Decomposing Multifractal Crossovers.

Zoltan Nagy,Andras Eke,Peter Mukli,Peter Herman

doi:10.3389/fphys.2017.00533

Abstract

Physiological processes—such as, the brain's resting-state electrical activity or hemodynamic fluctuations—exhibit scale-free temporal structuring. However, impacts common in biological systems such as, noise, multiple signal generators, or filtering by transport function, result in multimodal scaling that cannot be reliably assessed by standard analytical tools that assume unimodal scaling. Here, we present two methods to identify breakpoints or crossovers in multimodal multifractal scaling functions. These methods incorporate the robust iterative fitting approach of the focus-based multifractal formalism (FMF). The first approach (moment-wise scaling range adaptivity) 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 at decomposing signal constituents from multimodal scaling functions resulting from signal addition or co-sampling, such as, contamination by uncorrelated fractals. We demonstrated that these methods could handle multimodal, mono- or multifractal, and 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 blood oxygen level-dependent functional magnetic resonance imaging (fMRI-BOLD). The NIRS and fMRI-BOLD low-frequency fluctuations were 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 robust implementation of the SFD method should be regarded as essential in the 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.

Highlights

Fractal and multifractal concepts focus on characterizing scale-free properties in terms of scaling exponents—such as, spectral index (β) or Hurst exponent (H; Mandelbrot, 1982; Eke et al, 2002, 2012; Mukli et al, 2015)—of ideal or empirical signals
A similar scenario is seen with the impact of moment level (Figures 5B1,B2), where the actual scale-wise distribution of crossovers will be determined by the dynamics of the H(q) of the signal components
When applied to highdefinition empirical signals (EEG, near infrared spectroscopy (NIRS)) with high degrees of freedom, these methods performed in a robust manner

Summary

Introduction

Fractal and multifractal concepts focus on characterizing scale-free properties in terms of scaling exponents—such as, spectral index (β) or Hurst exponent (H; Mandelbrot, 1982; Eke et al, 2002, 2012; Mukli et al, 2015)—of ideal or empirical signals. D(h) captures the moment-wise distribution of the singularity strength of local roughness or multifractal scaling in the temporal process (Kantelhardt et al, 2002; Ihlen, 2012; Mukli et al, 2015). We recently demonstrated that standard moment-based multifractal analyses were susceptible to signal inhomogeneity leading to spurious estimates of the multifractal spectrum. We resolved this issue by developing focus-based multifractal formalism (FMF), which replaced the standard—essentially monofractal—analysis for H(q) by fitting an exact multifractal to the family of momentwise scaling functions all at once by enforcing an expected value at signal length (termed focus) as a guiding reference in the fitting procedure (Mukli et al, 2015). FMF explicitly relied on a previous observation on the focus (Kantelhardt et al, 2002) and can be related to some earlier multifractal approaches (Struzik, 1999; Struzik and Siebes, 2002)

Objectives

Methods

Results

Discussion

Conclusion