Estimator of a non-Gaussian parameter in multiplicative log-normal models

Multifractal nonlinearity as a robust estimator of multiplicative cascade dynamics

Natural and behavioral sciences increasingly model measurement time series as random multiplicative cascades-nonlinear processes that divide, branch, or aggregate structures across generations, creating multiplicative interactions that break ergodicity. Multifractal formalisms provide a framework for studying cascades, where multifractal spectrum width, Δα, fluctuates with the number of estimable power-law relationships. However, multifractality without surrogate comparison can be ambiguous, as the original measurement series' multifractal spectrum width, Δα_{original}, is sensitive to series length, ergodicity-breaking linear correlations (e.g., fractional Gaussian noise, fGn), or additive dynamics. To address these challenges, we constructed random cascades varying in length, noise type (additive white Gaussian noise, awGn, or fGn), mixtures of awGn and fGn across generations (progressively more awGn, progressively more fGn, or random mixing of awGn and fGn across generations), and noise operations (addition versus multiplication). We found that "multifractal nonlinearity," t_{MF} (a t-statistic comparing Δα_{original} to surrogate spectra width, Δα_{surrogates}), reliably identifies random multiplicative cascades regardless of series length or noise type-t_{MF} is more sensitive to interactivity and number of generations than series length. Unlike random additive cascades, random multiplicative cascades exhibit stronger ergodicity breaking due to their heavy histogram tails and show progressively greater ergodicity breaking with more correlated noise (e.g., fGn as opposed to awGn) and additional generations-independent of series length. Consequently, t_{MF} robustly identifies multiplicative cascade processes, quantifies the depth of cross-scale interactivity, and pinpoints sources of ergodicity breaking while remaining independent of ergodicity-breaking and series length. These properties highlight the potential of t_{MF} as a powerful metric for advancing our understanding of the mechanisms and origins of cascading processes in natural and behavioral sciences. The simulations further suggest its utility in extending findings from group-level analyses to the level of individuals.

Selective engagement of long-latency reflexes in postural control through wobble board training

Long-latency reflexes (LLRs) are critical precursors to intricate postural coordination of muscular adaptations that sustain equilibrium following abrupt disturbances. Both disturbances and adaptive responses reflect excursions of postural control from quiescent Gaussian stability under a narrow bell curve, excursions beyond Gaussianity unfolding at many timescales. LLRs slow with age, accentuating the risk of falls and undermining dexterity, particularly in settings with concurrent additional tasks. We investigated whether the wobble board could cultivate the engagement of LLRs selectively in healthy young participants executing a suprapostural Trail Making Task (TMT). A concurrent additional-task demand constituted visual precision predominantly along the anteroposterior (AP) axis and mechanical instability mainly along the mediolateral (ML) axis. We scrutinized planar center-of-pressure (CoP) trajectories to quantify postural non-Gaussianity across various temporal scales. Wobble board increased engagement of LLRs and decreased engagement of compensatory postural adjustments (CPAs), indicated by the peak in non-Gaussianity of CoP planar displacements over LLR-specific timescales (50-100 ms) and non-Gaussianity of CoP planar displacements progressively diminishing over CPA-specific timescales ([Formula: see text] ms). Engagement with TMT did not show any noticeable influence on non-Gaussian postural sway patterns. Despite aligning the unstable axis of the wobble board with participants' ML axis, thus rendering posture more unstable along the ML axis, the wobble board increased engagement of LLRs significantly more along the AP axis and reduced engagement of CPAs significantly more along the ML axis. These findings offer initial mechanistic insights into how wobble boards may bolster balance and potentially reduce the occurrence of falls by catalyzing the engagement of LLRs selectively.

Multifractal perturbations to multiplicative cascades promote multifractal nonlinearity with asymmetric spectra

