Pointwise Correlation Dimension Research Articles

Dissociative states and neural complexity

Recent findings indicate that neural mechanisms of consciousness are related to integration of distributed neural assemblies. This neural integration is particularly vulnerable to past stressful experiences that can lead to disintegration and dissociation of consciousness. These findings suggest that dissociation could be described as a level of neural disintegration reflecting a number of independent processes by means of neural complexity. In the present study measurement of dissociation, traumatic stress symptoms and neural complexity calculated using nonlinear analysis of EEG [during rest conditions], and electrodermal activity (EDA) [during rest and Stroop task] were performed in 52 university students (mean age 24.1). Neural complexity has been described using pointwise correlation dimension (PD2) calculated from EEG and EDA records. While no significant relationship was found between EEG complexity and dissociative symptoms, statistically significant relationship between EDA complexity and dissociative symptoms during rest, but not during the Stroop task, has been found. These results indicate that electrodermal complexity during rest may reflect a level of dissociative symptoms.

Electrodermal complexity during the Stroop Colour Word Test

Several recent studies suggest that quantitative description of signal complexity using algorithms of nonlinear analysis could uncover new information about the autonomic system that is not reflected using common methods applied to measures of autonomic activity. With this aim we have performed complexity analysis of electrodermal activity (EDA) assessed in 106 healthy university students during rest conditions and non-conflicting and conflicting Stroop task. Complexity analysis applied to EDA was performed using Skinner's algorithm for pointwise correlation dimension (PD2). Results have shown that EDA responses during the Stroop Colour Word test are related to significantly increased or decreased complexity. Particularly significant result is that PD2 has a unique ability to predict to an extent the change in EDA response to stress i.e. that subjects with low initial PD2 tended to respond to experimental stress by its increase and subjects with high initial PD2 values tended to respond by its decrease. This response was not found in EDA measures where increase of the EDA presented predominant response to experimental stress in majority of the subjects. These findings suggest that PD2 is more sensitive to subtle aspects of functionally and spatially distributed modulatory influences of various parts of the brain that are involved in the EDA modulation and provides novel information in comparison to traditional methods.

Effects of stress on heart rate complexity—A comparison between short-term and chronic stress

This study examined chronic and short-term stress effects on heart rate variability (HRV), comparing time, frequency and phase domain (complexity) measures in 50 healthy adults. The hassles frequency subscale of the combined hassles and uplifts scale (CHUS) was used to measure chronic stress. Short-term stressor reactivity was assessed with a speech task. HRV measures were determined via surface electrocardiogram (ECG). Because respiration rate decreased during the speech task ( p < .001), this study assessed the influence of respiration rate changes on the effects of interest. A series of repeated-measures analyses of covariance (ANCOVA) with Bonferroni adjustment revealed that short-term stress decreased HR D2 (calculated via the pointwise correlation dimension PD2) ( p < .001), but increased HR mean ( p < .001), standard deviation of R–R (SD RR) intervals ( p < .001), low (LF) ( p < .001) and high frequency band power (HF) ( p = .009). Respiratory sinus arrhythmia (RSA) and LF/HF ratio did not change under short-term stress. Partial correlation adjusting for respiration rate showed that HR D2 was associated with chronic stress ( r = −.35, p = .019). Differential effects of chronic and short-term stress were observed on several HRV measures. HR D2 decreased under both stress conditions reflecting lowered functionality of the cardiac pacemaker. The results confirm the importance of complexity metrics in modern stress research on HRV.

A New Potential Marker for Abnormal Cardiac Physiology in Depression

The purpose of the study was to determine if a nonlinear estimate of dimensional complexity, Pointwise correlation dimension (Pd2), could be used to identify abnormal cardiac physiology associated with depression in primary care outpatients. The subjects were 22 medical controls and 30 general medical outpatients with depression who wore a cardiac monitor for 24 h. There was a significant difference between depressed and control subjects for Pd2 based on the entire cardiac time-series, with depressed subjects exhibiting higher Pd2 values. A cutoff score reflecting high dimensional complexity was selected, and an odds ratio was calculated demonstrating that patients with Pd2 values above the cut-off were 8.8 times more likely to fall in the depressed group than the control group. Additional research is needed to determine if Pd2 is useful for identifying physiological markers of depression.

