False neighbors and false strands: a reliable minimum embedding dimension algorithm

Pathological Voice Detection Based on Phase Reconstitution and Convolutional Neural Network

The nonlinear dynamic features can effectively describe the acoustic characteristics of normal and pathological voice. In this paper, the phase space reconstruction and convolution neural network are used to classify the normal and pathological voice. The phase space information of normal and pathological voice is reconstructed using delay time and embedding dimension, the one-dimensional signal is converted to a two-dimensional matrix, and the reconstructed trajectory graph sample of the signal is generated. The trajectory graph samples are used as the input of the VGG-like convolutional neural network, and the graphical features are extracted to achieve a classification of normal and pathological voice. In order to overcome the lack of clinical data, a data enhancement scheme is used. The experiment which classifies the normal and pathological voice is carried out on three pathological databases respectively, i.e. the Massachusetts eye and ear infirmary (MEEI) database, Saarbrücken voice database (SVD) database, and a clinical database collected by the authors. Five-fold cross validation is used and the average recognition rates on the three databases are 99.42%, 97.30% and 95.88% respectively. The average recognition rates are 96.04% and 92.27% for normal, vocal fold paralysis and vocal fold non-paralysis voice in MEEI database and SVD database. The experimental results show that the method has high classification recognition rate and good robustness, and has certain universal applicability for the recognition of the normal and pathological voice.

Reservoir computing approaches to unsupervised concept drift detection in dynamical systems

While the assumption that dynamical systems are stationary is common for modeling purposes, in reality, this is rarely the case. Rather, these systems can change over time, a phenomenon referred to as concept drift in the modeling community. While there exist numerous statistics-based methods for concept drift detection on stochastic processes, approaches leveraging nonlinear time series analysis (NTSA) are rarer but seeing increased focus in cases where the processes are deterministic. In this work, we propose a novel approach to unsupervised concept drift detection in dynamical systems utilizing the embedding offered by a reservoir computing (RC) model. This approach is inspired by the performance of RC on supervised classification tasks that indicates a strong ability to characterize dynamical systems. We assess this method on a number of synthetic drifting data streams from dynamical systems as well as an experimental case concerning faulty ball bearing. Our results suggest that the RC based methods are able to generally outperform the existing NTSA methods across the test cases. We conclude our work with some comments regarding real-time implementation and the impact of hyper-parameters on the proposed algorithm.

Nonlinear time series analysis of state-wise COVID-19 in Malaysia using wavelet and persistent homology

The nonlinear progression of COVID-19 positive cases, their fluctuations, the correlations in amplitudes and phases across different regions, along with seasonality or periodicity, pose challenges to thoroughly examining the data for revealing similarities or detecting anomalous trajectories. To address this, we conducted a nonlinear time series analysis combining wavelet and persistent homology to detect the qualitative properties underlying COVID-19 daily infection numbers at the state level from the pandemic's onset to June 2024 in Malaysia. The first phase involved investigating the evolution of daily confirmed cases by state in the time-frequency domain using wavelets. Subsequently, a topological feature-based time series clustering is performed by reconstructing a higher-dimensional phase space through a delay embedding method. Our findings reveal a prominent 7-day periodicity in case numbers from mid-2021 to the end of 2022. The state-wise daily cases are moderately correlated in both amplitudes and phases during the Delta and Omicron waves. Biweekly averaged data significantly enhances the detection of topological loops associated with these waves. Selangor demonstrates unique case trajectories, while Pahang shows the highest similarity with other states. This methodological framework provides a more detailed understanding of epidemiological time series data, offering valuable insights for preparing for future public health crises.

Analysis of human ambulation as a chaotic time-series: with nonlinear dynamics tools

The aim of the present study is to investigate the complexity and stability of human ambulation and the implications on robotic prostheses control systems. Fourteen healthy individuals participate in two experiments, the first group run at three different speeds. The second group ascended and descended stairs of a five-level building block at a self-selected speed. All participants completed the experiment with seven inertial measurement units wrapped around the lower body segments and waist. The data were analyzed to determine the fractal dimension, spectral entropy, and the Lyapunov exponent (LyE). Two methods were used to calculate the long-term LyE, first LyE calculated using the full size of data sets. And the embedding dimensions were calculated using Average Mutual Information (AMI) and the False Nearest Neighbor (FNN) algorithm was used to find the time delay. Besides, a second approach was developed to find long-term LyE where the time delay was based on the average period of the gait cycle using adaptive event-based window. The average values of spectral entropy are 0.538 and 0.575 for stairs ambulation and running, respectively. The degree of uncertainty and complexity increases with the ambulation speed. The short term LyEs for tibia orientation have the minimum range of variation when it comes to stairs ascent and descent. Using two-way analysis of variance we demonstrated the effect of the ambulation speed and type of ambulation on spectral entropy. Moreover, it was shown that the fractal dimension only changed significantly with ambulation speed.

A robust time-delay selection criterion applied to convergent cross mapping

This work presents a heuristic for the selection of a time delay based on optimizing the global maximum of mutual information in orthonormal coordinates for embedding a dynamical system. This criterion is demonstrated to be more robust compared to methods that utilize a local minimum, as the global maximum is guaranteed to exist in the proposed coordinate system for any dynamical system. By contrast, methods using local minima can be ill-posed as a local minimum can be difficult to identify in the presence of noise or may simply not exist. The performance of the global maximum and local minimum methods are compared in the context of causality detection using convergent cross mapping using both a noisy Lorenz system and experimental data from an oscillating plasma source. The proposed heuristic for time lag selection is shown to be more consistent in the presence of noise and closer to an optimal uniform time lag selection.

