解稀疏欠定线性方程得到上面提到的可估计对象.以单个Lorenz系统及由200个 Lorenz系统组成的无标度网络为例说明方法的有效性. 仿真结果表明,提出的方法对噪声有很强的鲁棒性,收敛速度快,稳态误差极小, 克服了最小二乘估计方法收敛速度慢、 稳态误差大以及压缩传感估计方法对噪声鲁棒性不强的缺点.     

Topology-based signal separation

Traditional noise-filtering techniques are known to significantly alter features of chaotic data. In this paper, we present a noncausal topology-based filtering method for continuous-time dynamical systems that is effective in removing additive, uncorrelated noise from time-series data. Signal-to-noise ratios and Lyapunov exponent estimates are dramatically improved following the removal of the identified noisy points.     

Strong and weak constraint variational assimilations for reduced order fluid flow modeling

In this work we propose and evaluate two variational data assimilation techniques for the estimation of low order surrogate experimental dynamical models for fluid flows. Both methods are built from optimal control recipes and rely on proper orthogonal decomposition and a Galerkin projection of the Navier Stokes equation. The techniques proposed differ in the control variables they involve. The first one introduces a weak dynamical model defined only up to an additional uncertainty time-dependent function whereas the second one, handles a strong dynamical constraint in which the dynamical system’s coefficients constitute the control variables. Both choices correspond to different approximations of the relation between the reduced basis on which is expressed the motion field and the basis components that have been neglected in the reduced order model construction. The techniques have been assessed on numerical data and for real experimental conditions with noisy particle image velocimetry data.     

A heuristic method for identifying chaos from frequency content

The sign of the largest Lyapunov exponent is the fundamental indicator of chaos in a dynamical system. However, although the extraction of Lyapunov exponents can be accomplished with (necessarily noisy) the experimental data, this is still a relatively data-intensive and sensitive endeavor. This paper presents an alternative pragmatic approach to identifying chaos using response frequency characteristics and extending the concept of the spectrogram. The method is shown to work well on both experimental and simulated time series.     

A phase-synchronization and random-matrix based approach to multichannel time-series analysis with application to epilepsy

We present a general method to analyze multichannel time series that are becoming increasingly common in many areas of science and engineering. Of particular interest is the degree of synchrony among various channels, motivated by the recognition that characterization of synchrony in a system consisting of many interacting components can provide insights into its fundamental dynamics. Often such a system is complex, high-dimensional, nonlinear, nonstationary, and noisy, rendering unlikely complete synchronization in which the dynamical variables from individual components approach each other asymptotically. Nonetheless, a weaker type of synchrony that lasts for a finite amount of time, namely, phase synchronization, can be expected. Our idea is to calculate the average phase-synchronization times from all available pairs of channels and then to construct a matrix. Due to nonlinearity and stochasticity, the matrix is effectively random. Moreover, since the diagonal elements of the matrix can be arbitrarily large, the matrix can be singular. To overcome this difficulty, we develop a random-matrix based criterion for proper choosing of the diagonal matrix elements. Monitoring of the eigenvalues and the determinant provides a powerful way to assess changes in synchrony. The method is tested using a prototype nonstationary noisy dynamical system, electroencephalogram (scalp) data from absence seizures for which enhanced cortico-thalamic synchrony is presumed, and electrocorticogram (intracranial) data from subjects having partial seizures with secondary generalization for which enhanced local synchrony is similarly presumed.     

Counting unstable periodic orbits in noisy chaotic systems: A scaling relation connecting experiment with theory

The experimental detection of unstable periodic orbits in dynamical systems, especially those which yield short, noisy or nonstationary data sets, is a current topic of interest in many research areas. Unfortunately, for such data sets, only a few of the lowest order periods can be detected with quantifiable statistical accuracy. The primary observable is the number of encounters the general trajectory has with a particular orbit. Here we show that, in the limit of large period, this quantity scales exponentially with the period, and that this scaling is robust to dynamical noise. (c) 1998 American Institute of Physics.     

