MULTISCALE INTERMITTENCY IN PHYSICS AND PHYSIOLOGY

Transcription

1 MULTISCALE INTERMITTENCY IN PHYSICS AND PHYSIOLOGY V.M. Uritsky 1 and N.I. Muzalevskaya 2 1 University of Calgary, AB, Canada, vuritsky@phas.ucalgary.ca; 2 St. Petersburg State University, St Petersburg, Russia Introduction We review our recent results in multiscale intermittency analysis of correlated stochastic behavior in complex natural systems. The approaches discussed include spatial higher-order structure functions, fractal time-series analysis methods, as well as spatiotemporal decomposition of timedependent turbulent fields into sets of discrete dissipation events. These approaches are illustrated by several examples, including flaring activity in the solar corona, electron precipitation dynamics in the auroral zone, and multiscale fluctuations in human cardiovascular system. In each application, the proposed intermittency measures provide significant new information about the scaling regimes, correlation patterns, and the underlying thermodynamic states of studied systems. This paper summarizes recent advances in the applications of methods of dynamical complexity to complex physical and physiological systems. Due to the size limitation, it primarily focuses on original contributions by the authors. To compensate for this bias, references to more comprehensive review articles will be given throughout the text. The main outcome of our analyses is an innovative methodology relating statistical properties of intermittent processes in natural systems with their large-scale, low-dimensional behavior. This conceptual link provides an opportunity to better understand the relationship between stochastic and deterministic dynamics of complex systems, to classify their internal instabilities as well as the response to an external driver, and to predict future changes. The paper starts with a brief systematic overview of computational approaches for dealing with complex intermittent signals. Next, we present two examples of intermittent complexity and critical avalanching in turbulent space plasmas. Our third example (stochastic aspects of heart rate variability) features intermittency beyond physics, and shows diagnostic capabilities of multiscale complexity analysis in medical applications. Mathematical details will be kept at a minimum, with the emphasis placed on the interpretation of the processes under study. Spatial and temporal measures of intermittency Multiscale intermittency is a manifestation of dynamical complexity in driven spatially distributed nonequilibrium systems. Self-organized criticality (SOC) and intermittent turbulence (IT) represent two major paths to this dynamical state [3, 29]. In the classical fluid turbulence, scaling is often associated with a hierarchical structure of eddies extending over the inertial range, while in SOC, avalanches of localized instabilities organize the system toward a steady state exhibiting long-range correlations up to the system size. In both scenarios, intermittency implies strongly non-gaussian behavior of studied variables which undergo frequent and sudden changes often resulting in fat-tailed distributions functions such as those descried by the Levy statistics. The statistical structure of intermittent signals also involves long-range (nonexponential) autocorrelations observed over many decades of temporal and spatial scales. The hierarchy of memory effects behind this structure is usually described by fractal and multifractal models, and it tends to exhibit nonstationary properties when studied over restricted time intervals. The nontrivial statistical signatures of intermittent signals makes it difficult to obtain their quantitative parameters. Slow convergence of sample estimates, undefined statistical moments, unresolved low-frequency spectral components, poor reproducibility of results, broad confidence intervals, and other complications are very common when such signals are analyzed by standard statistical tools. These problems reflect principle limitations of classical probability theory which, according to its underlying axiomatics, is not intended to deal with cooperative stochastic processes with many interacting degrees of freedom, and can only tackle their simplified counterparts obtained as expansions about solutions that disregard the interactions. A more adequate framework for dealing with multiscale intermittent processes is offered by the modern theory of dynamical complexity which aims at a quantitative analysis and physical interpretation of correlated stochastic behaviors in nonlinear interactive systems. The data analysis tools designed in this actively growing research field have been successfully applied to a variety of problems unsolvable by classical methods. 349