Biological and psychological processes have been conceptualized as emerging from intricate multiplicative interactions among component processes across various spatial and temporal scales. Among the statistical models employed to approximate these intricate nonlinear interactions across scales, one prominent framework is that of cascades. Despite decades of empirical work using multifractal formalisms, several fundamental questions persist concerning the proper interpretations of multifractal evidence of nonlinear cross-scale interactivity. Does multifractal spectrum width depend on multiplicative interactions, constituent noise processes participating in those interactions, or both? We conducted numerical simulations of cascade time series featuring component noise processes characterizing a range of nonlinear temporal correlations: nonlinearly multifractal, linearly multifractal (obtained via the iterative amplitude adjusted wavelet transform of nonlinearly multifractal), phase-randomized linearity (obtained via the iterative amplitude adjustment Fourier transform of nonlinearly multifractal), and phase and amplitude randomized (obtained via shuffling of nonlinearly multifractal). Our findings show that the multiplicative interactions coordinate with the nonlinear temporal correlations of noise components to dictate emergent multifractal properties. Multiplicative cascades with stronger nonlinear temporal correlations make multifractal spectra more asymmetric with wider left sides. However, when considering multifractal spectral differences between the original and surrogate time series, even multiplicative cascades produce multifractality greater than in surrogate time series, even with linearized multifractal noise components. In contrast, additivity among component processes leads to a linear outcome. These findings provide a robust framework for generating multifractal expectations for biological and psychological models in which cascade dynamics flow from one part of an organism to another.

Spatial variability and directional shifts in postural control in Parkinson's disease

Individuals with Parkinson's disease exhibit tremors, rigidity, and bradykinesia, disrupting normal movement variability and resulting in postural instability. This comprehensive study aimed to investigate the link between the temporal structure of postural sway variability and Parkinsonism by analyzing multiple datasets from young and older adults, including individuals with Parkinson's disease, across various task conditions. We used the Oriented Fractal Scaling Component Analysis (OFSCA), which identifies minimal and maximal long-range correlations within the center of pressure time series, allowing for detecting directional changes in postural sway variability. The objective was to uncover the primary directions along which individuals exerted control during the posture. The results, as anticipated, revealed that healthy adults predominantly exerted control along two orthogonal directions, closely aligned with the anteroposterior (AP) and mediolateral (ML) axes. In stark contrast, older adults and individuals with Parkinson's disease exhibited control along suborthogonal directions that notably diverged from the AP and ML axes. While older adults and those with Parkinson's disease demonstrated a similar reduction in the angle between these two control directions compared to healthy older adults, their reliance on this suborthogonal angle concerning endogenous fractal correlations exhibited significant differences from the healthy aging cohort. Importantly, individuals with Parkinson's disease did not manifest the sensitivity to destabilizing task settings observed in their healthy counterparts, affirming the distinction between Parkinson's disease and healthy aging.

Multifractal foundations of biomarker discovery for heart disease and stroke

Any reliable biomarker has to be specific, generalizable, and reproducible across individuals and contexts. The exact values of such a biomarker must represent similar health states in different individuals and at different times within the same individual to result in the minimum possible false-positive and false-negative rates. The application of standard cut-off points and risk scores across populations hinges upon the assumption of such generalizability. Such generalizability, in turn, hinges upon this condition that the phenomenon investigated by current statistical methods is ergodic, i.e., its statistical measures converge over individuals and time within the finite limit of observations. However, emerging evidence indicates that biological processes abound with nonergodicity, threatening this generalizability. Here, we present a solution for how to make generalizable inferences by deriving ergodic descriptions of nonergodic phenomena. For this aim, we proposed capturing the origin of ergodicity-breaking in many biological processes: cascade dynamics. To assess our hypotheses, we embraced the challenge of identifying reliable biomarkers for heart disease and stroke, which, despite being the leading cause of death worldwide and decades of research, lacks reliable biomarkers and risk stratification tools. We showed that raw R-R interval data and its common descriptors based on mean and variance are nonergodic and non-specific. On the other hand, the cascade-dynamical descriptors, the Hurst exponent encoding linear temporal correlations, and multifractal nonlinearity encoding nonlinear interactions across scales described the nonergodic heart rate variability more ergodically and were specific. This study inaugurates applying the critical concept of ergodicity in discovering and applying digital biomarkers of health and disease.

Turing's cascade instability supports the coordination of the mind, brain, and behavior

Turing inspired a computer metaphor of the mind and brain that has been handy and has spawned decades of empirical investigation, but he did much more and offered behavioral and cognitive sciences another metaphor-that of the cascade. The time has come to confront Turing's cascading instability, which suggests a geometrical framework driven by power laws and can be studied using multifractal formalism and multiscale probability density function analysis. Here, we review a rapidly growing body of scientific investigations revealing signatures of cascade instability and their consequences for a perceiving, acting, and thinking organism. We review work related to executive functioning (planning to act), postural control (bodily poise for turning plans into action), and effortful perception (action to gather information in a single modality and action to blend multimodal information). We also review findings on neuronal avalanches in the brain, specifically about neural participation in body-wide cascades. Turing's cascade instability blends the mind, brain, and behavior across space and time scales and provides an alternative to the dominant computer metaphor.

