Investigation of microstructural, micromorphology, and surface plasmon resonance characteristics in Ni/Al, Ni/Cu, and Ni/SS thin films

As the first boundary between the environment and the material, the surface plays an important role in their interaction with each other, therefore, the use of appropriate tools and analysis to examine the mechanical properties and morphology of surfaces has particular importance in industry and research. In this research, a thin film of nickel was deposited on metal substrates made of aluminum, copper, and steel by using the RF magnetic cathode. Then, using a non-contact atomic force microscope, the morphological properties of the nickel film with static parameters, Minkowski functionals (MF's), fractal, and multifractal were extracted to be analyzed and studied. After that, using parameters such as root mean square (RMS) roughness, skewness, and kurtosis, it was determined how the surface roughness, distribution, and probability density of particles on the film surface alters with the change of the substrate. Next, by examining and analyzing the Δα and Δf parameters obtained from the multifractal section, the morphology of the produced film on the metal substrates was investigated. Then, the change in the surface plasmon resonance (SPR) peak position is changed for the prepared film in the range of the absorption spectrum due to the substrate effect and the microstructural properties of the formed film. HIGHLIGHTS: Ni film has been deposited by Rf magnetron sputtering. The effect of metal substrates on the topography, fractality, and optical properties was studied. Minkowski functionals were used to investigate the surface morphology of the samples. Substrate's material and the topography of the formed film can changed the surface plasmon resonance position.

Multifractal Multiscale Analysis of Human Movements during Cognitive Tasks

Continuous adaptations of the movement system to changing environments or task demands rely on superposed fractal processes exhibiting power laws, that is, multifractality. The estimators of the multifractal spectrum potentially reflect the adaptive use of perception, cognition, and action. To observe time-specific behavior in multifractal dynamics, a multiscale multifractal analysis based on DFA (MFMS-DFA) has been recently proposed and applied to cardiovascular dynamics. Here we aimed at evaluating whether MFMS-DFA allows identifying multiscale structures in the dynamics of human movements. Thirty-six (12 females) participants pedaled freely, after a metronomic initiation of the cadence at 60 rpm, against a light workload for 10 min: in reference to cycling (C), cycling while playing "Tetris" on a computer, alone (CT) or collaboratively (CTC) with another pedaling participant. Pedal revolution periods (PRP) series were examined with MFMS-DFA and compared to linearized surrogates, which attested to a presence of multifractality at almost all scales. A marked alteration in multifractality when playing Tetris was evidenced at two scales, τ ≈ 16 and τ ≈ 64 s, yet less marked at τ ≈ 16 s when playing collaboratively. Playing Tetris in collaboration attenuated these alterations, especially in the best Tetris players. This observation suggests the high sensitivity to cognitive demand of MFMS-DFA estimators, extending to the assessment of skill/demand interplay from individual behavior. So, by identifying scale-dependent multifractal structures in movement dynamics, MFMS-DFA has obvious potential for examining brain-movement coordinative structures, likely with sufficient sensitivity to find echo in diagnosing disorders and monitoring the progress of diseases that affect cognition and movement control.

A Dynamic Monitoring Method of Public Opinion Risk of Overseas Direct Investment-Based on Multifractal Situation Optimization

The negative public opinions and views on overseas direct investment (ODI) of a multinational enterprise (MNE) will damage the image of its brand and are likely to bring it serious economic and social losses. So, it is important for the MNE to understand the formation and spread mechanism of public opinion risk (POR) in order to effectively respond to and guide the public opinion. This research proposed a multifractal-based situation optimization method to explore the POR evolution based on the media-based negative sentiment on China's ODI. The sentiment measurement is obtained by a directed crawler for gathering the text of media reports corresponding to a certain ODI event using a URL knowledge base from the GDELT Event Database. Taking the public opinion crisis of the tax evasion incident of the local arm of China's MNE in India as an example, the experiments show that this method could dynamically monitor the POR event in real-time and help MNE guide the effective control and benign evolution of public opinion of the event.

Fractal Characterization of the Mass Loss of Bronze by Erosion-Corrosion in Seawater

The fractal approach is one of the nondestructive techniques for analyzing corrosion's effects on different materials. This article utilizes it to analyze the erosion-corrosion produced by cavitation on two types of bronze introduced into an ultrasonic cavitation field to investigate the differences between their behavior in saline water. The aim is to check the hypothesis that the fractal/multifractal measures significantly differ for the studied materials that belong to the same class (bronze) as a step in applying fractal techniques to distinguish between two materials. The study emphasizes the multifractal characteristics of both materials. While the fractal dimensions do not significantly differ, the highest multifractal dimensions correspond to the sample of bronze with Sn.

Microbial characterization based on multifractal analysis of metagenomes

Introduction

The species diversity of microbiomes is a cutting-edge concept in metagenomic research. In this study, we propose a multifractal analysis for metagenomic research.

Method and Results

Firstly, we visualized the chaotic game representation (CGR) of simulated metagenomes and real metagenomes. We find that metagenomes are visualized with self-similarity. Then we defined and calculated the multifractal dimension for the visualized plot of simulated and real metagenomes, respectively. By analyzing the Pearson correlation coefficients between the multifractal dimension and the traditional species diversity index, we obtain that the correlation coefficients between the multifractal dimension and the species richness index and Shannon diversity index reached the maximum value when q = 0, 1, and the correlation coefficient between the multifractal dimension and the Simpson diversity index reached the maximum value when q = 5. Finally, we apply our method to real metagenomes of the gut microbiota of 100 infants who are newborn and 4 and 12 months old. The results show that the multifractal dimensions of an infant's gut microbiomes can distinguish age differences.

Conclusion and Discussion

There is self-similarity among the CGRs of WGS of metagenomes, and the multifractal spectrum is an important characteristic for metagenomes. The traditional diversity indicators can be unified under the framework of multifractal analysis. These results coincided with similar results in macrobial ecology. The multifractal spectrum of infants' gut microbiomes are related to the development of the infants.