Characterization of synchrony with applications to epileptic brain signals

Measurement of synchrony in networks of complex or high-dimensional, nonstationary, and noisy systems such as the mammalian brain is technically difficult. We present a general method to analyze synchrony from multichannel time series. The idea is to calculate the phase-synchronization times and to construct a matrix. We develop a random-matrix-based criterion for proper choosing of the diagonal matrix elements. Monitoring of the eigenvalues and the determinant provides an effective way to assess changes in synchrony. The method is tested using a prototype nonstationary dynamical system, electroencephalogram (scalp) data from absence seizures for which enhanced synchrony is presumed, and electrocorticogram (intracranial) data from subjects having partial seizures with secondary generalization.     

Signal-to-noise ratio of a dynamical saturating system: Switching from stochastic resonator to signal processor

We consider an isolated dynamical saturating system for processing a noisy sinusoidal signal, and evaluate its performance with the measure of the signal-to-noise ratio. The considered system is linear for small inputs, but exhibits saturation in its response for large inputs. This nonlinearity displays the nonlinear phenomenon of stochastic resonance for a large biased sinusoid in appropriate system parameter regions. Without the stochastic resonance phenomenon, this dynamical saturating system can achieve a signal-to-noise ratio gain exceeding unity for a noisy unbiased sinusoid. These numerical results manifest the nonlinearities and the signal-processing ability of this system acting as a stochastic resonator or a signal processor.     

Tracking noisy limit cycle oscillation with nonlinear filters

For engineering systems, the dynamic state estimates provide valuable information for the detection and prediction of failure due to noise and vibration. From this perspective, nonlinear filtering techniques are applied to the problem of state estimation of dynamical systems undergoing noisy limit cycle oscillation. Specifically, the extended Kalman filter, ensemble Kalman filter and particle filter are used to track the limit cycle oscillations of a Duffing oscillator using noisy observational data. The noisy limit cycle oscillations feature highly non-Gaussian trends. The efficiency and limitations of the extended Kalman filter, ensemble Kalman filter and particle filter in tracking limit cycle oscillations are examined with respect to the model and measurement noise and sparsity of measurement data. For the limit cycle oscillations considered here, it is demonstrated that the ensemble Kalman filter and particle filter outperform the extended Kalman filter in the presence of sparse observational data or strong measurement noise. For moderate measurement noise and frequent measurement data, the ensemble Kalman filter and particle filter perform equally well in comparison to the extended Kalman filter.     

Some Problems of Information Neurodynamics

The goal of neural science is to understand the brain, how we perceive, move, think, and remember. All of these things are dynamical processes which are taking place in a complex, non-stationary and noisy environment. This means that these dynamical processes at all levels from small neural networks to behavior should be stable against perturbations but flexible and adaptive. The goal of neurodynamics is to formulate the main dynamical principles which can be a basis of such behavior and to predict the possible activities of neurons and neural ensembles using the tools of nonlinear dynamics. In this paper we discuss our last results related to the mostly challenging part of neurodynamics: information processing by dynamical neural ensembles.     

Pseudo-periodic surrogate test to sample time series in stochastic softening Duffing oscillator

Identification of typical noise-contaminated sample response is a hard task in a nonlinear system under stochastic background since irregularity of the sample response may come from measure noise, dynamical noise, or nonlinear effect, etc., and conventional dynamical methods are generally not useful. Here, the pseudo-periodic surrogate algorithm by Small is employed to test the sample time series in the softening Duffing oscillator under the Gaussian white noise excitation. The correlation dimensions of the noisy periodic and the noise-induced chaotic time series of the system are compared with those of their corresponding surrogate data respectively, the leading Lyapunov exponents by Rosenstein's algorithm are also presented for comparison.     

Optimal instruments and models for noisy chaos

Analysis of finite, noisy time series data leads to modern statistical inference methods. Here we adapt Bayesian inference for applied symbolic dynamics. We show that reconciling Kolmogorov's maximum-entropy partition with the methods of Bayesian model selection requires the use of two separate optimizations. First, instrument design produces a maximum-entropy symbolic representation of time series data. Second, Bayesian model comparison with a uniform prior selects a minimum-entropy model, with respect to the considered Markov chain orders, of the symbolic data. We illustrate these steps using a binary partition of time series data from the logistic and Henon maps as well as the R?ssler and Lorenz attractors with dynamical noise. In each case we demonstrate the inference of effectively generating partitions and kth-order Markov chain models.     