Insulin signaling shapes fractal scaling of C. elegans behavior

Fractal scaling in animal behavioral activity, where similar temporal patterns appear repeatedly over a series of magnifications among time scales, governs the complex behavior of various animal species and, in humans, can be altered by neurodegenerative diseases and aging. However, the mechanism underlying fractal scaling remains unknown. Here, we cultured C. elegans in a microfluidic device for 3 days and analyzed temporal patterns of C. elegans activity by fractal analyses. The residence-time distribution of C. elegans behaviors shared a common feature with those of human and mice. Specifically, the residence-time power-law distribution of the active state changed to an exponential-like decline at a longer time scale, whereas the inactive state followed a power-law distribution. An exponential-like decline appeared with nutrient supply in wild-type animals, whereas this decline disappeared in insulin-signaling-defective daf-2 and daf-16 mutants. The absolute value of the power-law exponent of the inactive state distribution increased with nutrient supply in wild-type animals, whereas the value decreased in daf-2 and daf-16 mutants. We conclude that insulin signaling differentially affects mechanisms that determine the residence time in active and inactive states in C. elegans behavior. In humans, diabetes mellitus, which is caused by defects in insulin signaling, is associated with mood disorders that affect daily behavioral activities. We hypothesize that comorbid behavioral defects in patients with diabetes may be attributed to altered fractal scaling of human behavior.

Assessment of autonomic function by long-term heart rate variability: beyond the classical framework of LF and HF measurements

In the assessment of autonomic function by heart rate variability (HRV), the framework that the power of high-frequency component or its surrogate indices reflects parasympathetic activity, while the power of low-frequency component or LF/HF reflects sympathetic activity has been used as the theoretical basis for the interpretation of HRV. Although this classical framework has contributed greatly to the widespread use of HRV for the assessment of autonomic function, it was obtained from studies of short-term HRV (typically 5‑10 min) under tightly controlled conditions. If it is applied to long-term HRV (typically 24 h) under free-running conditions in daily life, erroneous conclusions could be drawn. Also, long-term HRV could contain untapped useful information that is not revealed in the classical framework. In this review, we discuss the limitations of the classical framework and present studies that extracted autonomic function indicators and other useful biomedical information from long-term HRV using novel approaches beyond the classical framework. Those methods include non-Gaussianity index, HRV sleep index, heart rate turbulence, and the frequency and amplitude of cyclic variation of heart rate.

Hypothetical control of postural sway

Quiet standing exhibits strongly intermittent variability that has inspired at least two interpretations. First, variability can be intermittent through the alternating engagement and disengagement of complementary control processes at distinct scales. A second and perhaps deeper way to interpret this intermittency is through the possibility that postural control depends on cascade-like interactions across many timescales at once, suggesting specific non-Gaussian distributional properties at different timescales. Multiscale probability density function (PDF) analysis shows that quiet standing on a stable surface exhibits a crossover from low, increasing non-Gaussianity (consistent with exponential distributions) at shorter timescales, reflecting inertial control, towards higher non-Gaussianity. Feedback-based control at medium to longer timescales yields a linear decrease that is characteristic of cascade dynamics. Destabilizing quiet standing with an unstable surface or closed eyes serves to attenuate inertial control and to elicit more of the feedback-based control over progressively shorter timescales. The result was to strengthen the appearance of the linear decay indicating cascade dynamics. Finally, both linear and nonlinear indices of postural sway also govern the relative strength of crossover or of linear decay, suggesting that tempering of non-Gaussianity across log-timescale is a function of both extrinsic constraints and endogenous postural control. These results provide new evidence that cascading interactions across longer timescales supporting postural corrections can even recruit shorter timescale processes with novel task constraints that can destabilize posture.

Survival Predictors of Heart Rate Variability After Myocardial Infarction With and Without Low Left Ventricular Ejection Fraction

Background

Heart rate variability (HRV) and heart rate (HR) dynamics are used to predict the survival probability of patients after acute myocardial infarction (AMI), but the association has been established in patients with mixed levels of left ventricular ejection fraction (LVEF).

Objective

We investigated whether the survival predictors of HRV and HR dynamics depend on LVEF after AMI.

