Determining embedding dimension for phase-space reconstruction using a geometrical construction

Dynamical structure of center of pressure fluctuations in elderly people

This study investigates whether aging has an influence on the dynamics of the fluctuations of the displacement of the center of pressure (COP), during quiet standing. Two groups of healthy subjects were compared (11 young and 12 elderly) for two visual feedback conditions (eyes open and eyes closed). The data were analyzed using (i) classical stabilometric variables (length and surface) and (ii) recurrence quantification analysis (percentage of determinism and entropy), for the dynamical structure of COP signals. The length of the COP displacement was found to be the best discriminating stabilometric variable for both visual and aging effects. Visual feedback influenced recurrence quantification analysis (RQA) variables only for the elderly group. Both RQA outputs in the anterior posterior direction were sufficient to distinguish the young and elderly groups. The entropy estimation computed by RQA was significantly reduced for postural fluctuations in elderly people. We conclude that classical stabilometric variables and RQA outputs provide complementary information for the characterization of ageing effects on postural sway.

Comparison of different state space definitions for local dynamic stability analyses

Measures of local dynamic stability, such as the local divergence exponent (lambda*(s)) quantify how quickly small perturbations deviate from an attractor that defines the motion. When the governing equations of motion are unknown, an attractor can be reconstructed by defining an appropriate state space. However, state space definitions are not unique and accepted methods for defining state spaces have not been established for biomechanical studies. This study first determined how different state space definitions affected lambda*(s) for the Lorenz attractor, since exact theoretical values were known a priori. Values of lambda*(s) exhibited errors <10% for 7 of the 9 state spaces tested. State spaces containing redundant information performed the poorest. To examine these effects in a biomechanical context, 20 healthy subjects performed a repetitive sawing-like task for 5 min before and after fatigue. Local stability of pre- and post-fatigue shoulder movements was compared for 6 different state space definitions. Here, lambda*(s)decreased post-fatigue for all 6 state spaces. Differences were statistically significant for 3 of these state spaces. For state spaces defined using delay embedding, increasing the embedding dimension decreased lambda*(s) in both the Lorenz and experimental data. Overall, our findings suggest that direct numerical comparisons between studies that use different state space definitions should be made with caution. However, trends across experimental comparisons appear to persist. Biomechanical state spaces constructed using positions and velocities, or delay reconstruction of individual states, are likely to provide consistent results.

Nonlinear time series analysis of knee and ankle kinematics during side by side treadmill walking

Nonlinear time series analysis was used to estimate maximal Lyapunov exponents of select ankle and knee kinematics during three different conditions of treadmill walking: independent, side by side, and side by side with forced synchronization of stepping. Stride to stride variability was significantly increased for the condition in which individuals walked side by side and synchronized unintentionally when compared to the conditions of forced synchronization and independent walking. In addition, standard deviations of three kinematic variables of lower extremity movement were significantly increased during the condition in which unintentional synchronization occurred. No relationship was found between standard deviation and estimates of maximal Lyapunov exponents. An increase in kinematic variability during side by side walking for nonimpaired individuals who are not at risk of falling suggests that variability in certain aspects of performance might be indicative of a healthy system. Modeling this variability for an impaired individual to imitate may have beneficial effects on locomotor function. These results may therefore have implications for the rehabilitation of gait in humans by suggesting that a different functional outcome might be achieved by practicing side by side walking as opposed to more commonly used strategies involving independent walking.

Is slow walking more stable?

Several efforts have been made to study gait stability using measures derived from nonlinear time-series analysis. The maximum finite time Lyapunov exponent (lambda(max)) quantifies how a system responds to an infinitesimally small perturbation. Recent studies suggested that slow walking leads to lower lambda(max) values, and thus is more stable than fast walking, but these studies suffer from methodological limitations. We studied the effects of walking speed on the amount of kinematic variability and stability in human walking. Trunk motions of 15 healthy volunteers were recorded in 3D during 2 min of treadmill walking at different speeds. From those time series, maximum Lyapunov exponents, indicating short-term and long-term divergence (lambda(S-stride) and lambda(L-stride)), and mean standard deviation (MeanSD) were calculated. lambda(S-stride) showed a linear decrease with increasing speed for forward-backward (AP) movements and quadratic effects (inverted U-shaped) for medio-lateral (ML) and up-down (VT) movements. lambda(L-stride) showed a quadratic effect (inverted U-shaped) of walking speed for AP movements, a linear decrease for ML movements, and a linear increase for VT movements. Moreover, positive correlations between lambda(S) and MeanSD were found for all directions, while lambda(L-stride) and MeanSD were correlated negatively in the AP direction. The different effects of walking speed on lambda(S-stride) and lambda(L-stride) for the different planes suggest that slow walking is not necessarily more stable than fast walking. The absence of a consistent pattern of correlations between lambda(L-stride) and MeanSD over the three directions suggests that variability and stability reflect, at least to a degree, different properties of the dynamics of walking.

