Reconstructing embedding spaces of coupled dynamical systems from multivariate data

Learning Coupled Oscillators System with Reservoir Computing

In this paper, we reconstruct the dynamic behavior of the ring-coupled Lorenz oscillators system by reservoir computing. Although the reconstruction of various complex chaotic attractors has been well studied by using various neural networks, little attention has been paid to whether the spatio-temporal structure of some special attractors can be maintained in long-term prediction. Reservoir computing has been shown to be effective for model-free prediction, so we want to investigate whether reservoir computing can restore the rotational symmetry of the original ring-coupled Lorenz system. We find that although the state prediction of the trained reservoir computer will gradually deviate from the actual trajectory of the original system, the associated spatio-temporal structure is maintained in the process of reconstruction. Specifically, we show that the rotational symmetric structure of periodic rotating waves, quasi-periodic torus, and chaotic rotating waves is well maintained.

Predictive learning of multi-channel isochronal chaotic synchronization by utilizing parallel optical reservoir computers based on three laterally coupled semiconductor lasers with delay-time feedback

In this work, we utilize three parallel optical reservoir computers to model three optical dynamic systems, respectively. Here, the three laser-elements in the response laser array with both delay-time feedback and optical injection are utilized as nonlinear nodes to realize three optical chaotic reservoir computers (RCs). The nonlinear dynamics of three laser-elements in the driving laser array are predictively learned by these three parallel RCs. We show that these three parallel reservoir computers can reproduce the nonlinear dynamics of the three laser-elements in the driving laser array with self-feedback. Very small training errors for their predictions can be realized by the optimization of two key parameters such as the delay-time and the interval of the virtual nodes. Moreover, these three parallel RCs to be trained will well synchronize with three chaotic laser-elements in the driving laser array, respectively, even when there are some parameter mismatches between the response laser array and the driving laser array. Our findings show that optical reservoir computing approach possibly provide a successful path for the realization of the high-quality chaotic synchronization between the driving laser and the response laser when their rate-equations imperfectly match each other.

The extended Granger causality analysis for Hodgkin-Huxley neuronal models

How to extract directions of information flow in dynamical systems based on empirical data remains a key challenge. The Granger causality (GC) analysis has been identified as a powerful method to achieve this capability. However, the framework of the GC theory requires that the dynamics of the investigated system can be statistically linearized; i.e., the dynamics can be effectively modeled by linear regressive processes. Under such conditions, the causal connectivity can be directly mapped to the structural connectivity that mediates physical interactions within the system. However, for nonlinear dynamical systems such as the Hodgkin-Huxley (HH) neuronal circuit, the validity of the GC analysis has yet been addressed; namely, whether the constructed causal connectivity is still identical to the synaptic connectivity between neurons remains unknown. In this work, we apply the nonlinear extension of the GC analysis, i.e., the extended GC analysis, to the voltage time series obtained by evolving the HH neuronal network. In addition, we add a certain amount of measurement or observational noise to the time series to take into account the realistic situation in data acquisition in the experiment. Our numerical results indicate that the causal connectivity obtained through the extended GC analysis is consistent with the underlying synaptic connectivity of the system. This consistency is also insensitive to dynamical regimes, e.g., a chaotic or non-chaotic regime. Since the extended GC analysis could in principle be applied to any nonlinear dynamical system as long as its attractor is low dimensional, our results may potentially be extended to the GC analysis in other settings.

Neurophysiological Hallmarks of Neurodegenerative Cognitive Decline: The Study of Brain Connectivity as A Biomarker of Early Dementia