Methods

We studied 687 post-AMI patients including 147 with LVEF ≤35% and 540 with LVEF >35%, of which 23 (16%) and 22 (4%) died during the 25 month follow-up period, respectively. None had an implanted cardioverter-defibrillator. From baseline 24 h ECG, the standard deviation (SDNN), root mean square of successive difference (rMSSD), percentage of successive difference >50 ms (pNN50) of normal-to-normal R-R interval, ultra-low (ULF), very-low (VLF), low (LF), and high (HF) frequency power, deceleration capacity (DC), short-term scaling exponent (α₁), non-Gaussianity index (λ_25s), and the amplitude of cyclic variation of HR (Acv) were calculated.

Results

The predictors were categorized into three clusters; DC, SDNN, α₁, ULF, VLF, LF, and Acv as Cluster 1, λ_25s independently as Cluster 2, and rMSSD, pNN50, and HF as Cluster 3. In univariate analyses, mortality was best predicted by indices belonging to Cluster 1 regardless of LVEF. In multivariate analyses, however, mortality in patients with low LVEF was best predicted by the combinations of Cluster 1 predictors or Cluster 1 and 3 predictors, whereas in patients without low LVEF, it was best predicted by the combinations of Cluster 1 and 2 predictors.

Conclusion

The mortality risk in post-AMI patients with low LVEF is predicted by indices reflecting decreased HRV or HR responsiveness and cardiac parasympathetic dysfunction, whereas in patients without low LVEF, the risk is predicted by a combination of indices that reflect decreased HRV or HR responsiveness and indicator that reflects abrupt large HR changes suggesting sympathetic involvement.

Visual effort moderates postural cascade dynamics

Standing still and focusing on a visible target in front of us is a preamble to many coordinated behaviors (e.g., reaching an object). Hiding behind its apparent simplicity is a deep layering of texture at many scales. The task of standing still laces together activities at multiple scales: from ensuring that a few photoreceptors on the retina cover the target in the visual field on an extremely fine scale to synergies spanning the limbs and joints at smaller scales to the mechanical layout of the ground underfoot and optic flow in the visual field on the coarser scales. Here, we used multiscale probability density function (PDF) analysis to show that postural fluctuations exhibit similar statistical signatures of cascade dynamics as found in fluid flow. In participants asked to stand quietly, the oculomotor strain of visually fixating at different distances moderated postural cascade dynamics. Visually fixating at a comfortable viewing distance elicited posture with a similar cascade dynamics as posture with eyes closed. Greater viewing distances known to stabilize posture showed more diminished cascade dynamics. In contrast, nearest and farthest viewing distances requiring greater oculomotor strain to focus on targets elicited a dramatic strengthening of postural cascade dynamics, reflecting active postural adjustments. Critically, these findings suggest that vision stabilizes posture by reconfiguring the prestressed poise that prepares the body to interact with different spatial layouts.

Postural constraints recruit shorter-timescale processes into the non-Gaussian cascade processes

Healthy human postural sway exhibits strong intermittency, reflecting a richly interactive foundation of postural control. From a linear perspective, intermittent fluctuations might be interpreted as engagement and disengagement of complementary control processes at distinct timescales or from a nonlinear perspective, as cascade-like interactions across many timescales at once. The diverse control processes entailed by cascade-like multiplicative dynamics suggest specific non-Gaussian distributional properties at different timescales. Multiscale probability density function (PDF) analysis showed that when standing quietly while balancing a sand-filled tube with the two arms elicited non-Gaussianity profiles showing a negative-quadratic crossover between short and long timescales. A more stringent task of balancing a water-filled tube elicited simpler monotonic decreases in non-Gaussianity, that is, a positive-quadratic cancellation of the negative-quadratic crossover. Multiple known indices of postural sway governed the appearance or disappearance of the crossover. Finally, both tasks elicited lognormal distributions over progressively larger timescales. These results provide the first evidence that more stringent postural constraints recruit shorter-timescale processes into the non-Gaussian cascade processes, that indices of postural sway moderate this recruitment, and that more stringent postural constraints show stronger statistical hallmarks of cascade structure.

