Miroslaw Latka

Wavelet analysis of epileptic spikes.

Interictal spikes and sharp waves in human EEG are characteristic signatures of epilepsy. These potentials originate as a result of synchronous pathological discharge of many neurons. The reliable detection of such potentials has been the long standing problem in EEG analysis, especially after long-term monitoring became common in investigation of epileptic patients. The traditional definition of a spike is based on its amplitude, duration, sharpness, and emergence from its background. However, spike detection systems built solely around this definition are not reliable due to the presence of numerous transients and artifacts. We use wavelet transform to analyze the properties of EEG manifestations of epilepsy. We demonstrate that the behavior of wavelet transform of epileptic spikes across scales can constitute the foundation of a relatively simple yet effective detection algorithm.

Multifractality of cerebral blood flow

Scale invariance, the property relating time series across multiple scales, has provided a new perspective of physiological phenomena and their underlying control systems. The traditional “signal plus noise” paradigm of the engineer was first replaced with a model in which biological time series have a fractal structure in time (Fractal Physiology, Oxford University Press, Oxford, 1994). This new paradigm was subsequently shown to be overly restrictive when certain physiological signals were found to be characterized by more than one scaling parameter and therefore to belong to a class of more complex processes known as multifractals (Fractals, Plenum Press, New York, 1988). Here we demonstrate that in addition to heart rate (Nature 399 (1999) 461) and human gait (Phys. Rev. E, submitted for publication), the nonlinear control system for cerebral blood flow (CBF) (Phys. Rev. Lett., submitted for publication; Phys. Rev. E 59 (1999) 3492) is multifractal. We also find that this multifractality is greatly reduced for subjects with “serious” migraine and we present a simple model for the underlying control process to describe this effect.

Fractional Langevin model of gait variability

The stride interval in healthy human gait fluctuates from step to step in a random manner and scaling of the interstride interval time series motivated previous investigators to conclude that this time series is fractal. Early studies suggested that gait is a monofractal process, but more recent work indicates the time series is weakly multifractal. Herein we present additional evidence for the weakly multifractal nature of gait. We use the stride interval time series obtained from ten healthy adults walking at a normal relaxed pace for approximately fifteen minutes each as our data set. A fractional Langevin equation is constructed to model the underlying motor control system in which the order of the fractional derivative is itself a stochastic quantity. Using this model we find the fractal dimension for each of the ten data sets to be in agreement with earlier analyses. However, with the present model we are able to draw additional conclusions regarding the nature of the control system guiding walking. The analysis presented herein suggests that the observed scaling in interstride interval data may not be due to long-term memory alone, but may, in fact, be due partly to the statistics.

Fractal Rigidity in Migraine

Abstract We study the middle cerebral artery blood flow velocity in humans using transcranial Doppler ultrasonography. Scaling properties of time series of the axial flow velocity averaged over a cardiac beat interval may be characterized by two exponents. The short-time scaling exponent (STSE) determines the statistical properties of fluctuations of blood flow velocities in short-time intervals while the Hurst exponent describes the long-term fractal properties. In many migraineurs the value of the STSE is significantly reduced and may approach that of the Hurst exponent. This change in dynamical properties reflects the significant loss of short-term adaptability and the overall hyperexcitability of underlying cerebral blood flow control system. We call this effect fractal rigidity.

Detrended fluctuation analysis of scaling crossover effects

Detrended fluctuation analysis (DFA) is one of the most frequently used fractal time series algorithms. DFA has also become the tool of choice for analysis of the short-time fluctuations despite the fact that its validity in this domain has never been demonstrated. We adopt an Ornstein-Uhlenbeck Langevin equation to generate a time series which exhibits short-time power-law scaling and incorporates the fundamental property of physiological control systems ?negative feedback. To determine the scaling exponent, we derive the analytical expressions for the standard deviation of the solution X(t) of this equation using both the ensemble of statistically independent trajectories and the ensemble obtained by partitioning a single trajectory. The latter approach is used in DFA and many other physiological applications. Surprisingly, the formulas for the standard deviations are different for these two ensembles. We demonstrate that the partitioning amounts to building up deterministic trends that satisfy the trend?+?signal decomposition assumption which is characteristic of DFA. Consequently, the dependence of the rms of DFA residuals F(?) on the length ? of data window is the same for both ensembles. The growth of F(?) is significantly different from that of the standard deviation of X(t). While the DFA estimate of the short-time scaling exponent is correct, the polynomial detrending delays the approach of F(?) to the asymptotic value by as much as an order of magnitude. This delay may underlie the gradual change of the DFA scaling index typically observed in time series that exhibit crossover between the short- and long-time scaling.

