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.

In this study, functional near-infrared spectroscopy (fNIRS) and the graph theory approach were used to access the functional connectivity (FC) of the prefrontal cortex (PFC) in a resting state and during increased mental workload. For this very purpose, a pattern recognition-based test was developed, which elicited a strong response throughout the PFC during the test condition. FC parameters obtained during stimulation were found increased compared to those in a resting state after correlation based signal improvement (CBSI), which can attenuate those components of fNIRS signals which are unrelated to neural activity. These results indicate that the cognitive challenge increased the FC in the PFC and suggests a great potential in investigating FC in various cognitive states.

Dynamic functional connectivity (DFC) was established in the past decade as a potent approach to reveal non-trivial, time-varying properties of neural interactions – such as their multifractality or information content –, that otherwise remain hidden from conventional static methods. Several neuropsychiatric disorders were shown to be associated with altered DFC, with schizophrenia (SZ) being one of the most intensely studied among such conditions. Here we analyzed resting-state electroencephalography recordings of 14 SZ patients and 14 age- and gender-matched healthy controls (HC). We reconstructed dynamic functional networks from delta band (0.5–4 Hz) neural activity and captured their spatiotemporal dynamics in various global network topological measures. The acquired network measure time series were made subject to dynamic analyses including multifractal analysis and entropy estimation. Besides group-level comparisons, we built a classifier to explore the potential of DFC features in classifying individual cases. We found stronger delta-band connectivity, as well as increased variance of DFC in SZ patients. Surrogate data testing verified the true multifractal nature of DFC in SZ, with patients expressing stronger long-range autocorrelation and degree of multifractality when compared to controls. Entropy analysis indicated reduced temporal complexity of DFC in SZ. When using these indices as features, an overall cross-validation accuracy surpassing 89% could be achieved in classifying individual cases. Our results imply that dynamic features of DFC such as its multifractal properties and entropy are potent markers of altered neural dynamics in SZ and carry significant potential not only in better understanding its pathophysiology but also in improving its diagnosis. The proposed framework is readily applicable for neuropsychiatric disorders other than schizophrenia.

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.

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.