Evaluation of the dispersional analysis method for fractal time series

Multifractal Heart Rate Value Analysis: A Novel Approach for Diabetic Neuropathy Diagnosis

Type 2 diabetes mellitus (T2DM) is characterized by several complications, such as retinopathy, renal failure, cardiovascular disease, and diabetic neuropathy. Among these, neuropathy is the most severe complication, due to the challenging nature of its early detection. The linear Hearth Rate Variability (HRV) analysis is the most common diagnosis technique for diabetic neuropathy, and it is characterized by the determination of the sympathetic-parasympathetic balance on the peripheral nerves through a linear analysis of the tachogram obtained using photoplethysmography. We aimed to perform a multifractal analysis to identify autonomic neuropathy, which was not yet manifest and not detectable with the linear HRV analysis. We enrolled 10 healthy controls, 10 T2DM-diagnosed patients with not-full-blown neuropathy, and 10 T2DM diagnosed patients with full-blown neuropathy. The tachograms for the HRV analysis were obtained using finger photoplethysmography and a linear and/or multifractal analysis was performed. Our preliminary results showed that the linear analysis could effectively differentiate between healthy patients and T2DM patients with full-blown neuropathy; nevertheless, no differences were revealed comparing the full-blown to not-full-blown neuropathic diabetic patients. Conversely, the multifractal HRV analysis was effective for discriminating between full-blown and not-full-blown neuropathic T2DM patients. The multifractal analysis can represent a powerful strategy to determine neuropathic onset, even without clinical diagnostic evidence.

Increased regional Hurst exponent reflects response inhibition related neural complexity alterations in pediatric bipolar disorder patients during an emotional Go-Nogo task

Fractal patterns have been shown to change in resting- and task-state blood oxygen level-dependent signals in bipolar disorder patients. However, fractal characteristics of brain blood oxygen level-dependent signals when responding to external emotional stimuli in pediatric bipolar disorder remain unclear. Blood oxygen level-dependent signals of 20 PBD-I patients and 17 age- and sex-matched healthy controls were extracted while performing an emotional Go-Nogo task. Neural responses relevant to the task and Hurst exponent of the blood oxygen level-dependent signals were assessed. Correlations between clinical indices and Hurst exponent were estimated. Significantly increased activations were found in regions covering the frontal lobe, parietal lobe, temporal lobe, insula, and subcortical nuclei in PBD-I patients compared to healthy controls in contrast of emotional versus neutral distractors. PBD-I patients exhibited higher Hurst exponent in regions that involved in action control, such as superior frontal gyrus, inferior frontal gyrus, inferior temporal gyrus, and insula, with Hurst exponent of frontal orbital gyrus correlated with onset age. The present study exhibited overactivation, increased self-similarity and decreased complexity in cortical regions during emotional Go-Nogo task in patients relative to healthy controls, which provides evidence of an altered emotional modulation of cognitive control in pediatric bipolar disorder patients. Hurst exponent may be a fractal biomarker of neural activity in pediatric bipolar disorder.

Temporal fractal analysis of the rs-BOLD signal identifies brain abnormalities in autism spectrum disorder

Background

Brain connectivity in autism spectrum disorders (ASD) has proven difficult to characterize due to the heterogeneous nature of the spectrum. Connectivity in the brain occurs in a complex, multilevel and multi-temporal manner, driving the fluctuations observed in local oxygen demand. These fluctuations can be characterized as fractals, as they auto-correlate at different time scales. In this study, we propose a model-free complexity analysis based on the fractal dimension of the rs-BOLD signal, acquired with magnetic resonance imaging. The fractal dimension can be interpreted as measure of signal complexity and connectivity. Previous studies have suggested that reduction in signal complexity can be associated with disease. Therefore, we hypothesized that a detectable difference in rs-BOLD signal complexity could be observed between ASD patients and Controls.

Methods and findings

Anatomical and functional data from fifty-five subjects with ASD (12.7 ± 2.4 y/o) and 55 age-matched (14.1 ± 3.1 y/o) healthy controls were accessed through the NITRC database and the ABIDE project. Subjects were scanned using a 3T GE Signa MRI and a 32-channel RF-coil. Axial FSPGR-3D images were used to prescribe rs-BOLD (TE/TR = 30/2000ms) where 300 time points were acquired. Motion correction was performed on the functional data and anatomical and functional images were aligned and spatially warped to the N27 standard brain atlas. Fractal analysis, performed on a grey matter mask, was done by estimating the Hurst exponent in the frequency domain using a power spectral density approach and refining the estimation in the time domain with de-trended fluctuation analysis and signal summation conversion methods. Voxel-wise fractal dimension (FD) was calculated for every subject in the control group and in the ASD group to create ROI-based Z-scores for the ASD patients. Voxel-wise validation of FD normality across controls was confirmed, and non-Gaussian voxels were eliminated from subsequent analysis. To maintain a 95% confidence level, only regions where Z-score values were at least 2 standard deviations away from the mean (i.e. where |Z| > 2.0) were included in the analysis. We found that the main regions, where signal complexity significantly decreased among ASD patients, were the amygdala (p = 0.001), the vermis (p = 0.02), the basal ganglia (p = 0.01) and the hippocampus (p = 0.02). No regions reported significant increase in signal complexity in this study. Our findings were correlated with ADIR and ADOS assessment tools, reporting the highest correlation with the ADOS metrics.

Conclusions

Brain connectivity is best modeled as a complex system. Therefore, a measure of complexity as the fractal dimension of fluctuations in brain oxygen demand and utilization could provide important information about connectivity issues in ASD. Moreover, this technique can be used in the characterization of a single subject, with respect to controls, without the need for group analysis. Our novel approach provides an ideal avenue for personalized diagnostics, thus providing unique patient specific assessment that could help in individualizing treatments.