C. elegans episodic swimming is driven by multifractal kinetics

Fractal scaling is a common property of temporal change in various modes of animal behavior. The molecular mechanisms of fractal scaling in animal behaviors remain largely unexplored. The nematode C. elegans alternates between swimming and resting states in a liquid solution. Here, we report that C. elegans episodic swimming is characterized by scale-free kinetics with long-range temporal correlation and local temporal clusterization, namely consistent with multifractal kinetics. Residence times in actively-moving and inactive states were distributed in a power law-based scale-free manner. Multifractal analysis showed that temporal correlation and temporal clusterization were distinct between the actively-moving state and the inactive state. These results indicate that C. elegans episodic swimming is driven by transition between two behavioral states, in which each of two transition kinetics follows distinct multifractal kinetics. We found that a conserved behavioral modulator, cyclic GMP dependent kinase (PKG) may regulate the multifractal kinetics underlying an animal behavior. Our combinatorial analysis approach involving molecular genetics and kinetics provides a platform for the molecular dissection of the fractal nature of physiological and behavioral phenomena.

Redundancy among risk predictors derived from heart rate variability and dynamics: ALLSTAR big data analysis

BACKGROUND

Many indices of heart rate variability (HRV) and heart rate dynamics have been proposed as cardiovascular mortality risk predictors, but the redundancy between their predictive powers is unknown.

METHODS

From the Allostatic State Mapping by Ambulatory ECG Repository project database, 24-hr ECG data showing continuous sinus rhythm were extracted and SD of normal-to-normal R-R interval (SDNN), very-low-frequency power (VLF), scaling exponent α₁ , deceleration capacity (DC), and non-Gaussianity λ_25s were calculated. The values were dichotomized into high-risk and low-risk values using the cutoffs reported in previous studies to predict mortality after acute myocardial infarction. The rate of multiple high-risk predictors accumulating in the same person was examined and was compared with the rate expected under the assumption that these predictors are independent of each other.

RESULTS

Among 265,291 ECG data from the ALLSTAR database, the rates of subjects with high-risk SDNN, DC, VLF, α₁ , and λ_25s values were 2.95, 2.75, 5.89, 15.75, and 18.82%, respectively. The observed rate of subjects without any high-risk value was 66.68%, which was 1.10 times the expected rate (60.74%). The ratios of observed rate to the expected rate at which one, two, three, four, and five high-risk values accumulate in the same person were 0.73 times (24.10 and 32.82%), 1.10 times (6.56 and 5.99%), 4.26 times (1.87 and 0.44%), 47.66 times (0.63 and 0.013%), and 1,140.66 times (0.16 and 0.00014%), respectively.

CONCLUSIONS

High-risk predictors of HRV and heart rate dynamics tend to cluster in the same person, indicating a high degree of redundancy between them.

Gene Expression Is Not Random: Scaling, Long-Range Cross-Dependence, and Fractal Characteristics of Gene Regulatory Networks

Gene expression is a vital process through which cells react to the environment and express functional behavior. Understanding the dynamics of gene expression could prove crucial in unraveling the physical complexities involved in this process. Specifically, understanding the coherent complex structure of transcriptional dynamics is the goal of numerous computational studies aiming to study and finally control cellular processes. Here, we report the scaling properties of gene expression time series in Escherichia coli and Saccharomyces cerevisiae. Unlike previous studies, which report the fractal and long-range dependency of DNA structure, we investigate the individual gene expression dynamics as well as the cross-dependency between them in the context of gene regulatory network. Our results demonstrate that the gene expression time series display fractal and long-range dependence characteristics. In addition, the dynamics between genes and linked transcription factors in gene regulatory networks are also fractal and long-range cross-correlated. The cross-correlation exponents in gene regulatory networks are not unique. The distribution of the cross-correlation exponents of gene regulatory networks for several types of cells can be interpreted as a measure of the complexity of their functional behavior.

L/T-type calcium channel blocker reduces non-Gaussianity of heart rate variability in chronic kidney disease patients under preceding treatment with ARB

Introduction:

Increased sympathetic nerve activity has been suggested in patients with chronic kidney disease (CKD). Pathologic sympathetic activity can alter heart rate variability (HRV), and the altered HRV has prognostic importance, so that reducing sympathetic activity may be an important strategy. Novel nonlinear HRVs, including deceleration capacity (DC), have greater predictive power for mortality. We have recently proposed an increase in a non-Gaussianity index of HRV, λ_25s, which indicates the probability of volcanic heart rate deviations of departure from each standard deviation level, as a marker of sympathetic cardiac overdrive. L/T-type calcium channel blocker (L/T-CCB), azelnidipine, decreases sympathetic nerve activity in experimental and clinical studies.

Methods:

In 43 hypertensive patients with CKD under treatment with an angiotensin receptor blocker (ARB), we investigated whether 8-week add-on L/T-CCB treatment could restore HRV.

