## Abstract

Complex continuous wavelet transforms are used to study the dynamics of instantaneous phase difference Δφ between the fluctuations of arterial blood pressure (ABP) and cerebral blood flow velocity (CBFV) in a middle cerebral artery. For healthy individuals, this phase difference changes slowly over time and has an almost uniform distribution for the very low-frequency (0.02–0.07 Hz) part of the spectrum. We quantify phase dynamics with the help of the synchronization index γ = 〈sinΔφ〉^{2} + 〈cosΔφ〉^{2}that may vary between 0 (uniform distribution of phase differences, so the time series are statistically independent of one another) and 1 (phase locking of ABP and CBFV, so the former drives the latter). For healthy individuals, the group-averaged index γ has two distinct peaks, one at 0.11 Hz [γ = 0.59 ± 0.09] and another at 0.33 Hz (γ = 0.55 ± 0.17). In the very low-frequency range (0.02–0.07 Hz), phase difference variability is an inherent property of an intact autoregulation system. Consequently, the average value of the synchronization parameter in this part of the spectrum is equal to 0.13 ± 0.03. The phase difference variability sheds new light on the nature of cerebral hemodynamics, which so far has been predominantly characterized with the help of the high-pass filter model. In this intrinsically stationary approach, based on the transfer function formalism, the efficient autoregulation is associated with the positive phase shift between oscillations of CBFV and ABP. However, the method is applicable only in the part of the spectrum (0.1–0.3 Hz) where the coherence of these signals is high. We point out that synchrony analysis through the use of wavelet transforms is more general and allows us to study nonstationary aspects of cerebral hemodynamics in the very low-frequency range where the physiological significance of autoregulation is most strongly pronounced.

- cerebral blood flow
- transcranial Doppler sonography
- wavelets
- synchronization

the statistical properties of physiological fluctuations, such as those found in the time series for heartbeat dynamics (10, 28), respiration (1, 27, 40), human locomotion (11, 39), and posture control (4), have been the focus of interdisciplinary research for more than two decades. This research has underscored the significance of nonlinear and nonstationary aspects of intrinsic variability of many physiological phenomena. Such variability seems to indicate the adaptability of the underlying control systems. The change of paradigm, associated with how we view the dynamics of physiologic phenomena, has not, to date, significantly influenced the interpretation of fluctuations in cerebral hemodynamics. In particular, the mathematical analysis of the fluctuations in either intracranial pressure or cerebral blood flow (CBF) velocity (CBFV) in major arteries is largely confined to traditional spectral methods.

A healthy human brain is perfused with blood flowing laminarly through cerebral vessels, providing brain tissue with substrates such as oxygen and glucose. It turns out that CBF is relatively stable, with typical values between 45 and 65 ml·100 g brain tissue^{−1}·min^{−1}, despite variations in systemic pressure as large as 100 Torr. This phenomenon is known as cerebral autoregulation and has been thoroughly documented not only in humans, but also in animals (26).

The search for autoregulation tests that exploit the spontaneous fluctuations in arterial blood pressure (ABP) is the focus of the ongoing research reported herein. The goal of this paper is to quantify the interdependence of ABP and CBFV time series from the viewpoint of dynamic synchronization. In particular, we investigate the time evolution of the instantaneous phase difference between the two time series in healthy patients and compare this analysis with the more traditional transfer function approach. The importance of multiple scales, identified by Mitsis et al. (20, 23) in their two-scale neural network model, is herein extended through the use of continuous scales in the wavelet transforms. Although we do not propose a particular dynamic model for cerebral autoregulation as done, for example, in Ref. 38, we do determine the dependency of the time series on multiple time scales and consequently determine the multiscale nature of the underlying dynamics.

We hypothesize that the interdependence of the fluctuations in ABP and CBFV time series for normal healthy individuals is manifested differently at different scales. The phase difference between the two time series is synchronized at high frequencies but is essentially random at low frequencies.