Neurodegenerative processes of various types of dementia start years before symptoms, but the presence of a "neural reserve", which continuously feeds and supports neuroplastic mechanisms, helps the aging brain to preserve most of its functions within the "normality" frame. Mild cognitive impairment (MCI) is an intermediate stage between dementia and normal brain aging. About 50% of MCI subjects are already in a stage that is prodromal-to-dementia and during the following 3 to 5 years will develop clinically evident symptoms, while the other 50% remains at MCI or returns to normal. If the risk factors favoring degenerative mechanisms are modified during early stages (i.e., in the prodromal), the degenerative process and the loss of abilities in daily living activities will be delayed. It is therefore extremely important to have biomarkers able to identify-in association with neuropsychological tests-prodromal-to-dementia MCI subjects as early as possible. MCI is a large (i.e., several million in EU) and substantially healthy population; therefore, biomarkers should be financially affordable, largely available and non-invasive, but still accurate in their diagnostic prediction. Neurodegeneration initially affects synaptic transmission and brain connectivity; methods exploring them would represent a 1st line screening. Neurophysiological techniques able to evaluate mechanisms of synaptic function and brain connectivity are attracting general interest and are described here. Results are quite encouraging and suggest that by the application of artificial intelligence (i.e., learning-machine), neurophysiological techniques represent valid biomarkers for screening campaigns of the MCI population.

Synchronization of chaotic systems and their machine-learning models

Recent advances have demonstrated the effectiveness of a machine-learning approach known as "reservoir computing" for model-free prediction of chaotic systems. We find that a well-trained reservoir computer can synchronize with its learned chaotic systems by linking them with a common signal. A necessary condition for achieving this synchronization is the negative values of the sub-Lyapunov exponents. Remarkably, we show that by sending just a scalar signal, one can achieve synchronism in trained reservoir computers and a cascading synchronization among chaotic systems and their fitted reservoir computers. Moreover, we demonstrate that this synchronization is maintained even in the presence of a parameter mismatch. Our findings possibly provide a path for accurate production of all expected signals in unknown chaotic systems using just one observational measure.

Parsimonious description for predicting high-dimensional dynamics

When we observe a system, we often cannot observe all its variables and may have some of its limited measurements. Under such a circumstance, delay coordinates, vectors made of successive measurements, are useful to reconstruct the states of the whole system. Although the method of delay coordinates is theoretically supported for high-dimensional dynamical systems, practically there is a limitation because the calculation for higher-dimensional delay coordinates becomes more expensive. Here, we propose a parsimonious description of virtually infinite-dimensional delay coordinates by evaluating their distances with exponentially decaying weights. This description enables us to predict the future values of the measurements faster because we can reuse the calculated distances, and more accurately because the description naturally reduces the bias of the classical delay coordinates toward the stable directions. We demonstrate the proposed method with toy models of the atmosphere and real datasets related to renewable energy.

The Multivariate Largest Lyapunov Exponent as an Age-Related Metric of Quiet Standing Balance

The largest Lyapunov exponent has been researched as a metric of the balance ability during human quiet standing. However, the sensitivity and accuracy of this measurement method are not good enough for clinical use. The present research proposes a metric of the human body's standing balance ability based on the multivariate largest Lyapunov exponent which can quantify the human standing balance. The dynamic multivariate time series of ankle, knee, and hip were measured by multiple electrical goniometers. Thirty-six normal people of different ages participated in the test. With acquired data, the multivariate largest Lyapunov exponent was calculated. Finally, the results of the proposed approach were analysed and compared with the traditional method, for which the largest Lyapunov exponent and power spectral density from the centre of pressure were also calculated. The following conclusions can be obtained. The multivariate largest Lyapunov exponent has a higher degree of differentiation in differentiating balance in eyes-closed conditions. The MLLE value reflects the overall coordination between multisegment movements. Individuals of different ages can be distinguished by their MLLE values. The standing stability of human is reduced with the increment of age.

A Multistep Chaotic Model for Municipal Solid Waste Generation Prediction

In this study, a univariate local chaotic model is proposed to make one-step and multistep forecasts for daily municipal solid waste (MSW) generation in Seattle, Washington. For MSW generation prediction with long history data, this forecasting model was created based on a nonlinear dynamic method called phase-space reconstruction. Compared with other nonlinear predictive models, such as artificial neural network (ANN) and partial least square-support vector machine (PLS-SVM), and a commonly used linear seasonal autoregressive integrated moving average (sARIMA) model, this method has demonstrated better prediction accuracy from 1-step ahead prediction to 14-step ahead prediction assessed by both mean absolute percentage error (MAPE) and root mean square error (RMSE). Max error, MAPE, and RMSE show that chaotic models were more reliable than the other three models. As chaotic models do not involve random walk, their performance does not vary while ANN and PLS-SVM make different forecasts in each trial. Moreover, this chaotic model was less time consuming than ANN and PLS-SVM models.