Complexity of the dynamic QT variability and RR variability in patients with acute anterior wall myocardial infarction: A novel technique using a non-linear method

QT and RR intervals’ fluctuations are implicated in the development of malignant arrhythmias. Recent research has quantified repolarization lability using stochastic and linear methods. However, QT-RR intervals are nonlinearly coupled. To this end, QT and RR intervals were extracted from twenty four patients with acute myocardial infarction (AMI) and 13 controls, and a measure of local dimensional complexity (pointwise correlation dimension, PD2) was calculated. PD2 of QT intervals was significantly higher for the patients than for controls (4.83 ± 0.81 versus 3.40 ± 0.76, P = .0001), and vice versa for RR intervals (2.51 ± 0.62 versus 2.91 ± 0.42, P = .028). The RR and QT measures of complexity were highly correlated only among controls (r = 0.769, P = .0021). Our results support the presence of autonomic abnormalities during infarction and might complement existing tools for assessment of increased risk for sudden death after AMI.

Heart rate variability increases in elderly depressed patients who respond to electroconvulsive therapy

Objectives To evaluate the parasympathetic modulation in elderly inpatients with major depressive disorder (MDD) before and after electroconvulsive therapy (ECT) using both linear and nonlinear methods of heart rate variability (HRV) analysis. Method A measure of local dimensional complexity (pointwise correlation dimension, PD2), as well as spectral analysis measures (LF, low-frequency range; HF, high-frequency range) were calculated for the heart rate time series of 10 elderly inpatients with MDD (70±7 years) before and after ECT. Hamilton Depression Rating Scale (HAM-D) was evaluated concomitantly. Results Only the responders to ECT ( n=7; ≥50% reduction in HAM-D) exhibited a significant increase in PD2 ( P=.0035), which showed a tendency towards a correlation with symptom improvement ( r=.73, P=.06). Spectral analysis measures did not show a significant difference after ECT. Conclusion Elderly patients with MDD, who respond to ECT, might show increased vagal modulation. Since nonlinear HRV measures have been shown to be reduced by aging, similar to cholinergic deficit, they might shed light on the increased risk for cardiac mortality in depression.

Complexity of visual stimuli and non-linear EEG dynamics in humans

The effects of stimulus complexity on the nonlinear electrical brain (EEG) dynamics were investigated in a sample of 24 healthy volunteers. Stimuli used were either a single mechanical low-friction pendulum with a periodic movement (temporal frequency about 1 Hz) or a double-pendulum with a chaotic movement, which were observed for 2–3 min in each case. The prediction that a more complex visual stimulus (double-pendulum) increases the dimensional complexity of brain activity as compared to a simple visual stimulus (single-pendulum), was confirmed by determination of pointwise correlation dimension. Further, there was a significant decrease of alpha power in the double-pendulum compared to a single-pendulum condition. Moreover, a correlation analysis showed a positive correlation between EEG complexity and beta power over the whole cortex in the single- and, above all, in the double-pendulum condition, and also a positive correlation between dimensional complexity and alpha power in the double-pendulum condition only, particularly in the brain regions responsible for the ‘bottom-up’ sustained attention processes.

Nonlinear data analysis of experimental (EEG) data and comparison with theoretical (ANN) data

AbstractIn this article, nonlinear dynamical tools such as largest Lyapunov exponents (LE), fractal dimension, correlation dimension, pointwise correlation dimension will be used to analyze electroencephalogram (EEG) data obtained from healthy young subjects with eyes open and eyes closed condition with the view to compare brain complexity under this two condition. Results of similar calculations from some earlier works will be produced for comparison with present results. Also, a brief report on difference of opinion among coworkers regarding such tools will be reported; particularly applicability of LE will be reviewed. The issue of nonlinearity will be addressed by using surrogate data technique. We have extracted another data set that represented chaotic state of the system considered in our earlier work of mathematical modeling of artificial neural network. We further attempt to compare results to find nature of chaos arising from such theoretical models. © 2002 Wiley Periodicals, Inc.