Transfer function analysis (3, 7, 17, 25, 32, 41–43) sheds light on the dynamic properties of cerebral autoregulation from the unique perspective of linear response theory. Within this framework, cerebral autoregulation is considered to be a high-pass filter transmitting rapid changes in blood pressure but dampening and delaying low-frequency perturbations. Variations of blood pressure are the filter's input and cerebral velocities are its output, and this is presently the standard model for cerebral hemodynamics (9). More specifically, this type of physiological filtering is quantified for a given frequency as a phase shift between two signals. Phase shifts close to zero, at frequencies that are usually dampened, are interpreted as a loss of cerebral autoregulation (7, 34). However, in the low-frequency range, the low values of coherence between the changes in pressure and velocity indicate that the linearity and/or stationarity conditions, the prerequisites to the estimation of the linear transfer function gain and phase, may be violated. The measures based on Pearson's correlation coefficient between the averaged values of the ABP and CBFV time series (5, 6, 15, 29, 33), despite their appealing simplicity and proven clinical significance, are inherently incapable of taking into account nonlinear effects. In fact, Giller and Mueller (9), using four methods of comparison, conclude that there is strong evidence that cerebral autoregulation is nonlinear. Furthermore, Mitsis et al. (21) not only concur in the findings regarding nonlinearity but also conclude that cerebral autoregulation is nonstationary as well. The phase dynamics method introduced here provides a mathematical framework for analysis of both nonlinear and nonstationary effects in cerebral hemodynamics, thus overcoming the restrictions intrinsic to earlier methods.

## METHODS

Ten students (5 men and 5 women, mean age 24 ± 3 yr) of the Technical University of Wroclaw participated in the study that was approved by the Regional Ethics Committee in Opole, Poland. All volunteers gave their written informed consent. The subjects were free of cardiovascular, pulmonary, and cerebrovascular disorders. Measurements of ABP and CBFV of 45-min duration were preceded by 20 min of supine rest. Data were recorded during spontaneous, uncontrolled respiration. CBFV in the middle cerebral artery was monitored using a transcranial Doppler ultrasonograph (DWL MultiDop with 2-MHz probes placed over the temporal windows and fixed at a constant angle and position relative to the head). ABP was noninvasively measured by finger photoplethysmography (Finapres, Ohmeda). We divided the 45-min measurements into 15-min intervals to verify that stability of the Finapres setup (with deactivated servo mechanism) does not influence the outcome of data processing. Beat-to-beat average values of pressure and velocity were calculated via waveform integration of the corresponding signals sampled at 100 Hz and digitized at 12 bits. In numerical calculations, nonuniformly spaced time series were resampled at 2 Hz with the help of cubic spline interpolation.

Given our interest in the interdependence of physiological systems, let us consider two signals, *s*_{1}(*t*) and *s*_{2}(*t*), and their corresponding instantaneous phases, φ_{1} and φ_{2}. The phase synchronization takes place when *n*φ_{1} − *m*φ_{2} = constant, where *n* and *m* are integers indicating the ratios of possible frequency locking. Herein we consider only the simplest case, *n* = *m* = 1. Furthermore, as with most biological signals apparently contaminated by uncorrelated random fluctuations, we are forced to search for approximate rather than exact phase synchrony, i.e., φ_{1}(*t*) − φ_{2}(*t*) ≈ constant. Thus the studies of synchronization involve the introduction of some statistical measure of phase locking (16, 31) in addition to the determination of phases of signals.

The instantaneous phase φ(*t*) of a signal *s*(*t*) can be readily extracted by calculating its wavelet transform *W*_{s} (19): with a complex mother function ψ to obtain In the above formula we explicitly indicated the dependence of the instantaneous phase on the scale *a* of the wavelet transform. It is possible to associate a pseudofrequency *f*_{a} with the scale *a*: where *f*_{c} is the center frequency of the wavelet function and δ*t* is the sampling period. We employed the complex Morlet wavelet with *f*_{c} = 1 to determine the instantaneous phases of ABP and CBFV fluctuations. A more detailed description of wavelet transforms and their relation to traditional spectral analysis is discussed in appendix.

The distribution *P*(Δφ) of phase difference Δφ = φ_{1} − φ_{2} can be used to characterize the synchronization between the two time series of interest. A uniform distribution corresponds to the absence of synchronization (the 2 signals are statistically independent of one another), whereas a well-pronounced peak in the distribution *P*(Δφ) is a manifestation of phase locking, which means that one time series tracks the dynamics of the other. We quantify the stability of phase difference Δφ with the index: and the brackets denote a time average of the phase fluctuations. The synchronization index lies in the interval 0 ≤ γ ≤ 1 and varies with the scaling parameter *a*. When the distribution of the phase differences is uniform, the time average of both trigonometric functions in the definition of γ are zero, which leads to a vanishing synchronization index. It follows from the trigonometric identity that γ = 1 corresponds to perfect synchronization (phase locking of the 2 processes).