Scale-Free Functional Brain Networks Exhibit Increased Connectivity, Are More Integrated and Less Segregated in Patients with Parkinson's Disease following Dopaminergic Treatment

Dopaminergic treatment (DT), the standard therapy for Parkinson's disease (PD), alters the dynamics of functional brain networks at specific time scales. Here, we explore the scale-free functional connectivity (FC) in the PD population and how it is affected by DT. We analyzed the electroencephalogram of: (i) 15 PD patients during DT (ON) and after DT washout (OFF) and (ii) 16 healthy control individuals (HC). We estimated FC using bivariate focus-based multifractal analysis, which evaluated the long-term memory ( H ( 2 ) ) and multifractal strength ( Δ H 15 ) of the connections. Subsequent analysis yielded network metrics (node degree, clustering coefficient and path length) based on FC estimated by H ( 2 ) or Δ H 15 . Cognitive performance was assessed by the Mini Mental State Examination (MMSE) and the North American Adult Reading Test (NAART). The node degrees of the Δ H 15 networks were significantly higher in ON, compared to OFF and HC, while clustering coefficient and path length significantly decreased. No alterations were observed in the H ( 2 ) networks. Significant positive correlations were also found between the metrics of H ( 2 ) networks and NAART scores in the HC group. These results demonstrate that DT alters the multifractal coupled dynamics in the brain, warranting the investigation of scale-free FC in clinical and pharmacological studies.

A correspondence between the multifractal model of turbulence and the Navier-Stokes equations

The multifractal model of turbulence (MFM) and the three-dimensional Navier-Stokes equations are blended together by applying the probabilistic scaling arguments of the former to a hierarchy of weak solutions of the latter. This process imposes a lower bound on both the multifractal spectrum [Formula: see text], which appears naturally in the Large Deviation formulation of the MFM, and on [Formula: see text] the standard scaling parameter. These bounds respectively take the form: (i) [Formula: see text], which is consistent with Kolmogorov's four-fifths law ; and (ii) [Formula: see text]. The latter is significant as it prevents solutions from approaching the Navier-Stokes singular set of Caffarelli, Kohn and Nirenberg. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Morphologic characterization and fractal analysis of lapped and polished surfaces of quartz single crystals

Lapping and polishing are industrial processes sometimes used alternatively for surface finishing of hard and brittle materials. This article presents advanced image analysis of surfaces of quartz crystal blanks finished by lapping and polishing. Scanning electron micrographs were obtained from workpiece surfaces parallel to Y-, AT-, and Z-cut crystal planes treated with different normal stress and abrasive grit size, and stereometric and fractal/multifractal approaches were used to analyze the respective surfaces. Fractal dimensions and segmentation parameters were able to decode the effect of normal stress increasing on the surface roughness of lapped and polished samples. Moreover, the texture isotropy and the bifractal-hence agglomerated-nature of the surface patterns, suggest that both treatments dismiss the anisotropic signature of hardness and fracture toughness inherent to each crystal plane. This study provides promising results regarding the applicability of fractal analysis in the assessment of surfaces severely worn by the combined effect of brittle microcracking and plastic deformation mechanisms.

The Search for Atmospheric Laminar Channels: Experimental Results and Method Dissemination

In this paper, a practical application of theoretical developments found in our previous works is explored in relation to atmospheric lidar data. Multifractal structures, previously named "laminar channels", have been identified in atmospheric profiles-these exhibit cellular and self-structuring properties, and are spatially ordered across the atmospheric profile. Furthermore, these structures have been connected to the spontaneous emergence of turbulent behavior in the calm atmospheric flow. Calculating the location and occurrence of these channels can help identify features of atmospheric evolution, such as the development of the planetary boundary layer (PBL). Employing this theoretical background to atmospheric lidar data, attempts are made to confirm this suggestion and extract information about atmospheric structure and evolution by analyzing turbulent vortex scale dynamics and scale-corresponding Lyapunov exponents that form the basis of identifying the laminar channels in atmospheric lidar profiles. A parameter named "scale laminarity index" is then introduced, which quantifies the relation between vortex scale and chaoticity throughout the profile. Finally, the algorithmic methods employed in this study are described and distributed for future use.

[Scaling laws of altitudinal pattern of soil fauna diversity in Dongling Mountain, Beijing, China]

Identifying the complexity of diversity pattern of various taxa within a community is a challenge for ecologist. Scaling law is one of the suitable ways to detecting the complex ecological structure. In this study, we explored the scaling laws of soil fauna diversity pattern along an altitudinal gradient by multifractal analysis, and compared the difference of multifractal spectra between the litter and the soil layers. Consistent with results from plant communities in previous studies, there was power law scaling law for soil fauna diversity, i.e., richness, the exponential of Shannon's Diversity Index, and the inverse Simpson's Diversity Index. Moreover, power law scaling law also existed for the richness changes of different relative abundance species in both litter and soil layers. Although multifractal characteristics existed for both litter layer and soil layer of soil fauna diversity, the fractal structure of the diversity in the litter layer was more even than that in the soil layer, and the scaling properties of dominant and rare species showed different patterns in multifractal spectra between litter layer and soil layer. In conclusion, there were power law scaling laws for soil fauna diversity which had high richness and abundance along the altitudinal gradient, which would help us uncovering the spatial distribution mechanism of belowground biodiversity.

Study on Fractal Characteristics of Mineral Particles in Undisturbed Loess and Lime-Treated Loess

In order to explore the fractal characteristics of particle size distribution (PSD) of various minerals in loess and lime-treated loess, the Q4 undisturbed loess and lime-treated loess were studied. From the perspective of multi-scaled microstructure, the internal characteristics of loess were observed and the regularity statistics were carried out from a macroscopic view. Fractal theory was used to quantitatively study the distribution of mineral particles in undisturbed loess and lime-treated loess. It was found that the skeleton particles of undisturbed loess were obvious and the structure of soil was loose. While that of lime-treated loess decreased, the fine particles were connected with each other, and the structure of soil changed from loose to dense. The three mineral particles in the undisturbed loess and lime-treated loess did not accord with the single fractal distribution characteristics, but the total particles had fractal characteristics. The percentage content of the mineral particles in the soil varied greatly with the particle size. In addition, the non-uniform degrees of mineral particles in the two soils from large to small were carbonate minerals of lime-treated loess, carbonate minerals of undisturbed loess, quartz minerals of lime-treated loess, feldspar mineral of lime-treated loess, feldspar mineral of the undisturbed loess, and the quartz mineral of the undisturbed loess. This paper provided a basis for the future study of the different soil mechanical properties of undisturbed loess and lime-treated loess.