Results:

Means of all normal-to-normal intervals over 24 h (p<0.0001) and DC (p=0.002) increased, and λ_25s (p=0.001) decreased regardless of gender, age, renal function or blood pressure, while no significant changes were observed in the other HRVs.

Conclusions:

Reduction of λ_25s is useful to assess the effect of sympathoinhibitory treatment. Further studies are needed to investigate if the restoration of HRV is directly associated with the improvement of prognosis in patients with CKD.

Intermittent Feedback-Control Strategy for Stabilizing Inverted Pendulum on Manually Controlled Cart as Analogy to Human Stick Balancing

The stabilization of an inverted pendulum on a manually controlled cart (cart-inverted-pendulum; CIP) in an upright position, which is analogous to balancing a stick on a fingertip, is considered in order to investigate how the human central nervous system (CNS) stabilizes unstable dynamics due to mechanical instability and time delays in neural feedback control. We explore the possibility that a type of intermittent time-delayed feedback control, which has been proposed for human postural control during quiet standing, is also a promising strategy for the CIP task and stick balancing on a fingertip. Such a strategy hypothesizes that the CNS exploits transient contracting dynamics along a stable manifold of a saddle-type unstable upright equilibrium of the inverted pendulum in the absence of control by inactivating neural feedback control intermittently for compensating delay-induced instability. To this end, the motions of a CIP stabilized by human subjects were experimentally acquired, and computational models of the system were employed to characterize the experimental behaviors. We first confirmed fat-tailed non-Gaussian temporal fluctuation in the acceleration distribution of the pendulum, as well as the power-law distributions of corrective cart movements for skilled subjects, which was previously reported for stick balancing. We then showed that the experimental behaviors could be better described by the models with an intermittent delayed feedback controller than by those with the conventional continuous delayed feedback controller, suggesting that the human CNS stabilizes the upright posture of the pendulum by utilizing the intermittent delayed feedback-control strategy.

Multifractal analysis of time series generated by discrete Ito equations

In this study, we show that discrete Ito equations with short-tail Gaussian marginal distribution function generate multifractal time series. The multifractality is due to the nonlinear correlations, which are hidden in Markov processes and are generated by the interrelation between the drift and the multiplicative stochastic forces in the Ito equation. A link between the range of the generalized Hurst exponents and the mean of the squares of all averaged net forces is suggested.

Log-amplitude statistics for Beck-Cohen superstatistics

As a possible generalization of Beck-Cohen superstatistical processes, we study non-Gaussian processes with temporal heterogeneity of local variance. To characterize the variance heterogeneity, we define log-amplitude cumulants and log-amplitude autocovariance and derive closed-form expressions of the log-amplitude cumulants for χ(2), inverse χ(2), and log-normal superstatistical distributions. Furthermore, we show that χ(2) and inverse χ(2) superstatistics with degree 2 are closely related to an extreme value distribution, called the Gumbel distribution. In these cases, the corresponding superstatistical distributions result in the q-Gaussian distribution with q=5/3 and the bilateral exponential distribution, respectively. Thus, our finding provides a hypothesis that the asymptotic appearance of these two special distributions may be explained by a link with the asymptotic limit distributions involving extreme values. In addition, as an application of our approach, we demonstrated that non-Gaussian fluctuations observed in a stock index futures market can be well approximated by the χ(2) superstatistical distribution with degree 2.

Parameter estimation and forecasting for multiplicative log-normal cascades

We study the well-known multiplicative log-normal cascade process in which the multiplication of Gaussian and log normally distributed random variables yields time series with intermittent bursts of activity. Due to the nonstationarity of this process and the combinatorial nature of such a formalism, its parameters have been estimated mostly by fitting the numerical approximation of the associated non-Gaussian probability density function to empirical data, cf. Castaing et al. [Physica D 46, 177 (1990)]. More recently, alternative estimators based upon various moments have been proposed by Beck [Physica D 193, 195 (2004)] and Kiyono et al. [Phys. Rev. E 76, 041113 (2007)]. In this paper, we pursue this moment-based approach further and develop a more rigorous generalized method of moments (GMM) estimation procedure to cope with the documented difficulties of previous methodologies. We show that even under uncertainty about the actual number of cascade steps, our methodology yields very reliable results for the estimated intermittency parameter. Employing the Levinson-Durbin algorithm for best linear forecasts, we also show that estimated parameters can be used for forecasting the evolution of the turbulent flow. We compare forecasting results from the GMM and Kiyono et al.'s procedure via Monte Carlo simulations. We finally test the applicability of our approach by estimating the intermittency parameter and forecasting of volatility for a sample of financial data from stock and foreign exchange markets.

