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The Hurst Exponent of Heart Rate Variability in Neonatal Stress, Based The Hurst exponent of heart rate variability in neonatal stress, based on a mean-reverting fractional Lévy stable motion Matej Šapina, Matthieu Garcin, Karolina Kramarić, Krešimir Milas, Dario Brdarić, Marko Pirić To cite this version: Matej Šapina, Matthieu Garcin, Karolina Kramarić, Krešimir Milas, Dario Brdarić, et al.. The Hurst exponent of heart rate variability in neonatal stress, based on a mean-reverting fractional Lévy stable motion. 2017. hal-01649280v2 HAL Id: hal-01649280 https://hal.archives-ouvertes.fr/hal-01649280v2 Preprint submitted on 31 Oct 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Hurst exponent of heart rate variability in neonatal stress, based on a mean-reverting fractional L´evy stable motion Matej Sapina,ˇ MDa,b,c,∗, Matthieu Garcin, PhDd;∗, Karolina Kramari´c,MSca,b,c, Kreˇsimir Milas, MDa,b, Dario Brdari´c,PhDe, Marko Piri´c,MDb October 31, 2018 Abstract 1 Introduction In the evolutionary context, acute stress is life-saving We aim at detecting stress in newborns when evading a predator. However, exposure to by observing heart rate variability. The chronic stress is harmful [32]. Stressors are various heart rate variability features nonlinearities. Fractal dynamics are a usual way to model nonspecific stimuli which cause a stress response vari- them and the Hurst exponent summarizes ety [51]. What does not seem to be stressful to adults, the fractal information. In our framework, might have significant effects on a term-, moreover, we have observations of short duration, for on a preterm newborn [30, 20]. which usual estimators of the Hurst expo- If exposed to acute stress, like a simple heel prick nent, like DFA, are not adapted. Moreover, blood drawing, the newborn goes through signifi- we observe that the Hurst exponent does not vary much between stress and rest phases, cant physiological changes. The infants start cry- but its decomposition in memory and under- ing, but in behind the curtains the activation of the lying probability distribution leads to satis- hypothalamic-pituitary-adrenocortical (HPA) system factory diagnostic tools. This decomposition goes on [21, 10]. Serum catecholamine and glu- of the Hurst exponent is in addition embed- cocorticoid levels rise, the sympathetic of the au- ded in a mean-reverting model. The result- tonomic nervous system rises, both the heart and ing model is a mean-reverting fractional L´evy blood pressure increase, the heart rate variability, stable motion. We estimate it and use its respiration rate and respiration movements change parameters as diagnostic tools of neonatal etc. [20, 38, 49, 45, 29]. stress: the value of the speed of reversion pa- rameter as well as the evolution of both pa- Much effort is given towards the prevention of stress rameters in which the Hurst exponent is de- in human term and preterm newborns [6]. Due to composed are significant indicators of stress, their immaturity, preterm infants are more vulnera- whereas the Hurst exponent itself does not ble than term infants to acute stress. Having already bear useful information. a compromised autoregulation of cerebral blood flow, its fluctuations caused by various stressors might dis- turb normal neurodevelopment [30, 55]. Keywords { fractals, heart rate variability, mean The long-term effects of stress exposure are related to reversion, stable process, Hurst exponent the prolonged effects of glucocorticoids on every body ∗These authors contributed equally to this study. Corresponding author: [email protected]. a University hospital Osijek, Pediatric Clinic, J. Huttlera 3, 31000 Osijek, Croatia. b Medical faculty Osijek, Cara Hadrijana 10E, 31000 Osijek, Croatia. c Faculty of Dental Medicine and Health, Crkvena 21, 31000 Osijek, Croatia. d L´eonard de Vinci P^ole Universitaire, Research center, 92916 Paris La D´efense, France - LabEx ReFi e Institute of Public Health for the Osijek-Baranya County, Drinska 8, 31000 Osijek, Croatia. 