Multifractal Functional Connectivity Analysis of Electroencephalogram Reveals Reorganization of Brain Networks in a Visual Pattern Recognition Paradigm

The human brain consists of anatomically distant neuronal assemblies that are interconnected via a myriad of synapses. This anatomical network provides the neurophysiological wiring framework for functional connectivity (FC), which is essential for higher-order brain functions. While several studies have explored the scale-specific FC, the scale-free (i.e., multifractal) aspect of brain connectivity remains largely neglected. Here we examined the brain reorganization during a visual pattern recognition paradigm, using bivariate focus-based multifractal (BFMF) analysis. For this study, 58 young, healthy volunteers were recruited. Before the task, 3-3 min of resting EEG was recorded in eyes-closed (EC) and eyes-open (EO) states, respectively. The subsequent part of the measurement protocol consisted of 30 visual pattern recognition trials of 3 difficulty levels graded as Easy, Medium, and Hard. Multifractal FC was estimated with BFMF analysis of preprocessed EEG signals yielding two generalized Hurst exponent-based multifractal connectivity endpoint parameters, H(2) and ΔH ₁₅; with the former indicating the long-term cross-correlation between two brain regions, while the latter captures the degree of multifractality of their functional coupling. Accordingly, H(2) and ΔH ₁₅ networks were constructed for every participant and state, and they were characterized by their weighted local and global node degrees. Then, we investigated the between- and within-state variability of multifractal FC, as well as the relationship between global node degree and task performance captured in average success rate and reaction time. Multifractal FC increased when visual pattern recognition was administered with no differences regarding difficulty level. The observed regional heterogeneity was greater for ΔH ₁₅ networks compared to H(2) networks. These results show that reorganization of scale-free coupled dynamics takes place during visual pattern recognition independent of difficulty level. Additionally, the observed regional variability illustrates that multifractal FC is region-specific both during rest and task. Our findings indicate that investigating multifractal FC under various conditions - such as mental workload in healthy and potentially in diseased populations - is a promising direction for future research.

Multifractal Analysis of a Seismic Moment Distribution Obtained From InSAR Inversion

Interferometric synthetic aperture radar (InSAR) interferograms contain valuable information about the fault systems hidden beneath the surface of the Earth. In a new approach, we aim to fit InSAR ground deformation data using a distribution of multiple seismic point sources whose parameters are found by a genetic algorithm. The resulting source distribution could provide another useful tool in solving the difficult problem of accurately mapping earthquake faults. We apply the algorithm to an ALOS-2 InSAR interferogram and perform a multifractal analysis on the resulting distribution, finding that it exhibits multifractal properties. We report first results and discuss advantages and disadvantages of this approach.

Multifractal Behaviors of Stock Indices and Their Ability to Improve Forecasting in a Volatility Clustering Period

The financial market is a complex system, which has become more complicated due to the sudden impact of the COVID-19 pandemic in 2020. As a result there may be much higher degree of uncertainty and volatility clustering in stock markets. How does this “black swan” event affect the fractal behaviors of the stock market? How to improve the forecasting accuracy after that? Here we study the multifractal behaviors of 5-min time series of CSI300 and S&P500, which represents the two stock markets of China and United States. Using the Overlapped Sliding Window-based Multifractal Detrended Fluctuation Analysis (OSW-MF-DFA) method, we found that the two markets always have multifractal characteristics, and the degree of fractal intensified during the first panic period of pandemic. Based on the long and short-term memory which are described by fractal test results, we use the Gated Recurrent Unit (GRU) neural network model to forecast these indices. We found that during the large volatility clustering period, the prediction accuracy of the time series can be significantly improved by adding the time-varying Hurst index to the GRU neural network.

Identification of Electroencephalogram Signals in Alzheimer's Disease by Multifractal and Multiscale Entropy Analysis

Alzheimer's disease (AD) is the most common form of dementia and is a progressive neurodegenerative disease that primarily develops in old age. In recent years, it has been reported that early diagnosis of AD and early intervention significantly delays disease progression. Hence, early diagnosis and intervention are emphasized. As a diagnostic index for AD patients, evaluating the complexity of the dependence of the electroencephalography (EEG) signal on the temporal scale of Alzheimer's disease (AD) patients is effective. Multiscale entropy analysis and multifractal analysis have been performed individually, and their usefulness as diagnostic indicators has been confirmed, but the complemental relationship between these analyses, which may enhance diagnostic accuracy, has not been investigated. We hypothesize that combining multiscale entropy and fractal analyses may add another dimension to understanding the alteration of EEG dynamics in AD. In this study, we performed both multiscale entropy and multifractal analyses on EEGs from AD patients and healthy subjects. We found that the classification accuracy was improved using both techniques. These findings suggest that the use of multiscale entropy analysis and multifractal analysis may lead to the development of AD diagnostic tools.

Retinal microvascular complexity comparing mono- and multifractal dimensions in relation to cardiometabolic risk factors in a Middle Eastern population

PURPOSE

Metrics that capture changes in the retinal microvascular structure are relevant in the context of cardiometabolic disease development. The microvascular topology is typically quantified using monofractals, although it obeys more complex multifractal rules. We study mono- and multifractals of the retinal microvasculature in relation to cardiometabolic factors.

METHODS

The cross-sectional retrospective study used data from 3000 Middle Eastern participants in the Qatar Biobank. A total of 2333 fundus images (78%) passed quality control and were used for further analysis. The monofractal (D_f ) and five multifractal metrics were associated with cardiometabolic factors using multiple linear regression and were studied in clinically relevant subgroups.

RESULTS

D_f and multifractals are lowered in function of age, and D_f is lower in males compared to females. In models corrected for age and sex, D_f is significantly associated with BMI, insulin, systolic blood pressure, glycated haemoglobin (HbA1c), albumin, LDL and total cholesterol concentrations. Multifractals are negatively associated with systolic and diastolic blood pressure, glucose and the WHO/ISH cardiovascular risk score. D_f was higher, and multifractal curve asymmetry was lower in diabetic patients (HbA1c > 6.5%) compared to healthy individuals (HbA1c < 5.7%). Insulin resistance (insulin ≥ 23 mcU/mL) was associated with significantly lower D_f values.