2 Spatial intermittency. A large group of complexity methods is based on a generalization of the classical turbulence theory to the case when the dissipation field is represented by an inhomogeneous spatially correlated multifractal set [1, 29]. ollowing the ideas originated by A.N.Kolmogorov (1941), such irregular turbulent fields can be characterized by a collection of (unsigned) structure functions S q (l) = A(x) A(x + l) q x, l =l, where A is the variable under study (velocity, magnetic field, Elsässer variables, or a relevant passive scalar), x spatial coordinate, l displacement vector, q the order of the structure function. It is expected that each structure function varies with the spatial scale as S q ~ l ζ (q). The ζ (q) dependence plays an important role in identifying the nature of the turbulent fluid. It takes the linear ζ =q/3 form for the non-intermittent (homogeneous) 3D turbulence, and exhibits a more complicated scaling in intermittent systems [19]. In certain cases, the so-called extended self-similarity (ESS) analysis is also applied in which S q is plotted versus a reference structure function (e.g. S q (S 3 )). Such normalization yields relative values of ζ exponents, which is sufficient for validating many turbulent models. Temporal intermittency. The second group of tools that are commonly used to study intermittent systems is fractal and multifracal time series analysis methods. A time-series generalization of the structure function analysis is straightforward, and it provides a spectrum of temporal ς exponents. Some other methods are the detrended fluctuation analysis, wavelet transforms, methods relying on singular value decomposition of temporal signals in the vector space, a variety of fractal and multifractal tools, and adaptations of ourier power spectral analysis for processing nonstationary signals [8, 11, 22]. The intermittency measures provided by these methods are scaling exponents and scaling functions describing the relationship between statistical memory effects across different time scales. Some of these methods are applicable to spatially distributed data fields and can be used as auxiliary tools for examining inhomogeneous scaling of turbulent processes. Intermittency in space-time. The third group of methods addresses the coupling between spatial and temporal aspects of the behavior of the complex system. This coupling is important because in many situations, complex processes simultaneously evolve in space and in time, and the interaction between the two is farily nontrivial. Paradigmatic examples of spatiotemporal intermittency in nonlinear systems with spatially extended degrees of freedom are critical avalanches of instabilities in sandpile SOC models and bursty localized energy dissipation in high-reynolds number fluids. In our earlier works [23, 26], we have developed an approach for quantifying such manifestations of complex intermittent behavior which are abundant in nature and simulations. The approach is based on spatiotemporal decomposition of a continuous time-dependent turbulent field into a collection of discrete dissipation events composed of contiguous spatial regions of propagating activity. This nonlinear decomposition provides a detailed representation of the intermittent component in the studied dynamics in terms of its most essential statistical and topological features, while significantly reducing the amount of stored information. It also allows to selectively address different classes of intermittent disturbances based on multidimensional filtering criteria. Example 1: Solar corona Dissipation mechanisms in the solar corona are activated by changes in the configuration of its magnetic field which has distinct IT signatures [1]. Convection of magnetic fields leads to radiative transients, plasma jets, and explosive events known as flares. The latter are associated with spatially concentrated release of magnetic energy accompanied by localized plasma heating up to temperatures of 10 7 K, and can be observed by short-wavelength light emission. lares tend to appear at irregular times and locations and exhibit broadband energy, size, and lifetime statistics with no obvious characteristics scales. This behavior is often interpreted as a signature of SOC [5]. We have studied time series of full-disk digital images of the corona taken by the extreme ultraviolet imaging telescope (EIT) on board the SOHO spacecraft in the 195A wavelength band corresponding to the e XII emission. The data included two observation periods: 3240 images from a solar minimum period and 4407 images from a solar minimum period, with a typical time resolution of 13.3 min. The EIT luminosity was analyzed as a function of time and position on the image plane. To characterized spatial intermittency of SOHO EIT images, we computed higher-order structure functions in which the luminosity was used as the relevant field variable A. To identify SOC avalanches, we used the spatiotemporal decomposition method [23] resolving concurrent events. Avalanching regions were identified by applying an activity threshold representing a background EUV flux. Contiguous spatial regions above the threshold were treated as pieces of evolving dissipation events, and their statistics have been evaluated. Our results suggest that the intermittency in the corona has a fundamental impact on the dissipation mechanism in this system (ig. 1). The main energy resource for the flaring activity is the photospheric magnetic field. Its complex, highly fragmented spatial geometry contributes to the intermittent scaling of the 350