Brain-machine interactions for assessing the dynamics of neural systems

A critical advance for brain–machine interfaces is the establishment of bi-directional communications between the nervous system and external devices. However, the signals generated by a population of neurons are expected to depend in a complex way upon poorly understood neural dynamics. We report a new technique for the identification of the dynamics of a neural population engaged in a bi-directional interaction with an external device. We placed in vitro preparations from the lamprey brainstem in a closed-loop interaction with simulated dynamical devices having different numbers of degrees of freedom. We used the observed behaviors of this composite system to assess how many independent parameters − or state variables − determine at each instant the output of the neural system. This information, known as the dynamical dimension of a system, allows predicting future behaviors based on the present state and the future inputs. A relevant novelty in this approach is the possibility to assess a computational property – the dynamical dimension of a neuronal population – through a simple experimental technique based on the bi-directional interaction with simulated dynamical devices. We present a set of results that demonstrate the possibility of obtaining stable and reliable measures of the dynamical dimension of a neural preparation.

Does well-harmonized homeostasis exist in heart rate fluctuations? Time series analysis and model simulations

Analyzing heart rate variability from electrocardiographic recordings has been an important method for assessing cardiovascular autonomic regulation. Researchers have conducted extensive analyses on normal as well as pathological hearts, however, it is still unclear whether increasing or decreasing the complexity of heart rate variability is a characteristic of healthy systems. In this study, we find the existence of well-harmonized homeostasis in heart rate fluctuations, in particular, the evidence is verified among different individuals including healthy subjects, ICU patients, and one child with brainstem dysfunction. The methodology we used is composed of two parts, in which one is the consideration of reduction of cardiorespiratory fluctuations inherited in the original R-R intervals and the other is based upon the concept of nonlinear dynamics to construct the low-dimensional trajectory in the angle plot. The cross-correlation measure between the theoretical angle map and the numerically derived angle trajectory is used to separate recovery (0.73+/-0.13) from deterioration (0.25+/-0.08) of ICU patients. In addition, a simple physiologic (deterministic) model of the interaction between the cardiovascular system and baroreceptor control of arterial pressure is used to explain why homeostasis can exist in heart rate fluctuations. Our study provides a potential link between the clinical data and circulatory system.

Denoising of surface EMG with a modified Wiener filtering approach

The correlation dimension D(2) yields good results in several biomedical fields. Nonetheless, no clinical application to electromyography has been developed yet. One reason is the high electromagnetic noise typical of clinical environments. This noise is characterized by sharp spectral lines of variable intensity and frequency. The filtering techniques commonly implemented in electromyographs can efficiently deal with this kind of noise. They allow a safe estimate of linear quantities like the root mean square (r.m.s.) or the median frequency (MF). Their performance is not as good for nonlinear purposes. The filters may modify the nonlinear properties of the signal, leading to unacceptable estimates of D(2). We consider a simple procedure based on a modified Wiener filter. Its performance is compared with that from a bandpass followed by multiple notch filters. Our procedure leads to a gain in precision and accuracy when estimating D(2). The two filtering approaches are also compared with respect to a biomedical application proposed by others. Using data from 12 healthy subjects, the modified Wiener procedure raises the percentage of successes in that application from 17% to 83%. New experimental data suggest D(2) carries information not carried by r.m.s. or MF.

Nonlinear analysis and modeling of cortical activation and deactivation patterns in the immature fetal electrocorticogram

An approach combining time-continuous nonlinear stability analysis and a parametric bispectral method was introduced to better describe cortical activation and deactivation patterns in the immature fetal electroencephalogram (EEG). Signal models and data-driven investigations were performed to find optimal parameters of the nonlinear methods and to confirm the occurrence of nonlinear sections in the fetal EEG. The resulting measures were applied to the in utero electrocorticogram (ECoG) of fetal sheep at 0.7 gestation when organized sleep states were not developed and compared to previous results at 0.9 gestation. Cycling of the nonlinear stability of the fetal ECoG occurred already at this early gestational age, suggesting the presence of premature sleep states. This was accompanied by cycling of the time-variant biamplitude which reflected ECoG synchronization effects during premature sleep states associated with nonrapid eye movement sleep later in gestation. Thus, the combined nonlinear and time-variant approach was able to provide important insights into the properties of the immature fetal ECoG.