1 system [32]. Abnormal perinatal conditions have also of HRV. been associated with various behavioral and emo- The rest of the paper is structured as follows. We tional deviations, lower cognitive capabilities, and first present the mean-reverting fLsm model and the might cause epigenetic modifications [45, 53, 16, 22]. method for estimating its parameters. Then, we In this paper, we propose a new method to detect present the results of our analysis on 40 patients. Fi- stress in newborns. This method is based on statisti- nally, we discuss these results and conclude. cal properties of the heart rate variability (HRV). We focus on the fractal scaling properties of the HRV, in particular its Hurst exponent in the framework of a 2 Methodology new model: a mean-reverting fractional L´evy stable motion. We show how to interpret the parameters of this model in the case of neonatal stress. In this section, we present our model, a mean- reverting fractional L´evy stable motion, and we pro- In the literature, the autocorrelation of HRV has pose a method to estimate all its parameters. After- been depicted by several models and statistical meth- wards, we will focus on quantifying the significance ods [1], like spectral analysis [50, 4, 44, 15], entropy of each estimated value by the mean of a statistical measures [42, 43, 46] and tools of the chaos theory bootstrap. We finish this section by presenting the such as the correlation dimension [37, 3], Poincar´e study protocol. plots [26, 11, 49], recurrence plots [36, 35] or Lya- punov exponents [2, 54]. Fractal dynamics are an- other popular family of models of HRV. For these 2.1 The model: a mean-reverting dynamics, the most relevant parameter is the Hurst fractional L´evy stable motion exponent, which quantifies the time scaling of the stochastic process or time series: by definition, if we 2.1.1 Rationale of the model note H the Hurst exponent of a stochastic process St, an increment of duration d, St+d − St, has the H We first focus on the two specificities introduced in same probability distribution as d (St+1 − St), for all time t. The Hurst exponent can be estimated this model: the mean-reversion and the decomposi- by several methods. For the analysis of the HRV, tion of the scaling rule. it has been estimated by the method of absolute moments [9], rescaled range analysis (R/S) [31, 34] . It has been noted in the literature based on and detrended fluctuation analysis (DFA), in which DFA that the Hurst exponent for the HRV sig- the HRV signal is detrended by a piecewise linear nal is in fact scale-dependent. This implies that trend [41, 40, 13, 52, 48] or by a moving average [5]. the fBm is not well specified for HRV. Multi- Among these three techniques, our preference goes to fractal dynamics are thus more accurate in this the method of absolute moments [39, 18] because it framework [25, 33]. But the fact that fBm is not does not need a dataset as big as for computing R/S well suited to HRV can be explained by many analysis or DFA, which are based on regressions on other model specifications than the sole multi- many subsamples. fractality of the process, like a time-dependent Hurst exponent [18] or a Lamperti transform But Hurst exponent is not only a statistical indica- of a fBm [19], which generalizes the mean- tor. It can indeed be directly linked to a well-known reverting Ornstein-Uhlenbeck process. A rapid stochastic process, the fractional Brownian motion look at some healthy and at rest HRV time se- (fBm), which is the Gaussian process consistent with ries shows besides a mean reversion [12, 24]. As the scaling rule based on Hurst exponent. However, a consequence, we add in our model a mean re- HRV does not look like a fBm and we now introduce version to the fractional process. in this paper a dynamic featuring two other speci- ficities: a mean-reversion and a more detailed scaling . In the fBm model, one can link the Hurst ex- rule than the traditional Hurst exponent, based on a ponent H to the autocorrelation of the pro- memory term and on a parameter of a stable distri- cess. Thus, one usually considers that H > 1=2 bution. These specificities enable a more accurate fit corresponds to a long-memory feature, whereas of the shape of HRV time series. They result in some H < 1=2 corresponds to a negative autocor- additional parameters, which can easily be estimated relation and H = 1=2 to an absence of auto- and interpreted. These specificities of the model con- correlation. This is true for the fBm but not stitute an innovation in the description and analysis for all fractional processes. If the increments 2 of the process are not Gaussian, another in- 2.1.2 Definition of the model terpretation should be made [8]. Indeed, the Hurst exponent is a scaling parameter and the We observe the interbeat interval process Bt between scaling rule is established by two causes: the time 0 and time T . We translate it first to get a non-conditional probability distribution of the centred process Xt: increments of the process and the dependence T between these increments. In particular, if 1 Z Xt = Bt − Bsds: we consider symmetric α-stable increments, of T 0 which the standard Gaussian variable is a par- ticular case,1 we can build a much more gen- We assume that Xt is the solution of the following eral process called fractional L´evy stable mo- equation, which defines our mean-reverting fLsm: tion (fLsm) [56].
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