Nonlinear Dynamics Estimation of EEG Signals Accompanying Self-Paced Goal-Directed Movements

The study was aimed at comparison of several nonlinear characteristics (NC) computed in time for one and the same EEG record accompanying voluntary goal-directed movements. This allowed us to reveal different aspects of the nonlinear behavior of the process underlying the self-paced movement organization. Parallel alterations of these characteristics during the task performance supported the hypothesis that cognitive task performance is reflected by the changes in the nonlinear dynamics of the EEG activity. Four NCs of scalar EEG time series were estimated: Point-wise Correlation Dimension (PD2), Kolmogorov Entropy (K2), and Largest Lyapunov Exponent (LLE) as a function of time, and Nonlinear Prediction (NP) for successive EEG segments. The time evolution of these characteristics exhibited several transients between chaos-like states to almost periodic states during the task performance: the gradual increase to higher values indicating chaos are probably related to the onset of the successive phases of brain organization of the movement. The results suggest that even in short periods an EEG signal changes its dynamic structure and these periods could be determined precisely enough using different methods of nonlinear dynamics.

Early detection of acute allograft rejection by linear and nonlinear analysis of heart rate variability

Objective: The first months after orthotopic heart transplantation are associated with the highest risk of acute allograft rejection. This study explores the utility and reliability of linear and novel nonlinear metrics of heart rate variability as predictors of graft rejection. The underlying hypothesis is that the transplanted heart, in response to inflammatory mediators, alters the dynamic properties of its rhythm-generating system. Methods: In a cross-sectional study of 45 patients who had undergone heart transplantation, spanning a period of 4 months after the operation, heart rate variability was examined by time- and frequency-domain analysis. The nonlinear features of heart rate variability were studied by computing a pointwise correlation dimension of R-R interval time series. The results of heart rate variability analysis were compared with those of endomyocardial surveillance biopsy studies using the International Society for Heart and Lung Transplantation scoring system. Results: Duration of heart transplantation itself exhibited a significant (P < .05) association with the onset of rejection. Specific predictors of acute rejection based on heart rate variability were identified, including shortening of the R-R interval (from 700 ± 68 to 648 ± 72 ms), an increase in the ratio of low-frequency (0.04-0.15 Hz) to high-frequency (0.15-0.40 Hz) spectral power (from 0.3 ± 0.2 to 0.6 ± 0.4), and a decrease in pointwise correlation dimension values (from 1.7 ± 0.7 to 0.9 ± 0.3 units). Multivariable logistic regression analysis (R2 = 0.4) revealed that the only significant independent risk predictors were pointwise correlation dimension (odds ratio, 2.2 per 0.1 unit) and duration of heart transplantation (odds ratio, 1.7 per week). Conclusion: Nonlinear measures of heart rate variability provide noninvasive means for identifying patients undergoing cardiac transplantation with acute rejection, thereby enabling the assessment of the time-dependent adaptive response of the donor heart to its host. (J Thorac Cardiovasc Surg 2000;120:737-45)