Hidden Signals-The History and Methods of Heart Rate Variability

The understanding of heart rate variability (HRV) has increased parallel with the development of modern physiology. Discovered probably first in 1847 by Ludwig, clinical applications evolved in the second part of the twentieth century. Today HRV is mostly used in cardiology and research settings. In general, HRV can be measured over shorter (e.g., 5-10 min) or longer (12 or 24 h) periods. Since 1996, most measurements and calculations are made according to the standard of the Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology. As the first step, the series of times between successive R-peaks in the ECG are in milliseconds. It is crucial, however, to identify and remove extrasystoles and artifacts according to standard protocols. The series of QRS distances between successive heartbeats can be analyzed with simple or more sophisticated algorithms, beginning with standard deviation (SDNN) or by the square root of the mean of the sum of squares of differences between adjacent normal RR (rMSSD). Short-term HRV is frequently analyzed with the help of a non-parametric fast Fourier transformation quantifying the different frequency bands during the measurement period. In the last decades, various non-linear algorithms have been presented, such as different entropy and fractal measures or wavelet analysis. Although most of them have a strong theoretical foundation, their clinical relevance is still debated.

Dynamic properties of calcium-activated chloride currents in Xenopus laevis oocytes

Chloride is the most abundant permeable anion in the cell, and numerous studies in the last two decades highlight the great importance and broad physiological role of chloride currents mediated anion transport. They participate in a multiplicity of key processes, as for instance, the regulation of electrical excitability, apoptosis, cell cycle, epithelial secretion and neuronal excitability. In addition, dysfunction of Cl⁻ channels is involved in a variety of human diseases such as epilepsy, osteoporosis and different cancer types. Historically, chloride channels have been of less interest than the cation channels. In fact, there seems to be practically no quantitative studies of the dynamics of chloride currents. Here, for the first time, we have quantitatively studied experimental calcium-activated chloride fluxes belonging to Xenopus laevis oocytes, and the main results show that the experimental Cl⁻ currents present an informational structure characterized by highly organized data sequences, long-term memory properties and inherent “crossover” dynamics in which persistent correlations arise at short time intervals, while anti-persistent behaviors become dominant in long time intervals. Our work sheds some light on the understanding of the informational properties of ion currents, a key element to elucidate the physiological functional coupling with the integrative dynamics of metabolic processes.

Using Continuous Glucose Monitoring Data and Detrended Fluctuation Analysis to Determine Patient Condition: A Review

Patients admitted to critical care often experience dysglycemia and high levels of insulin resistance, various intensive insulin therapy protocols and methods have attempted to safely normalize blood glucose (BG) levels. Continuous glucose monitoring (CGM) devices allow glycemic dynamics to be captured much more frequently (every 2-5 minutes) than traditional measures of blood glucose and have begun to be used in critical care patients and neonates to help monitor dysglycemia. In an attempt to obtain a better insight relating biomedical signals and patient status, some researchers have turned toward advanced time series analysis methods. In particular, Detrended Fluctuation Analysis (DFA) has been a topic of many recent studies in to glycemic dynamics. DFA investigates the "complexity" of a signal, how one point in time changes relative to its neighboring points, and DFA has been applied to signals like the inter-beat-interval of human heartbeat to differentiate healthy and pathological conditions. Analyzing the glucose metabolic system with such signal processing tools as DFA has been enabled by the emergence of high quality CGM devices. However, there are several inconsistencies within the published work applying DFA to CGM signals. Therefore, this article presents a review and a "how-to" tutorial of DFA, and in particular its application to CGM signals to ensure the methods used to determine complexity are used correctly and so that any relationship between complexity and patient outcome is robust.

Nonlinear complexity analysis of brain FMRI signals in schizophrenia

We investigated the differences in brain fMRI signal complexity in patients with schizophrenia while performing the Cyberball social exclusion task, using measures of Sample entropy and Hurst exponent (H). 13 patients meeting diagnostic and Statistical Manual of Mental Disorders, 4th Edition (DSM IV) criteria for schizophrenia and 16 healthy controls underwent fMRI scanning at 1.5 T. The fMRI data of both groups of participants were pre-processed, the entropy characterized and the Hurst exponent extracted. Whole brain entropy and H maps of the groups were generated and analysed. The results after adjusting for age and sex differences together show that patients with schizophrenia exhibited higher complexity than healthy controls, at mean whole brain and regional levels. Also, both Sample entropy and Hurst exponent agree that patients with schizophrenia have more complex fMRI signals than healthy controls. These results suggest that schizophrenia is associated with more complex signal patterns when compared to healthy controls, supporting the increase in complexity hypothesis, where system complexity increases with age or disease, and also consistent with the notion that schizophrenia is characterised by a dysregulation of the nonlinear dynamics of underlying neuronal systems.

A comparative analysis of spectral exponent estimation techniques for 1/f(β) processes with applications to the analysis of stride interval time series

BACKGROUND

The time evolution and complex interactions of many nonlinear systems, such as in the human body, result in fractal types of parameter outcomes that exhibit self similarity over long time scales by a power law in the frequency spectrum S(f)=1/f(β). The scaling exponent β is thus often interpreted as a "biomarker" of relative health and decline.

NEW METHOD

This paper presents a thorough comparative numerical analysis of fractal characterization techniques with specific consideration given to experimentally measured gait stride interval time series. The ideal fractal signals generated in the numerical analysis are constrained under varying lengths and biases indicative of a range of physiologically conceivable fractal signals. This analysis is to complement previous investigations of fractal characteristics in healthy and pathological gait stride interval time series, with which this study is compared.

RESULTS

The results of our analysis showed that the averaged wavelet coefficient method consistently yielded the most accurate results.

COMPARISON WITH EXISTING METHODS

Class dependent methods proved to be unsuitable for physiological time series. Detrended fluctuation analysis as most prevailing method in the literature exhibited large estimation variances.