TRANSITIONS FROM STABLE EQUILIBRIA TO CHAOS, AND BACK, IN AN EXPERIMENTAL FOOD WEB

The question of whether deterministic chaos occurs in natural populations has been discussed since the 1970s following the discovery that simple population models can generate chaotic dynamics. Natural populations undergo a diverse mixture of deterministic and stochastic processes that define population dynamics. In most habitats populations are also exposed to changes in biotic and abiotic parameters. Models predict that shifts in ecological parameters may lead to a transition between deterministic chaos, stable equilibria, and limit cycles, yet clear examples from empirical studies are rare. However, such transitions should be considered when discussing the occurrence of chaos in nature because ecological time series are in general short and have large sampling intervals. Here we document short-term transitions in population dynamics to and from chaos in an experimental system. Manipulation of only one experimental parameter (chemostat dilution rate) in a multi-species food web of two bacteria and a bacterivorous ciliate showed that switching between different dynamic behaviors occured with surprising rapidity in the microbial populations. Thus, short periods of chaotic dynamics may easily be overlooked in field observations.

Recurrence quantification analysis of surface electromyographic signal: sensitivity to potentiation and neuromuscular fatigue

This study aimed to assess the capacity of recurrence quantification analysis (RQA) to detect potentiation and to determine the fatigue components to which RQA is sensitive. Fifteen men were divided in two groups [8 endurance-trained athletes (END) and 7 power-trained athletes (POW)]. They performed a 10-min intermittent (5s contraction, 5s rest) knee extension exercise at 50% of their maximal voluntary isometric contraction. Muscular fatigue and potentiation were evaluated with neurostimulation technique. Mechanical (peak torque, Pt) and electrophysiological (M-wave) responses following electrical stimulation of the femoral nerve were measured at rest and every 10s throughout exercise. Vastus lateralis muscle activity (root mean square, RMS) was recorded during each contraction, and RMS was normalized to M-wave area (RMS/M). During contraction, muscle activity was analyzed with RQA to obtain the percentage of determinism (%Det). At the beginning of exercise, a significant Pt increase (+52%, P<0.001) was observed in both groups, indicating potentiation. At this time, %Det remained constant in both groups, indicating that RQA did not detect potentiation. Thereafter, Pt decreased in POW from 5min 30s of exercise (-30%, P<0.001), reflecting impairment in excitation-contraction coupling, and %Det increased from 3min 30s (P<0.01). In END, Pt remained high and %Det was unchanged. These two results indicated that RQA detected the peripheral component of fatigue. Conversely, RQA did not detect central adaptation to fatigue since %Det remained constant when a significant increase in RMS/M (P<0.01) appeared in END.

Verification of chaotic behavior in an experimental loudspeaker

The dynamics of an experimental electrodynamic loudspeaker is studied by using the tools of chaos theory and time series analysis. Delay time, embedding dimension, fractal dimension, and other empirical quantities are determined from experimental data. Particular attention is paid to issues of stationarity in a system in order to identify sources of uncertainty. Lyapunov exponents and fractal dimension are measured using several independent techniques. Results are compared in order to establish independent confirmation of low dimensional dynamics and a positive dominant Lyapunov exponent. We thus show that the loudspeaker may function as a chaotic system suitable for low dimensional modeling and the application of chaos control techniques.

Time series analysis of spontaneous upper-extremity movements of premature infants with brain injuries

BACKGROUND AND PURPOSE

Comparisons of spontaneous movements of premature infants with brain injuries and those without brain injuries can provide insights into normal and abnormal processes in the ontogeny of motor development. In this study, the characteristics of spontaneous upper-extremity movements of premature infants with brain injuries and those without brain injuries were examined with time series analysis.

SUBJECTS

Participants were 7 premature infants with brain injuries and 7 matched, low-risk, premature infants at the age of 1 month after term.

METHODS

A triaxial accelerometer was used to measure upper-extremity limb acceleration in 3-dimensional space. Acceleration signals were recorded from the right wrist when the infant was in an active, alert state and lying in the supine position. The recording time was 200 seconds. The acceleration signal was sampled at a rate of 200 Hz. The acceleration time series data were analyzed by nonlinear analysis as well as linear analysis.

RESULTS

The nonlinear time series analysis indicated that spontaneous movements of premature infants have nonlinear, chaotic, dynamic characteristics. The movements of the infants with brain injuries were characterized by larger dimensionality, and they were more unstable and unpredictable than those of infants without brain injuries.

DISCUSSION AND CONCLUSION

As determined by nonlinear analysis, the spontaneous movements of the premature infants with brain injuries had the characteristics of increased disorganization compared with those of the infants without brain injuries. Infants with brain injuries may manifest problems with self-organization as a function of the coordination of subsystems. Physical therapists should be able to support interactions among the subsystems and promote self-organization of motor learning through the individualized provision of various sensorimotor experiences for infants.