3 radiated UV flux described by a nonlinear ζ (q) dependence. The model by Müller & Biskamp [19] which provides a reasonable fit to our data describes IT in an ideal MHD plasma resulting from direct energy cascades. On the hand, the power-law statistics of energy release events suggests that a significant fraction of plasma energy is liberated in the form of inverse cascades characteristic of SOC systems. The apparent contradiction cannot be resolved in frames of existing coronal heating theories, and can be a starting point of building a more general framework which would incorporate a bi-directional energy cascade associated with collaborative IT and SOC scenarios. (a) (b) (c) ig. 1. Coexisting signatures of IT and SOC in the activity of solar corona [26]. (a) Higher-order spatial structure functions of the EIT luminosity. The inset shows ESS plots exhibiting broad-band power-law scaling. (b) Relative (ESS-based) structure function exponents compared to turbulence models due to Kolmogorov (K41), She & Leveque (SL) and Müller & Biskamp (MB) [19]. (c) Energy distributions of coronal avalanches during periods of solar minimum and maximum showing robust power-law scaling indicative of SOC. The solar min distributions are shifted for easier comparison. Different colors represent several activity thresholds used to identify the avalanches. rom a more practical point of view, it is evident that the process of coronal dissipation can not be predicted without taking into account its stochastic intermittent component which seems to control the primary energy conversion in the turbulent solar plasma [1]. One way to model future dynamical transitions in the corona is to reconstruct its magnetic network and to run a SOC algorithm that would reconnect magnetic loops thus producing flaring events. Such simulation would help reveal unstable magnetic topologies responsible for major flares, coronal mass ejections, and other space weather phenomena. Example 2: Earth s magnetosphere The necessity of using complexity tools in magnetospheric research has fundamental reasons. Unlike solar wind turbulence which at the distance 1 AU from the coronal source can be considered as "fully developed", many, if not all, magnetospheric processes are usually in a highly intermittent transient state [4, 28]. Sporadic bursts of energy dissipation, localized acceleration processes, non-steady driving, strongly inhomogeneous fluctuations involving both kinetic and MHD domains, and other forms of transient stochastic activity are fairly common in magnetospheric plasma. This messy, non-steady turbulent dynamics can not be adequately described by any of the established turbulence theories. However, it naturally fits in a more general framework of multiscale dynamical complexity providing a rich variety of methods for dealing with non-classical stochastic processes such as those observed in Earth's magnetosphere [7, 27]. Magnetospheric substorms are accompanied by a variety of intermitted processes in the auroral zone. Soon after the development of the basic substorm phenomenology, it has been realized that the nighttime auroral oval is not a simple latitudinally bound distribution of emission brightness and electric currents. The activity of this part of the ionosphere is extremely complex, and it incorporates a multitude of effects reflecting different conditions on the solar wind - magnetosphere - ionosphere coupling system. Examples of these are substorm expansion onsets, pseudobreakups, steady magnetospheric convection events with or without substorm activity, bursty bulk flows, sawtooth events, and other processes [2, 6, 21]. Despite a remarkable diversity of physical phenomena involved in the magnetospheric response to the solar wind driver, the output energy dissipation flux as estimated from particle precipitations in the nighttime aurora tends to cluster in intermittent spatiotemporal bursts obeying simple and nearly universal scale-free statistics [10, 12, 23]. 351

4 The term scale-free has been coined in statistical mechanics of turbulent and/or critical phenomena to describe correlated perturbations with no characteristic scales other than the scales dictated by the finite size of the system, as opposed to scale-dependent perturbations reflecting physical conditions that vary across different scales [29]. Considered in the context of other geophysical processes, the nighttime auroral activity provides one of the most impressive examples of scale-free behavior in Nature. Thus, the energy probability distribution of electron emission regions as seen by the POLAR satellite exhibits power-law shape over about 6 orders of magnitude [23]. By combining POLAR data with ground-based TV observations [10], the power-law scaling range has been extended up to 11 orders of magnitude (ig. 2). The consistency of the power-law slopes obtained