CONCLUSIONS

The comparative numerical analysis and experimental applications provide a thorough basis for determining an appropriate and robust method for measuring and comparing a physiologically meaningful biomarker, the spectral index β. In consideration of the constraints of application, we note the significant drawbacks of detrended fluctuation analysis and conclude that the averaged wavelet coefficient method can provide reasonable consistency and accuracy for characterizing these fractal time series.

Statistical persistence and timing characteristics of repetitive circle drawing in children with ASD

OBJECTIVE

Standardized tests to assess movement difficulties in autism spectrum disorder (ASD) evaluate motor output, but do not provide information about underlying dynamics. The objective of this research is to study the statistical persistence and temporal dynamics of a circle drawing task in children with ASD.

METHODS

For this study 15 children diagnosed with ASD, aged 4-8 years, were asked to draw circles under various conditions using a computerized tablet. We then assessed fractal dynamics and global temporal dynamics (mean and coefficient of variation) and compared these quantities to those of typically developing (TD) controls.

RESULTS

No difference in statistical persistence was found between children with ASD and TD children. Temporal measures showed increased variability in the ASD population in the discontinuous task.

CONCLUSION

Results support the hypothesis that children with ASD have an intact ability to consistently produce continuous movements, but increased variability in production of discontinuous movements.

Pitfalls in Fractal Time Series Analysis: fMRI BOLD as an Exemplary Case

This article will be positioned on our previous work demonstrating the importance of adhering to a carefully selected set of criteria when choosing the suitable method from those available ensuring its adequate performance when applied to real temporal signals, such as fMRI BOLD, to evaluate one important facet of their behavior, fractality. Earlier, we have reviewed on a range of monofractal tools and evaluated their performance. Given the advance in the fractal field, in this article we will discuss the most widely used implementations of multifractal analyses, too. Our recommended flowchart for the fractal characterization of spontaneous, low frequency fluctuations in fMRI BOLD will be used as the framework for this article to make certain that it will provide a hands-on experience for the reader in handling the perplexed issues of fractal analysis. The reason why this particular signal modality and its fractal analysis has been chosen was due to its high impact on today’s neuroscience given it had powerfully emerged as a new way of interpreting the complex functioning of the brain (see “intrinsic activity”). The reader will first be presented with the basic concepts of mono and multifractal time series analyses, followed by some of the most relevant implementations, characterization by numerical approaches. The notion of the dichotomy of fractional Gaussian noise and fractional Brownian motion signal classes and their impact on fractal time series analyses will be thoroughly discussed as the central theme of our application strategy. Sources of pitfalls and way how to avoid them will be identified followed by a demonstration on fractal studies of fMRI BOLD taken from the literature and that of our own in an attempt to consolidate the best practice in fractal analysis of empirical fMRI BOLD signals mapped throughout the brain as an exemplary case of potentially wide interest.

Multifractal analysis of nonlinear complexity of sacral skin blood flow oscillations in older adults

The objective of this study was to investigate the relationship between cutaneous vasodilatory function and nonlinear complexity of blood flow oscillations (BFO) in older people. A non-painful fast local heating protocol was applied to the sacral skin in 20 older subjects with various vasodilatory functions. Laser Doppler flowmetry was used to measure skin blood oscillations. The complexity of the characteristic frequencies (i.e., metabolic (0.0095-0.02 Hz), neurogenic (0.02-0.05 Hz), myogenic (0.05-0.15 Hz), respiratory (0.15-0.4 Hz), and cardiac (0.4-2 Hz)) of BFO was quantified using the multifractal detrended fluctuation analysis. Compared with the 65-75 years group, the complexity of metabolic BFO in the 75-85 years group was significantly lower at the baseline (P < 0.05) and the second peak (P < 0.001). Compared with baseline BFO, subjects in the 65-75 years group had a significant increase in the complexity of metabolic BFO (P < 0.01) in response to local heating; while subjects in the 75-85 years group did not. Our findings support the use of multifractal analysis to assess aging-related microvascular dysfunction.

Quantitative analysis of cellular metabolic dissipative, self-organized structures

One of the most important goals of the postgenomic era is understanding the metabolic dynamic processes and the functional structures generated by them. Extensive studies during the last three decades have shown that the dissipative self-organization of the functional enzymatic associations, the catalytic reactions produced during the metabolite channeling, the microcompartmentalization of these metabolic processes and the emergence of dissipative networks are the fundamental elements of the dynamical organization of cell metabolism. Here we present an overview of how mathematical models can be used to address the properties of dissipative metabolic structures at different organizational levels, both for individual enzymatic associations and for enzymatic networks. Recent analyses performed with dissipative metabolic networks have shown that unicellular organisms display a singular global enzymatic structure common to all living cellular organisms, which seems to be an intrinsic property of the functional metabolism as a whole. Mathematical models firmly based on experiments and their corresponding computational approaches are needed to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the cellular catalytic reactions and also to help comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise. Understanding the molecular mechanisms responsible for the metabolic dissipative structures is crucial for unraveling the dynamics of cellular life.

Exercise modulation of cardiorespiratory variability in humans

Cardiorespiratory variability is the product of the integration of centrally generated rhythms with feedback from central and peripheral sensors. To quantify the effect of increased central drive on scaling patterns of cardiorespiratory activity, breath-to-breath interval (BBI) and heartbeat-to-heartbeat interval (RRI) were recorded from 17 female and 17 male adult subjects at rest and at two levels of mild exercise. Temporal scaling of BBI and RRI was quantified with detrended fluctuation analysis. Relative to a resting state, exercise induced a decrease in the short-term scaling of BBI (p=0.022), an increase in the long-term scaling of RRI (p=0.006), and abolished a significant positive linear relationship in females subjects (p=0.024) and a significant negative relationship in male subjects (p=0.025) in the short-term scaling of BBI and RRI. In conclusion, exercise has opposing effects on the control of breathing and heart rate, and modulates a divergent gender-based coupling of the temporal scaling of cardiorespiratory function.