## RESULTS

Figure 1 displays the analysis of phase synchronization between fluctuations of ABP (Fig. 1*A*) and CBFV (Fig. 1*B*) time series in the middle cerebral artery for a healthy subject. The results depicted in this figure are qualitatively the same for all members of the cohort of healthy subjects. The density map (Fig. 1*D*) represents the time evolution of the normalized phase difference Δφ/2π for 100 integer values of the wavelet scale *a*. The color bar reveals the assignment of colors to the various synchronization levels. To facilitate interpretation of phase dynamics, Fig. 1*E* displays the variation of the normalized phase difference Δφ/2π for the scale *a* = 50 (*f*_{a} = 0.04 Hz). In the very low-frequency part of the spectrum (0.02–0.07 Hz), the phase difference slowly evolves over time and has an almost uniform distribution, as unambiguously indicated by the low value of the synchronization parameter γ for *a* > 30 (cf. Fig. 1*C*).

The conclusions derived from the case study are corroborated by the analysis of the group-averaged synchronization index γ depicted in Fig. 2 using the 10 members of the control group. The value of the synchronization parameter averaged over scales 30 to 100, γ_{30:100} = 0.13 ± 0.03, indicates the physiological absence of phase locking in the very low-frequency region of the spectrum. The low value of standard deviation of γ_{30:100} reflects the absence of synchronization in this frequency range for all members of the control group. This is also confirmed by the uniform distribution of averaged phase difference shown in the right polar density plot in Fig. 3. However, for smaller scales (higher frequencies) the entrainment of ABP and CBF velocities is much stronger. In particular, the group-averaged index γ has two distinct peaks, one at 0.11 Hz (γ = 0.59 ± 0.09) and another at 0.33 Hz (γ = 0.55 ± 0.17). These peaks correspond to the maxima of the distribution function of group-averaged phase difference (cf. the left polar density plot in Fig. 3).

## DISCUSSION

It is apparent that the interplay of the fluctuations of ABP and CBFV for high frequencies (>0.07 Hz) is entirely different from that in the low-frequency part of the spectrum. Let us focus on the peak of the synchronization curve at 0.33 Hz (Fig. 2). The strong reactivity of cerebral vessels to carbon dioxide arterial content (26) might underlie the respiratory origin of this peak. However, a number of studies have shown that the CBF response to a step increase in CO_{2} is not instantaneous but trails the induced change by several seconds. Poulin et al. (30) described the rising phase of CBFV by a simple exponential equation involving a time constant, an amplitude factor, and a time delay of the order of 6 s. A more elaborate nonlinear model of the interaction between cerebral autoregulation and the CO_{2} reactivity was formulated by the Ursino and Lodi (38). Mitsis et al. (21) have also emphasized the nonlinearities in the dynamics of the end-tidal CO_{2}-blood flow velocity relationship. It is worth pointing out that the dynamic effect of respiration on cerebral autoregulation may also be described in terms of the quasi-periodic driving. Periodic driving of nonlinear systems leads to a number of interesting dynamic effects, entrainment being one of the possibilities (2). Further studies should verify whether in fact the peak at the respiration frequency may have this type of dynamic origin.

The interpretation of the very well-pronounced maximum at 0.11 Hz is even more subtle. Despite some controversy, the current evidence suggests that the baroreflex substantially contributes to low-frequency, nonrespiratory (≥0.1 Hz) oscillations in blood pressure, duration of R-R intervals, and peripheral sympathetic nerve activity to muscle (MSNA) (see, for example, Ref. 14 and references therein).

Strong susceptibility of the brain tissue to even short periods of ischemia underlies the physiological significance of autoregulation in the low-frequency part of the spectrum. Phase synchrony analysis provides a clear-cut distinction of low-frequency dynamics. The abrupt fall of the synchronization parameter γ at ∼0.07 Hz in Fig. 2 marks the transitions from the strong high-frequency synchronization to low-frequency phase difference variability. We can see in Fig. 1*E* hat the CBFV and ABP periodically become phase locked for short times (the phase difference becomes very small). This phenomenon has been referred to as intermittent autoregulatory failure (9). However, this transient effect stems naturally from the phase dynamics and does not have a pathological origin. The insight provided by the synchrony analysis may also allow to put in different perspective the reports of the very low values of ARI autoregulation index in healthy individuals (18). The low values of this index would suggest the absence of autoregulatory response in physiological condition.

