Andras Eke

Fractal characterization of complexity in temporal physiological signals

This review first gives an overview on the concept of fractal geometry with definitions and explanations of the most fundamental properties of fractal structures and processes like self-similarity, power law scaling relationship, scale invariance, scaling range and fractal dimensions. Having laid down the grounds of the basics in terminology and mathematical formalism, the authors systematically introduce the concept and methods of monofractal time series analysis. They argue that fractal time series analysis cannot be done in a conscious, reliable manner without having a model capable of capturing the essential features of physiological signals with regard to their fractal analysis. They advocate the use of a simple, yet adequate, dichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). They demonstrate the importance of incorporating a step of signal classification according to the fGn/fBm model prior to fractal analysis by showing that missing out on signal class can result in completely meaningless fractal estimates. Limitation and precision of various fractal tools are thoroughly described and discussed using results of numerical experiments on ideal monofractal signals. Steps of a reliable fractal analysis are explained. Finally, the main applications of fractal time series analysis in biomedical research are reviewed and critically evaluated.

Physiological time series: distinguishing fractal noises from motions

Abstract. Many physiological signals appear fractal, in having self-similarity over a large range of their power spectral densities. They are analogous to one of two classes of discretely sampled pure fractal time signals, fractional Gaussian noise (fGn) or fractional Brownian motion (fBm). The fGn series are the successive differences between elements of a fBm series; they are stationary and are completely characterized by two parameters, σ2, the variance, and H, the Hurst coefficient. Such efficient characterization of physiological signals is valuable since H defines the autocorrelation and the fractal dimension of the time series. Estimation of H from Fourier analysis is inaccurate, so more robust methods are needed. Dispersional analysis (Disp) is good for noise signals while bridge detrended scaled windowed variance analysis (bdSWV) is good for motion signals. Signals whose slopes of their power spectral densities lie near the border between fGn and fBm are difficult to classify. A new method using signal summation conversion (SSC), wherein an fGn is converted to an fBm or an fBm to a summed fBm and bdSWV then applied, greatly improves the classification and the reliability of Ĥ, the estimates of H, for the times series. Applying these methods to laser-Doppler blood cell perfusion signals obtained from the brain cortex of anesthetized rats gave Ĥ of 0.24±0.02 (SD, n=8) and defined the signal as a fractional Brownian motion. The implication is that the flow signal is the summation (motion) of a set of local velocities from neighboring vessels that are negatively correlated, as if induced by local resistance fluctuations.

The modified Beer-Lambert law revisited.

The modified Beer-Lambert law (MBLL) is the basis of continuous-wave near-infrared tissue spectroscopy (cwNIRS). The differential form of MBLL (dMBLL) states that the change in light attenuation is proportional to the changes in the concentrations of tissue chromophores, mainly oxy- and deoxyhaemoglobin. If attenuation changes are measured at two or more wavelengths, concentration changes can be calculated. The dMBLL is based on two assumptions: (1) the absorption of the tissue changes homogeneously, and (2) the scattering loss is constant. It is known that absorption changes are usually inhomogeneous, and therefore dMBLL underestimates the changes in concentrations (partial volume effect) and every calculated value is influenced by the change in the concentration of other chromophores (cross-talk between chromophores). However, the error introduced by the second assumption (cross-talk of scattering changes) has not been assessed previously. An analytically treatable special case (semi-infinite, homogeneous medium, with optical properties of the cerebral cortex) is utilized here to estimate its order of magnitude. We show that the per cent change of the transport scattering coefficient and that of the absorption coefficient have an approximately equal effect on the changes of attenuation, and a 1% increase in scattering increases the estimated concentration changes by about 0.5 microM.

Pitfalls in fractal time series analysis: fMRI BOLD as an exemplary case

This article will be positioned on our previous work demonstrating the importance of adhering to a carefully selected set of criteria when choosing the suitable method from those available ensuring its adequate performance when applied to real temporal signals, such as fMRI BOLD, to evaluate one important facet of their behavior, fractality. Earlier, we have reviewed on a range of monofractal tools and evaluated their performance. Given the advance in the fractal field, in this article we will discuss the most widely used implementations of multifractal analyses, too. Our recommended flowchart for the fractal characterization of spontaneous, low frequency fluctuations in fMRI BOLD will be used as the framework for this article to make certain that it will provide a hands-on experience for the reader in handling the perplexed issues of fractal analysis. The reason why this particular signal modality and its fractal analysis has been chosen was due to its high impact on today’s neuroscience given it had powerfully emerged as a new way of interpreting the complex functioning of the brain (see “intrinsic activity”). The reader will first be presented with the basic concepts of mono and multifractal time series analyses, followed by some of the most relevant implementations, characterization by numerical approaches. The notion of the dichotomy of fractional Gaussian noise and fractional Brownian motion signal classes and their impact on fractal time series analyses will be thoroughly discussed as the central theme of our application strategy. Sources of pitfalls and way how to avoid them will be identified followed by a demonstration on fractal studies of fMRI BOLD taken from the literature and that of our own in an attempt to consolidate the best practice in fractal analysis of empirical fMRI BOLD signals mapped throughout the brain as an exemplary case of potentially wide interest.