Global self-regulation of the cellular metabolic structure

BACKGROUND

Different studies have shown that cellular enzymatic activities are able to self-organize spontaneously, forming a metabolic core of reactive processes that remain active under different growth conditions while the rest of the molecular catalytic reactions exhibit structural plasticity. This global cellular metabolic structure appears to be an intrinsic characteristic common to all cellular organisms. Recent work performed with dissipative metabolic networks has shown that the fundamental element for the spontaneous emergence of this global self-organized enzymatic structure could be the number of catalytic elements in the metabolic networks.

METHODOLOGY/PRINCIPAL FINDINGS

In order to investigate the factors that may affect the catalytic dynamics under a global metabolic structure characterized by the presence of metabolic cores we have studied different transitions in catalytic patterns belonging to a dissipative metabolic network. The data were analyzed using non-linear dynamics tools: power spectra, reconstructed attractors, long-term correlations, maximum Lyapunov exponent and Approximate Entropy; and we have found the emergence of self-regulation phenomena during the transitions in the metabolic activities.

CONCLUSIONS/SIGNIFICANCE

The analysis has also shown that the chaotic numerical series analyzed correspond to the fractional Brownian motion and they exhibit long-term correlations and low Approximate Entropy indicating a high level of predictability and information during the self-regulation of the metabolic transitions. The results illustrate some aspects of the mechanisms behind the emergence of the metabolic self-regulation processes, which may constitute an important property of the global structure of the cellular metabolism.

The effect of wavelet-based filtering and data set length on the fractal scaling of cardiorespiratory variability

The effect of filtering and data set length on the accuracy of the quantification of fractal characteristics of cardiorespiratory activity remains unclear. Breath-to-breath interval (BBI) and heartbeat-to-heartbeat interval (RRI) were recorded from 8 healthy human subjects during a quiet seated posture. Movement artifact was filtered from the raw respiratory data using a simple low-pass (LP) or a wavelet-based (WB) filter. The RRI data was segmented into three sets of 256, 512, and 1024 sequential data points. BBI and RRI fractal scaling was quantified using detrended fluctuation analysis and a wavelet-based estimation of fractal dimension. No significant difference in the calculation of fractal behavior of BBI was identified after using a LP or a WB filter. Furthermore, there was no significant difference in fractal measurements among the different RRI data set lengths. In conclusion, filtering of physiologic data with standard LP or WB techniques or data set length, between 256 and 1024 sequential points, does not significantly affect the calculation of fractal behavior.

Local scale exponents of blood pressure and heart rate variability by detrended fluctuation analysis: effects of posture, exercise, and aging

Heart rate self-affinity is often assessed by detrended fluctuations analysis, obtaining two coefficients only: a short-term (alpha(1)) exponent and a long-term (alpha(2)) exponent. Our aim is to show the limits of this approach and alternatively propose the estimation of the whole spectrum of local exponents alpha(n) for heart rate and blood pressure. To illustrate the advantages of this approach, we assess the effects of autonomic activations and age on alpha(n). We measured ECG and arterial pressure in 60 volunteers for 10 min, considering three conditions at increasing sympathetic activation: supine rest, sitting, and sitting during exercise. We computed alpha(n) of R-R intervals and systolic, mean, and diastolic blood pressures, as the slope of the detrended fluctuations function in a log-log plot. Volunteers were divided into age groups and compared. Results indicate that: 1) alpha(1) cannot be defined because short-term coefficients decrease with n, while alpha(2) cannot be defined only for blood pressure during supine rest; 2) heart rate and blood pressure scaling structures differ during supine rest but not during exercise; and 3) age effects appear mainly in supine rest, explaining discrepant results in literature. In conclusion, we recommend estimating the whole alpha(n) spectrum before possibly providing the "two-exponent" description only.

Fractal characterization of complexity in dynamic signals: application to cerebral hemodynamics

We introduce the concept of spatial and temporal complexity with emphasis on how its fractal characterization for 1D, 2D or 3D hemodynamic brain signals can be carried out. Using high-resolution experimental data sets acquired in animal and human brain by noninvasive methods - such as laser Doppler flowmetry, laser speckle, near infrared, or functional magnetic resonance imaging - the spatiotemporal complexity of cerebral hemodynamics is demonstrated. It is characterized by spontaneous, seemingly random (that is disorderly) fluctuation of the hemodynamic signals. Fractal analysis, however, proved that these fluctuations are correlated according to the special order of self-similarity. The degree of correlation can be assessed quantitatively either in the temporal or the frequency domain respectively by the Hurst exponent (H) and the spectral index (beta). The values of H for parenchymal regions of white and gray matter of the rat brain cortex are distinctly different. In human studies, the values of beta were instrumental in identifying age-related stiffening of cerebral vasculature and their potential vulnerability in watershed areas of the brain cortex such as in borderline regions between frontal and temporal lobes. Biological complexity seems to be present within a restricted range of H or beta values which may have medical significance because outlying values can indicate a state of pathology.

A blind method for the estimation of the Hurst exponent in time series: theory and application

Nowadays many methods for the estimation of self-similarity (Hurst coefficient, H) in time series are available. Most of them, even if very effective, need some a priori information to be applied. We analyzed the eight most used methods for H estimation (working both in time and in frequency). We tested these methods on data generated with four kinds of time series models (fBm and fGn generated iteratively with Feder algorithm, 1f(alpha), and the fractional autoregressive integrated moving-average) in the range 0.1<or=H<or=0.9. We evaluated the performances of each method in terms of accuracy (bias) and precision [standard deviation (STD)] of the deviation from the expected value. The paper proposes a procedure useful for a reliable estimation of H, using these existing methods, without any assumptions on the stationarity/nonstationarity of the time series, where for these types of processes the "nonstationarity" is mainly caused by the divergence of the variance with time. This procedure suggests that one performs, as a first step, the detrended fluctuations analysis, which provides an indication about stationarity of the series and is related to the properties of self-similarity and long correlations. The procedure then identifies the best method for the estimation of H, depending on this first estimation. As an example application, we use our procedure to evaluate the Hurst coefficient in microelectrode array neuronal recordings.