CONCLUSION

One or more fractal metrics are in association with sex, age, BMI, systolic and diastolic blood pressure and biochemical blood measurements in a Middle Eastern population study. Follow-up studies aiming at investigating retinal microvascular changes in relation to cardiometabolic risk should analyse both monofractal and multifractal metrics for a more comprehensive microvascular picture.

Multifractal Characteristics of Geomagnetic Field Fluctuations for the Northern and Southern Hemispheres at Swarm Altitude

This paper explores the spatial variations of the statistical scaling features of low to high latitude geomagnetic field fluctuations at Swarm altitude. The data for this study comes from the vector field magnetometer onboard Swarm A satellite, measured at low resolution (1 Hz) for one year (from 9 March 2016, to 9 March 2017). We estimated the structure-function scaling exponents using the p-leaders discrete wavelet multifractal technique, from which we obtained the singularity spectrum related to the magnetic fluctuations in the North-East-Center (NEC) coordinate system. From this estimation, we retain just the maximal fractal subset, associated with the Hurst exponent H. Here we present thresholding for two levels of the Auroral Electrojet index and almost the whole northern and southern hemispheres, the Hurst exponent, the structure-function scaling exponent of order 2, and the multifractal p-exponent width for the geomagnetic fluctuations. The latter quantifies the relevance of the multifractal property. Sometimes, we found negative values of H, suggesting a behavior similar to wave breaking or shocklet-like propagating front. Furthermore, we found some asymmetries in the magnetic field turbulence between the northern and southern hemispheres. These estimations suggest that different turbulent regimes of the geomagnetic field fluctuations exist along the Swarm path.

A Theoretical Model for Release Dynamics of an Antifungal Agent Covalently Bonded to the Chitosan

The aim of the study was to create a mathematical model useful for monitoring the release of bioactive aldehydes covalently bonded to the chitosan by reversible imine linkage, considered as a polymer-drug system. For this purpose, two hydrogels were prepared by the acid condensation reaction of chitosan with the antifungal 2-formyl-phenyl-boronic acid and their particularities; influencing the release of the antifungal aldehyde by shifting the imination equilibrium to the reagents was considered, i.e., the supramolecular nature of the hydrogels was highlighted by polarized light microscopy, while scanning electron microscopy showed their microporous morphology. Furthermore, the in vitro fungicidal activity was investigated on two fungal strains and the in vitro release curves of the antifungal aldehyde triggered by the pH stimulus were drawn. The theoretical model was developed starting from the hypothesis that the imine-chitosan system, both structurally and functionally, can be assimilated, from a mathematical point of view, with a multifractal object, and its dynamics were analyzed in the framework of the Scale Relativity Theory. Thus, through Riccati-type gauges, two synchronous dynamics, one in the scale space, associated with the fungicidal activity, and the other in the usual space, associated with the antifungal aldehyde release, become operational. Their synchronicity, reducible to the isomorphism of two SL(2R)-type groups, implies, by means of its joint invariant functions, bioactive aldehyde compound release dynamics in the form of "kink-antikink pairs" dynamics of a multifractal type. Finally, the theoretical model was validated through the experimental data.

Scale-Free Coupled Dynamics in Brain Networks Captured by Bivariate Focus-Based Multifractal Analysis

While most connectivity studies investigate functional connectivity (FC) in a scale-dependent manner, coupled neural processes may also exhibit broadband dynamics, manifesting as power-law scaling of their measures of interdependence. Here we introduce the bivariate focus-based multifractal (BFMF) analysis as a robust tool for capturing such scale-free relations and use resting-state electroencephalography (EEG) recordings of 12 subjects to demonstrate its performance in reconstructing physiological networks. BFMF was employed to characterize broadband FC between 62 cortical regions in a pairwise manner, with all investigated connections being tested for true bivariate multifractality. EEG channels were also grouped to represent the activity of six resting-state networks (RSNs) in the brain, thus allowing for the analysis of within- and between- RSNs connectivity, separately. Most connections featured true bivariate multifractality, which could be attributed to the genuine scale-free coupling of neural dynamics. Bivariate multifractality showed a characteristic topology over the cortex that was highly concordant among subjects. Long-term autocorrelation was higher in within-RSNs, while the degree of multifractality was generally found stronger in between-RSNs connections. These results offer statistical evidence of the bivariate multifractal nature of functional coupling in the brain and validate BFMF as a robust method to capture such scale-independent coupled dynamics.

Is SARS CoV-2 a Multifractal?—Unveiling the Fractality and Fractal Structure

A first report of unveiling the fractality and fractal nature of severe acute respiratory syndrome coronavirus (SARS CoV-2) responsible for the pandemic disease widely known as coronavirus disease 2019 (COVID 19) is presented. The fractal analysis of the electron microscopic and atomic force microscopic images of 40 coronaviruses (CoV), by the normal and differential box-counting method, reveals its fractal structure. The generalised dimension indicates the multifractal nature of the CoV. The higher value of fractal dimension and lower value of Hurst exponent (H) suggest higher complexity and greater roughness. The statistical analysis of generalised dimension and H is understood through the notched box plot. The study on CoV clusters also confirms its fractal nature. The scale-invariant value of the box-counting fractal dimension of CoV yields a value of 1.820. The study opens the possibility of exploring the potential of fractal analysis in the medical diagnosis of SARS CoV-2.

