Power-law Scaling Behavior Research Articles

Investigating Equatorial Plasma Depletions through CSES-01 Satellite Data

Ionospheric plasma density irregularities, which are one of the primary sources of disturbance for the Global Navigation Satellite System, significantly impact the propagation of electromagnetic signals, leading to signal degradation and potential interruptions. In the equatorial ionospheric F region after sunset, certain plasma density irregularities, identified as equatorial plasma bubbles, encounter optimal conditions for their formation and development. The energy spectra of electron density fluctuations associated with these irregularities exhibit a power-law scaling behavior qualitatively similar to the Kolmogorov power law observed in fluid turbulence theory. This intriguing similarity raises the possibility that these plasma density irregularities may possess turbulent characteristics. In this study, we analyzed electron density, temperature, and pressure data obtained from the China Seismo-Electromagnetic Satellite (CSES-01) to delve into the spectral properties of equatorial plasma depletions in the ionospheric F region at an altitude of about 500 km. This research marks the first exploration of these properties utilizing CSES-01 data and focuses on 14 semi-orbits that crossed the equator after midnight (01:00–03:00 LT), characterized by a geomagnetic quiet condition (Kp < 1). The analysis of electron temperature, density and pressure within equatorial plasma depletions revealed power-law scaling behavior for all the selected parameters. Notably, the spectral index values of these parameters are different from each other. The significance of these findings in terms of investigating plasma depletions via magnetic field signatures, as well as their relationship to the occurrence of Rayleigh–Taylor convective turbulence, is examined and discussed.

Contact Angle Modulation: In Situ Polymer Deposition during Sessile Drop Evaporation.

The drying kinetics of a sessile drop on a solid surface are a widely studied phenomenon because of their relevance to various fields such as coating, printing, medical diagnostics, sensing, and microfluidic technology. Typically, the drop undergoes drying either at a constant contact radius (R) with a decrease in the three-phase contact angle or at a constant contact angle (θ) with a reduction in the radius with time. These two drying modes are referred to as CCR and CCA, respectively. It is not uncommon where both R and θ may decrease during drying, especially in the penultimate stage of drying. In this work, we report a scenario wherein the θ increases while R decreases during the drying process of an aqueous polymer solution on a high surface energy substrate. This behavior is observed across different polymer systems (such as poly(ethylene oxide) and polyvinyl pyrrolidine), varying molecular weights, and polymer concentrations. As the drop dries, the polymer gets deposited at the three-phase contact line, thus reducing the surface energy of the substrate and leading to an increase in the contact angle. The drop responds by attempting to reach a new equilibrium contact angle through slipping. The temporal increase in contact angle follows a power law scaling behavior. This study demonstrates an in situ modulation of contact angle facilitated by evaporation and polymer deposition, showcasing unconventional drying dynamics.

Hyperuniformity classes of quasiperiodic tilings via diffusion spreadability.

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the vicinity of the origin, e.g., S(k)∼|k|^{α} in cases where S(k) varies continuously with k as k→0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)10.1103/PhysRevE.104.054102] that, for media with finite α, the long-time behavior of the excess spreadability S(∞)-S(t) can be fit to a power law of the form t^{-(d-α)/2}, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ[over ̃]_{_{V}}(k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the one-dimensional (1D) limit-periodic period-doubling chain (α=1) and the 1D quasicrystalline Fibonacci chain (α=3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α=5.97±0.06 for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that α is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of α for an infinite self-similar quasicrystal from a modestly sized finite sample. The methods described here offer a simple and general procedure to characterize accurately the large-scale translational order present in quasicrystalline and limit-periodic media in any space dimension that are self-similar. Moreover, the scattering information extracted from these two-phase media encoded in χ[over ̃]_{_{V}}(k), can be used to estimate their physical properties, such as their effective dynamic dielectric constants, effective dynamic elastic constants, and fluid permeabilities.

Diffusive persistence on disordered lattices and random networks.