Fractal fluctuations in breath number, period, and amplitude are independently controlled in awake, healthy humans

The present study was designed to characterize respiratory fluctuations in awake, healthy humans under resting conditions. Specifically, we used Allan factor and dispersional analysis to test whether the fluctuations in breath number, respiratory period, and breath amplitude were fractal or random in nature. The results can be summarized as follows. Fluctuations in all three parameters were fractal in nine subjects. There were four subjects in whom only the fluctuations in breath number and amplitude were fractal, three subjects in whom only fluctuations in breath number were fractal, and two subjects in whom only fluctuations in breath number and respiratory period were fractal. Fluctuations in the three parameters occurred randomly in the remaining two subjects. The data suggest that fractal fluctuations in breath number, respiratory period, and breath amplitude are controlled by separate processes.

Using 1/f noise to examine planning and control in a discrete aiming task

The present study used 1/f noise to examine how spatial, physical, and timing constraints affect planning and control processes in aiming. Participants moved objects of different masses to different distances at preferred speed (Experiment 1) and as quickly as possible (Experiment 2). Power spectral density, standardized dispersion, rescaled range, and an autoregressive fractionally integrated moving average (ARFIMA) model selection procedure were used to quantify 1/f noise. Measures from all four analyses were in reasonable agreement, with more ARFIMA (long-range) models selected at peak velocity in Experiment 1 and fewer selected at peak velocity in Experiment 2. There also was a nonsignificant trend where, at preferred speed, of those participants who showed 1/f noise, more tended to show 1/f noise at peak velocity, when planning and control would overlap most. This trend disappeared for fast movements, where planning and control would have less time to overlap. Summing short-range processes at different timescales can produce 1/f-like noise. As planning is a slower-moving process and control faster, present results suggest that, with enough time for both planning and control, 1/f noise in aiming may arise from a similar summation of processes. Potential limitations of time series length in the present task are discussed.

Fractal dimensions of pacing and grip force in drawing and handwriting production

We performed a repeated measures experiment to show that the pacing of a cyclic, ballistic drawing task has a fractal dimension. We also estimated the dimensionality of the force used to grip the drawing implement. Finally, we present an analysis of pediatric data to show that grip force has a fractal dimension in an actual handwriting task. In our experiment, subjects drew circles of varying sizes and at varying rates on a digitizing tablet, using a pen instrumented to measure radial force applied to its barrel. Subjects also drew circles in synchrony with a metronome. We found strong evidence for fractal scaling of both drawing period and grip force in the circle-drawing study. The dimensionality ranged from fractal Gaussian noise (fGn) to fractal Brownian motion, with Hurst coefficients clustering around the value for 1/f noise. When the subjects were required to synchronize their drawing with a metronome, the Hurst coefficient for the drawing period decreased, while the coefficient for grip force did not. This result indicates that independent processes control the variations in pacing and grip force. Grip force in the handwriting study also displayed fractal properties, with Hurst coefficients in the range of correlated fGn. We draw parallels between our handwriting measurements and studies of human gait.

Beta- and gamma-frequency coupling between olfactory and motor brain regions prior to skilled, olfactory-driven reaching

A major question in neuroscience concerns how widely separated brain regions coordinate their activity to produce unitary cognitive states or motor actions. To investigate this question, we employed multisite, multielectrode recording in rats to study how olfactory and motor circuits are coupled prior to the execution of an olfactory-driven, GO/NO-GO variant of a skilled, rapidly executed (approximately 350-600 ms) reaching task. During task performance, we recorded multi-single units and local field potentials (LFPs) simultaneously from the rats' olfactory cortex (specifically, the posterior piriform cortex) and from cortical and subcortical motor sites (the caudal forepaw M1, and the magnocellular red nucleus, respectively). Analyses on multi-single units across areas revealed an increase in beta-frequency spiking (12-30 Hz) during a approximately 100 ms window surrounding the Final Sniff of the GO cue before lifting the arm (the "Sniff-GO window") that was seldom seen when animals sniffed the NO-GO cue. Also during the Sniff-GO window, LFPs displayed a striking increase in beta, low-gamma, and high-gamma energy (12-30, 30-50, and 50-100 Hz, respectively), and oscillations in the high gamma band appeared to be coherent across the recorded sites. These results indicate that transient, multispectral coherence across cortical and subcortical brain sites is part of the coordination process prior to sensory-guided movement initiation.

Long-term invariant parameters obtained from 24-h Holter recordings: a comparison between different analysis techniques

Over the last two decades, a large number of different methods had been used to study the fractal-like behavior of the heart rate variability (HRV). In this paper some of the most used techniques were reviewed. In particular, the focus is set on those methods which characterize the long memory behavior of time series (in particular, periodogram, detrended fluctuation analysis, rescale range analysis, scaled window variance, Higuchi dimension, wavelet-transform modulus maxima, and generalized structure functions). The performances of the different techniques were tested on simulated self-similar noises (fBm and fGn) for values of alpha, the slope of the spectral density for very small frequency, ranging from -1 to 3 with a 0.05 step. The check was performed using the scaling relationships between the various indices. DFA and periodogram showed the smallest mean square error from the expected values in the range of interest for HRV. Building on the results obtained from these tests, the effective ability of the different methods in discriminating different populations of patients from RR series derived from Holter recordings, was assessed. To this extent, the Noltisalis database was used. It consists of a set of 30, 24-h Holter recordings collected from healthy subjects, patients suffering from congestive heart failure, and heart transplanted patients. All the methods, with the exception at most of rescale range analysis, were almost equivalent in distinguish between the three groups of patients. Finally, the scaling relationships, valid for fBm and fGn, when empirically used on HRV series, also approximately held.