Fractal and noisy CBV dynamics in humans : influence of age and gender

The complexity of spontaneous cerebral blood volume (CBV) fluctuations can emerge from random, fractal, or chaotic processes. Our aims were to define the contribution of these patterns to the observed complexity and to evaluate the effect of age and gender on it. The total hemoglobin content as the measure of CBV was monitored by near-infrared spectroscopy on volunteers (men n = 19, age = 20 to 78 years; women n = 23, age = 21 to 79 years). Random and fractal patterns were distinguished by the spectral index (β). Chaos was identified by surrogate analysis of the correlation dimension (a static chaotic parameter, the dimension of the correlation integral) and the largest Lyapunov exponent (a dynamic chaotic parameter, the rate of exponential divergence of the system states from a perturbed initial condition over the chaotic attractor). In spontaneous CBV fluctuations, both fast random and slow fractal dynamics are present separately in their spectra by a cutoff frequency, f′. Below f′ the pattern is fractal, in that power rises inversely with frequency as 1/fβ. f′ decreases with age in men and women alike (F1: up to 0.12 ± 0.06 Hz versus F2: up to 0.05 ± 0.04 Hz at P = 0.015, and M1: up to 0.16 ± 0.05 Hz versus M2: up to 0.11 ± 0.04 Hz at P = 0.044). Neither pre- nor postmenopausal age groups (1 and 2, respectively) showed a lowβ gender difference. Surrogate analysis showed that CBV dynamics cannot be characterized on the grounds of deterministic chaos. Cerebral blood volume fluctuates in a complex, bimodal manner in humans, in that the fast dynamics has no structure, while the slow dynamics exhibits a self-similar, that is, fractal temporal structure. The range of fluctuation amplitudes produced by fractal dynamics is always larger than that of random fluctuations, and it shrinks with an altered structuring in aging women only.

Fractal analysis of spontaneous fluctuations of the BOLD signal in rat brain

Analysis of task-evoked fMRI data ignores low frequency fluctuations (LFF) of the resting-state the BOLD signal, yet LFF of the spontaneous BOLD signal is crucial for analysis of resting-state connectivity maps. We characterized the LFF of resting-state BOLD signal at 11.7T in α-chloralose and domitor anesthetized rat brain and modeled the spontaneous signal as a scale-free (i.e., fractal) distribution of amplitude power (|A|²) across a frequency range (f) compatible with an |A(f)|² ∝ 1/f(β) model where β is the scaling exponent (or spectral index). We compared β values from somatosensory forelimb area (S1FL), cingulate cortex (CG), and caudate putamen (CPu). With α-chloralose, S1FL and CG β values dropped from ~0.7 at in vivo to ~0.1 at post mortem (p<0.0002), whereas CPu β values dropped from ~0.3 at in vivo to ~0.1 at post mortem (p<0.002). With domitor, cortical (S1FL, CG) β values were slightly higher than with α-chloralose, while subcortical (CPu) β values were similar with α-chloralose. Although cortical and subcortical β values with both anesthetics were significantly different in vivo (p<0.002), at post mortem β values in these regions were not significantly different and approached zero (i.e., range of -0.1 to 0.2). Since a water phantom devoid of susceptibility gradients had a β value of zero (i.e., random), we conclude that deoxyhemoglobin present in voxels post-sacrifice still impacts tissue water diffusion. These results suggest that in the anesthetized rat brain the LFF of BOLD signal at 11.7T follow a general 1/f(β) model of fractality where β is a variable responding to physiology. We describe typical experimental pitfalls which may elude detection of fractality in the resting-state BOLD signal.

Nonlinear analysis of blood cell flux fluctuations in the rat brain cortex during stepwise hypotension challenge