Fractal organization of the pointwise correlation dimension of the heart rate

Objective: To depict and quantify the degree of organization of the heart rate variability (HRV) in normal subjects. Methods: A modified algorithm was created to estimate series of ‘point-dimensions’ ( PD2) from interbeat (R-R) interval series of 10 healthy subjects (21–56 years). Our innovation is twofold: (i) we quantified instances of low-dimensional chaos, random fluctuations, and those for which our method failed to provide either (due to poor statistics); (ii) consecutive subepochs of PD2s underwent a relative dispersion (RD) analysis, yielding an index ( D) which quantifies the dynamical organization of the heart rate generator. Results: The mean values of PD2 series varied between 4.58 and 5.88 (mean ± SD= 5.21 ± 0.41, n = 10). For group 1 (21–30 years, n = 6) we found an averaged PD2 of 5.49 ± 0.27, while for group 2 (47–56 years, n = 4) PD2 averaged 4.79 ± 0.17. The RD analysis performed for subepochs of PD2s yielded both instances obeying fractal scaling (D < 1.5) and stochasticity (D > 1.5). The average D for group 1 was 1.39 ± 0.04 (14 subepochs) and for group 2, 1.20 ± 0.008 (8 subepochs). Paired t-test and Hartley F-max test for comparison between D values and homogeneity of variance between the two groups were performed, yielding P-values 0.004 and 0.02, respectively. Conclusions: The complexity of the HRV seems to be modulated by a non-random fractal mechanism of a ‘hyperchaotic’ system, i.e. it can be hypothesized to contain more than one attractor. Also, our results support the ‘chaos hypothesis’ put forth recently, namely, the complexity of the cardiovascular dynamics is reduced with aging. The index of relative dispersion of the dimensional complexity has to be tested in various clinico-pathological settings, in order to corroborate its value as a potential new physiological measure.

Evolution in functional complexity of heart rate dynamics: a measure of cardiac allograft adaptability.

The capacity of self-organized systems to adapt is embodied in the functional organization of intrinsic control mechanisms. Evolution in functional complexity of heart rate variability (HRV) was used as measure of the capacity of the transplanted heart to express newly emergent regulatory order. In a cross-sectional study of 100 patients after (0-10 yr) heart transplantation (HTX), heart rate dynamics were assessed using pointwise correlation dimension (PD2) analysis. A new observation is that, commencing with the acute event of allograft transplantation, the dynamics of rhythm formation proceed through complex phase transitions. At implantation, the donor heart manifested metronome-like chronotropic behavior (PD2 approximately 1.0). At 11-100 days, dimensional complexity of HRV reached a peak (PD2 approximately 2.0) associated with resurgence in the high-frequency component (0.15-0.5 Hz) of the power spectral density. Subsequent dimensional loss to PD2 approximately 1.0 at 20-30 mo after HTX was followed by a progressive near-linear gain in system complexity, reaching PD2 approximately 3.0 7-10 yr after HTX. The "dynamic reorganization" in the allograft rhythm-generating system, seen in the first 100 days, is a manifestation of the adaptive capacity of intrinsic control mechanisms. The loss of HRV 2 yr after HTX implies a withdrawal of intrinsic autonomic control and/or development of an entrained dynamic pattern characteristic of extrinsic sympathetic input. The subsequent long-term progressive rise in dimensional complexity of HRV can be attributed to the restoration of a functional order patterning parasympathetic control. The recognition that the decentralized heart can restitute the multidimensional state space of HR generator dynamics independent of external autonomic signaling may provide a new perspective on principles that constitute homeodynamic regulation.

Nonlinear analysis of absence seizures by pointwise correlation dimension estimates

Nonlinear analysis of absence seizures by pointwise correlation dimension estimates

Failure of dimension analysis in a simple five-dimensional system

Dynamic systems may be characterized by their fractal dimension. The classical Grassberger-Procaccia algorithm is widely used to analyze time series. However, if this method is used beyond its intrinsic limitations it may cause incorrect classification of systems. We found that a simple deterministic five-dimensional system leads to erroneous dimension values around 5.5 if the following methods are used uncritically: The classical Grassberger-Procaccia algorithm, a pointwise correlation dimension algorithm, and an algorithm for calculation of the information dimension yielded this erroneous result for a wide range of numbers of data points (N=30 000--${10}^{6}$) and various delay times. Estimates of dimensions are only reliable if long plateaus of the local slope of the correlation integrals exist for small distances; these were not found in our example. This example suggests that a correlation dimension of 5 is too high to be recognized using even one million noise-free data points.

Variance structure in the human cardiovascular system—periodicity, chaos, or old-fashioned noise?