Sampling variability of computer-aided fractal-corrected measures of liver fibrosis in needle biopsy specimens

AIM: To assess the sampling variability of computer-aided, fractal-corrected measures of fibrosis in liver biopsies.

METHODS: Samples were derived from six to eight different parts of livers removed from 12 patients with clinically and histologically proven cirrhosis undergoing orthotopic liver transplantation. Sirius red-stained sections with a thickness of 2 μm were digitized using a computer-aided image analysis system that automatically measures the surface of fibrosis, as well as its outline perimeter, fractal surface and outline dimensions, wrinkledness, and Hurst coefficient.

RESULTS: We found a high degree of inter-sample variability in the measurements of the surface [coefficient of variation (CV) = 43% ± 13%] and wrinkledness (CV = 28% ± 9%) of fibrosis, but the inter-sample variability of Hurst’s exponent was low (CV = 14% ± 2%).

CONCLUSION: This study suggests that Hurst’s exponent might be used in clinical practice as the best histological estimate of fibrosis in the whole organ, and evidences the fact that biopsy sections, which are fundamental for the qualitative diagnosis of chronic hepatitis, play a key role in the quantitative estimate of architectural changes in liver tissue.

Utility of traditional and alternative EMG-based measures of fatigue during low-moderate level isometric efforts

Traditional electromyographic (EMG) measures (e.g., amplitude, mean and median frequencies of the power spectra) have demonstrated inconsistent abilities in monitoring localized muscle fatigue at relatively low effort levels. In the present study, several alternative EMG-based fatigue indices were evaluated, derived using a logarithmic representation of the power spectrum, the fractal dimension of the raw signal, and a Poisson distribution fit to the power spectrum. These methods, along with traditional approaches, were applied to EMG data obtained from three separate experiments. In the first two experiments, 24 participants performed sustained isometric shoulder abductions and torso extensions at 30% of maximum voluntary strength (MVC). In the third experiment, another group of 12 participants performed similar shoulder exercises at 15% and 30% MVC, with repeatability assessed at 15% MVC. Both traditional and alternative EMG measures were analyzed for their 'utility', in terms of sensitivity to fatigue, variability, repeatability, and predictive ability. Results demonstrated that parameters derived from fractal analysis and the Poisson distribution demonstrated high utility. These alternative approaches appear promising as fatigue indices for low level isometric tasks.

Weaning From Mechanical Ventilation: A Retrospective Analysis Leading to a Multimodal Perspective

Practitioners' decision for mechanical aid discontinuation is a challenging task that involves a complete knowledge of a great number of clinical parameters, as well as its evolution in time. Recently, an increasing interest on respiratory pattern variability as an extubation readiness indicator has appeared. Reliable assessment of this variability involves a set of signal processing and pattern recognition techniques. This paper presents a suitability analysis of different methods used for breathing pattern complexity assessment. The contribution of this analysis is threefold: 1) to serve as a review of the state of the art on the so-called weaning problem from a signal processing point of view; 2) to provide insight into the applied processing techniques and how they fit into the problem; 3) to propose additional methods and further processing in order to improve breathing pattern regularity assessment and weaning readiness decision. Results on experimental data show that sample entropy outperforms other complexity assessment methods and that multidimensional classification does improve weaning prediction. However, the obtained performance may be objectionable for real clinical practice, a fact that paves the way for a multimodal signal processing framework, including additional high-quality signals and more reliable statistical methods.

Computer-aided morphometry of liver inflammation in needle biopsies

AIM

To introduce a computer-aided morphometric method for quantifying the necro-inflammatory phase in liver biopsy specimens using fractal geometry and Delaunay's triangulation.

METHODS

Two-micrometer thick biopsy sections taken from 78 chronic hepatitis C virus-infected patients were immunohistochemically treated to identify the inflammatory cells. An automatic computer-aided image analysis system was used to define the inflammatory cell network defined on the basis of Delaunay's triangulation, and the inflammatory cells were geometrically classified as forming a cluster (an aggregation of a minimum of three cells) or as being irregularly distributed within the tissue. The phase of inflammatory activity was estimated using Hurst's exponent.

RESULTS

The proposed automatic method was rapid and objective. It could not only provide rigorous results expressed by scalar numbers, but also allow the state of the whole organ to be represented by Hurst's exponent with an error of no more than 12%.

CONCLUSION

The availability of rigorous metrical measures and the reasonable representativeness of the status of the organ as a whole raise the question as to whether the indication for hepatic biopsy should be revised by establishing clear rules concerning the contraindications suggested by its invasiveness and subjective interpretation.

Liver fibrosis and tissue architectural change measurement using fractal-rectified metrics and Hurst’s exponent

AIM: To provide the accurate alternative metrical means of monitoring the effects of new antiviral drugs on the reversal of newly formed collagen.

METHODS: Digitized histological biopsy sections taken from 209 patients with chronic C virus hepatitis with different grade of fibrosis or cirrhosis, were measured by means of a new, rapid, user-friendly, fully computer-aided method based on the international system meter rectified using fractal principles.

RESULTS: The following were described: geometric perimeter, area and wrinkledness of fibrosis; the collation of the Knodell, Sheuer, Ishak and METAVIR scores with fractal-rectified metric measurements; the meaning of the physical composition of fibrosis in relation to the magnitude of collagen islets; the intra- and inter-biopsy sample variability of these parameters; the “staging” of biopsy sections indicating the pathway covered by fibrosis formation towards its maximum known value; the quantitative liver tissue architectural changes with the Hurst exponent.

CONCLUSION: Our model provides the first metrical evaluations of the geometric properties of fibrosis and the quantitative architectural changes of the liver tissue. The representativeness of histological sections of the whole liver is also discussed in the light of the results obtained with the Hurst coefficient.