Blood flow through a region of interest in the brain cortex (cerebral blood flow (CBF)) as measured by laser-Doppler flowmetry (LDF) shows a complex temporal pattern, which can be either merely random or a manifestation of segmental chaotic dynamics of vasomotion-induced flowmotion in the arterial tree, a deterministic phenomenon; or that of a fractal, self-similar correlation order that emerges from the set of segmental perfusion events on statistical ground. Fractal content (F%) was determined by coarse-graining spectral analysis and their self-similar exponent, H, estimated by bridge-detrended Scaled Windowed Variance (bdSWV) method and a variant of the power Spectral Density method (lowPSDw,e). Chaotic dynamics were assessed by computing the correlation dimension (Dcorr) and the largest Lyapunov exponent (Λmax) on unfiltered raw and surrogate datasets. In 10 Sprague–Dawley rats anesthetized by halothane, CBF was measured through the thinned calvarium by the LDF method. Blood pressure (BP) was reduced from 100 to 40 mm Hg in steps of 20 mm Hg maintained for 2 mins by the lower body negative pressure method. Fractal and chaotic patterns coexisted in tissue perfusion. CBF did not show autoregulation. At every BP step, F% remained high (72% to 88%) and was independent of BP. Laser-Doppler flowmetry signals proved to be nonstationary fractional Brownian motions. Their Hs by bdSWV and lowPSDw,e (0.29 ± 0.006 and 0.25 ± 0.012, respectively) were independent of BP. Neither Dcorr nor Λmax varied with hypotension. Their values were characteristic of a chaotic system, but surrogate data analysis rendered some of them inconclusive. Hence, CBF fluctuations can be regarded as a robust phenomenon that is not abolished even by sustained hypotension at 40 mm Hg.

Fractal Analysis of Spontaneous Fluctuations in Human Cerebral Hemoglobin Content and its Oxygenation Level Recorded by NIRS

The NIRS technology provides a non-invasive tool to monitor hemodynamics from the human brain cortex (Jobsis 1977; Wyatt et al., 1986). Our interest recently has turned toward studies utilizing this technology in understanding the complexity of hemodynamics in the human brain as captured in high resolution hemoglobin time series. In our definition a time series or signal is complex when it is difficult to recognize a pattern such as it is being stable or periodic in its structuring. A complex signal cannot be adequately characterized by descriptive statistical measures such as its mean, standard deviation, etc., or by its frequency spectrum because by doing so other relevant information that is present in these signals are discarded or not revealed. Hence, adequate models, and methods need to be identified for such signals to be characterized in their entirety. Similar to our previous study on red blood cell flux time series acquired from the brain cortex of the rat (Eke et al., 1997) we have applied the fractal model as implemented in a combination of the power spectral density (PSD) and the scaled windowed variance (SWV) methods in the analysis of human cerebrocortical hemoglobin signals in an attempt to assess their temporal pattern.

Fractal Characterization of Complexity in Dynamic Signals: Application to Cerebral Hemodynamics

We introduce the concept of spatial and temporal complexity with emphasis on how its fractal characterization for 1D, 2D or 3D hemodynamic brain signals can be carried out. Using high-resolution experimental data sets acquired in animal and human brain by noninvasive methods - such as laser Doppler flowmetry, laser speckle, near infrared, or functional magnetic resonance imaging - the spatiotemporal complexity of cerebral hemodynamics is demonstrated. It is characterized by spontaneous, seemingly random (that is disorderly) fluctuation of the hemodynamic signals. Fractal analysis, however, proved that these fluctuations are correlated according to the special order of self-similarity. The degree of correlation can be assessed quantitatively either in the temporal or the frequency domain respectively by the Hurst exponent (H) and the spectral index (beta). The values of H for parenchymal regions of white and gray matter of the rat brain cortex are distinctly different. In human studies, the values of beta were instrumental in identifying age-related stiffening of cerebral vasculature and their potential vulnerability in watershed areas of the brain cortex such as in borderline regions between frontal and temporal lobes. Biological complexity seems to be present within a restricted range of H or beta values which may have medical significance because outlying values can indicate a state of pathology.

Temporal Fluctuations in Regional Red Blood Cell Flux in the Rat Brain Cortex is a Fractal Process

Temporal fluctuations ever since it was observed and recognized as an elementary feature of blood flow in the microcirculation by Krogh in 1929 (Krogh, 1929) became the essence of concepts termed “vasomotion” and “flow motion”. Such fluctuations in flow are common in various microcirculatory beds including that of the brain and can be continuously monitored by laser-Doppler flowmetry (Stern, 1975; Nilsson et al., 1980; Rosenblum et al., 1987; Fasano et al., 1988; Hudetz et al., 1992; Morita-Tsuzuki et al., 1992). When challenged, they often show slow (6–12 cycles/minute) oscillatory pattern, which allow for a relatively simple characterization if the higher frequencies are eliminated from the signal (Hudetz et al., 1992; Morita-Tsuzuki et al., 1992). Temporal variation in microcirculatory flow is however multifactorial and the interactions among these factors can manifest in a complex structuring of the time series recorded by high-resolution laserDoppler flowmetry (LDF). With no apparent dominating frequency present (Fig. 3), they are much too complex to be analyzed in specific terms by conventional descriptive statistics, amplitude and frequency measures. Thus we have used fractal methods genuinely holistic in nature to gain statistical insight into random signals of this kind and to determine if under unchallenged control conditions they represent disorganized behavior or show long-range correlations (West and Goldberger, 1987; Bassingthwaighte, 1988; West and Shlesinger, 1990; Weibel, 1991; Bassingthwaighte et al., 1994).