The phase difference variability sheds new light on the nature of cerebral hemodynamics, which so far has been predominantly characterized using the high-pass filter model. In the latter framework efficient autoregulation is associated with the positive transfer function phase shift between oscillations of CBFV and ABP. Herein we propose to base the assessment of autoregulation on a truly dynamic quantity such as synchronization strength.

To emphasize the fundamental differences between the two approaches, let us compare the outcome of the wavelet analysis (Fig. 4*A*) with that of the transfer function (Fig. 4, *B*–*D*). In particular, we analyze two monochromatic signals with frequency 0.04 Hz and the unit amplitude. One signal mimics the input and the other output of the regulatory system. In Fig. 4*A* we can see that at *t*_{0} = 500 s, the phase difference between signals jumps from 50° to 150°. The application of the complex Morlet wavelet (*f*_{c} = 1, *f*_{b} = 1, *a* = 50) allows us to accurately determine the time evolution of phase difference. On the other hand, the transfer function analysis yields the value of the phase difference equivalent to 100°, i.e., the average value of phase difference over the entire interval. The change of the Hamming window's length does not affect this average value but does influence the spectral resolution. We can see in Fig. 4*C* that for *L* = 128, the transfer function phase is equal to 100° across the entire frequency range even though the analysis concerns the monochromatic signals with frequency 0.04 Hz. The longer window (*L* = 512) provides better spectral resolution. Although the amplitudes of both signals are equal, the gain in Fig. 4*B* is ∼0.6. This error exemplifies the difficulties the transfer function analysis encounters while dealing with low-coherence signals.

In Fig. 5, we present the estimates of transfer function gain, phase, and coherence for the experimental data. We can see in Fig. 5*C* that the frequency region where the coherence is >0.5 extends from approximately 0.08 Hz to 0.38 Hz, in good agreement with the original estimate of Zhang et al. (41). For frequencies <0.08 Hz, coherence is considerably below 0.50. In Fig. 5*B* we can see that in the low-frequency region the phase difference between the ABP and the CBFV rather weakly depends on frequency and oscillates ∼50°. This is obviously at variance with the outcome of the wavelet analysis, which yields fluctuations of phase difference. To trace the source of the apparent inconsistency, in Fig. 6 we present a model calculation. In Fig. 6 we compare the frequency dependence of the transfer function phase, determined using a Hamming window with *L* = 128 (the value used in the calculations presented in Fig. 5) with that corresponding to the window with *L* = 512. For the sampling frequency 2 Hz, these windows are equivalent to 64 and 256 s, respectively. We can see that for the larger window the group-averaged phase strongly fluctuates in the region where the coherence is low. These fluctuations may originate as a result of the transfer function's averaging of the nonstationary phase difference, the effect elucidated by Fig. 4*C*. When the length of the window is comparable to the time it takes the phase difference to sweep through 2π, the outcome of such averaging is unpredictable (cf. Fig. 1*E*). It must be emphasized that the problem cannot be solved by decreasing *L* because the length of the window also determines the lowest frequency in the power spectrum estimates. In fact the smooth dependence of the phase difference on frequency for *L* = 128 may, to a large extent, be an artifact of low spectral resolution. This effect can clearly be seen for the model calculations in Fig. 4*C*.

One could possibly argue that the phase difference variability reported herein has its source in the measurement noise. In other words, in the absence of such noise the phase difference between the fluctuations of ABP and CBFV would be constant. However, the high values of coherence for the intermediate frequencies (0.1–0.3 Hz) found in this and several other independent studies makes this hypothesis unlikely. Moreover, we observe the loss of low-frequency variability in patients with brain injury-related impaired autoregulation. Such loss could not have been detected if noise had dominated the experimental signal. The detailed discussion of phase dynamics in patients with brain injuries will be presented elsewhere.

It is worth pointing out that quite likely the relation between the CBFV and ABP time series in the low-frequency region is not entirely determined by autoregulation. The variability of the phase difference may also reflect the generation of cerebral slow waves independent of ABP fluctuations or spontaneous vasomotion. In fact, the existence of spontaneous CBFV fluctuations was a major driving force behind the research exploring the nonlinear properties of CBF [see, for example, Giller and Mueller (9)].