Non-gaussianity of low frequency heart rate variability and sympathetic activation: lack of increases in multiple system atrophy and Parkinson disease

The correlates of indices of long-term ambulatory heart rate variability (HRV) of the autonomic nervous system have not been completely understood. In this study, we evaluated conventional HRV indices, obtained from the daytime (12:00–18:00) Holter recording, and a recently proposed non-Gaussianity index (λ; Kiyono et al., 2008) in 12 patients with multiple system atrophy (MSA) and 10 patients with Parkinson disease (PD), known to have varying degrees of cardiac vagal and sympathetic dysfunction. Compared with the age-matched healthy control group, the MSA patients showed significantly decreased HRV, most probably reflecting impaired vagal heart rate control, but the PD patients did not show such reduced variability. In both MSA and PD patients, the low-to-high frequency (LF/HF) ratio and the short-term fractal exponent α₁, suggested to reflect the sympathovagal balance, were significantly decreased, as observed in congestive heart failure (CHF) patients with sympathetic overdrive. In contrast, the analysis of the non-Gaussianity index λ showed that a marked increase in intermittent and non-Gaussian HRV observed in the CHF patients was not observed in the MSA and PD patients with sympathetic dysfunction. These findings provide additional evidence for the relation between the non-Gaussian intermittency of HRV and increased sympathetic activity.

Increased non-gaussianity of heart rate variability predicts cardiac mortality after an acute myocardial infarction

Non-Gaussianity index (λ) is a new index of heart rate variability (HRV) that characterizes increased probability of the large heart rate deviations from its trend. A previous study has reported that increased λ is an independent mortality predictor among patients with chronic heart failure. The present study examined predictive value of λ in patients after acute myocardial infarction (AMI). Among 670 post-AMI patients, we performed 24-h Holter monitoring to assess λ and other HRV predictors, including SD of normal-to-normal interval, very-low frequency power, scaling exponent α(1) of detrended fluctuation analysis, deceleration capacity, and heart rate turbulence (HRT). At baseline, λ was not correlated substantially with other HRV indices (|r| < 0.4 with either indices) and was decreased in patients taking β-blockers (P = 0.04). During a median follow-up period of 25 months, 45 (6.7%) patients died (32 cardiac and 13 non-cardiac) and 39 recurrent non-fatal AMI occurred among survivors. While all of these HRV indices but λ were significant predictors of both cardiac and non-cardiac deaths, increased λ predicted exclusively cardiac death (RR [95% CI], 1.6 [1.3-2.0] per 1 SD increment, P < 0.0001). The predictive power of increased λ was significant even after adjustments for clinical risk factors, such as age, diabetes, left ventricular function, renal function, prior AMI, heart failure, and stroke, Killip class, and treatment ([95% CI], 1.4 [1.1-2.0] per 1 SD increment, P = 0.01). The prognostic power of increased λfor cardiac death was also independent of all other HRV indices and the combination of increased λ and abnormal HRT provided the best predictive model for cardiac death. Neither λ nor other HRV indices was an independent predictor of AMI recurrence. Among post-AMI patients, increased λ is associated exclusively with increased cardiac mortality risk and its predictive power is independent of clinical risk factors and of other HRV predictors.

Near scale-free dynamics in neural population activity of waking/sleeping rats revealed by multiscale analysis

A neuron embedded in an intact brain, unlike an isolated neuron, participates in network activity at various spatial resolutions. Such multiple scale spatial dynamics is potentially reflected in multiple time scales of temporal dynamics. We identify such multiple dynamical time scales of the inter-spike interval (ISI) fluctuations of neurons of waking/sleeping rats by means of multiscale analysis. The time scale of large non-Gaussianity in the ISI fluctuations, measured with the Castaing method, ranges up to several minutes, markedly escaping the low-pass filtering characteristics of neurons. A comparison between neural activity during waking and sleeping reveals that non-Gaussianity is stronger during waking than sleeping throughout the entire range of scales observed. We find a remarkable property of near scale independence of the magnitude correlations as the primary cause of persistent non-Gaussianity. Such scale-invariance of correlations is characteristic of multiplicative cascade processes and raises the possibility of the existence of a scale independent memory preserving mechanism.