Non-linear Amplification of Variability Through Interaction Across Scales Supports Greater Accuracy in Manual Aiming: Evidence From a Multifractal Analysis With Comparisons to Linear Surrogates in the Fitts Task

Movement coordination depends on directing our limbs to the right place and in the right time. Movement science can study this central requirement in the Fitts task that asks participants to touch each of two targets in alternation, as accurately and as fast as they can. The Fitts task is an experimental attempt to focus on how the movement system balances its attention to speed and to accuracy. This balance in the Fitts task exhibits a hierarchical organization according to which finer details (e.g., kinematics of single sweeps from one target to the other) change with relatively broader constraints of task parameters (e.g., distance between targets and width of targets). The present work seeks to test the hypothesis that this hierarchical organization of movement coordination reflects a multifractal tensegrity in which non-linear interactions across scale support stability. We collected movement series data during a easy variant of the Fitts task to apply just such a multifractal analysis with surrogate comparison to allow clearer test of non-linear interactions across scale. Furthermore, we test the role of visual feedback both in potential and in fact, i.e., by manipulating both whether experimenters instructed participants that they might potentially have to close their eyes during the task and whether participants actually closed their eyes halfway through the task. We predict that (1) non-linear interactions across scales in hand movement series will produce variability that will actually stabilize aiming in the Fitts task, reducing standard deviation of target contacts; (2) non-linear interactions across scales in head sway will stabilize aiming following the actual closing eyes; and (3) non-linear interactions across scales in head sway and in hand movements will interact to support stabilizing effects of expectation about closing eyes. In sum, this work attempts to make the case that the multifractal-tensegrity hypothesis supports more accurate aiming behavior in the Fitts task.

Multifractal Spectrum Curvature of RR Tachograms of Healthy People and Patients with Congestive Heart Failure, a New Tool to Assess Health Conditions

We calculate the multifractal spectra of heartbeat RR-interval time series (tachograms) of healthy subjects and patients with congestive heart failure (CHF). From these time series, we obtained new subseries of 6 h durations when healthy persons and patients were asleep and awake respectively. For each time series and subseries, we worked out the multifractal spectra with the Chhabra and Jensen method and found that their graphs have different shapes for CHF patients and healthy persons. We suggest to measure two parameters: the curvature around the maximum and the symmetry for all these multifractal spectra graphs, because these parameters were different for healthy and CHF subjects. Multifractal spectra of healthy subjects tend to be right skewed especially when the subjects are asleep and the curvature around the maximum is small compared with the curvature around the maximum of the CHF multifractal spectra; that is, the spectra of patients tend to be more pointed around the maximum. In CHF patients, we also have encountered differences in the curvature of the multifractal spectra depending on their respective New York Heart Association (NYHA) index.

About Universality and Thermodynamics of Turbulence

This paper investigates the universality of the Eulerian velocity structure functions using velocity fields obtained from the stereoscopic particle image velocimetry (SPIV) technique in experiments and direct numerical simulations (DNS) of the Navier-Stokes equations. It shows that the numerical and experimental velocity structure functions up to order 9 follow a log-universality (Castaing et al. Phys. D Nonlinear Phenom. 1993); this leads to a collapse on a universal curve, when units including a logarithmic dependence on the Reynolds number are used. This paper then investigates the meaning and consequences of such log-universality, and shows that it is connected with the properties of a “multifractal free energy”, based on an analogy between multifractal and thermodynamics. It shows that in such a framework, the existence of a fluctuating dissipation scale is associated with a phase transition describing the relaminarisation of rough velocity fields with different Hölder exponents. Such a phase transition has been already observed using the Lagrangian velocity structure functions, but was so far believed to be out of reach for the Eulerian data.

Multifractal and lacunarity analyses of microvascular morphology in eyes with diabetic retinopathy: A projection artifact resolved optical coherence tomography angiography study

OBJECTIVE

To evaluate the degree of microvascular impairment in DR using multifractal and lacunarity analyses and to compare the diagnostic ability between traditional Euclidean measures (fovea avascular zone area and vessel density) and fractal geometric features.

METHODS

This retrospective cross-sectional study included a total of 143 eyes of 94 patients with different stages of DR. The retinal microvasculature was imaged by projection removed OCTA. We examined the degree of association between fractal metrics of the retinal microvasculature and DR severity. The area under the ROC curve was used to estimate the diagnostic performance.

RESULTS

With increasing DR severity, the multifractal spectrum shifted toward the left bottom and exhibited less left skewness and asymmetry. The vessel density, multifractal features, and lacunarity measured from the DCP were strongly associated with DR severity. The multifractal feature D₅ showed the highest diagnostic ability. The combination of multifractal features further improved the discriminating power.

CONCLUSIONS

Multifractal and lacunarity analyses can be potentially valuable tools for assessment of microvascular impairments in DR. Multifractal geometric parameters exhibit a better discriminatory performance than Euclidean measures, particularly for detection of the early stages of DR.

Fractal and Multifractal Properties of Electrographic Recordings of Human Brain Activity: Toward Its Use as a Signal Feature for Machine Learning in Clinical Applications

The quantification of brain dynamics is essential to its understanding. However, the brain is a system operating on multiple time scales, and characterization of dynamics across time scales remains a challenge. One framework to study such dynamics is that of fractal geometry; and currently there exist several methods for the study of brain dynamics using fractal geometry. We aim to highlight some of the practical challenges of applying fractal geometry to brain dynamics—and as a putative feature for machine learning applications, and propose solutions to enable its wider use in neuroscience. Using intracranially recorded electroencephalogram (EEG) and simulated data, we compared monofractal and multifractal methods with regards to their sensitivity to signal variance. We found that both monofractal and multifractal properties correlate closely with signal variance, thus not being a useful feature of the signal. However, after applying an epoch-wise standardization procedure to the signal, we found that multifractal measures could offer non-redundant information compared to signal variance, power (in different frequency bands) and other established EEG signal measures. We also compared different multifractal estimation methods to each other in terms of reliability, and we found that the Chhabra-Jensen algorithm performed best. Finally, we investigated the impact of sampling frequency and epoch length on the estimation of multifractal properties. Using epileptic seizures as an example event in the EEG, we show that there may be an optimal time scale (i.e., combination of sampling frequency and epoch length) for detecting temporal changes in multifractal properties around seizures. The practical issues we highlighted and our suggested solutions should help in developing robust methods for the application of fractal geometry in EEG signals. Our analyses and observations also aid the theoretical understanding of the multifractal properties of the brain and might provide grounds for new discoveries in the study of brain signals. These could be crucial for the understanding of neurological function and for the developments of new treatments.