Faster walking speeds increase local instability among people with peripheral neuropathy

Individuals with peripheral neuropathy (PN) may compensate for decreased somatosensation by reducing walking speed. Predisposition to falls may therefore arise from an inability to adapt to challenging walking speeds. The purpose of this study was to examine the effects of PN on the magnitude of variability and local instability on walking at different speeds. Twelve individuals with PN and 12 controls completed a 6-min walk test to determine fast walking speed (FWS). Sagittal plane hip, knee, and ankle joint angles were then calculated during 3 min of treadmill walking at 100%, 80%, and 60% FWS. The magnitudes of stride duration variability (SDvar), joint angle variability (JTvar), and both short- and long-term Lyapunov exponents (used to estimate local instability) were calculated. The PN group walked slower than the control group (p<.001). With groups combined, walking faster led to increased local instability and increased variability (p<.001). The PN group exhibited increased variability (SDvar, p=.02; JTvar, p=.01) over all speeds, and exaggerated local instability (p<.05) when walking at the fastest speed. PN leads to increased walking variability and local instability, particularly when walking at challenging speeds. These results are important to consider in future patient education and rehabilitation programs.

Modelling of polysomnographic respiratory measurements for artefact detection and signal restoration

Polysomnography (PSG), which incorporates measures of sleep with measures of EEG arousal, air flow, respiratory movement and oxygenation, is universally regarded as the reference standard in diagnosing sleep-related respiratory diseases such as obstructive sleep apnoea syndrome. Over 15 channels of physiological signals are measured from a subject undergoing a typical overnight PSG session. The signals often suffer from data losses, interferences and artefacts. In a typical sleep scoring session, artefact-corrupted signal segments are visually detected and removed from further consideration. This is a highly time-consuming process, and subjective judgement is required for the job. During typical sleep scoring sessions, the target is the detection of segments of diagnostic interest, and signal restoration is not utilized for distorted segments. In this paper, we propose a novel framework for artefact detection and signal restoration based on the redundancy among respiratory flow signals. We focus on the air flow (thermistor sensors) and nasal pressure signals which are clinically significant in detecting respiratory disturbances. The method treats the respiratory system and other organs that provide respiratory-related inputs/outputs to it (e.g., cardiovascular, brain) as a possibly nonlinear coupled-dynamical system, and uses the celebrated Takens embedding theorem as the theoretical basis for signal prediction. Nonlinear prediction across time (self-prediction) and signals (cross-prediction) provides us with a mechanism to detect artefacts as unexplained deviations. In addition to detection, the proposed method carries the potential to correct certain classes of artefacts and restore the signal. In this study, we categorize commonly occurring artefacts and distortions in air flow and nasal pressure measurements into several groups and explore the efficacy of the proposed technique in detecting/recovering them. The results we obtained from a database of clinical PSG signals indicated that the proposed technique can detect artefacts/distortions with a sensitivity>88.3% and specificity>92.4%. This work has the potential to simplify the work done by sleep scoring technicians, and also to improve automated sleep scoring methods.

Differential effects of plantar desensitization on locomotion dynamics

Reduced plantar sensation secondary to chronic diffuse polyneuropathy (PN) is believed to reduce locomotor stability, especially when walking at non-preferred speeds. However, the contribution of plantar sensation to the maintenance of locomotor stability is not entirely clear. The purpose of this study was to examine the effects of acute loss of plantar sensation on the stability-related kinematic properties of walking at different speeds. Lower-extremity joint kinematics were acquired as healthy young adults walked on a treadmill at their preferred walking speed (PWS) and three predetermined speeds (0.8, 1.0, and 1.2m/s) under both normal and desensitized conditions. Desensitization of the foot soles was induced by ice-exposure, and plantar pressure sensation was assessed by a 5.07 monofilament. The average magnitude of stride duration variability (SDvar) and lower-extremity joint angle variability (JTvar), as well as short- and long-term "finite-time" Lyapunov exponents (lambda(ST)(*), lambda(LT)(*)) associated with lower-extremity joint angles were computed. Ice-induced plantar desensitization led to increased lambda(ST)(*) ( approximately 40%) and lambda(LT)(*) (approximately 8%) values but did not affect SDvar or JTvar. Higher treadmill speed led to greater lambda(ST)(*) and lambda(LT)(*) values, but the speed effects were not influenced by plantar desensitization.While acute loss of plantar sensation does not appear to influence the magnitude of spatial or temporal variability, it did attenuate the state-space trajectory divergence caused by stride-to-stride variability (i.e., lambda(ST)( *) and lambda(LT)(*)). However, as opposed to walking at PWS, otherwise healthy locomotor systems do not appear to place increased reliance on plantar sensation when walking at non-preferred treadmill speeds.