from high-resolution ground-based auroral observations and those characterizing POLAR ultra-violet imager (UVI) data reveals an extremely wide range of power-law scaling of energy dissipation in the nighttime magnetosphere. (a) (b) ig. 2. (a) A diagram explaining the idea of spatiotemporal tracking of auroral emission regions in time series of POLAR UVI frames (LBH-long filter). (b) Power-law probability distributions of electron emission areas obtained from ground-based all-sky camera data (triangles) and the POLAR UVI observations (squares) [10, 23]. It is worth noting that these scale-free statistics represent long-term ensembles-averaged properties of nighttime magnetospheric disturbances, and they can mask a more complex dynamics on the level of specific plasma sheet structures responsible for the generation of various forms of auroral precipitations. Our recent results [24, 25] confirm the causal relationship between the auroral precipitation statistics and the nonuniform morphology of the central plasma sheet. They show that the inner and the outer plasma sheet regions are responsible for distinct scaling modes of the auroral precipitation dynamics which can a manifestation of two competing substrm scenarios represented by the current disruption and the midtail magnetic reconnection models [13, 20]. Exploring such second-order scaling effects could help build a more solid theoretical link between the statistical and dynamical plasma descriptions, evaluate predictability of different classes of magnetospheric disturbances, and obtain statistical guidelines for designing future space missions targeted at multiscale plasma phenomena. Example 3: Human heart rate variability This section illustrates an intermittent stochastic behavior in a quite different system the system of human homeostasis, monitored by the low-frequency component of heart rate variability (HRV) [9,17]. HRV is the temporal variability of the beat-to-beat RR-interval in human electrocardiogram which exhibits distinct intermittent properties in the frequency range Hz [14]. In many cases, this variability is described by the power-law 1/f β dependence of ourier power spectral density on the frequency f [9]. Typically, 1/f β HRV spectra with constant β are observed in healthy people, whereas pathologies and malfunctions are associated with more complex forms of spectral behavior. The connection between the broken fractality and the disease [15-18, 22] indicates a possibility of using 1/f β fluctuations for the purposes of clinical diagnostics, and stimulates further investigation of this phenomenon. In our previous studies, we have explored fractal and multifractal properties of HRV using a variety of time series analysis tools [15,17,18]. The results have confirmed that the low-frequency HRV is a sensitive marker of homeostatic processes. Here we present new results showing that intermittency measures of HRV can be used for early identification of pathological conditions. In addition to power-law spectral exponent β, we consider two intermittency parameters (a and T) describing nonstationary behavior of standard deviation in HRV signals as explained in ig. 3. In medical applications, extreme abnormal values of σ and β are informative state parameters. The interpretation of σ is largely empirical and is carried out on the individual basis for different sets of symptoms. Earlier, we have proposed an interpretative system for σ understanding measurements in the SOC region of HRV regulation 352

5 where the standard deviation plays a role of stochastic magnitude of dissipation losses under the conditions of fractal symmetry of homeostatic dynamics (σ i = σ =σ ). Since the constant magnitude is a signature of the stationary balance between the supplied and dissipated regulatory energies, we suggest that in general, the parameter σ can be used as a sensitive statistical marker of the current amount of available regulatory energy characterizing fractal self-organization of HRV for a broad range of physiological conditions. Distortion of fractal symmetry of HRV occurring outside the SOC region [17] reduces the scaling range of fractal R-R fluctuations, decreases available energy resource, and increases nonstationarity and intermittency in HRV, which requires a substantially different approach to interpreting σ and β measurements. σ = i β ( ) σ = min{ σ i S( f ) f } 1 ti + N RR t RR 2 t [ ti ti N N ],, t= t Intermittency measures : i σ σ a = ; σ T = a RR norm σ σ σ 1 RR norm ig. 3. Analysis of intermittency of HRV signals. The HRV sample is divided into subintervals of length N characterized by different degree of intermittency as measured by local estimates σ i of the standard deviation σ. The subinterval with the smallest σ =σ describes the laminar fractal component of HRV and is used to evaluate the intermittency indices a and T. RR norm is the normalized age-adjusted average value of the R-R interval. Table 1. Intermittency measures of HRV for several cardiovascular disorders ( n number of cases ) n a <RR> norm T Physiological characteristics