Spectral and Multifractal Signature of Cortical Spreading Depolarisation in Aged Rats

Cortical spreading depolarisation (CSD) is a transient disruption of ion balance that propagates along the cortex. It has been identified as an important factor in the progression of cerebral damage associated with stroke or traumatic brain injury. We analysed local field potential signals during CSD in old and young rats to look for age-related differences. We compared CSDs elicited under physiological conditions (baseline), during ischaemia and during reperfusion. We applied short-time Fourier transform and a windowed implementation of multifractal detrended fluctuation analysis to follow the electrophysiological signature of CSD. Both in the time-dependent spectral profiles and in the multifractal spectrum width, CSDs appeared as transient dips, which we described on the basis of their duration, depression and recovery slope and degree of drop and rise. The most significant age-related difference we found was in the depression slope, which was significantly more negative in the beta band and less negative in the delta band of old animals. In several parameters, we observed an attenuation-regeneration pattern in reaction to ischaemia and reperfusion, which was absent in the old age group. The age-related deviation from the pattern took two forms: the rise parameter did not show any attenuation in ischaemic conditions for old animals, whilst the depression slope in most frequency bands remained attenuated during reperfusion and did not regenerate in this age group. Though the multifractal spectrum width proved to be a reliable indicator of events like CSDs or ischaemia onset, we failed to find any case where it would add extra detail to the information provided by the Fourier description.

Bringing the Nonlinearity of the Movement System to Gestural Theories of Language Use: Multifractal Structure of Spoken English Supports the Compensation for Coarticulation in Human Speech Perception

Coarticulation is the tendency for speech vocalization and articulation even at the phonemic level to change with context, and compensation for coarticulation (CfC) reflects the striking human ability to perceive phonemic stability despite this variability. A current controversy centers on whether CfC depends on contrast between formants of a speech-signal spectrogram-specifically, contrast between offset formants concluding context stimuli and onset formants opening the target sound-or on speech-sound variability specific to the coordinative movement of speech articulators (e.g., vocal folds, postural muscles, lips, tongues). This manuscript aims to encode that coordinative-movement context in terms of speech-signal multifractal structure and to determine whether speech's multifractal structure might explain the crucial gestural support for any proposed spectral contrast. We asked human participants to categorize individual target stimuli drawn from an 11-step [ga]-to-[da] continuum as either phonemes "GA" or "DA." Three groups each heard a specific-type context stimulus preceding target stimuli: either real-speech [al] or [a], sine-wave tones at the third-formant offset frequency of either [al] or [aɹ], and either simulated-speech contexts [al] or [aɹ]. Here, simulating speech contexts involved randomizing the sequence of relatively homogeneous pitch periods within vowel-sound [a] of each [al] and [aɹ]. Crucially, simulated-speech contexts had the same offset and extremely similar vowel formants as and, to additional naïve participants, sounded identical to real-speech contexts. However, randomization distorted original speech-context multifractality, and effects of spectral contrast following speech only appeared after regression modeling of trial-by-trial "GA" judgments controlled for context-stimulus multifractality. Furthermore, simulated-speech contexts elicited faster responses (like tone contexts do) and weakened known biases in CfC, suggesting that spectral contrast depends on the nonlinear interactions across multiple scales that articulatory gestures express through the speech signal. Traditional mouse-tracking behaviors measured as participants moved their computer-mouse cursor to register their "GA"-or-"DA" decisions with mouse-clicks suggest that listening to speech leads the movement system to resonate with the multifractality of context stimuli. We interpret these results as shedding light on a new multifractal terrain upon which to found a better understanding in which movement systems play an important role in shaping how speech perception makes use of acoustic information.

Three-dimensional multifractal analysis of trabecular bone under clinical computed tomography

PURPOSE

An adequate understanding of bone structural properties is critical for predicting fragility conditions caused by diseases such as osteoporosis, and in gauging the success of fracture prevention treatments. In this work we aim to develop multiresolution image analysis techniques to extrapolate high-resolution images predictive power to images taken in clinical conditions.

METHODS

We performed multifractal analysis (MFA) on a set of 17 ex vivo human vertebrae clinical CT scans. The vertebrae failure loads (FFailure) were experimentally measured. We combined bone mineral density (BMD) with different multifractal dimensions, and BMD with multiresolution statistics (e.g., skewness, kurtosis) of MFA curves, to obtain linear models to predict FFailure. Furthermore we obtained short- and long-term precisions from simulated in vivo scans, using a clinical CT scanner. Ground-truth data - high-resolution images - were obtained with a High-Resolution Peripheral Quantitative Computed Tomography (HRpQCT) scanner.

RESULTS

At the same level of detail, BMD combined with traditional multifractal descriptors (Lipschitz-Hölder exponents), and BMD with monofractal features showed similar prediction powers in predicting FFailure (87%, adj. R² ). However, at different levels of details, the prediction power of BMD with multifractal features raises to 92% (adj. R2) of FFailure. Our main finding is that a simpler but slightly less accurate model, combining BMD and the skewness of the resulting multifractal curves, predicts 90% (adj. R2) of FFailure.

CONCLUSIONS

Compared to monofractal and standard bone measures, multifractal analysis captured key insights in the conditions leading to FFailure. Instead of raw multifractal descriptors, the statistics of multifractal curves can be used in several other contexts, facilitating further research.

Analysis of normal human retinal vascular network architecture using multifractal geometry

AIM

To apply the multifractal analysis method as a quantitative approach to a comprehensive description of the microvascular network architecture of the normal human retina.

METHODS

Fifty volunteers were enrolled in this study in the Ophthalmological Clinic of Cluj-Napoca, Romania, between January 2012 and January 2014. A set of 100 segmented and skeletonised human retinal images, corresponding to normal states of the retina were studied. An automatic unsupervised method for retinal vessel segmentation was applied before multifractal analysis. The multifractal analysis of digital retinal images was made with computer algorithms, applying the standard box-counting method. Statistical analyses were performed using the GraphPad InStat software.