To better understand the temporal characteristics and the lifetime of fluctuations in stochastic processes in networks, we investigated diffusive persistence in various graphs. Global diffusive persistence is defined as the fraction of nodes for which the diffusive field at a site (or node) has not changed sign up to time t (or, in general, that the node remained active or inactive in discrete models). Here we investigate disordered and random networks and show that the behavior of the persistence depends on the topology of the network. In two-dimensional (2D) disordered networks, we find that above the percolation threshold diffusive persistence scales similarly as in the original 2D regular lattice, according to a power law P(t,L)∼t^{-θ} with an exponent θ≃0.186, in the limit of large linear system size L. At the percolation threshold, however, the scaling exponent changes to θ≃0.141, as the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. Moreover, studying finite-size effects for 2D lattices at and above the percolation threshold, we find that at the percolation threshold, the long-time asymptotic value obeys a power law P(t,L)∼L^{-zθ} with z≃2.86 instead of the value of z=2 normally associated with finite-size effects on 2D regular lattices. In contrast, we observe that in random networks without a local regular structure, such as Erdős-Rényi networks, no simple power-law scaling behavior exists above the percolation threshold.

Transport and localization properties of excitations in one-dimensional lattices with diagonal disordered mosaic modulations

We present a numerical study of the transport and localization properties of excitations in one-dimensional lattices with diagonal disordered mosaic modulations. The model is characterized by the modulation period κ and the disorder strength W. We calculate the disorder averages , , and , where T is the transmittance and P is the participation ratio, as a function of energy E and system size L, for different values of κ and W. For excitations at quasiresonance energies determined by κ, we find power-law scaling behaviors of the form , , and , as L increases to a large value. In the strong disorder limit, the exponents are seen to saturate at the values , , and , regardless of the quasiresonance energy value. This behavior is in contrast to the exponential localization behavior occurring at all other energies. The appearance of sharp peaks in the participation ratio spectrum at quasiresonance energies provides additional evidence for the existence of an anomalous power-law localization phenomenon. The corresponding eigenstates demonstrate multifractal behavior and exhibit unique node structures. In addition, we investigate the time-dependent wave packet dynamics and calculate the mean square displacement , spatial probability distribution, participation number, and return probability. When the wave packet’s initial momentum satisfies the quasiresonance condition, we observe a subdiffusive spreading of the wave packet, characterized by where η is always less than 1. We also note the occurrence of partial localization at quasiresonance energies, as indicated by the saturation of the participation number and a nonzero value for the return probability at long times.

Persistent post-COVID headache is associated with suppression of scale-free functional brain dynamics in non-hospitalized individuals.

Post-acute coronavirus disease 2019 (COVID-19) syndrome (PACS) is a growing concern, with headache being a particularly debilitating symptom with high prevalence. The long-term effects of COVID-19 and post-COVID headache on brain function remain poorly understood, particularly among non-hospitalized individuals. This study focused on the power-law scaling behavior of functional brain dynamics, indexed by the Hurst exponent (H). This measure is suppressed during physiological and psychological distress and was thus hypothesized to be reduced in individuals with post-COVID syndrome, with greatest reductions among those with persistent headache. Resting-state blood oxygenation level-dependent (BOLD) functional magnetic resonance imaging data were collected for 57 individuals who had COVID-19 (32 with no headache, 14 with ongoing headache, 11 recovered) and 17 controls who had cold and flu-like symptoms but tested negative for COVID-19. Individuals were assessed an average of 4-5 months after COVID testing, in a cross-sectional, observational study design. No significant differences in H values were found between non-headache COVID-19 and control groups., while those with ongoing headache had significantly reduced H values, and those who had recovered from headache had elevated H values, relative to non-headache groups. Effects were greatest in temporal, sensorimotor, and insular brain regions. Reduced H in these regions was also associated with decreased BOLD activity and local functional connectivity. These findings provide new insights into the neurophysiological mechanisms that underlie persistent post-COVID headache, with reduced BOLD scaling as a potential biomarker that is specific to this debilitating condition.

Tunable oleosome-based oleogels: Effect of polysaccharide architecture on polymer bridging-based structuring

A simple method for structuring natural oleosome emulsions by polymer-bridging mechanism is proposed. Polymer bridging of oleosome droplets was induced by the addition of two different adsorbing polymers. Over a range of polymer/oleosome ratios, the mixture results in the formation of a particle gel network of aggregated oleosome droplets. It is found that polymer bridging ability is heavily influenced by the strength of binding between polymer and oleosome surface where sodium alginate interacted stronger to oleosome surface than ι-carrageenan. These effects are associated with the different molecular architecture and physical differences between the two hydrocolloids. Alginate has a co-block arrangement of charged and uncharged units. The polymer promotes stronger adsorption to the oleosome surface, in contrast to ι-carrageenan, where the negative charges are distributed uniformly along its chain. The polymer bridging ability will influence the resulting microstructure and therefore rheological properties. Confocal scanning laser microscopy showed that the difference in microstructure is mainly in the extent of heterogeneity over different length scales where sodium alginate produced the most heterogeneous microstructures. Bridging-flocculated emulsions showed power-law scaling behavior of the storage modulus with the oleosome concentration, which was explained using percolation theory.