Diagnostics of unmanned aerial vehicle with recurrence based approach of piezo-element voltage signals

This work experimentally addresses damage calibration of an unmanned aerial vehicle in operational condition. A wide range of damage level and types are simulated and controlled by an electric motor via pulse width modulation in this regard. The measurement is carried out via established protocols of using a piezo-patch on one of the 8 arms, utilising the vibration sensitivity and flexibility of the arms, demonstrating repeatability of such protocol. Subsequently, recurrence analysis on the voltage time series data is performed for detection of damage. Quantifiers of damage extent are then created for the full range of damage conditions, including the extreme case of complete loss of power. Experimental baseline condition for no damage condition is also established in this regard. Both diagonal-line and vertical-line based indicators from recurrence analysis are sensitive to the quantitative estimates of damage levels and a statistical test of significance analysis confirms that it is possible to automate distinguishing the levels of damage. The damage quantifiers proposed in this paper are useful for rapid monitoring of unmanned aerial vehicle operations of connection.

Model adaptive phase space reconstruction

Phase space reconstruction (PSR) methods allow for the analysis of low-dimensional data with methods from dynamical systems theory, but their application to prediction models, such as those from machine learning (ML), is limited. Therefore, we here present a model adaptive phase space reconstruction (MAPSR) method that unifies the process of PSR with the modeling of the dynamical system. MAPSR is a differentiable PSR based on time-delay embedding and enables ML methods for modeling. The quality of the reconstruction is evaluated by the prediction loss. The discrete-time signal is converted into a continuous-time signal to achieve a loss function, which is differentiable with respect to the embedding delays. The delay vector, which stores all potential embedding delays, is updated along with the trainable parameters of the model to minimize prediction loss. Thus, MAPSR does not rely on any threshold or statistical criterion for determining the dimension and the set of delay values for the embedding process. We apply the MAPSR method to uni- and multivariate time series stemming from chaotic dynamical systems and a turbulent combustor. We find that for the Lorenz system, the model trained with the MAPSR method is able to predict chaotic time series for nearly seven to eight Lyapunov time scales, which is found to be much better compared to other PSR methods [AMI-FNN (average mutual information-false nearest neighbor) and PECUZAL (Pecora-Uzal) methods]. For the univariate time series from the turbulent combustor, the long-term cumulative prediction error of the MAPSR method for the regime of chaos stays between other methods, and for the regime of intermittency, MAPSR outperforms other PSR methods.

Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology

Delay embedding methods are a staple tool in the field of time series analysis and prediction. However, the selection of embedding parameters can have a big impact on the resulting analysis. This has led to the creation of a large number of methods to optimize the selection of parameters such as embedding lag. This paper aims to provide a comprehensive overview of the fundamentals of embedding theory for readers who are new to the subject. We outline a collection of existing methods for selecting embedding lag in both uniform and non-uniform delay embedding cases. Highlighting the poor dynamical explainability of existing methods of selecting non-uniform lags, we provide an alternative method of selecting embedding lags that includes a mixture of both dynamical and topological arguments. The proposed method, Significant Times on Persistent Strands (SToPS), uses persistent homology to construct a characteristic time spectrum that quantifies the relative dynamical significance of each time lag. We test our method on periodic, chaotic, and fast-slow time series and find that our method performs similar to existing automated non-uniform embedding methods. Additionally, n-step predictors trained on embeddings constructed with SToPS were found to outperform other embedding methods when predicting fast-slow time series.

Identification of Gait Unbalance and Fallers Among Subjects with Cerebellar Ataxia by a Set of Trunk Acceleration-Derived Indices of Gait

This study aimed to assess the ability of 25 gait indices to characterize gait instability and recurrent fallers among persons with primary degenerative cerebellar ataxia (pwCA), regardless of gait speed, and investigate their correlation with clinical and kinematic variables. Trunk acceleration patterns were acquired during the gait of 34 pwCA, and 34 age- and speed-matched healthy subjects (HS_matched) using an inertial measurement unit. We calculated harmonic ratios (HR), percent recurrence, percent determinism, step length coefficient of variation, short-time largest Lyapunov exponent (sLLE), normalized jerk score, log-dimensionless jerk (LDLJ-A), root mean square (RMS), and root mean square ratio of accelerations (RMSR) in each spatial direction for each participant. Unpaired t-tests or Mann-Whitney tests were performed to identify significant differences between the pwCA and HS_matched groups. Receiver operating characteristics were plotted to assess the ability to characterize gait alterations in pwCA and fallers. Optimal cutoff points were identified, and post-test probabilities were calculated. The HRs showed to characterize gait instability and pwCA fallers with high probabilities. They were correlated with disease severity and stance, swing, and double support duration, regardless of gait speed. sLLEs, RMSs, RMSRs, and LDLJ-A were slightly able to characterize the gait of pwCA but failed to characterize fallers.