RESULTS

The architecture of normal human retinal microvascular network was able to be described using the multifractal geometry. The average of generalized dimensions (Dq ) for q=0, 1, 2, the width of the multifractal spectrum (Δα=αmax - αmin ) and the spectrum arms' heights difference (|Δf|) of the normal images were expressed as mean±standard deviation (SD): for segmented versions, D0 =1.7014±0.0057; D1 =1.6507±0.0058; D2 =1.5772±0.0059; Δα=0.92441±0.0085; |Δf|= 0.1453±0.0051; for skeletonised versions, D0 =1.6303±0.0051; D1 =1.6012±0.0059; D2 =1.5531±0.0058; Δα=0.65032±0.0162; |Δf|= 0.0238±0.0161. The average of generalized dimensions (Dq ) for q=0, 1, 2, the width of the multifractal spectrum (Δα) and the spectrum arms' heights difference (|Δf|) of the segmented versions was slightly greater than the skeletonised versions.

CONCLUSION

The multifractal analysis of fundus photographs may be used as a quantitative parameter for the evaluation of the complex three-dimensional structure of the retinal microvasculature as a potential marker for early detection of topological changes associated with retinal diseases.

Prognostic biomarker value of binary and grayscale breast tumor histopathology images

AIM

Breast cancer prognosis is in the spotlight owing to its potentially major clinical importance in effective therapeutic management. Following our recent prognostic establishment of the fractal features calculated on binary breast tumor histopathology images, this study aimed to accomplish the first optimization of this methodology by direct comparison of monofractal, multifractal and co-occurrence algorithms in analysis of binary versus grayscale image formats.

PATIENTS & METHODS

The study included 93 patients with invasive breast cancer, without systemic treatment and a long median follow-up of 150 months.

RESULTS

Grayscale images provided a better prognostic source in comparison to binary, while monofractal, multifractal and co-occurrence image analysis algorithms exerted a comparable performance.

CONCLUSION

The critical prognostic importance of the grayscale texture is revealed.

Exploring multifractal-based features for mild Alzheimer's disease classification

PURPOSE

Multifractal applications to resting state functional MRI (rs-fMRI) time series for diagnosing Alzheimer's disease (AD) are still limited. We aim to address two issues: (I) if and what multifractal features are sufficiently discriminative to detect AD from the healthy; (II) if AD classification could be further improved by combining multifractal features with traditional features in this field.

METHODS

Rs-fMRI data of 25 AD patients and 38 normal controls were analyzed. A set of multifractal features were systematically investigated. Traditional features in monofractal, linear, and network-based categories were also extracted for comparison and combination. Both support vector machines and multiple kernel learning (MKL) were used to perform classification with individual and combined features.

RESULTS

We identified a multifractal feature, Δf, which has the strongest discriminative power among all the features in our study. Moreover, we found that the classification accuracy could be significantly improved from 69% (by Δf only) to up to 76%, when nonsparse MKL is used to combine Δf with the monofractal feature, Hurst. Finally, we showed that incorporating other multifractal features, α(0), Δα and Pc, could also improve traditional-feature-based AD classification.

CONCLUSION

Our work demonstrated the potential usefulness of multifractal analysis for AD research, especially when combining with the traditional rs-fMRI features. It contributes to distinguishing AD from NC subjects. Magn Reson Med 76:259-269, 2016. © 2015 Wiley Periodicals, Inc.

Objective differentiation of neonatal EEG background grades using detrended fluctuation analysis

A quantitative and objective assessment of background electroencephalograph (EEG) in sick neonates remains an everyday clinical challenge. We studied whether long range temporal correlations quantified by detrended fluctuation analysis (DFA) could be used in the neonatal EEG to distinguish different grades of abnormality in the background EEG activity. Long-term EEG records of 34 neonates were collected after perinatal asphyxia, and their background was scored in 1 h epochs (8 h in each neonate) as mild, moderate or severe. We applied DFA on 15 min long, non-overlapping EEG epochs (n = 1088) filtered from 3 to 8 Hz. Our formal feasibility study suggested that DFA exponent can be reliably assessed in only part of the EEG epochs, and in only relatively short time scales (10-60 s), while it becomes ambiguous if longer time scales are considered. This prompted further exploration whether paradigm used for quantifying multifractal DFA (MF-DFA) could be applied in a more efficient way, and whether metrics from MF-DFA paradigm could yield useful benchmark with existing clinical EEG gradings. Comparison of MF-DFA metrics showed a significant difference between three visually assessed background EEG grades. MF-DFA parameters were also significantly correlated to interburst intervals quantified with our previously developed automated detector. Finally, we piloted a monitoring application of MF-DFA metrics and showed their evolution during patient recovery from asphyxia. Our exploratory study showed that neonatal EEG can be quantified using multifractal metrics, which might offer a suitable parameter to quantify the grade of EEG background, or to monitor changes in brain state that take place during long-term brain monitoring.

Evaluation of scaling and root planing effect in generalized chronic periodontitis by fractal and multifractal analysis

BACKGROUND AND OBJECTIVE

Fractal and multifractal analysis are useful additional non-invasive methods for quantitative description of complex morphological features. However, the quantitative and qualitative assessment of morphologic changes within human gingival cells and tissues are still unexplored. The aim of this work is to assess the structural gingival changes in patients with generalized chronic periodontitis (GCP), before and after scaling and root planing (SRP) by using fractal and multifractal analysis.

MATERIAL AND METHODS

Twelve adults with untreated chronic periodontitis were treated only by SRP. At baseline and after SRP, gingivomucosal biopsies were collected for histopathological examination. Fractal and multifractal analysis of digital images of the granular, spinous and basal and conjunctive layers structure, using the standard box-counting method was performed. The fractal dimension was determined for cell membrane, nuclear membrane of cell and nucleolus membrane of cell.

RESULTS

In GCP a higher fractal dimension corresponds to a higher geometric complexity of cells contour, as its values increase when the contour irregularities increase. The generalized fractal dimensions were determined for the conjunctive layer structure of patients with GCP and patients with GCP and SRP. The fractal and multifractal analysis of gingival biopsies confirmed earlier findings that SRP reduces gingival injury in patients with GCP.

CONCLUSION

It has been shown that fractal and multifractal analysis of tissue images as a non-invasive technique could be used to measure contrasting morphologic changes within human gingival cells and tissues and can provide detailed information for investigation of healthy and diseased gingival mucosa from patients with GCP.