Pathoadaptation dynamics prior to onsets of cardiac arrhythmias ± ± 0.30 One week before an atrial fibrillation (A) event ± ± days before A ± ± day before A ± ± 0.40 Prior to atrial palpitation (AP), sinus tachycardia ± ± 0.30 Prior to AP, sinus bradycardia ± ± 0.20 Same, 1 day before AP event Myocardial infarction, case Acute condition, first week weeks later months later Rehabilitation Myocardial infarction, case > >10 Acute condition Intensive care (reanimation) > >10 Critical condition Ischemic brain stroke, case ± ± 0.60 Acute condition ± ± 0.29 Before discharge from hospital Ischemic brain stroke, case ± ± 0.73 Acute condition ± ± 0.90 Coma To evaluate the intermittency of HRV signals, we use two constituents of the standard deviation its stationary fractal component σ as well as the nonstationary component σ σ. Each component has its own diagnostic value, while their balance which is reflected by the definitions of a and T (see ig.3) is a sensitive measure of turbulent intermittency in the HRV signal. Both a and T are small (of the order of 0.1) for healthy unperturbed homeostatic regulation, and they gradually increase with an increase of mental concentration and emotional load. Pathological conditions (Table 1) are characterized by much higher levels of a and T 353

6 indicating a significant increase of HRV intermittency in disease. or T >10, self-organization homeostatic processes necessary to maintain normal HRV are effectively replaced by random intermittency, a signature of a severe medical condition. Our new findings strongly suggest that the intermittency measures a and T can be used as empirical proxies for the available energy resource of the adaptation system. In pathological conditions, this resource significantly decreases as reflected by abnormal a and T values. Considering inherent nonstationarity of HRV signals observed for such conditions, it is also evident that the physiological interpretation of the standard deviation of R-R intervals the quantity most widely used in clinical applications [14] must be adjusted for different diseases and adaptation scenarios, depending on the relative contribution of the fractal (1/f β ) and the intermittent (a, T) variability to the studied stochastic signal. Concluding remarks We have provided several illustrative examples in which intermittency measures of seemingly random signals carry new information about system-level properties of studied processes. The common element of the complexity techniques that have been invoked in this context is their ability to characterize multiscale hierarchy of studied physical or physiological processes. This methodological advantage proves quite valuable when the macroscopic behavior is critically dependent on cross-scale interactions. The latter can be implemented in a real-space of spatially distributed geophysical systems, or in an abstract state-space of of complex dynamical system such as the system of human homeostasis. In both cases, adequately chosen intermittency measures can be used to obtain significant new information about the scaling regimes, predictability, correlation pattern, and functional stability of the studied system. References 1. Abramenko,V. I. and V. B. Yurchyshyn, Astrophys. J., 597: 1135, Angelopoulos, V. et al. Phys. Plasmas, 6(11): , Bak, P. et al., Phys. Rev. Lett., 59: 381, Borovsky, J. E. et al. J. Plasma Phys., 57:1 34, Charbonneau, P. et al. Sol. Phys., 203(2): , Donovan, E. et al. J. Atmos. Sol. Terr. Phys., 68(13): , reeman, M. P. and N. W. Watkins. Science, 298(5595): , Glenny, R.W. et la. J. Applied Physiology, 70(6): , Kobayashi, M. and T. Musha. IEEE Trans. Biomed. Eng.,29: , Kozelov, B.V. et al. Geophysical Research Lett., 31(20 ), Kumar, P, and oufoulageorgiou, E. Reviews of Geophysics, 35(4): , Lui, A. T. Y. J. Atmos. Sol.-Terr. Phys., 64(2): , Lui, A. T. Y. Space Science Rev., 95(1-2): , Malik, M. et al. Circulation, 93: , Muzalevskaya, N.I and V.G. Kamenskaya, Human Physiology, 33(2): , Muzalevskaya, N.I et al., Zhurnal Nnevropatologii i Psikhiatrii Imeni Korsakova, 7: 54-58, Muzalevskaya, N. I. and V. Uritsky, In: Telemedicine: the 21st Century Informational Technologies, St. Petersburg: Anatolia Press, , Muzalevskaya, N.I. and V.M. Uritsky, In: Longevity, Aging and Degradation Models in Reliability, Public Health, Medicine and Biology, 1: , St.Petersburg: SPbGTU Press, Müller, W.-C. and D. Biskamp, Phys. Rev. Lett., 84: 475, Ohtani, S. I. Space Science Rev., 113(1-2):77 96, Sergeev, V. A. et al. J. Geophys. Res. Space Phys., 98(A10): , Stanley, H. E. et al. Physica A, 270: , Uritsky, V. M. et al. J. Geophys. Res. Space Phys., 107(A12):1426, Uritsky, V. M. et al. Ann. Geophys. (submitted), Uritsky, V. M. et al. Geophysical Research Lett., 35:L21101, Uritsky, V.M.. et al., Phys. Rev. Lett., 99: , Valdivia, J. A. et al. Advances in Space Research, 35(5): , Voros, Z. et al. Space Science Rev., 122(1-4): , Warhaft, Z. Proc. National Acad. Sciences United States Am., 99: , Waldrop, M. M. Complexity: The Emerging Science at the Edge of Order and Chaos. New York: Simon & Schuster,