The comprehensive study of 168 patients with severe carotid stenosis or occlusion provided a unique opportunity to thoroughly evaluate clinical applicability of transfer function theory to the noninvasive assessment of cerebral autoregulation (32). Two standard approaches based either on spontaneous oscillations (SPO) of ABP of ∼0.1 Hz or respiratory-induced oscillations, during deep breathing (DB), at a rate 6 breaths/min, were compared. Transfer function phase and gain in low (0.06–0.12 Hz)- and high-frequency (0.20–0.30 Hz) regions were employed to characterize cerebral hemodynamics. Reinhard et al. (34) point out that spectral analysis for DB is unambiguous because in most cases there is a distinct peak at the target frequency of 0.1 Hz. The selection of a single frequency for extraction of transfer function phase and gain is problematic in the SPO approach, both in terms of physiological validity and interobserver agreement. The authors continue to argue that transfer function calculations are reliable only when coherence is significant. Consequently, the point of maximum coherence within the selected frequency range is a logical choice (12). However, in as many as 13% of the patients with carotid stenosis or occlusion, the low coherence makes transfer function analysis inapplicable (32). Moreover, Giller (8) suggested that low coherence may, in fact, be just a manifestation of efficient autoregulation.

The instantaneous phase dynamics seem to circumvent the transfer function dilemma. One may choose a frequency range in which pathological manifestations are most strongly pronounced and where the consequences of autoregulation impairment are most severe. Figure 2 indicates that the low-frequency region is particularly interesting because of both the lack of synchronization at the physiological condition and the susceptibility of brain tissue to extended periods of ischemia. Panerai et al. (24) have recently brought up the issue of the significant intersubject variability of currently used dynamic autoregulation indexes. These authors identify nonlinear and multivariate character of CBF control mechanisms as well as the influence of various metabolites on the vasomotor reactivity as the plausible sources of the variability of CBFV. The measures derived from synchronization theory, such as the synchronization index γ, are applicable in the presence of both nonstationarities and nonlinearities. Consequently, such measures open up new possibilities for improving diagnostic reliability of cerebral autoregulation assessment.

The influence of autonomic neural control on cerebral circulation is not fully understood. The transfer function analysis of autoregulation before and after ganglion blockade with trimethapan has shed light on this complex problem (42). In the very low-frequency part of the spectrum (0.02–0.07 Hz), the blockade resulted in the increase of gain, reduction of phase difference between CBFV and ABP time series, and interestingly enough, significant reduction of coherence (to as low as 0.25). Zhang et al. (42) point out that nonlinearities of the cerebral circulation might have contributed to the low coherence and conclude that “…the optimal techniques for accurate quantification of these nonlinearities has yet to be determined.” The phase dynamics analysis presented herein contributes to the solution of this dilemma because it enables us to properly account for both nonlinear and nonstationary properties of CBF. We are convinced that better understanding of cerebral hemodynamics will inevitably lead to more clinically significant measures of dynamic autoregulation.

It is worth pointing out that low-frequency fluctuations were observed in intracranial pressure, CBFV, cerebral hemoglobin oxygenation, tissue Po_{2}, the cortical cytochrome *c* oxidase, and the cortical blood volume. The cerebral microvasculature shows slow rhythmic activity known as vasomotion. It remains unclear whether these different microcirculatory and metabolic oscillations are localized to meet varying metabolic demands and whether they are synchronized by unknown mechanisms or are induced by systemic hemodynamic factors (35). The phase dynamics approach presented herein is particularly well suited to study of such complex problems.

## APPENDIX

The transfer function has been successfully used for decades to characterize the influence of one time series on another. However, transfer functions are limited in their domain of application because they are only rigorously defined for linear processes, such that one signal is the output and the other the input to the convolution (A1) Using the convolution theorem, one may express the frequency-dependent transfer function *H*(*f*) between the two time series as (A2) where averaging, denoted by the angle brackets, is carried out over an ensemble of realizations of the fluctuations and *ŝ*_{1}(*f*) and *ŝ*_{2}(*f*) denote the Fourier amplitudes of the corresponding signals. *S*_{12}(*f*) in *Eq. A2* is the cross-spectrum of the changes in the two time series and *S*_{11}(*f*) is the autospectrum, both of which are frequency dependent. The magnitude of the transfer function |*H*(*f*)| is called the gain, and the phase of the transfer function is defined by (A3) the real and imaginary parts of *H*, so that, when taken together, these two quantities quantify the relationship between the two signals. When the relation between the two time series is linear and the fluctuations are stationary, the transfer function is constant in time, and, consequently, so is the phase *Φ(f)*, the phase being the phase difference between the two time series. Welch's periodogram method is commonly used as the numerical basis of the transfer function calculations (13). One divides the dataset of length *N* into segments of length *L* usually overlapped by half (the smaller overlapping ratios may also be used). The segments are detrended and multiplied by a Hamming window. We used *N* = 2,048 and either *L* = 128 or *L* = 512.