Dynamical analysis in real time: detecting perturbations to team communication

Dynamical systems methods characterise patterns of change over time. Typically, such methods are applied only after data collection is complete. However, brief disturbances - perturbations - can occur as a process unfolds and can result in undesirable outcomes if not acted on. The application of dynamics in real time would be useful for detecting these sudden changes. Real-time analysis was accomplished by updating dynamical estimates simultaneously across different window sizes. We calculated the largest Lyapunov exponent, a measure of dynamical stability, to detect a perturbation to team communication in a simulated uninhabited air vehicle (UAV) reconnaissance mission. The perturbation consisted of information demands from a confederate that occurred unexpectedly during performance of a UAV mission. We demonstrate the use of real-time methods in detecting that perturbation as it occurred. In application, this technique would have enabled real-time intervention. Extensions of the real-time dynamical method to other domains of psychological inquiry are discussed.

PRACTITIONER SUMMARY

A real-time dynamical analysis method that was developed to detect unexpected perturbations in team communication is described. The use of the method is demonstrated on perturbed communication from a three-person uninhabited air vehicle command-and-control team. The generalisability of the method is considered with respect to physiological and motor coordination dynamics.

Nonuniform state-space reconstruction and coupling detection

We investigate the state space reconstruction from multiple time series derived from continuous and discrete systems and propose a method for building embedding vectors progressively using information measure criteria regarding past, current, and future states. The embedding scheme can be adapted for different purposes, such as mixed modeling, cross-prediction and Granger causality. In particular, we apply this method in order to detect and evaluate information transfer in coupled systems. As a practical application, we investigate in records of scalp epileptic EEG the information flow across brain areas.

Modeling dynamics from only output data

We examine the problem of reconstructing input-output systems from time series data. Although the method of delays has already been used in the case where both input and output are measured, in some cases, the inputs cannot be measured, and hence, the method of delays cannot be used. On the basis of ideas derived from existing embedding theorems, we propose to build models by using delays of multivariate observations of output data. Assuming that the inputs are few, we use several observations for obtaining information about the inputs, and the remaining observations for obtaining information about the state of the system. Numerical examples on a discrete map and a continuous-time system show that input-output systems can indeed be identified by using multivariate observations of output data only. We also discuss the application of this method to the analysis of coupled systems or complex networks, by partitioning such large systems and analyzing each subsystem separately. The models used in this paper are nonpredictive models; thus, they cannot be used to predict the future behavior of the system. However, since they model the dynamics of the system, they have other possible applications such as change detection and noise reduction.

Neighborhood detection for the identification of spatiotemporal systems

Neighborhood detection and local state vector construction for the identification of spatiotemporal systems is considered in this paper. Determining the neighborhood size both in the space and time domain can considerably reduce the complexity of the set of candidate model terms for the identification of coupled map lattice models. The computation requirements of the model identification algorithm can also be greatly reduced instead of the more direct identification approach of searching over the entire spatiotemporal neighborhood in the original space. In this paper, a new neighborhood detection method is introduced based on embedding theory for nonlinear dynamical systems to produce an initial spatiotemporal neighborhood for the identification of spatiotemporal systems. Numerical examples are provided to demonstrate the feasibility and applicability of the new neighborhood detection method.

Model-based detector and extraction of weak signal frequencies from chaotic data

Detecting a weak signal from chaotic time series is of general interest in science and engineering. In this work we introduce and investigate a signal detection algorithm for which chaos theory, nonlinear dynamical reconstruction techniques, neural networks, and time-frequency analysis are put together in a synergistic manner. By applying the scheme to numerical simulation and different experimental measurement data sets (Henon map, chaotic circuit, and NH(3) laser data sets), we demonstrate that weak signals hidden beneath the noise floor can be detected by using a model-based detector. Particularly, the signal frequencies can be extracted accurately in the time-frequency space. By comparing the model-based method with the standard denoising wavelet technique as well as supervised principal components analysis detector, we further show that the nonlinear dynamics and neural network-based approach performs better in extracting frequencies of weak signals hidden in chaotic time series.