Differences in Lower Extremity Coordination Patterns as a Function of Sports Specialization

The practice of early sport specialization, defined as intense year-round training in a single sport at the exclusion of others, is increasing in youth athletics. Despite potential benefits, sport specialization may be detrimental to the health of young athletes, as specialization may increase the risk of musculoskeletal injuries-particularly overuse injuries. However, there remains limited knowledge about how sports specialization uniquely alters underlying sports-related motor behavior. The purpose of this study was to compare the variability of movement patterns exhibited by highly sports specialized youth athletes to that of nonspecialized athletes during performance of a sport-specific, virtual reality based cutting task. It was hypothesized that highly specialized athletes would display different patterns of movement coordination compared to nonspecialized athletes during both the run-up phase and cut-and-decelerate phase. In support of the hypothesis, specialized athletes exhibited both intra- and inter-limb coordination that were significantly different than unspecialized athletes. Overall, the results indicate that the highly specialized athletes tended to exhibit greater degrees of coordination but also the ability to break the coordinated patterns of joint angle changes to execute a cutting maneuver, which requires asymmetric demands on the lower extremities while planting on one leg and changing direction.

Feature Selection and Uncertainty Analysis for Bubbling Fluidized Bed Oxy-Fuel Combustion Data

This paper presents a novel feature extraction and validation technique for data-driven prediction of oxy-fuel combustion emissions in a bubbling fluidized bed experimental facility. The experimental data were analyzed and preprocessed to minimize the size of the data set while preserving patterns and variance and to find an optimal configuration of the feature vector. The Boruta Feature Selection Algorithm (BFSA) finds feature vector’s configuration and the Multiscale False Neighbours Analysis (MSFNA) is newly extended and proposed to validate the BFSA’s design for emission prediction to assure minimal uncertainty in mapping between feature vectors and corresponding outputs. The finding is that the standalone BFSA does not reflect various sampling period setups that appeared significantly influencing the false neighborhood in the design of feature vectors for possible emission prediction, and MSFNA resolves that.

Ability of a Set of Trunk Inertial Indexes of Gait to Identify Gait Instability and Recurrent Fallers in Parkinson's Disease

The aims of this study were to assess the ability of 16 gait indices to identify gait instability and recurrent fallers in persons with Parkinson’s disease (pwPD), regardless of age and gait speed, and to investigate their correlation with clinical and kinematic variables. The trunk acceleration patterns were acquired during the gait of 55 pwPD and 55 age-and-speed matched healthy subjects using an inertial measurement unit. We calculated the harmonic ratios (HR), percent recurrence, and percent determinism (RQAdet), coefficient of variation, normalized jerk score, and the largest Lyapunov exponent for each participant. A value of ≤1.50 for the HR in the antero-posterior direction discriminated between pwPD at Hoehn and Yahr (HY) stage 3 and healthy subjects with a 67% probability, between pwPD at HY 3 and pwPD at lower HY stages with a 73% probability, and it characterized recurrent fallers with a 77% probability. Additionally, HR in the antero-posterior direction was correlated with pelvic obliquity and rotation. RQAdet in the antero-posterior direction discriminated between pwPD and healthy subjects with 67% probability, regardless of the HY stage, and was correlated with stride duration and cadence. Therefore, HR and RQA_det in the antero-posterior direction can both be used as age- and-speed-independent markers of gait instability.

Attention-based multi-scale features fusion for unobtrusive atrial fibrillation detection using ballistocardiogram signal

BACKGROUND

Atrial fibrillation (AF) represents the most common arrhythmia worldwide, related to increased risk of ischemic stroke or systemic embolism. It is critical to screen and diagnose AF for the benefits of better cardiovascular health in lifetime. The ECG-based AF detection, the gold standard in clinical care, has been restricted by the need to attach electrodes on the body surface. Recently, ballistocardiogram (BCG) has been investigated for AF diagnosis, which is an unobstructive and convenient technique to monitor heart activity in daily life. However, here is a lack of high-dimension representation and deep learning analysis of BCG.

METHOD

Therefore, this paper proposes an attention-based multi-scale features fusion method by using BCG signal. The 1-D morphology feature extracted from Bi-LSTM network and 2-D rhythm feature extracted from reconstructed phase space are integrated by means of CNN network to improve the robustness of AF detection. To the best of our knowledge, this is the first study where the phase space trajectory of BCG is conducted.

RESULTS

2000 segments (AF and NAF) of BCG signals were collected from 59 volunteers suffering from paroxysmal AF in this survey. Compared to the classical time and frequency features and the state-of-the-art energy features with the popular machine learning classifiers, AF detection performance of the proposed method is superior, which has 0.947 accuracy, 0.935 specificity, 0.959 sensitivity, and 0.937 precision, for the same BCG dataset. The experimental results show that combined feature could excavate more potential characteristics, and the attention mechanism could enhance the pertinence for AF recognition.

CONCLUSIONS

The proposed method can provide an innovative solution to capture the diverse scale descriptions of BCG and explore ways to involve the deep learning method to accurately screen AF in routine life.

Dual-Task Gait Stability after Concussion and Subsequent Injury: An Exploratory Investigation