Long-range correlations in rabbit brain neural activity

We have analyzed the presence of persistence properties in rabbit brain electrical signals by means of non-equilibrium statistical physics tools. To measure long-memory properties of these experimental signals, we have first determined whether the data are fractional Gaussian noise (fGn) or fractional Brownian motion (fBm) by calculating the slope of the power spectral density plot of the series. The results show that the series correspond to fBm. Then, the data were studied by means of the bridge detrended scaled windowed variance analysis, detecting long-term correlation. Three different types of experimental signals have been studied: neural basal activity without stimulation, the response induced by a single flash light stimulus and the average of the activity evoked by 200 flash light stimulations. Analysis of the series revealed the existence of persistent behavior in all cases. Moreover, the results also exhibited an increasing correlation in the level of long-term memory from recordings without stimulation, to one sweep recording or 200 sweeps averaged recordings. Thus, brain neural electrical activity is affected not only by its most recent states, but also by previous states much more distant in the past.

Self-similarity behavior characterization of fetal heart rate signal in healthy and intrauterine growth retarded fetuses

In this paper we deal with the problem of the interpretation of the fetal heart rate (FHR) signal. From literature is known that FHR contains both linear and non linear components. Starting from this consideration we analyzed FHR as a fractal time series and we evaluated its self similarity behavior using the Hurst's coefficient (H). We first evaluated the stationarity of FHR time series and then we estimated H with Detrend fluctuation analysis (DFA) method. We calculated Hurst's coefficient for healthy fetuses and for fetuses affected by Intrauterine grow retardation (IUGR). Results provided H = 0.350 +/- 0.064 (avg +/- std) for healthy patients and H = 0.461 +/- 0.059 for IUGR. It is also shown that IUGR patients exhibit a "less non-stationary" and longer-memory behavior than normals with a reduced information content of FHR signal. We propose for this phenomenon a physiological explanation connected with the abnormal autonomic nervous system development of IUGR patients.

Assessment of the Autonomic Control of Heart Rate Variability in Healthy and Spinal-Cord Injured Subjects: Contribution of Different Complexity-Based Estimators

We investigated how complexity-based estimators of heart rate variability can detect changes in cardiovascular autonomic drive with respect to traditional measures of variability. This was done by analyzing healthy subjects and paraplegic patients with different autonomic impairment due to low (vascular impairment only) or high (cardiac and vascular impairment) spinal cord injury, during progressive autonomic activations. While traditional techniques only quantified the effects of the autonomic activation, not distinguishing the effects of the lesion level, some recently proposed complexity estimators could also reveal the pathologic alterations in the autonomic control of heart rate. These estimators included the detrended fluctuation analysis coefficient (sensitive to both low and high autonomic lesions), sample entropy (sensitive to low-level lesions) and the largest Lyapunov exponent (sensitive to high-level lesions). Thus complexity-based methods provide information on the autonomic function from the heart rate dynamics that cannot be obtained by traditional techniques. This finding supports the combined use of both complexity-based and traditional methods to investigate the autonomic cardiovascular control from a more comprehensive perspective.

Fractal fluctuations in human respiration

The present study was designed to characterize respiratory fluctuations in awake, healthy adult humans under resting conditions. For this purpose, we recorded respiratory movements with a strain-gauge pneumograph in 20 subjects. We then used Allan factor, Fano factor, and dispersional analysis to test whether the fluctuations in the number of breaths, respiratory period, and breath amplitude were fractal (i.e., time-scale-invariant) or random in occurrence. Specifically, we measured the slopes of the power laws in the Allan factor, Fano factor, and dispersional analysis curves for original time series and compared these with the slopes of the curves for surrogates (randomized data sets). In addition, the Hurst exponent was calculated from the slope of the power law in the Allan factor curve to determine whether the long-range correlations among the fluctuations in breath number were positively or negatively correlated. The results can be summarized as follows. Fluctuations in all three parameters were fractal in nine subjects. There were four subjects in whom only the fluctuations in number of breaths and breath amplitude were fractal, three subjects in whom only the fluctuations in number of breaths were fractal, and two subjects in whom only fluctuations in breath number and respiratory period were fractal. Time-scale-invariant behavior was absent in the two remaining subjects. The results indicate that, in most cases, apparently random fluctuations in respiratory pattern are, in fact, correlated over more than one time scale. Moreover, the data suggest that fractal fluctuations in breath number, respiratory period, and breath amplitude are controlled by separate processes.

Detrended fluctuation analysis: a suitable method for studying fetal heart rate variability?

We evaluate the suitability of an enhanced detrended fluctuation analysis for studying fetal heart rate series involving imperfect quality of information. Our results indicate that to explore persistent long-range correlations, or fractality, the collection requirements of the data can be relaxed by allowing the possibility of using averaged fetal heart rate series. In addition, it also appears feasible to employ, without producing major alterations in the long-range scaling behaviour, fragmented fetal heart rate series involving up to 50% of random missing values, or up to 50 min of consecutive missing samples in recordings of approximately equal to 8 h length. These are crucial advantages to overcome the often variable quality of fetal data. Consequently, these findings may open the possibility of obtaining information concerning the development of neural processes from fetal heart rate series, despite their non-stationary and fragmented nature.

Fractal properties of human muscle sympathetic nerve activity

Muscle sympathetic nerve activity (MSNA) in resting humans is characterized by cardiac-related bursts of variable amplitude that occur sporadically or in clusters. The present study was designed to characterize the fluctuations in the number of MSNA bursts, interburst interval, and burst amplitude recorded from the peroneal nerve of 15 awake, healthy human subjects. For this purpose, we used the Allan and Fano factor analysis and dispersional analysis to test whether the fluctuations were time-scale invariant (i.e., fractal) or random in occurrence. Specifically, we measured the slopes of the power laws in the Allan factor, Fano factor, and dispersional analysis curves. In addition, the Hurst exponent was calculated from the slope of the power law in the Allan factor curve. Whether the original time series contained fractal fluctuations was decided on the basis of a comparison of the values of these parameters with those for surrogate data blocks. The results can be summarized as follows. Fluctuations in the number of MSNA bursts and interburst interval were fractal in each of the subjects, and fluctuations in burst amplitude were fractal in four of the subjects. We also found that fluctuations in the number of heartbeats and heart period (R-R interval) were fractal in each of the subjects. These results demonstrate for the first time that apparently random fluctuations in human MSNA are, in fact, dictated by a time-scale-invariant process that imparts “long-term memory” to the sequence of cardiac-related bursts. Whether sympathetic outflow to the heart also is fractal and contributes to the fractal component of heart rate variability remains an open question.

