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1 Institute of Physics, Wroclaw University of Technology, Wroclaw, Poland
2 Department of Neurosurgery, Opole Regional Medical Center, Opole, Poland
3 Mathematical & Information Sciences Directorate, Army Research Office, Research Triangle, N.C., USA
* To whom correspondence should be addressed. E-mail: Miroslaw.Latka{at}pwr.wroc.pl.
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< cos >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 arterial blood pressure and cerebral flow velocity, 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 to 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 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 non-stationary aspects of cerebral hemodynamics in the very low-frequency range where the physiological significance of autoregulation is most strongly pronounced.
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