Zbigniew R. Struzik

Multifractality in Human Heartbeat Dynamics.

There is evidence that physiological signals under healthy conditions may have a fractal temporal structure. Here we investigate the possibility that time series generated by certain physiological control systems may be members of a special class of complex processes, termed multifractal, which require a large number of exponents to characterize their scaling properties. We report onevidence for multifractality in a biological dynamical system, the healthy human heartbeat, and show that the multifractal character and nonlinear properties of the healthy heart rate are encoded in the Fourier phases. We uncover a loss of multifractality for a life-threatening condition, congestive heart failure.

From 1'f noise to multifractal cascades in heartbeat dynamics

We explore the degree to which concepts developed in statistical physics can be usefully applied to physiological signals. We illustrate the problems related to physiologic signal analysis with representative examples of human heartbeat dynamics under healthy and pathologic conditions. We first review recent progress based on two analysis methods, power spectrum and detrended fluctuation analysis, used to quantify long-range power-law correlations in noisy heartbeat fluctuations. The finding of power-law correlations indicates presence of scale-invariant, fractal structures in the human heartbeat. These fractal structures are represented by self-affine cascades of beat-to-beat fluctuations revealed by wavelet decomposition at different time scales. We then describe very recent work that quantifies multifractal features in these cascades, and the discovery that the multifractal structure of healthy dynamics is lost with congestive heart failure. The analytic tools we discuss may be used on a wide range of physiologic signals. (c) 2001 American Institute of Physics.

The Haar Wavelet Transform in the Time Series Similarity Paradigm

Similarity measures play an important role in many data mining algorithms. To allow the use of such algorithms on non-standard databases, such as databases of financial time series, their similarity measure has to be defined. We present a simple and powerful technique which allows for the rapid evaluation of similarity between time series in large data bases. It is based on the orthonormal decomposition of the time series into the Haar basis. We demonstrate that this approach is capable of providing estimates of the local slope of the time series in the sequence of multi-resolution steps. The Haar representation and a number of related represenations derived from it are suitable for direct comparison, e.g. evaluation of the correlation product. We demonstrate that the distance between such representations closely corresponds to the subjective feeling of similarity between the time series. In order to test the validity of subjective criteria, we test the records of currency exchanges, finding convincing levels of correlation.

DETERMINING LOCAL SINGULARITY STRENGTHS AND THEIR SPECTRA WITH THE WAVELET TRANSFORM

We present a robust method of estimating the effective strength of singularities (the effective Holder exponent) locally at an arbitrary resolution. The method is motivated by the multiplicative cascade paradigm, and implemented on the hierarchy of singularities revealed with the wavelet transform modulus maxima (WTMM) tree. In addition, we illustrate the direct estimation of the scaling spectrum of the effective singularity strength, and we link it to the established partition function-based multifractal formalism. We motivate both the local and the global multifractal analysis by showing examples of computer-generated and real-life time series.

Wavelet methods in (financial) time-series processing

We briefly describe the major advantages of using the wavelet transform for the processing of financial time series on the example of the SP Fractals 8 (2) (2000) 163). We use it to display the local spectral (multifractal) contents of the S&P index. In addition to this, we analyse the collective properties of the local correlation exponent as perceived by the trader, exercising various time horizon analyses of the index. We observe an intriguing interplay between such (different) time horizons. Heavy oscillations at shorter time horizons, which seem to be accompanied by a steady decrease of correlation level for longer time horizons, seem to be characteristic patterns before the biggest crashes of the index. We find that this way of local presentation of scaling properties may be of economic importance.

Of Mice and Men — Universality and Breakdown of Behavioral Organization

Mental or cognitive brain functions, and the effect on them of abnormal psychiatric diseases, are difficult to approach through molecular biological techniques due to the lack of appropriate assay systems with objective measures. We therefore study laws of behavioral organization, specifically how resting and active periods are interwoven throughout daily life, using objective criteria, and first discover that identical laws hold both for healthy humans subject to the full complexity of daily life, and wild-type mice subject to maximum environmental constraints. We find that active period durations with physical activity counts successively above a predefined threshold, when rescaled with individual means, follow a universal stretched exponential (gamma-type) cumulative distribution, while resting period durations below the threshold obey a universal power-law cumulative distribution with identical parameter values for both of the mammalian species. Further, by analyzing the behavioral organization of mice with a circadian clock gene (Period2) eliminated, and humans suffering from major depressive disorders, we find significantly lower parameter values (power-law scaling exponents) for the resting period durations in both these cases. Such a universality and breakdown of the behavioral organization of mice and humans, revealed through objective measures, is expected to facilitate the understanding of the molecular basis of the pathophysiology of neurobehavioral diseases, including depression, and lay the foundations for formulating a range of neuropsychiatric behavioral disorder models.