Granger causality and cross recurrence plots in rheochaos

Our stress relaxation measurements on wormlike micelles using a Rheo-SALS (rheology + small angle light scattering) apparatus allow simultaneous measurements of the stress and the scattered depolarized intensity. The latter is sensitive to orientational ordering of the micelles. To determine the presence of causal influences between the stress and the depolarized intensity time series, we have used the technique of linear and nonlinear Granger causality. We find there exists a feedback mechanism between the two time series and that the orientational order has a stronger causal effect on the stress than vice versa. We have also studied the phase space dynamics of the stress and the depolarized intensity time series using the recently developed technique of cross recurrence plots (CRPs). The presence of diagonal line structures in the CRPs unambiguously proves that the two time series share similar phase space dynamics.

Reconstructing state spaces from multivariate data using variable delays

We study two methods for constructing a nonuniform embedding for multivariate data. A nonuniform embedding is a state space reconstruction which is more flexible than the common delay coordinates with fixed delays since it contains variable delays. Using these methods, we can extract causal relationships among many variables in a more suitable way. We demonstrate that the proposed methods can give more precise predictions and simpler models than some previous methods.

Using chaotic forcing to detect damage in a structure

In this work we develop a numerical test for Holder continuity and apply it and another test for continuity to the difficult problem of detecting damage in structures. We subject a thin metal plate with incremental damage to the plate changes, its filtering properties, and therefore the phase space trajectories of the response chaotic excitation of various bandwidths. Damage to the plate changes its filtering properties and therefore the phase space of the response. Because the data are multivariate (the plate is instrumented with multiple sensors) we use a singular value decomposition of the set of the output time series to reduce the embedding dimension of the response time series. We use two geometric tests to compare an attractor reconstructed from data from an undamaged structure to that reconstructed from data from a damaged structure. These two tests translate to testing for both generalized and differentiable synchronization between responses. We show loss of synchronization of responses with damage to the structure.

Synchronization of non-phase-coherent chaotic electrochemical oscillations

Experiments on phase and generalized synchronization of two coupled, nonidentical chaotic electrochemical oscillations are presented. We adapt measures of characterizing synchronization of a non-phase-coherent chaotic behavior and compare its properties and physicochemical mechanism to those of a phase-coherent behavior. Phase synchronization sets in along with the onset of generalized synchronization for the non-phase-coherent oscillations in contrast to phase-coherent oscillations in which the phase synchronization usually occurs at a weaker coupling strength.

Synchronization of spatially extended chaotic systems in the presence of asymmetric coupling

In a recent paper [Phys. Rev. Lett. 91, 064103 (2003)]] we described the effects of asymmetric coupling configurations on the synchronization of spatially extended systems. In this paper, we report the consequences induced by the presence of asymmetries in the coupling scheme on the synchronization process of a pair of one-dimensional fields obeying complex Ginzburg-Landau equations. While synchronization always occurs for large enough coupling strengths, asymmetries have the effect of enhancing synchronization and play a crucial role in setting the threshold for the appearance of the synchronized dynamics, as well as in selecting the statistical and dynamical properties of the synchronized motion. We analyze the process of synchronization in the presence of asymmetries when the dynamics is affected by the presence of phase singularities, and show that defects tend to anchor one system to the other. In addition, asymmetry controls the number of synchronized defects that are present in the dynamics. Possible consequences of such asymmetry induced effects in biological and natural systems are discussed.

Detecting local synchronization in coupled chaotic systems

We introduce a technique to detect and quantify local functional dependencies between coupled chaotic systems. The method estimates the fraction of locally synchronized configurations, in a pair of signals with an arbitrary state of global synchronization. Application to a pair of interacting Rössler oscillators shows that our method is able to quantify the number of dynamical configurations where a local prediction task is possible, as well as in the absence of global synchronization features.

Assessment of damage in an eight-oscillator circuit using dynamical forcing

We employ chaotic interrogation of a circuit simulation of a structure in order to test for damage to the structure. The circuit simulation provides a realistic test of our attractor-based method and permits close control over parameters in the structure. In this circuit, simulating an eight-degree-of-freedom spring-mass system, we were able to detect changes of as little as 2% in the coupling between two oscillators in the circuit. This corresponded to detection of a 2% loss in stiffness to one spring in the modeled system.