Persistent gait alterations can occur after concussion and may underlie future musculoskeletal injury risk. We compared dual-task gait stability measures among adolescents who did/did not sustain a subsequent injury post-concussion, and uninjured controls. Forty-seven athletes completed a dual-task gait evaluation. One year later, they reported sport-related injuries and sport participation volumes. There were three groups: concussion participants who sustained a sport-related injury (n = 8; age =15.4 ± 3.5 years; 63% female), concussion participants who did not sustain a sport-related injury (n = 24; 14.0 ± 2.6 years; 46% female), and controls (n = 15; 14.2 ± 1.9 years; 53% female). Using cross-recurrence quantification, we quantified dual-task gait stability using diagonal line length, trapping time, percent determinism, and laminarity. The three groups reported similar levels of sports participation (11.8 ± 5.8 vs. 8.6 ± 4.4 vs. 10.9 ± 4.3 hours/week; p = 0.37). The concussion/subsequent injury group walked slower (0.76 ± 0.14 vs. 0.65 ± 0.13 m/s; p = 0.008) and demonstrated higher diagonal line length (0.67 ± 0.08 vs. 0.58 ± 0.05; p = 0.02) and trapping time (5.3 ± 1.5 vs. 3.8 ± 0.6; p = 0.006) than uninjured controls. Dual-task diagonal line length (hazard ratio =1.95, 95% CI = 1.05–3.60), trapping time (hazard ratio = 1.66, 95% CI = 1.09–2.52), and walking speed (hazard ratio = 0.01, 95% CI = 0.00–0.51) were associated with subsequent injury. Dual-task gait stability measures can identify altered movement that persists despite clinical concussion recovery and is associated with future injury risk.

The Clinical Application of EEG-Signals Recurrence Analysis as a Measure of Functional Connectivity: Comparative Case Study of Patients with Various Neuropsychiatric Disorders

Background. An electroencephalogram (EEG) is a simple and widely used assessment tool that allows one to analyze the bioelectric activity of the brain. As a result, one can observe brain waves with different frequencies and amplitudes that correspond to the temporary synchronization of different parts of the brain. Synchronization patterns may be changed by almost any type of pathological conditions, such as psychiatric diseases and structural abnormalities of the brain tissue. In various neuropsychiatric disorders, the coordination of cortical activity may be decreased or enhanced as a result of neurobiological compensatory mechanisms. Methods. In this paper, we analyzed the EEG signals in resting-state condition, with reference to three patients with a similar set of psychopathological symptoms typical for the first psychotic episode, but with different functional and structural neural basis of the disease. Additionally, those patients were compared with a demographically matched healthy individual. We used the non-linear method of time series analysis based on the recurrences of states, to verify whether functional connectivity configurations assessed with recurrence method will qualitatively distinguish patients from a healthy subject, but also differentiate patients from each other. Results. Obtained results confirmed that the connectivity architecture mapped with the recurrence analysis substantially differentiated all participants from each other. An applied analysis additionally showed the specificity of cortical desynchronization and over-synchronization matched to the psychiatric or neurological basis of the disease. Despite this encouraging finding, group-oriented studies are needed to corroborate our qualitative results, based only on a series of clinical case studies.

Electrocortical dynamics differentiate athletes exhibiting low- and high- ACL injury risk biomechanics

Anterior cruciate ligament (ACL) injuries are physically and emotionally debilitating for athletes,while motor and biomechanical deficits that contribute to ACL injury have been identified, limited knowledge about the relationship between the central nervous system (CNS) and biomechanical patterns of motion has impeded approaches to optimize ACL injury risk reduction strategies. In the current study it was hypothesized that high-risk athletes would exhibit altered temporal dynamics in their resting state electrocortical activity when compared to low-risk athletes. Thirty-eight female athletes performed a drop vertical jump (DVJ) to assess their biomechanical risk factors related to an ACL injury. The athletes' electrocortical activity was also recorded during resting state in the same visit as the DVJ assessment. Athletes were divided into low- and high-risk groups based on their performance of the DVJ. Recurrence quantification analysis was used to quantify the temporal dynamics of two frequency bands previously shown to relate to sensorimotor and attentional control. Results revealed that high-risk participants showed more deterministic electrocortical behavior than the low-risk group in the frontal theta and central/parietal alpha-2 frequency bands. The more deterministic resting state electrocortical dynamics for the high-risk group may reflect maladaptive neural behavior-excessively stable deterministic patterning that makes transitioning among functional task-specific networks more difficult-related to attentional control and sensorimotor processing neural regions.

Phase Space Reconstruction from a Biological Time Series: A Photoplethysmographic Signal Case Study

In the analysis of biological time series, the state space is comprised of a framework for the study of systems with presumably deterministic and stationary properties. However, a physiological experiment typically captures an observable that characterizes the temporal response of the physiological system under study; the dynamic variables that make up the state of the system at any time are not available. Only from the acquired observations should state vectors be reconstructed to emulate the different states of the underlying system. This is what is known as the reconstruction of the state space, called the phase space in real-world signals, in many cases satisfactorily resolved using the method of delays. Each state vector consists of m components, extracted from successive observations delayed a time τ . The morphology of the geometric structure described by the state vectors, as well as their properties depends on the chosen parameters τ and m. The real dynamics of the system under study is subject to the correct determination of the parameters τ and m. Only in this way can be deduced features have true physical meaning, revealing aspects that reliably identify the dynamic complexity of the physiological system. The biological signal presented in this work, as a case study, is the photoplethysmographic (PPG) signal. We find that m is five for all the subjects analyzed and that τ depends on the time interval in which it is evaluated. The Hénon map and the Lorenz flow are used to facilitate a more intuitive understanding of the applied techniques.