Unbiased reconstruction of the dynamics underlying a noisy chaotic time series

Refined methods for the construction of a deterministic dynamical system which can consistently reproduce observed aperiodic data are discussed. The determination of the dynamics underlying a noisy chaotic time series suffers strongly from two systematic errors: One is a consequence of the so-called "error-in-variables problem." Standard least-squares fits implicitly assume that the independent variables are noise free and that the dependent variable is noisy. We show that due to the violation of this assumption one receives considerably wrong results for moderate noise levels. A straightforward modification of the cost function solves this problem. The second problem consists in a mutual inconsistency between the images of a point under the model dynamics and the corresponding observed values. For an improved fit we therefore introduce a multistep prediction error which exploits the information stored in the time series in a better way. The performance is demonstrated by several examples, including experimental data. (c) 1996 American Institute of Physics.     

Efficient noncausal noise reduction for deterministic time series

We present a simple noncausal noise reduction algorithm for time series that consist of noisy measurements of the state vectors of a deterministic (chaotic) nonlinear system. The underlying dynamical system is assumed to be known and to operate in discrete time. The noise reduction algorithm is an iterative scheme for finding exact deterministic orbits close to the measured noisy orbits. Furthermore, we discuss cases where the solution is not the original orbit but homoclinic to it. (c) 2001 American Institute of Physics.     

Control of spatiotemporal chaos: A study with an autocatalytic reaction-diffusion system

The characterization of chaotic spatiotemporal dynamics has been studied for a representative nonlinear autocatalytic reaction mechanism coupled with diffusion. This has been carried out by an analysis of the Lyapunov spectrum in spatiallylocalised regions. The linear scaling relationships observed in the invariant measures as a function of thesub-system size have been utilized to assess the controllability, stability and synchronization properties of the chaotic dynamics. The dynamical synchronization properties of this high-dimensional system has been analyzed using suitable Lyapunov functionals. The possibility of controlling spatiotemporal chaos for relevant objectives using available noisy scalar time-series data with simultaneous self-adaptation of the control parameter(s) has also been discussed.     

Error Covariance Matrix Estimation of Noisy and Dynamically Coupled Time Series

We estimate the covariance matrix of the errors in several dynamically coupled time series corrupted by measurement errors. We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. The estimation of the covariance matrix of the errors is made using a noise reduction algorithm that efficiently exploits the information contained jointly in the dynamically coupled noisy time series. The method is particularly powerful for short length time series with high uncertainties.     

Comparison of stochastic resonance in static and dynamical nonlinearities

We compare the stochastic resonance (SR) effects in parallel arrays of static and dynamical nonlinearities via the measure of output signal-to-noise ratio (SNR). For a received noisy periodic signal, parallel arrays of both static and dynamical nonlinearities can enhance the output SNR by optimizing the internal noise level. The static nonlinearity is easily implementable, while the dynamical nonlinearity has more parameters to be tuned, at the risk of not exploiting the beneficial role of internal noise components. It is of interest to note that, for an input signal buried in the external Laplacian noise, we show that the dynamical nonlinearity is superior to the static nonlinearity in obtaining a better output SNR. This characteristic is assumed to be closely associated with the kurtosis of noise distribution.     

Mean field Ising model with fluctuating interaction parameter. Organization in the presence of noise

A dynamical model describing a system of spins near thermodynamic equilibrium is considered. Relaxation properties of the system are briefly discussed by assuming a noisy character of spin interactions. The possibility of spontaneous organization of the system is presented for a particular range of correlation time of the noise.     

Transition to chaos in continuous-time random dynamical systems

We consider situations where, in a continuous-time dynamical system, a nonchaotic attractor coexists with a nonattracting chaotic saddle, as in a periodic window. Under the influence of noise, chaos can arise. We investigate the fundamental dynamical mechanism responsible for the transition and obtain a general scaling law for the largest Lyapunov exponent. A striking finding is that the topology of the flow is fundamentally disturbed after the onset of noisy chaos, and we point out that such a disturbance is due to changes in the number of unstable eigendirections along a continuous trajectory under the influence of noise.