As we mentioned, the transfer function formalism is applicable only when the interaction between the two processes is predominantly linear. In that case a magnitude-squared coherence (MSC) function (A4) approaches unity. Low values of this function in a given frequency range may result from nonlinear effects, substantial noise, nonstationarity of the time series, or independence of the signals. The value MSC = 0.5 is frequently used as a threshold below which transfer function theory is no longer applicable because this value implies 50% shared variance between the two signals. The determination of coherence threshold based on an *F-*test is discussed, for example, in Ref. 36. However, herein we do not address the rather subtle issue of whether, for a given choice of the parameters of power spectrum estimate, ABP and CBFV are incoherent. We instead examine the reliability of the transfer function phase difference calculations in low-coherence regions. This emphasis is justified by the clinical significance commonly attributed to this phase difference in evaluation of integrity of cerebral autoregulation.

When the time series are either nonstationary or nonlinearly related to one another, the justification for transfer function theory is lost. One needs only to recall that nonstationary time series give rise to time-dependent spectra, so that both the gain and phase of the transfer function become ill-defined quantities. Consequently, in the text we introduce a procedure that overlaps with transfer function theory in the linear domain but that was developed to treat nonstationary time series. This is the method of wavelet transforms and its application does not demand that we know in advance if the relation between the two time series is linear or not.

A wavelet transform is an integral transform that employs basis functions, known as wavelets, that are localized both in time and frequency (19), unlike the infinitely long wavetrains required for Fourier transforms. Such wavelets are constructed from a single mother wavelet ψ(*t*) by means of translations and dilations. The instantaneous phase φ(*t*) of a signal *s*(*t*) can be readily determined by calculating the signal's wavelet transform *W*_{s}(*a*,*t*_{0}) (A5) with a complex mother function to obtain the phase (A6) Here we see that the phase explicitly depends on the scale *a* and on the time index *t*_{0}, where the latter sweeps through the time series determining the local properties of the signal's phase. Note that this time-dependence of the phase enters naturally through the definition of the wavelet transform, unlike the case of the Fourier transform where the time dependence of the phase actually violates the conditions of the transform. The dual localization of wavelets in time and frequency enables us to associate a pseudofrequency *f*_{a} with the scale *a* (19): (A7) where *f*_{c} is the center frequency and δ*t* is the sampling period. In instantaneous phase calculations we employ the complex Morlet wavelet: (A8) and set both the center frequency *f*_{c} and the bandwidth parameter *f*_{b} to 1.

The continuous wavelet transform of a discrete time series of length *N* and equal spacing δ*t* is defined as (A9) (see for example Ref. 37). The above convolution can be evaluated for any of *N* values of the time index *n*. However, the convolution theorem allows us to compute all *N* convolutions (*Eq. A9*) simultaneously in Fourier space using a discrete Fourier transform: (A10) where *k* = 0,…,*N* − 1 is the frequency index. If one notes that the Fourier transform of a function ψ(*t*/*a*) is *a*ψ̂(*af*), then by the convolution theorem (A11) where frequencies *f*_{k} are defined in the conventional way. *Equation A11* can be efficiently calculated using the standard fast Fourier transform routine. It is worth pointing out that formally *Eq. A11* does not yield the discrete linear convolution corresponding to *Eq. A9*, but rather a discrete circular convolution in which the shift *n* − *n*′ is taken modulo *N*. Consequently, the border effects should be handled properly. The numerical implementation of continuous wavelet transforms may be found on the following websites, http://paos.colorado.edu/research/wavelets/ or http://www-stat.stanford.edu/∼wavelab/. The Mathworks' MATLAB Wavelet Toolbox provides a host of wavelet-related routines.

## GRANTS

We gratefully acknowledge the financial support of the U.S. Army Research Office (Grant DAAD19–03-1–0349).

## Acknowledgments

The hemodynamic studies of healthy volunteers were performed at Wroclaw Military Hospital. We thank Professor P. Ponikowski for his help and stimulating discussion.

## Footnotes

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- Copyright © 2005 by the American Physiological Society