We discuss various traditional and new methods of modeling time series data to reveal dynamic characteristics. We consider cases in which measurements represent signals from a deterministic generator contaminated by additive or multiplicative noise. The properties of chaotic dynamics (nonlinear deterministic mechanisms) are also discussed. Fluctuations in cardiovascular data, as demonstrated in 24-hour ambulatory monitoring records of blood pressures, heart rate and core temperature, are then examined from these various perspectives, including spectral analysis. The pointwise correlation dimension calculated on R - R intervals on the 24-hour electrocardiogram (Holter record) is shown to reveal variations in physiological state that cannot be seen as clearly by spectral analysis. We conclude that both (near-) periodic models (simple systems) and chaotic models (complex systems) are useful, in that different groups of subjects can be compared by either analysis. Neither model by itself, using standard algorithms, has the capability of identifying the mechanism actually generating the time history. Thus, we answer the question posed in the title of this paper as follows: it is difficult to decide on the basis of models of data only because of the ubiquity of quasi-periodic motions based on noise modulation. For that reason, the model-free “regularity statistic” of Pincus may be a preferred descriptor of noisy data. (A preliminary version of this paper has appeared [1].)

A technique for measuring fractal dimensions from time series on a real-time scale

A dedicated electronic instrument for measuring fractal dimension of chaotic systems called “dimensiometer” is described. The pointwise correlation dimension can be obtained from experimental time series on a real-time scale. This technique employs a discrete sequence of the time intervals between the successive intersections of a single variable x( t) and an arbitrary level c. The “dimensiometer” is tested by means of a simple electronic chaos oscillator. The experimental results are compared with the numerical calculations carried out for the corresponding dynamical model.

Slope filtered pointwise correlation dimension algorithm and its evaluation with prefibrillation heart rate data.

Various studies have shown that a low variability in heart rate is associated with increased risk of ventricular fibrillation. Low chaotic (correlation) dimension in the heart rate also appears to predict fibrillation risk. However, these results have been based on intergroup comparisons and have not been found useful for predicting when a patient may fibrillate with any degree of sensitivity, specificity, or temporal accuracy. There are two primary limitations in using dimensional analysis to predict imminent fibrillation. The first is that the standard algorithms (for correlation dimension) assume stationarity of the system. The second limitation is that these algorithms require 10,000-50,000 data points to achieve good accuracy. Thus, even if stationarity were not an issue, there would be a lag of 2.4-12 hours to warn of impending fibrillation. An algorithm has been developed to calculate an accurate pointwise correlation dimension of heart rate data. The slope filtered pointwise correlation dimension algorithm requires as few as 1,000 points of data. Using this algorithm, it was found that the correlation dimension dropped from 2.50 +/- 0.81 to 1.07 +/- 0.18 in the minute before fibrillation in conscious pigs with an occluded coronary artery. In clinical studies, Holter tapes from patients that had suffered fatal fibrillation were also analyzed along with healthy controls and nonfibrillation ventricular patients. The fibrillation patients all had excursions of low dimension (less than 1.5), while the majority of the others did not. In the minutes before fibrillation, the correlation dimension dropped to a steady range of 0.8-1.3. Drops in the slope filtered pointwise correlation dimension appear to predict fibrillation in animals and patients.

Variability of pressures and rates in the human cardiovascular system — Is it chaos or old fashioned noise?

We discuss various traditional and new methods of modelling time series data to reveal dynamic characteristics. We consider cases in which measurements represent signals from a deterministic generator contaminated by additive noise or multiplicative noise. The properties of chaotic dynamics (nonlinear deterministic mechanisms) are also discussed. Fluctuations in cardiovascular data, as demonstrated in 24-hour ambulatory monitoring records of blood pressures, heart rate and core temperature, are then examined from these various perspectives, including spectral analysis. The pointwise correlation dimension calculated on R-R intervals on the 24-hour electrocardiogram (Holter record) is shown to reveal variations in physiological state that cannot be seen as clearly by spectral analysis. We conclude that both periodic models (simple systems) and chaotic models (complex systems) are useful in that different groups of subjects can be compared by either analysis. Neither model by itself, using standard algorithms, has the capability of identifying the mechanism actually generating the time history. Thus, we answer the question posed in the title of this paper as follows: It is difficult to decide on the basis of models of data only.