Temperature-dependent Scaling Behavior of Magnetic Hysteresis Loop of CoFeB/Pd Multilayer Film

We report our experimental investigation on the temperature-dependent hysteresis loops and magnetic domain behaviors of CoFeB (0.4 nm)/Pd (1 nm) multilayer on polyimide substrate with a perpendicular magnetic anisotropy, by means of magneto-optical Kerr microscopy. Hysteresis characteristics such as saturation magnetization, coercivity, and hysteresis loop area with respect to the temperature have been analyzed, where powerlaw scaling behaviors are observed with microscopic magnetic domain patterns mediated by numerous nucleation sites.

Nanoparticle geometrical effects on percolation, packing density, and magnetoresistive properties in ferromagnet-superconductor-insulator nanocomposites

The influence of nanoparticle geometrical and percolation effects on the electrical transport and magnetotransport properties of the system are investigated in a number of binary networks composed of superconducting (${\mathrm{MgB}}_{2}$), ferromagnetic (${\mathrm{CrO}}_{2}$ or ${\mathrm{La}}_{1/3}{\mathrm{Sr}}_{2/3}{\mathrm{MnO}}_{3}$), and insulating (${\mathrm{LiCoO}}_{2}$ or ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$) nanoparticles. For all-metal ${\mathrm{CrO}}_{2}/{\mathrm{MgB}}_{2}$ binary composites an anomalously high resistance state with two distinct percolation thresholds is found within a narrow range of the constituent composition, as a result of the strong suppression of heterointerfacial conductance between superconducting and half-metallic (${\mathrm{CrO}}_{2}$) nanoparticles and an unexpectedly large value of the percolation threshold of ${\mathrm{MgB}}_{2}$. The latter can be attributed to the geometric mismatch between the constituent nanoparticles and the related particle packing and excluded volume effects. The filling factor in this binary composite attains a maximum near the same percolation threshold and exhibits a power-law scaling behavior, which would indicate another geometric phase transition. The geometric effect of constituent particles is also verified through the large changes of percolation threshold in different types of combinations of binary components. Interestingly, the percolation transitions in these binary nanocomposites, either in the case of single or double percolation, are found to be governed by an unconventional scaling behavior of resistance just below the percolation threshold, due to mechanisms determined by heterogeneous interfaces of particles that are absent in conventional single-component percolation systems. In addition, magnetotransport is investigated in ${\mathrm{CrO}}_{2}/{\mathrm{MgB}}_{2}$ junctions, demonstrating nonhysteretic low-temperature magnetoresistance at the half-metal/Type II superconductor interface with the highest values around $42%$ at liquid helium temperatures obtained at volume fraction between the two percolation thresholds. The results show large variations of magnetoresistance that are highly sensitive to the constituent composition and the sample temperature near the percolation thresholds, demonstrating the interrelation between the structural or geometrical property of the system and materials functionality.

Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processes

We analyze the statistical properties of a temporal point process driven by a confined fractional Brownian motion. The event count distribution and power spectral density of this non-Markovian point process exhibit power-law scaling. We show that a nonlinear Markovian point process can reproduce the same scaling behavior. This result indicates a possible link between nonlinearity and apparent non-Markovian behavior.

The Protein-Protein Interaction Network of Hereditary Parkinsonism Genes Is a Hierarchical Scale-Free Network.