Comparison of tests for embeddings

It is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test. The classical tests, which depend on real numbers (fractal dimensions, Lyapunov exponents) averaged over an attractor, are compared with a topological test that depends on integers. The comparison can only be done for mappings into three dimensions. We find that the classical tests fail to predict when a mapping is an embedding and when it is not. We point out the reasons for this failure, which are not restricted to three dimensions.

Automatic selection of the threshold value R for approximate entropy

Calculation of approximate entropy (ApEn) requires a priori determination of two unknown parameters, m and r. While the recommended values of r, in the range of 0.1-0.2 times the standard deviation of the signal, have been shown to be applicable for a wide variety of signals, in certain cases, r values within this prescribed range can lead to an incorrect assessment of the complexity of a given signal. To circumvent this limitation, we recently advocated finding the maximum ApEn value by assessing all values of r from 0 to 1, and found that maximum ApEn does not always occur within the prescribed range of r values. Our results indicate that finding the maximum ApEn leads to the correct interpretation of a signal's complexity. One major limitation, however, is that the calculation of all choices of r values is often impractical due to the computational burden. Our new method, based on a heuristic stochastic model, overcomes this computational burden, and leads to the automatic selection of the maximum ApEn value for any given signal. Based on Monte Carlo simulations, we derive general equations that can be used to estimate the maximum ApEn with high accuracy for a given value of m. Application to both synthetic and experimental data confirmed the advantages claimed with the proposed approach.

Complex dynamics in the Oregonator model with linear delayed feedback

The Belousov-Zhabotinsky (BZ) reaction can display a rich dynamics when a delayed feedback is applied. We used the Oregonator model of the oscillating BZ reaction to explore the dynamics brought about by a linear delayed feedback. The time-delayed feedback can generate a succession of complex dynamics: period-doubling bifurcation route to chaos; amplitude death; fat, wrinkled, fractal, and broken tori; and mixed-mode oscillations. We observed that this dynamics arises due to a delay-driven transition, or toggling of the system between large and small amplitude oscillations, through a canard bifurcation. We used a combination of numerical bifurcation continuation techniques and other numerical methods to explore the dynamics in the strength of feedback-delay space. We observed that the period-doubling and quasiperiodic route to chaos span a low-dimensional subspace, perhaps due to the trapping of the trajectories in the small amplitude regime near the canard; and the trapped chaotic trajectories get ejected from the small amplitude regime due to a crowding effect to generate chaotic-excitable spikes. We also qualitatively explained the observed dynamics by projecting a three-dimensional phase portrait of the delayed dynamics on the two-dimensional nullclines. This is the first instance in which it is shown that the interaction of delay and canard can bring about complex dynamics.

Detecting anomalous phase synchronization from time series

Modeling approaches are presented for detecting an anomalous route to phase synchronization from time series of two interacting nonlinear oscillators. The anomalous transition is characterized by an enlargement of the mean frequency difference between the oscillators with an initial increase in the coupling strength. Although such a structure is common in a large class of coupled nonisochronous oscillators, prediction of the anomalous transition is nontrivial for experimental systems, whose dynamical properties are unknown. Two approaches are examined; one is a phase equational modeling of coupled limit cycle oscillators and the other is a nonlinear predictive modeling of coupled chaotic oscillators. Application to prototypical models such as two interacting predator-prey systems in both limit cycle and chaotic regimes demonstrates the capability of detecting the anomalous structure from only a few sets of time series. Experimental data from two coupled Chua circuits shows its applicability to real experimental system.

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.

Nonlinear analysis of electroencephalogram in schizophrenia patients with persistent auditory hallucination

OBJECTIVE

The recent nonlinear analyses of electroencephalogram (EEG) data have shown that the correlation dimension (D2) reflects the degree of integration of information processing in the brain. There is now considerable evidence that auditory hallucination (AH) reflects dysfunctional gamma and beta frequency oscillations. Gamma oscillations are thought to reflect internally driven representations of objects, and the occurrence of subsequent beta oscillations can reflect the modification of the neuronal circuitry used to encode the sensory perception. The purpose of this study was to test whether AH in schizophrenia patients is reflected in abnormalities in D2 in their EEG, especially in the gamma and beta frequency bands.

METHODS

Twenty-five schizophrenia patients with a history of treatment-refractory AH over at least the past 2 years, and 23 schizophrenia patients with no AH (N-AH) within the past 2 years were recruited for the study. Artifact-free 30-s EEG epochs during rest were examined for D2.

