Decomposing Multifractal Crossovers - Nagy, Zoltan; Mukli, Peter; Herman, Peter; Eke, Andras (Frontiers in Physiology, 2017)
Physiological processes—such as brain’s resting-state electrical activity or hemodynamic fluctuations—exhibit scale-free temporal structuring. However impacts, such as noise, multiple signal generators or filtering by transport function, common in biological systems results in multimodal scaling that cannot be reliably assessed by standard analytical tools as these assume unimodal scaling. Here we present two methods to identify breakpoints or crossovers in multimodal multifractal scaling functions. These methods incorporated the robust iterative fitting approach of the focus-based multifractal formalism. The first approach (moment-wise scaling range adaptivity, qSRA) allows for a breakpoint-based adaptive treatment that analyzes segregated scale-invariant ranges. The second method (scaling function decomposition method, SFD) is a crossover-based design aimed to decompose signal constituents from multimodal scaling functions resulted from signal addition or co-sampling, such as contamination by uncorrelated fractal. We demonstrate that they can handle multimodal; mono- or multifractal; exact or empirical signals, alike. Their precision was numerically characterized on ideal signals and a robust performance was demonstrated on exemplary empirical signals capturing resting-state brain dynamics by near infrared spectroscopy (NIRS), electroencephalography (EEG) and functional magnetic resonance imaging for blood oxygen level dependent signal (fMRI-BOLD). The NIRS and fMRI-BOLD low-frequency fluctuations were found dominated by a multifractal component over an underlying biologically relevant random noise thus forming a bimodal signal. The crossover between the EEG signal components was found at the boundary between the δ and θ bands suggesting an independent generator for the multifractal δ rhythm. The reported robust implementation of the SFD method should be regarded essential in a seamless processing of large volumes of bimodal fMRI-BOLD imaging data for the topology of multifractal metrics free of the masking effect of the underlying random noise.