Forecasting of landslide displacements using a chaos theory based wavelet analysis-Volterra filter model

Landslide displacement time series can directly reflects landslide deformation and stability characteristics. Hence, forecasting of the non-linear and non-stationary displacement time series is necessary and significant for early warning of landslide failure. Traditionally, conventional machine learning methods are adopted as forecasting models, these forecasting models mainly determine the input and output variables experientially and does not address the non-stationary characteristics of displacement time series. However, it is difficult for these conventional machine learning methods to obtain appropriate input-output variables, to determine appropriate model parameters and to acquire satisfied prediction performance. To deal with these drawbacks, this study proposes the wavelet analysis (WA) to decompose the displacement time series into low- and high-frequency components to address the non-stationary characteristics; then proposes thee chaos theory to obtain appropriate input-output variables of forecasting models, and finally proposes Volterra filter model to construct the forecasting model. The GPS monitoring cumulative displacement time series, recorded on the Shuping and Baijiabao landslides, distance measuring equipment monitoring displacements on the Xintan landslide in Three Gorges Reservoir area of China, are used as test data of the proposed chaotic WA-Volterra model. The chaotic WA-support vector machine (SVM) model and single chaotic Volterra model without WA method, are used as comparisons. The results show that there are chaos characteristics in the GPS monitoring displacement time series, the non-stationary characteristics of landslide displacements are captured well by the WA method, and the model input-output variables are selected suitably using chaos theory. Furthermore, the chaotic WA-Volterra model has higher prediction accuracy than the chaotic WA-SVM and single chaotic Volterra models.

Human cognition involves the dynamic integration of neural activity and neuromodulatory systems

The human brain integrates diverse cognitive processes into a coherent whole, shifting fluidly as a function of changing environmental demands. Despite recent progress, the neurobiological mechanisms responsible for this dynamic system-level integration remain poorly understood. Here we investigated the spatial, dynamic, and molecular signatures of system-wide neural activity across a range of cognitive tasks. We found that neuronal activity converged onto a low-dimensional manifold that facilitates the execution of diverse task states. Flow within this attractor space was associated with dissociable cognitive functions, unique patterns of network-level topology, and individual differences in fluid intelligence. The axes of the low-dimensional neurocognitive architecture aligned with regional differences in the density of neuromodulatory receptors, which in turn relate to distinct signatures of network controllability estimated from the structural connectome. These results advance our understanding of functional brain organization by emphasizing the interface between neural activity, neuromodulatory systems, and cognitive function.

Brain Network Changes in Fatigued Drivers: A Longitudinal Study in a Real-World Environment Based on the Effective Connectivity Analysis and Actigraphy Data

The analysis of neurophysiological changes during driving can clarify the mechanisms of fatigue, considered an important cause of vehicle accidents. The fluctuations in alertness can be investigated as changes in the brain network connections, reflected in the direction and magnitude of the information transferred. Those changes are induced not only by the time on task but also by the quality of sleep. In an unprecedented 5-month longitudinal study, daily sampling actigraphy and EEG data were collected during a sustained-attention driving task within a near-real-world environment. Using a performance index associated with the subjects' reaction times and a predictive score related to the sleep quality, we identify fatigue levels in drivers and investigate the shifts in their effective connectivity in different frequency bands, through the analysis of the dynamical coupling between brain areas. Study results support the hypothesis that combining EEG, behavioral and actigraphy data can reveal new features of the decline in alertness. In addition, the use of directed measures such as the Convergent Cross Mapping can contribute to the development of fatigue countermeasure devices.

Beyond HRV: attractor reconstruction using the entire cardiovascular waveform data for novel feature extraction

Advances in monitoring technology allow blood pressure waveforms to be collected at sampling frequencies of 250-1000 Hz for long time periods. However, much of the raw data are under-analysed. Heart rate variability (HRV) methods, in which beat-to-beat interval lengths are extracted and analysed, have been extensively studied. However, this approach discards the majority of the raw data.

OBJECTIVE

Our aim is to detect changes in the shape of the waveform in long streams of blood pressure data.

APPROACH

Our approach involves extracting key features from large complex data sets by generating a reconstructed attractor in a three-dimensional phase space using delay coordinates from a window of the entire raw waveform data. The naturally occurring baseline variation is removed by projecting the attractor onto a plane from which new quantitative measures are obtained. The time window is moved through the data to give a collection of signals which relate to various aspects of the waveform shape.

MAIN RESULTS

This approach enables visualisation and quantification of changes in the waveform shape and has been applied to blood pressure data collected from conscious unrestrained mice and to human blood pressure data. The interpretation of the attractor measures is aided by the analysis of simple artificial waveforms.

SIGNIFICANCE

We have developed and analysed a new method for analysing blood pressure data that uses all of the waveform data and hence can detect changes in the waveform shape that HRV methods cannot, which is confirmed with an example, and hence our method goes 'beyond HRV'.

Trophic and tectonic limits to the global increase of marine invertebrate diversity

The marine invertebrate fossil record provides the most comprehensive history of how the diversity of animal life has evolved through time. One of the main features of this record is a modest rise in diversity over nearly a half-billion years. The long-standing view is that ecological interactions such as resource competition and predation set upper limits to global diversity, which, in the absence of external perturbations, is maintained indefinitely at equilibrium. However, the effect of mechanisms associated with the history of the seafloor, and their influence on the creation and destruction of marine benthic habitats, has not been explored. Here we use statistical methods for causal inference to investigate the drivers of marine invertebrate diversity dynamics through the Phanerozoic. We find that diversity dynamics responded to secular variations in marine food supply, substantiating the idea that global species richness is regulated by resource availability. Once diversity was corrected for changes in food resource availability, its dynamics were causally linked to the age of the subducting oceanic crust. We suggest that the time elapsed between the formation (at mid-ocean ridges) and destruction (at subduction zones) of ocean basins influences the diversity dynamics of marine invertebrates and may have contributed to constrain their diversification.