RESULTS

The AH patients showed significantly increased gamma frequency D2 in Fp2 and decreased beta frequency D2 in the P3 region compared with the N-AH patients. These results imply that gamma frequency D2 in the right prefrontal cortex is more chaotic and that beta frequency D2 in the left parietal cortex is more coherent (less chaotic) in AH patients than in N-AH patients.

CONCLUSION

Our study supports the previous evidence indicating that gamma and beta oscillations are pivotal to AH, and also shows the distinctive dimensional complexity between the right prefrontal and left parietal cortexes as the underlying biological correlates of AH in schizophrenia patients.

Using recurrence plot for determinism analysis of EEG recordings in genetic absence epilepsy rats

OBJECTIVE

Understanding the transition of brain activity towards an absence seizure is a challenging task. In this paper, we use recurrence quantification analysis to indicate the deterministic dynamics of EEG series at the seizure-free, pre-seizure and seizure states in genetic absence epilepsy rats.

METHODS

The determinism measure, DET, based on recurrence plot, was applied to analyse these three EEG datasets, each dataset containing 300 single-channel EEG epochs of 5-s duration. Then, statistical analysis of the DET values in each dataset was carried out to determine whether their distributions over the three groups were significantly different. Furthermore, a surrogate technique was applied to calculate the significance level of determinism measures in EEG recordings.

RESULTS

The mean (+/-SD) DET of EEG was 0.177+/-0.045 in pre-seizure intervals. The DET values of pre-seizure EEG data are significantly higher than those of seizure-free intervals, 0.123+/-0.023, (P<0.01), but lower than those of seizure intervals, 0.392+/-0.110, (P<0.01). Using surrogate data methods, the significance of determinism in EEG epochs was present in 25 of 300 (8.3%), 181 of 300 (60.3%) and 289 of 300 (96.3%) in seizure-free, pre-seizure and seizure intervals, respectively.

CONCLUSIONS

Results provide some first indications that EEG epochs during pre-seizure intervals exhibit a higher degree of determinism than seizure-free EEG epochs, but lower than those in seizure EEG epochs in absence epilepsy.

SIGNIFICANCE

The proposed methods have the potential of detecting the transition between normal brain activity and the absence seizure state, thus opening up the possibility of intervention, whether electrical or pharmacological, to prevent the oncoming seizure.

Local dynamic stability in turning and straight-line gait

Successful community and household ambulation require the ability to navigate corners and maneuver around obstacles, posing unique challenges compared to straight-line walking. The challenges associated with turning may contribute to an increased incidence of falling and the occurrence of fall-related injuries. A measure of stability applied to turning gait may be able to quantify a system's response to naturally occurring disturbances associated with turning and identify subjects at greater risk of falling. An index of stability has been used previously to assess the rate of kinematic separation (local dynamic stability) during straight-line gait. The purpose of this study was to determine if local dynamic stability during constant speed turning is reduced compared to straight-line treadmill walking. Maximum finite-time Lyapunov exponents (lambda) were used to estimate the local stability of able-bodied subjects' (n=19) sagittal plane hip, knee, and ankle trajectories for turning compared to straight-line walking at two different walking speeds. Turning lambda was greater than straight lambda for the hip, right knee, and ankle (p<0.05). Turning lambda for the left knee angle was similar to straight lambda. There were no differences in lambda between left and right limbs for the hip and ankle and also no differences between the inside and outside limbs during turning for all joints. These findings indicate able-bodied subjects' hip, right knee, and ankle kinematics are less locally stable while turning than walking in a straight line and may be used as a comparative tool for determining the efficacy of therapeutic interventions for mobility-impaired populations.

Quantitative complexity analysis in multi-channel intracranial EEG recordings form epilepsy brains

Epilepsy is a brain disorder characterized clinically by temporary but recurrent disturbances of brain function that may or may not be associated with destruction or loss of consciousness and abnormal behavior. Human brain is composed of more than 10 to the power 10 neurons, each of which receives electrical impulses known as action potentials from others neurons via synapses and sends electrical impulses via a sing output line to a similar (the axon) number of neurons. When neuronal networks are active, they produced a change in voltage potential, which can be captured by an electroencephalogram (EEG). The EEG recordings represent the time series that match up to neurological activity as a function of time. By analyzing the EEG recordings, we sought to evaluate the degree of underlining dynamical complexity prior to progression of seizure onset. Through the utilization of the dynamical measurements, it is possible to classify the state of the brain according to the underlying dynamical properties of EEG recordings. The results from two patients with temporal lobe epilepsy (TLE), the degree of complexity start converging to lower value prior to the epileptic seizures was observed from epileptic regions as well as non-epileptic regions. The dynamical measurements appear to reflect the changes of EEG's dynamical structure. We suggest that the nonlinear dynamical analysis can provide a useful information for detecting relative changes in brain dynamics, which cannot be detected by conventional linear analysis.