Stochastic qualifier of gel and glass transitions in laponite suspensions

The existence of the important similarities between gelation and glass transition makes it hard to distinguish between the two types of nonergodic states experimentally. Here, we report on a stochastic analysis of the scattered light intensity through a colloidal particles suspension during the gel and glass formation. In this analysis, we exploit the methods developed for complex hierarchical systems, such as turbulence. Using the multiplicative log-normal cascade models, we provide a criterion to distinguish gels from glasses.

Multifractal analysis of light scattering-intensity fluctuations

We provide a simple interpretation of non-Gaussian nature of the light scattering-intensity fluctuations from an aging colloidal suspension of Laponite using the multiplicative cascade model, Markovian method, and volatility correlations. The cascade model and Markovian method enable us to reproduce most of recent empirical findings: long-range volatility correlations and non-Gaussian statistics of intensity fluctuations. We provide evidence that the intensity increments Deltax(tau)=I(t+tau)-I(t), upon different delay time scales tau, can be described as a Markovian process evolving in tau. Thus, the tau dependence of the probability density function p(Deltax,tau) on the delay time scale tau can be described by a Fokker-Planck equation. We also demonstrate how drift and diffusion coefficients in the Fokker-Planck equation can be estimated directly from the data.

Log-amplitude statistics of intermittent and non-Gaussian time series

In studies of hydrodynamic turbulence and nonequilibrium systems, it has been demonstrated that the observed non-Gaussian probability density functions are often described effectively by a superposition of Gaussian distributions with fluctuating variances. Based on this framework, we propose a general method to characterize intermittent and non-Gaussian time series. In our approach, an observed time series is assumed to be described by the multiplication of Gaussian and amplitude random variables, where the amplitude variable describes the variance fluctuation. It is shown analytically that statistical properties of the log-amplitude fluctuations can be estimated using the logarithmic-absolute moments of the observed time series. This method is applicable to a wide variety of symmetric unimodal distributions with heavy tails in order to quantify the deviation from a Gaussian distribution. By analyzing random cascade-type processes and superstatistical non-Gaussian models with power-law tails, we demonstrate that our method can provide detailed characterization in a wide range of non-Gaussian fluctuations.

Turbulencelike behavior of seismic time series

We report on a stochastic analysis of Earth's vertical velocity time series by using methods originally developed for complex hierarchical systems and, in particular, for turbulent flows. Analysis of the fluctuations of the detrended increments of the series reveals a pronounced transition in their probability density function from Gaussian to non-Gaussian. The transition occurs 5-10 hours prior to a moderate or large earthquake, hence representing a new and reliable precursor for detecting such earthquakes.

Non-Gaussian heart rate as an independent predictor of mortality in patients with chronic heart failure

BACKGROUND

Morbidity and mortality due to chronic heart failure remain unacceptably high despite effective drug therapies, and the search for a better risk predictor is ongoing. Statistics derived from beat-to-beat fluctuations in heart rate or heart rate variability (HRV) have been used for this purpose, but the current predictability level is low or moderate at best.

OBJECTIVE

The purpose of this study was to evaluate whether a recently proposed non-Gaussian index of HRV is a significant and independent mortality predictor in patients with congestive heart failure (CHF).

METHODS

Twenty-four-hour Holter ECGs from 108 CHF patients were evaluated. Thirty-nine (36.1%) of the patients died during the follow-up period of 33 +/- 17 months. Cox proportional hazards regression analysis was performed to determine factors related to all-cause mortality. The factors evaluated derived from clinical information, including plasma brain natriuretic peptide, conventional time- and frequency-domain and fractal HRV measures, and a recently proposed non-Gaussian index lambda of HRV.

RESULTS

The short-term (<40 beats) non-Gaussian index lambda(40) (hazard ratio per increment of unit standard deviation 1.64, 95% confidence interval [1.23, 2.18], P <.001) and the long-term (<1,000 beats) index lambda(1000) (hazard ratio 1.42, 95% confidence interval [1.07, 2.18], P <.02), together with brain natriuretic peptide (hazard ratio 2.26, 95% confidence interval [1.45, 3.53], P <.001), are significant univariate risk predictors of mortality. In a multivariate model, lambda(40) (1.49, [1.13, 1.96], P <.005) and brain natriuretic peptide (2.39, [1.53, 3.75], P <.001) are independent predictors of the survival statistics of patients. None of the conventional HRV measures have predicted the mortality of patients in a significant and independent manner.

CONCLUSION

The results of this study indicate the usefulness of the short-term non-Gaussian index of HRV for risk prediction in patients with CHF.