Stochastic model for heart-rate fluctuations

A normal human heart rate shows complex fluctuations in time, which is natural, because the heart rate is controlled by a large number of different feedback control loops. These unpredictable fluctuations have been shown to display fractal dynamics, long-term correlations, and 1/f noise. These characterizations are statistical and they have been widely studied and used, but much less is known about the detailed time evolution (dynamics) of the heart-rate control mechanism. Here we show that a simple one-dimensional Langevin-type stochastic difference equation can accurately model the heart-rate fluctuations in a time scale from minutes to hours. The model consists of a deterministic nonlinear part and a stochastic part typical to Gaussian noise, and both parts can be directly determined from the measured heart-rate data. Studies of 27 healthy subjects reveal that in most cases, the deterministic part has a form typically seen in bistable systems: there are two stable fixed points and one unstable one.

The spectra and periodograms of anti-correlated discrete fractional Gaussian noise

Discrete fractional Gaussian noise (dFGN) has been proposed as a model for interpreting a wide variety of physiological data. The form of actual spectra of dFGN for frequencies near zero varies as f(1-2H), where 0 < H < 1 is the Hurst coefficient; however, this form for the spectra need not be a good approximation at other frequencies. When H approaches zero, dFGN spectra exhibit the 1 - 2H power-law behavior only over a range of low frequencies that is vanishingly small. When dealing with a time series of finite length drawn from a dFGN process with unknown H, practitioners must deal with estimated spectra in lieu of actual spectra. The most basic spectral estimator is the periodogram. The expected value of the periodogram for dFGN with small H also exhibits non-power-law behavior. At the lowest Fourier frequencies associated with a time series of N values sampled from a dFGN process, the expected value of the periodogram for H approaching zero varies as f(0) rather than f(1-2H). For finite N and small H, the expected value of the periodogram can in fact exhibit a local power-law behavior with a spectral exponent of 1 - 2H at only two distinct frequencies.

Fractal properties of sympathetic nerve discharge

Fano factor analysis and dispersional analysis were used to characterize time series of single and multifiber spikes recorded from the preganglionic cervical sympathetic nerve and cardiac-related slow-wave activity of the whole postganglionic sympathetic vertebral nerve (VN) in anesthetized cats. Fluctuations in spike counts and interspike intervals for single preganglionic fibers proved to be fractal (i.e., time-scale invariant), as reflected by a power law relationship between indices of the variance of these properties and the window size used to make the measurements. Importantly, random shuffling of the data eliminated the power law relationships. Fluctuations in spike counts in preganglionic multifiber activity also were fractal, as were fluctuations in the height and of the area of cardiac-related slow waves recorded from the whole postganglionic VN. These fractal fluctuations were persistent (i.e., positively correlated), as reflected by a Hurst exponent significantly >0.5. Although fluctuations in the interval between cardiac-related VN slow waves were random, those in the interval between heart beats were fractal and persistent. These results demonstrate for the first time that apparently random fluctuations in sympathetic nerve discharge are, in fact, dictated by a complex deterministic process that imparts "long-term" memory to the system. Whether such time-scale invariant behavior plays a role in generating the fractal component of heart rate variability remains to be determined.

Fractal characterization of complexity in temporal physiological signals

This review first gives an overview on the concept of fractal geometry with definitions and explanations of the most fundamental properties of fractal structures and processes like self-similarity, power law scaling relationship, scale invariance, scaling range and fractal dimensions. Having laid down the grounds of the basics in terminology and mathematical formalism, the authors systematically introduce the concept and methods of monofractal time series analysis. They argue that fractal time series analysis cannot be done in a conscious, reliable manner without having a model capable of capturing the essential features of physiological signals with regard to their fractal analysis. They advocate the use of a simple, yet adequate, dichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). They demonstrate the importance of incorporating a step of signal classification according to the fGn/fBm model prior to fractal analysis by showing that missing out on signal class can result in completely meaningless fractal estimates. Limitation and precision of various fractal tools are thoroughly described and discussed using results of numerical experiments on ideal monofractal signals. Steps of a reliable fractal analysis are explained. Finally, the main applications of fractal time series analysis in biomedical research are reviewed and critically evaluated.

Nonlinear methods of biosignal analysis in assessing terbutaline-induced heart rate and blood pressure changes

The aim of this study was to characterize how different nonlinear methods characterize heart rate and blood pressure dynamics in healthy subjects at rest. The randomized, placebo-controlled crossover study with intravenous terbutaline was designed to induce four different stationary states of cardiovascular regulation system. The R-R interval, systolic arterial blood pressure, and heart rate time series were analyzed with a set of methods including approximate entropy, sample entropy, Lempel-Ziv entropy, symbol dynamic entropy, cross-entropy, correlation dimension, fractal dimensions, and stationarity test. Results indicate that R-R interval and systolic arterial pressure subsystems are mutually connected but have different dynamic properties. In the drug-free state the subsystems share many common features. When the strength of the baroreflex feedback loop is modified with terbutaline, R-R interval and systolic blood pressure lose mutual synchrony and drift toward their inherent state of operation. In this state the R-R interval system is rather complex and irregular, but the blood pressure system is much simpler than in the drug-free state.