Generation of very low frequency cerebral blood flow fluctuations in humans

BACKGROUND Slow oscillations of cerebral blood flow induced by synchronous variations of arterial blood pressure (ABP) are often used for clinical assessment of cerebral autoregulation. In the alternative scenario, spontaneous cerebral vasocycling may produce waves in cerebral blood flow that are, to a large extent, independent of ABP fluctuations. We use wavelet analysis to test the latter hypothesis. METHODS The wavelet variability V(f), defined as the time averaged moduli of frequency dependent wavelet coefficients, is employed to analyze the relation between dynamics of arterial blood pressure and that of cerebral blood flow velocity in middle cerebral artery (MCA). FINDINGS In the very low frequency (VLF, 0.02-0.07 Hz) band the variability in traumatic brain injury (TBI) patients with low intracranial pressure (V(ABP) = 0.36 +/- 0.28) is significantly smaller than that of the volunteers (V(ABP) = 0.70 +/- 0.25) with p = 7 x 10(-5). Interestingly, the corresponding variabilities of MCA flow velocity for both cohorts are comparable. V(MCA) = 0.83 +/- 0.65 of the brain injury patients is not statistically different from that of the volunteers V(MCA) = 1.06 +/- 0.41 (p = 0.11). CONCLUSIONS In TBI patients without cerebral hypertension, the VLF oscillations must have been spontaneously generated within intracranial volume to compensate for the reduced ABP variability. Vasomotion is identified as a plausible physiological mechanism underlying such oscillations. We argue that vasomotion may be beneficial for brain tissue oxygenation especially during periods of critically low perfusion.

Wavelet assessment of cerebrospinal compensatory reserve and cerebrovascular reactivity

We introduce a wavelet transfer model to relate spontaneous arterial blood pressure (ABP) fluctuations to intracranial pressure (ICP) fluctuations. We employ a complex continuous wavelet transform to develop a consistent mathematical framework capable of parametrizing both cerebral compensatory reserve and cerebrovascular reactivity. The frequency-dependent gain and phase of the wavelet transfer function are introduced because of the non-stationary character of the ICP and ABP time series. The gain characterizes the dampening of spontaneous ABP fluctuations and is interpreted as a novel measure of cerebrospinal compensatory reserve. For a group of 12 patients who died as a result of cerebral lesions (Glasgow Outcome Scale (GOS) = 1) the average gain in the low-frequency (0.02- 0.07 Hz) range was 0.51 +/- 0.13 and significantly exceeded that of 17 patients with GOS = 2 having an average gain of 0.26 +/- 0.11 with p = 1x10(-4) (Kruskal-Wallis test). A time-averaged synchronization index (which may vary from 0 to 1) defined in terms of the wavelet transfer function phase yields information about the stability of the phase difference of the ABP and ICP signals and is used as a cerebrovascular reactivity index. A low value of synchronization index reflects a normally reactive vascular bed, while a high value indicates pathological entrainment of ABP and ICP fluctuations. Such entrainment is strongly pronounced in patients with fatal outcome (for this group the low-frequency synchronization index was 0.69 +/- 0.17). The gain and synchronization parameters define a cerebral hemodynamic state space (CHS) in which the patients with GOS = 1 are to large extent partitioned away from those with GOS = 2. The concept of CHS elucidates the interplay of vascular and compensatory mechanisms.

Scaling Properties of Cerebral Hemodynamics

Cerebral autoregulation (CA) is a vital protective mechanism that maintains relatively stable cerebral blood flow despite variations in systemic pressure as large as 100 Torr. It is commonly perceived to operate as a high-pass filter which transmits rapid changes in blood pressure but strongly attenuates and delays low-frequency perturbations. The ongoing search for clinically significant measures of CA integrity fuels the study of relations between the statistical properties of arterial blood pressure fluctuations (ABP) and those of blood flow velocity in major cerebral arteries, for example in middle cerebral artery (MCA). Using the method of averaged wavelet coefficients (AWC) we find that in the healthy subjects the scaling properties of both time series may be characterized by two exponents. The short time scaling exponent determines the statistical properties of fluctuations in short-time intervals while the Hurst exponent H describes the long-term fractal properties. Surprisingly, the group-averaged Hurst exponents coincide: H ABP = H MCA = 1 . To explain this effect, we employ complex continuous wavelet transforms to characterize autoregulation in terms of the wavelet gain and instantaneous phase difference between the arterial blood pressure and cerebral flow velocity. In the very low frequency (0.02–0.07 Hz) part of the spectrum, where autoregulation is most strongly pronounced, the damping of ABP slow oscillations weakly depends on frequency. In this frequency range phase difference evolves slowly over time and has an almost uniform distribution. Thus, CA not only dampens low frequency oscillations but also randomizes their phases. However, phase randomization of fractional Brownian motion does not affect its scaling properties. Consequently, fractal dynamics of arterial pressure is essentially carried over to cerebral blood flow.

Asymmetry of short-term control of spatio-temporal gait parameters during treadmill walking

Optimization of energy cost determines average values of spatio-temporal gait parameters such as step duration, step length or step speed. However, during walking, humans need to adapt these parameters at every step to respond to exogenous and/or endogenic perturbations. While some neurological mechanisms that trigger these responses are known, our understanding of the fundamental principles governing step-by-step adaptation remains elusive. We determined the gait parameters of 20 healthy subjects with right-foot preference during treadmill walking at speeds of 1.1, 1.4 and 1.7 m/s. We found that when the value of the gait parameter was conspicuously greater (smaller) than the mean value, it was either followed immediately by a smaller (greater) value of the contralateral leg (interleg control), or the deviation from the mean value decreased during the next movement of ipsilateral leg (intraleg control). The selection of step duration and the selection of step length during such transient control events were performed in unique ways. We quantified the symmetry of short-term control of gait parameters and observed the significant dominance of the right leg in short-term control of all three parameters at higher speeds (1.4 and 1.7 m/s).