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INNOVATIVE METHODOLOGY
1Institute of Physics, Wroclaw University of Technology, Wroclaw; and 2Department of Neurosurgery, Opole Regional Medical Center, Opole, Poland; and 3Mathematical and Information Sciences Directorate, Army Research Office, Research Triangle, North Carolina
Submitted 30 December 2004 ; accepted in final form 21 June 2005
Complex continuous wavelet transforms are used to study the dynamics of instantaneous phase difference 
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 
= 
sin
2 + cos2that 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 has two distinct peaks, one at 0.11 Hz [ = 0.59 ± 0.09] and another at 0.33 Hz ( = 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
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