Locally embedded presages of global network bursts

Spontaneous, synchronous bursting of neural population is a widely observed phenomenon in nervous networks, which is considered important for functions and dysfunctions of the brain. However, how the global synchrony across a large number of neurons emerges from an initially nonbursting network state is not fully understood. In this study, we develop a state-space reconstruction method combined with high-resolution recordings of cultured neurons. This method extracts deterministic signatures of upcoming global bursts in "local" dynamics of individual neurons during nonbursting periods. We find that local information within a single-cell time series can compare with or even outperform the global mean-field activity for predicting future global bursts. Moreover, the intercell variability in the burst predictability is found to reflect the network structure realized in the nonbursting periods. These findings suggest that deterministic local dynamics can predict seemingly stochastic global events in self-organized networks, implying the potential applications of the present methodology to detecting locally concentrated early warnings of spontaneous seizure occurrence in the brain.

Untangling Brain-Wide Dynamics in Consciousness by Cross-Embedding

Brain-wide interactions generating complex neural dynamics are considered crucial for emergent cognitive functions. However, the irreducible nature of nonlinear and high-dimensional dynamical interactions challenges conventional reductionist approaches. We introduce a model-free method, based on embedding theorems in nonlinear state-space reconstruction, that permits a simultaneous characterization of complexity in local dynamics, directed interactions between brain areas, and how the complexity is produced by the interactions. We demonstrate this method in large-scale electrophysiological recordings from awake and anesthetized monkeys. The cross-embedding method captures structured interaction underlying cortex-wide dynamics that may be missed by conventional correlation-based analysis, demonstrating a critical role of time-series analysis in characterizing brain state. The method reveals a consciousness-related hierarchy of cortical areas, where dynamical complexity increases along with cross-area information flow. These findings demonstrate the advantages of the cross-embedding method in deciphering large-scale and heterogeneous neuronal systems, suggesting a crucial contribution by sensory-frontoparietal interactions to the emergence of complex brain dynamics during consciousness.

Dimensionality reduction and dynamical filtering: Stimulated Brillouin scattering in optical fibers

Stimulated Brillouin scattering (SBS) is a noise-driven nonlinear interaction between acoustical and optical waves. In optical fibers, SBS can be observed at relatively low optical powers and can severely limit signal transmission. Although SBS is initiated by high dimensional noise, it also exhibits many of the hallmarks of a complex nonlinear dynamical system. We report here a comprehensive experimental and numerical study of the fluctuations in the reflected Stokes wave produced by SBS in optical fibers. Using time series analysis, we demonstrate a reduction of dimensionality and dynamical filtering of the Stokes wave. We begin with a careful comparison of the measured average transmitted and reflected intensities from below the SBS threshold to saturation of the transmitted power. Initially the power spectra and correlation functions of the time series of the reflected wave fluctuations at the SBS threshold and above are measured and simulated. Much greater dynamical insight is provided when we study the scaling behavior of the intensity fluctuations using Hurst exponents and detrended fluctuation analysis for time scales extending over six orders of magnitude. At the highest input powers, we notice the emergence of three distinct dynamical scaling regimes: persistent, Brownian, and antipersistent. Next, we explore the Hilbert phase fluctuations of the intensity time series and amplitude-phase coupling. Finally, time-delay embedding techniques reveal a gradual reduction in dimensionality of the spatiotemporal dynamics as the laser input is increased toward saturation of the transmitted power. Through all of these techniques, we find a transition from noisier to smoother dynamics with increasing input power. We find excellent agreement between our experimental measurements and simulations.

Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

If data are generated by a system with a d-dimensional attractor, then Takens’ theorem guarantees that reconstruction that is diffeomorphic to the original attractor can be built from the single time series in 2d+1-dimensional phase space. However, under certain conditions, reconstruction is possible even in a space of smaller dimension. This topic is very important because the size of the reconstruction space relates to the effectiveness of the whole subsequent analysis. In this paper, the false nearest neighbour (FNN) methods are revisited to estimate the optimum embedding parameters and the most appropriate observables for state space reconstruction. A modification of the false nearest neighbour method is introduced. The findings contribute to evidence that the length of the embedding time window (TW) is more important than the reconstruction delay time and the embedding dimension (ED) separately. Moreover, if several time series of the same system are observed, the choice of the one that is used for the reconstruction could also be critical. The results are demonstrated on two chaotic benchmark systems.

Two chaotic global models for cereal crops cycles observed from satellite in northern Morocco

The dynamics underlying cereal crops in the northern region of Morocco is investigated using a global modelling technique applied to a vegetation index time series derived from satellite measurements, namely, the normalized difference vegetation index from 1982 to 2008. Two three-dimensional chaotic global models of reduced size (14-term and 15-term models) are obtained. The model validation is performed by comparing their horizons of predictability with those provided in previous studies. The attractors produced by the two global models have a complex foliated structure-evidenced in a Poincaré section-rending a topological characterization difficult to perform. Thus, the Kaplan-Yorke dimension is estimated from the synthetic data produced by our global models. Our results suggest that cereal crops in the northern Morocco are governed by a weakly dissipative three-dimensional chaotic dynamics.