Influence of embedding parameters and noise in center of pressure recurrence quantification analysis

Recurrence quantification analysis (RQA) can extract the dynamics of postural control from center of pressure (CoP) data by quantifying the system's repeatability, complexity, and local dynamic stability through several variables. Computation of these variables requires the selection of suitable embedding parameters for state space reconstruction (i.e. time delay and embedding dimension); however, it is unclear how the parameters influence RQA variables when examining noisy CoP data. This study evaluated the sensitivity of RQA variables to embedding parameter values and noise level, and assessed methods of selecting embedding parameters for CoP data. Five healthy male subjects maintained quiet stance for 30s while the anterior-posterior CoP was measured. The effect of noise was evaluated by adding uniform white noise of increasing amplitude to the raw CoP signal. The magnitude of all RQA variables decreased with increasing noise amplitude for all subjects. A sensitivity analysis was performed by systematically altering the embedding parameters for the raw data with and without a selected level of added noise. The key result was that, for all subjects, the RQA variables were sensitive to the embedding parameter values and the level of noise in the CoP data. Finally, the performance of false nearest neighbors and average displacement algorithms for choosing embedding parameters was evaluated. Both methods gave clear and consistent results for all subjects with either raw or noisy data. The results suggest that careful selection of embedding parameters is essential when using RQA to examine postural control based on noisy CoP data.

Observability of nonlinear dynamics: normalized results and a time-series approach

This paper investigates the observability of nonlinear dynamical systems. Two difficulties associated with previous studies are dealt with. First, a normalized degree observability is defined. This permits the comparison of different systems, which was not generally possible before. Second, a time-series approach is proposed based on omnidirectional nonlinear correlation functions to rank a set of time series of a system in terms of their potential use to reconstruct the original dynamics without requiring the knowledge of the system equations. The two approaches proposed in this paper and a former method were applied to five benchmark systems and an overall agreement of over 92% was found.

Comparison of nonlinear Granger causality extensions for low-dimensional systems

The identification of hidden interdependences among the parts of a complex system is a fundamental issue. Typically, the objective is, given only a sequence of scalar measurements, to infer as much as possible about the internal dynamics of the system and about the interactions between its subsystems. In general, such interactions are not only nonlinear but also asymmetric. Constraints on the estimation of hidden relationships are further posed by noise and by the length of signals sampled from real world systems. The focus of this paper is causal dependences between bivariate time series. We especially focus on the nonlinear extension of Granger causality with polynomial terms of the conventional embedding vector. In this paper, we study the performance of this measure in comparison with three alternative methods proposed recently in low-dimensional and low-order-nonlinearity systems. Those methods are tested with three different artificial chaotic maps with several noise contamination setups. As a result, we find that the polynomial embedding technique successfully detects asymmetric (causal) dependences between bivariate time series in many low-dimensional cases.

Bifurcations and chaos in register transitions of excised larynx experiments

Experimental data from an excised larynx are analyzed in the light of nonlinear dynamics. The excised larynx provides an experimental framework that enables artificial control and direct observation of the vocal fold vibrations. Of particular interest in this experiment is the coexistence of two distinct vibration patterns, which closely resemble chest and falsetto registers of the human voice. Abrupt transitions between the two registers are typically accompanied by irregular vibrations. Two approaches are presented for the modeling of the excised larynx experiment; one is the nonlinear predictive modeling of the experimental time series and the other is the biomechanical modeling (three-mass model) that takes into account basic mechanisms of the vocal fold vibrations. The two approaches show that the chest and falsetto vibrations correspond to two coexisting limit cycles, which jump to each other with a change in the bifurcation parameter. Irregular vibrations observed at the register jumps are due to chaos that exists near the two limit cycles. This provides an alternative mechanism to generate chaotic vibrations in excised larynx experiment, which is different from the conventionally known mechanisms such as strong asymmetry between the left and right vocal folds or excessively high subglottal pressure.

Recurrence plots for dynamical analysis of non-invasive mechanical ventilation

Quantifiers were introduced to convert recurrence plots into a statistical analysis of dynamical properties. It is shown that the Shannon entropy, if properly computed, increases as the chaotic regime is developed as expected. Recurrence plots and a new estimator for the Shannon entropy are then used to identify asynchronisms in non-invasive mechanical ventilation. It is thus shown that the phase coherence-easily identified using a Shannon entropy-is relevant in the quality of the mechanical ventilation. In particular, some patients with chronic respiratory diseases or healthy subjects can have a high rate of asynchronisms but a regular breathing rhythm.