Twenty-four hour variation in heart rate variability indices derived from fractional differintegration

Assuming that RR time-series behave as a fractionally differintegrated Gaussian process, García-González et al. (2003) recently proposed new indices for quantifying variability and structure in RR data. One of these was the 'fractional noise quantifier' (fnQ), measuring the departure of an RR time-series from a monofractal structure (i.e. a measure of its multifractality). Sixty-nine participants (aged = 34·5 ± 12·4 years, body mass index (BMI) = 23·9 ± 2·9 kg m(-2) , maximal oxygen uptake rate (V˙O2peak ) = 42·4 ± 10·9 ml min(-1)  kg(-1) , 39 males) provided continuous beat-to-beat ECG recordings for a 24-h period. Fractional differintegration was used to quantify fnQ, and heart rate variability was calculated in the time domain. All variables were evaluated during consecutive 1-h periods and also during four 6-h blocks corresponding to morning, afternoon, evening and night periods. Apart from RR, circadian trends in all variables were independent of gender (P = 0·11-0·59). Apart from fnQ, all variables exhibited circadian variation (0·0005<P<0·012). Although fnQ was statistically uniform during the 24-h period, it showed a trend towards elevated values during evening and night. The main finding of this study was that fnQ was elevated by around 10% during the evening and night, although this was not statistically significant. This suggests that the structure of RR time-series in healthy individuals is most strongly 'multifractal' during evening and night periods. fnQ appears to be a plausible surrogate measure of multifractality in RR time-series.

Multifractal detrended fluctuation analysis of human gait diseases

In this paper multifractal detrended fluctuation analysis (MFDFA) is used to study the human gait time series for normal and diseased sets. It is observed that long range correlation is primarily responsible for the origin of multifractality. The study reveals that the degree of multifractality is more for normal set compared to diseased set. However, the method fails to distinguish between the two diseased sets.

Influence of age and aerobic fitness on the multifractal characteristics of electrocardiographic RR time-series

Multifractal properties of electrocardiographic inter-beat (RR) time-series offer insight into its long-term correlation structure, independently of RR variability. Here we quantify multifractal characteristics of RR data during 24-h diurnal-nocturnal activity in healthy participants. We tested the hypotheses that (1) age, gender and aerobic fitness influence RR multifractal properties, and that (2) these are influenced by circadian variation. Seventy adults (39 males) aged 19–58 years and of various fitness levels were monitored using 24-h ECG. Participants were dichotomized by median age and fitness for sub-group analysis. Gender and fitness were independent of age (p = 0.1, p > 0.5). Younger/older group ages were substantially different (p < 0.0005) and were independent of gender and fitness. Multifractality was quantified using the probability spectrum of Hölder exponents (h), from which modal h (h^*) and the full-width and half-widths at half-maximum measures (FWHM, HWHM+, and HWHM−) were derived. FWHM decreased (p = 0.004) and h^* increased (p = 0.011) in older people, indicating diminished long-range RR correlations and weaker anti-persistent behavior. Anti-persistent correlation (h^*) was strongest in the youngest/fittest individuals and weakest in the oldest/least fit individuals (p = 0.015). Long-range correlation (HWHM+/FWHM) was strongest in the fittest males and weakest in the least fit females (p = 0.007–0.033). Multifractal RR characteristics in our healthy participants showed strong age-dependence, with diminished long-range anti-persistent correlation in older people. Circadian variation of these characteristics was influenced by fitness and gender: fitter males and females of all ages had the greatest degree of multifractality or long-range order. Multifractal characterization appears to be a useful method for exploring the physiological basis of long-term correlation structure in RR time-series as well as the benefits thereon of physical fitness training.

Introduction to multifractal detrended fluctuation analysis in matlab

Fractal structures are found in biomedical time series from a wide range of physiological phenomena. The multifractal spectrum identifies the deviations in fractal structure within time periods with large and small fluctuations. The present tutorial is an introduction to multifractal detrended fluctuation analysis (MFDFA) that estimates the multifractal spectrum of biomedical time series. The tutorial presents MFDFA step-by-step in an interactive Matlab session. All Matlab tools needed are available in Introduction to MFDFA folder at the website www.ntnu.edu/inm/geri/software. MFDFA are introduced in Matlab code boxes where the reader can employ pieces of, or the entire MFDFA to example time series. After introducing MFDFA, the tutorial discusses the best practice of MFDFA in biomedical signal processing. The main aim of the tutorial is to give the reader a simple self-sustained guide to the implementation of MFDFA and interpretation of the resulting multifractal spectra.

Scaling in cognitive performance reflects multiplicative multifractal cascade dynamics

Self-organized criticality purports to build multi-scaled structures out of local interactions. Evidence of scaling in various domains of biology may be more generally understood to reflect multiplicative interactions weaving together many disparate scales. The self-similarity of power-law scaling entails homogeneity: fluctuations distribute themselves similarly across many spatial and temporal scales. However, this apparent homogeneity can be misleading, especially as it spans more scales. Reducing biological processes to one power-law relationship neglects rich cascade dynamics. We review recent research into multifractality in executive-function cognitive tasks and propose that scaling reflects not criticality but instead interactions across multiple scales and among fluctuations of multiple sizes.

Effects of accuracy feedback on fractal characteristics of time estimation

The current experiment investigated the effect of visual accuracy feedback on the structure of variability of time interval estimates in the continuation tapping paradigm. Participants were asked to repeatedly estimate a 1-s interval for a prolonged period of time by tapping their index finger. In some conditions, participants received accuracy feedback after every estimate, whereas in other conditions, no feedback was given. Also, the likelihood of receiving visual feedback was manipulated by adjusting the tolerance band around the 1-s target interval so that feedback was displayed only if the temporal estimate deviated from the target interval by more than 50, 100, or 200 ms respectively. We analyzed the structure of variability of the inter-tap intervals with fractal and multifractal methods that allow for a quantification of complex long-range correlation patterns in the timing performance. Our results indicate that feedback changes the long-range correlation structure of time estimates: Increased amounts of feedback lead to a decrease in fractal long-range correlations, as well to a decrease in the magnitude of local fluctuations in the performance. The multifractal characteristics of the time estimates were not impacted by the presence of accuracy feedback. Nevertheless, most of the data sets show significant multifractal signatures. We interpret these findings as showing that feedback acts to constrain and possibly reorganize timing performance. Implications for mechanistic and complex systems-based theories of timing behavior are discussed.