A model-based method for computation of correlation dimension, Lyapunov exponents and synchronization from depth-EEG signals

In order to predict epileptic seizures many precursory features, extracted from the EEG signals, have been introduced. Before checking out the performance of features in detection of pre-seizure state, it is required to see whether these features are accurately extracted. Evaluation of feature estimation methods has been less considered, mainly due to the lack of a ground truth for the real EEG signals' features. In this paper, some simulated long-term depth-EEG signals, with known state spaces, are generated via a realistic neural mass model with physiological parameters. Thanks to the known ground truth of these synthetic signals, they are suitable for evaluating different algorithms used to extract the features. It is shown that conventional methods of estimating correlation dimension, the largest Lyapunov exponent, and phase coherence have non-negligible errors. Then, a parameter identification-based method is introduced for estimating the features, which leads to better estimation results for synthetic signals. It is shown that the neural mass model is able to reproduce real depth-EEG signals accurately; thus, assuming this model underlying real depth-EEG signals, can improve the accuracy of features' estimation.

Sensitivity vector fields in time-delay coordinate embeddings: theory and experiment

Identifying changes in the parameters of a dynamical system can be vital in many diagnostic and sensing applications. Sensitivity vector fields (SVFs) are one way of identifying such parametric variations by quantifying their effects on the morphology of a dynamical system's attractor. In many cases, SVFs are a more effective means of identification than commonly employed modal methods. Previously, it has only been possible to construct SVFs for a given dynamical system when a full set of state variables is available. This severely restricts SVF applicability because it may be cost prohibitive, or even impossible, to measure the entire state in high-dimensional systems. Thus, the focus of this paper is constructing SVFs with only partial knowledge of the state by using time-delay coordinate embeddings. Local models are employed in which the embedded states of a neighborhood are weighted in a way referred to as embedded point cloud averaging. Application of the presented methodology to both simulated and experimental time series demonstrates its utility and reliability.

Polynomial search and global modeling: Two algorithms for modeling chaos

Global modeling aims to build mathematical models of concise description. Polynomial Model Search (PoMoS) and Global Modeling (GloMo) are two complementary algorithms (freely downloadable at the following address: http://www.cesbio.ups-tlse.fr/us/pomos_et_glomo.html) designed for the modeling of observed dynamical systems based on a small set of time series. Models considered in these algorithms are based on ordinary differential equations built on a polynomial formulation. More specifically, PoMoS aims at finding polynomial formulations from a given set of 1 to N time series, whereas GloMo is designed for single time series and aims to identify the parameters for a selected structure. GloMo also provides basic features to visualize integrated trajectories and to characterize their structure when it is simple enough: One allows for drawing the first return map for a chosen Poincaré section in the reconstructed space; another one computes the Lyapunov exponent along the trajectory. In the present paper, global modeling from single time series is considered. A description of the algorithms is given and three examples are provided. The first example is based on the three variables of the Rössler attractor. The second one comes from an experimental analysis of the copper electrodissolution in phosphoric acid for which a less parsimonious global model was obtained in a previous study. The third example is an exploratory case and concerns the cycle of rainfed wheat under semiarid climatic conditions as observed through a vegetation index derived from a spatial sensor.

Using filtering effects to identify objects

Reflecting signals off of targets is a method widely used to locate objects, but the reflected signal also contains information that can be used to identify the object. In radar or sonar, the signal amplitudes used are small enough that only linear effects are present, so we can consider the effect of the target on the signal as a linear filter. Using the known effects of linear filters on chaotic signals, we can create a reference that allows us to match a particular target to a particular reflected signal. Furthermore, if some parts of this "filter" vary only slowly as the aspect angle of the object changes, we can produce a reference that averages out the parts that are highly angle dependent so that one reference can be used to identify the target over a range of angles.

Evidence-based modeling of network discharge dynamics during periodic pacing to control epileptiform activity

Deep brain stimulation (DBS) is a promising therapeutic approach for epilepsy treatment. Recently, research has focused on the implementation of stimulation protocols that would adapt to the patients need (adaptive stimulation) and deliver electrical stimuli only when it is most useful. A formal mathematical description of the effects of electrical stimulation on neuronal networks is a prerequisite for the development of adaptive DBS algorithms. Using tools from non-linear dynamic analysis, we describe an evidence-based, mathematical modeling approach that (1) accurately simulates epileptiform activity at time-scales of single and multiple ictal discharges, (2) simulates modulation of neural dynamics during epileptiform activity in response to fixed, low-frequency electrical stimulation, (3) defines a mapping from real-world observations to model state, and (4) defines a mapping from model state to real-world observations. We validate the real-world utility of the model's properties by statistical comparison between the number, duration, and interval of ictal-like discharges observed in vitro and those simulated in silica under conditions of repeated stimuli at fixed-frequency. These validation results confirm that the evidence-based modeling approach captures robust, informative features of neural network dynamics of in vitro epileptiform activity under periodic pacing and support its use for further implementation of adaptive DBS protocols for epilepsy treatment.

Nearest neighbors, phase tubes, and generalized synchronization

In this paper we report on the necessity of the refinement of the concept of generalized chaotic synchronization. We show that the state vectors of the interacting chaotic systems being in the generalized synchronization regime are related to each other by the functional, but not the functional relation as it was assumed until now. We propose the phase tube approach explaining the essence of generalized synchronization and allowing the detection and the study of this regime in many relevant physical circumstances. The finding discussed in this Brief Report provides great potential for different applications.

Optimal reconstruction of dynamical systems: a noise amplification approach

In this work we propose an objective function to guide the search for a state space reconstruction of a dynamical system from a time series of measurements. These statistics can be evaluated on any reconstructed attractor, thereby allowing a direct comparison among different approaches: (uniform or nonuniform) delay vectors, PCA, Legendre coordinates, etc. It can also be used to select the most appropriate parameters of a reconstruction strategy. In the case of delay coordinates this translates into finding the optimal delay time and embedding dimension from the absolute minimum of the advocated cost function. Its definition is based on theoretical arguments on noise amplification, the complexity of the reconstructed attractor, and a direct measure of local stretch which constitutes an irrelevance measure. The proposed method is demonstrated on synthetic and experimental time series.