Critical Scale Invariance in a Healthy Human Heart Rate

We demonstrate the robust scale-invariance in the probability density function (PDF) of detrended healthy human heart rate increments, which is preserved not only in a quiescent condition, but also in a dynamic state where the mean level of the heart rate is dramatically changing. This scale-independent and fractal structure is markedly different from the scale-dependent PDF evolution observed in a turbulentlike, cascade heart rate model. These results strongly support the view that a healthy human heart rate is controlled to converge continually to a critical state.

1/f scaling in heart rate requires antagonistic autonomic control

We present systematic evidence for the origins of 1/f -type temporal scaling in human heart rate. The heart rate is regulated by the activity of two branches of the autonomic nervous system: the parasympathetic (PNS) and the sympathetic (SNS) nervous systems. We examine alterations in the scaling property when the balance between PNS and SNS activity is modified, and find that the relative PNS suppression by congestive heart failure results in a substantial increase in the Hurst exponent H towards random-walk scaling 1/ f(2) and a similar breakdown is observed with relative SNS suppression by primary autonomic failure. These results suggest that 1/f scaling in heart rate requires the intricate balance between the antagonistic activity of PNS and SNS.

Dynamics of sleep stage transitions in healthy humans and patients with chronic fatigue syndrome

Physiological and/or pathological implications of the dynamics of sleep stage transitions have not, to date, been investigated. We report detailed duration and transition statistics between sleep stages in healthy subjects and in others with chronic fatigue syndrome (CFS); in addition, we also compare our data with previously published results for rats. Twenty-two healthy females and 22 female patients with CFS, characterized by complaints of unrefreshing sleep, underwent one night of polysomnographic recording. We find that duration of deep sleep (stages III and IV) follows a power-law probability distribution function; in contrast, stage II sleep durations follow a stretched exponential and stage I, and REM sleep durations follow an exponential function. These stage duration distributions show a gradually increasing departure from the exponential form with increasing depth of sleep toward a power-law type distribution for deep sleep, suggesting increasing complexity of regulation of deeper sleep stages. We also find a substantial number of REM to non-REM sleep transitions in humans, while this transition is reported to be virtually nonexistent in rats. The relative frequency of this REM to non-REM sleep transition is significantly lower in CFS patients than in controls, resulting in a significantly greater relative transition frequency of moving from both REM and stage I sleep to awake. Such an alteration in the transition pattern suggests that the normal continuation of sleep in light or REM sleep is disrupted in CFS. We conclude that dynamic transition analysis of sleep stages is useful for elucidating yet-to-be-determined human sleep regulation mechanisms with pathophysiological implications.

Plasma Cytokine Fluctuations over Time in Healthy Controls and Patients with Fibromyalgia

We examined the pattern of cytokine secretion across the 24-hr day for women with widespread pain and tenderness having the diagnosis of fibromyalgia (FM) and matched healthy controls. Subjects were given time to habituate to being in a clinical research laboratory environment and then were sampled for cytokines without their being disturbed for a 24-hr period including an 8-hr sleep period. Cytokine levels were uniformly low but characterized by bursts of secretion. Bursting occurred either in singlets or in doublets with a range from 88 to 131 mins between doublet bursts. There was an element of synchronization of these bursts with most occurring at the beginning of sampling. FM patients showed a shift to increased IL-10 in the nighttime compared to controls. The relation between this anti-inflammatory cytokine to the pro-inflammatory cytokines studied also differed between groups: FM patients showed an increased ratio of IL-10 burst amplitude to that of pro-inflammatory cytokines IL-1β, IL-8, and TNF-α. We interpret this to indicate a skew away from the normal balance favoring pro-inflammatory cytokines in controls toward one favoring an anti-inflammatory response in FM. These changes toward anti-inflammatory predominance in FM may explain their common complaint of disturbed sleep because these cytokines are known to disrupt sleep.