Detecting direction of causal interactions between dynamically coupled signals

The problem of temporal localization and directional mapping of the dynamic interdependencies between parts of a complex system is addressed. We present a technique that weights the sampled values so as to minimize the mutual prediction error between pairs of measured signals. The reliability of the detected intermittent causal interactions is maximized by (a) smoothing the weight landscape through regularization, and (b) using a nonlinear (polynomial) variant of the conventional embedding vector. The effectiveness of the proposed technique is demonstrated by studying three numerical examples of dynamically coupled chaotic maps and by comparing it with two other measures of causal dependency.

Nonlinear dynamical analysis of the neonatal EEG time series: the relationship between neurodevelopment and complexity

OBJECTIVE

To investigate the relationship between the complexity of sleep EEG time series and neurodevelopment for premature or full-term neonates.

METHODS

Nonlinear dynamical analysis of neonatal sleep EEG time series is used to compute the correlation dimension D2 which is an index of the complexity of the dynamics of the developing brain. The dimensional complexity is estimated using Theiler's modification of the Grassberger-Procaccia algorithm for two different values of Theiler's w parameter. The hypothesis that neonatal EEG data during sleep contains nonlinear features is verified by means of surrogate data testing.

RESULTS

The dimensional complexity of the neonatal EEG increases with neurodevelopment and brain maturation. There is furthermore a statistically significant difference between the dimensional complexity of the EEG for neonates born prematurely when compared to full-term neonates at the same postmenstrual age (PMA). The neonatal EEG time series data used in this study proved to contain nonlinear features where the 'null hypothesis' of surrogate data testing is rejected with p<<0.0001.

CONCLUSIONS

A relationship between neurodevelopment and brain maturation and the complexity of the dynamics of the brain as measured by the dimensional complexity of the sleep EEG time series has been established. In particular, the dimensional complexity tends to increase with neurodevelopment and maturation as indicated by their PMA and birth status (premature or full-term). In particular, the brain dynamics of neonates born prematurely is less complex than the brain dynamics of neonates born full-term even at the same PMA. We attribute this to differences in the neurodevelopment between these two cohorts. We propose that the dimensional complexity can be used as an index for quantifying neurodevelopment.

SIGNIFICANCE

The dimensional complexity as measured by the correlation dimension of the sleep EEG time series may potentially be a useful measure for quantifying neurodevelopment in neonates. Future work is directed at the analysis of other EEG channels to understand the relationship between complexity in different regions of the brain and maturation and neurodevelopment, along with the utility of complexity to relate to neurodevelopment at older ages as measured by the Bayley score.

Finding alternatives and reduced formulations for process-based models

This paper addresses the problem of model complexity commonly arising in constructing and using process-based models with intricate interactions. Apart from complex process details the dynamic behavior of such systems is often limited to a discrete number of typical states. Thus, models reproducing the system's processes in all details are often too complex and over-parameterized. In order to reduce simulation times and to get a better impression of the important mechanisms, simplified formulations are desirable. In this work a data adaptive model reduction scheme that automatically builds simple models from complex ones is proposed. The method can be applied to the transformation and reduction of systems of ordinary differential equations. It consists of a multistep approach using a low dimensional projection of the model data followed by a Genetic Programming/Genetic Algorithm hybrid to evolve new model systems. As the resulting models again consist of differential equations, their process-based interpretation in terms of new state variables becomes possible. Transformations of two simple models with oscillatory dynamics, simulating a mathematical pendulum and predator-prey interactions respectively, serve as introductory examples of the method's application. The resulting equations of force indicate the predator-prey system's equivalence to a nonlinear oscillator. In contrast to the simple pendulum it contains driving and damping forces that produce a stable limit cycle.

Irregular firing of isolated cortical interneurons in vitro driven by intrinsic stochastic mechanisms

Pharmacologically isolated GABAergic irregular spiking and stuttering interneurons in the mouse visual cortex display highly irregular spike times, with high coefficients of variation approximately 0.9-3, in response to a depolarizing, constant current input. This is in marked contrast to cortical pyramidal cells, which spike quite regularly in response to the same current injection. We applied time-series analysis methods to show that the irregular behavior of the interneurons was not a consequence of low-dimensional, deterministic processes. These methods were also applied to the Hindmarsh and Rose neuronal model to confirm that the methods are adequate for the types of data under investigation. This result has important consequences for the origin of fluctuations observed in the cortex in vivo.