Detecting variation in chaotic attractors

If the output of an experiment is a chaotic signal, it may be useful to detect small changes in the signal, but there are a limited number of ways to compare signals from chaotic systems, and most known methods are not robust in the presence of noise. One may calculate dimension or Lyapunov exponents from the signal, or construct a synchronizing model, but all of these are only useful in low noise situations. I introduce a method for detecting small variations in a chaotic attractor based on directly calculating the difference between vector fields in phase space. The differences are found by comparing close strands in phase space, rather than close neighbors. The use of strands makes the method more robust to noise and more sensitive to small attractor differences.

Modifications of the Takens-Ellner algorithm for medium- and high-dimensional signals

The paper presents the modifications of the classic Takens-Ellner (TE) algorithm that are necessary to estimate the dimensional complexity (d) of the medium- and high-dimensional signals. The main idea is the fitting of the fourth-degree polynomial to the relation d=fn(W) (where W is window width) in order to find the point of minimum slope in this relation. This point corresponds to the plateau area being the feature of the low-dimensional signals and representing the range of the optimal W used in the calculations. The point of minimum slope is represented by the local minimum in the first derivative function (third-degree polynomial). The exclusion of the attractor pairs of points lying approximately closer than the autocorrelation time removes the tendency to down-estimate d observed in the classic TE algorithm. The procedure of the choice of the embedding parameters was modified: The lag (L) was calculated for the given embedding dimension (m) and W was calculated by using the formula L=W/(m-1) in order to obtain the desired values of W for generating the precise relation d=fn(W) being the basis for the polynomial fitting. The cubic interpolation of the signal is proposed for the noninteger L's. Three signals possessing the dimensional complexities d(A)≈4, d(B)≈6, and d(C)≈8 and being the sums of two, three, and four Lorenz signals, respectively, were analyzed. Their lengths were 65,536 points. The applied algorithm gives a precise estimation of d for every signal A, B, and C. Also presented are the results of (a) the method for the estimation of the cubic interpolation error, (b) the analysis for signals contaminated with the white noise, and (c) the comparison between the original signals and the surrogates.

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.

Globally enumerating unstable periodic orbits for observed data using symbolic dynamics

The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an observed time series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from observed time series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.

A nonlinear dynamics method for signal identification

When a radio frequency signal is radiated by a transmitter, the properties of the transmitter itself affect the properties of the signal. These transmitter-induced changes are known as unintentional modulation, to differentiate them from intentional modulation used to add information to the signal. The unintentional modulation can be used to identify which transmitter produced a signal. This paper shows how phase space analysis based on nonlinear dynamics ideas can be used to determine which of several amplifiers produced a signal.

A unified approach to attractor reconstruction

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e., attractor reconstruction. The process has focused primarily on intuitive, heuristic, and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several longstanding, but common problems unresolved in which the standard approaches produce inferior results or give no guidance at all. We view the current reconstruction process as unnecessarily broken into separate problems. We propose an alternative approach that views the problem of choosing all embedding parameters as being one and the same problem addressable using a single statistical test formulated directly from the reconstruction theorems. This allows for varying time delays appropriate to the data and simultaneously helps decide on embedding dimension. A second new statistic, undersampling, acts as a check against overly long time delays and overly large embedding dimension. Our approach is more flexible than those currently used, but is more directly connected with the mathematical requirements of embedding. In addition, the statistics developed guide the user by allowing optimization and warning when embedding parameters are chosen beyond what the data can support. We demonstrate our approach on uni- and multivariate data, data possessing multiple time scales, and chaotic data. This unified approach resolves all the main issues in attractor reconstruction.

Statistically relaxing to generating partitions for observed time-series data

We introduce a relaxation algorithm to estimate approximations to generating partitions for observed dynamical time series. Generating partitions preserve dynamical information of a deterministic map in the symbolic representation. Our method optimizes an essential property of a generating partition: avoiding topological degeneracies. We construct an energy-like functional and use a nonequilibrium stochastic minimization algorithm to search through configuration space for the best assignment of symbols to observed data. As each observed point may be assigned a symbol, the partitions are not constrained to an arbitrary parametrization. We further show how to select particular generating partition solutions which also code low-order unstable periodic orbits in a given way, hence being able to enumerate through a number of potential generating partition solutions.

Improved statistical test for nonstationarity using recurrence time statistics

We have recently introduced a measure for nonstationarity using a recurrence time statistic to assess stationarity. In this paper we propose an extension of this method based on a detailed study of the statistics for the case of stationary systems. We derive a simple scheme that allows us to estimate the effective number of degrees of freedom relevant for this statistic. This substantially improves the statistical significance of the method and can be used to improve the significance of various other nonlinear statistics.

Estimating good discrete partitions from observed data: symbolic false nearest neighbors

A symbolic analysis of observed time series requires a discrete partition of a continuous state space containing the dynamics. A particular kind of partition, called "generating," preserves all deterministic dynamical information in the symbolic representation, but such partitions are not obvious beyond one dimension. Existing methods to find them require significant knowledge of the dynamical evolution operator. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition, avoiding topological degeneracies, by minimizing the number of "symbolic false nearest neighbors." It requires only the observed time series and is sensible even in the presence of noise when no truly generating partition is possible.