PurposeHereditary parkinsonism genes consist of causative genes of familial Parkinson’s disease (PD) with a locus symbol prefix (PARK genes) and hereditary atypical parkinsonian disorders that present atypical features and limited responsiveness to levodopa (non-PARK genes). Although studies have shown that hereditary parkinsonism genes are related to idiopathic PD at the phenotypic, gene expression, and genomic levels, no study has systematically investigated connectivity among the proteins encoded by these genes at the protein-protein interaction (PPI) level.Materials and MethodsTopological measurements and physical interaction enrichment were performed to assess PPI networks constructed using some or all the proteins encoded by hereditary parkinsonism genes (n=96), which were curated using the Online Mendelian Inheritance in Man database and literature.ResultsNon-PARK and PARK genes were involved in common functional modules related to autophagy, mitochondrial or lysosomal organization, catecholamine metabolic process, chemical synapse transmission, response to oxidative stress, neuronal apoptosis, regulation of cellular protein catabolic process, and vesicle-mediated transport in synapse. The hereditary parkinsonism proteins formed a single large network comprising 51 nodes, 83 edges, and three PPI pairs. The probability of degree distribution followed a power-law scaling behavior, with a degree exponent of 1.24 and a correlation coefficient of 0.92. LRRK2 was identified as a hub gene with the highest degree of betweenness centrality; its physical interaction enrichment score was 1.28, which was highly significant.ConclusionBoth PARK and non-PARK genes show high connectivity at the PPI and biological functional levels.

Structural, mechanical, and vibrational properties of particulate physical gels.

Our lives are surrounded by a rich assortment of disordered materials. In particular, glasses are well known as dense, amorphous materials, whereas gels exist in low-density, disordered states. Recent progress has provided a significant step forward in understanding the material properties of glasses, such as mechanical, vibrational, and transport properties. In contrast, our understanding of particulate physical gels is still highly limited. Here, using molecular dynamics simulations, we study a simple model of particulate physical gels, the Lennard-Jones (LJ) gels, and provide a comprehensive understanding of their structural, mechanical, and vibrational properties, all of which are markedly different from those of LJ glasses. First, the LJ gels show sparse, heterogeneous structures, and the length scale ξs of the structures grows as the density is lowered. Second, the LJ gels are extremely soft, with both shear G and bulk K moduli being orders of magnitude smaller than those of LJ glasses. Third, many low-frequency vibrational modes are excited, which form a characteristic plateau with the onset frequency ω* in the vibrational density of states. Structural, mechanical, and vibrational properties, characterized by ξs, G, K, and ω*, respectively, show power-law scaling behaviors with the density, which establishes a close relationship between them. Throughout this work, we also reveal that LJ gels are multiscale, solid-state materials: (i) homogeneous elastic bodies at long lengths, (ii) heterogeneous elastic bodies with fractal structures at intermediate lengths, and (iii) amorphous structural bodies at short lengths.

An analysis of temporal scaling behaviour of extreme rainfall in Germany based on radar precipitation QPE data

Abstract. We investigated the depth–duration relationship of maximum rainfall over all of Germany based on 16 years of radar-derived quantitative precipitation estimates (namely, RADKLIM-YW, German Meteorological Service) with a space–time resolution of 1 km2 and 5 min. Contrary to the long-term historic records that identified a smooth power law scaling behaviour between the maximum rainfall depth and duration, our analysis revealed three distinct scaling regimes of which boundaries are approximately 1 h and 1 d. A few extraordinary events dominated a wide range of durations and deviate to the usual power law. Furthermore, the shape of the depth–duration relationship varied with the sample size of randomly selected radar pixels. A smooth scaling behaviour was identified when the sample size was small (e.g. 10 to 100), but the original three distinct scaling regimes became more apparent as the sample size increases (e.g. 1000 to 10 000). Lastly, a pixel-wise classification of the depth–duration relationship of the maximum rainfall at all individual pixels in Germany revealed three distinguishable types of scaling behaviour, clearly determined by the temporal structure of the extreme rainfall events at a pixel. Thus, the relationship might change with longer time series and can be improved once available.

Temperature dependent 89Y NMR study on multiferroic YCrO3

In this manuscript, we report the 89Y NMR measurement as a function of temperature on single phase and pure polycrystalline YCrO3 sample to study the magnetism and relaxation times on a microscopic level across the magnetic transition (T N ≃ 141 K) from paramagnetic to antiferromagnetic state. The NMR peak width broadens abruptly upon crossing T N due to the onset of internal magnetic fields, while peakshift slight decreases. A slight increase and subsequent anomalous decrease in the NMR peak intensity is observed on approaching T N from 300 K. There is also a significant increase in peak width. The temperature dependence of the 89Y NMR spin–lattice relaxation rates 1/T 1 indicates a phase transition at T N which is of magnetic origin due to Cr3+ ions, with an anomalously rise of fluctuations below T N. Above T N, this spin–lattice relaxation rate can be fitted to a power-law scaling behavior 1/T 1 ∼ T β with an exponent factor β ≈ 0.8, indicates low energy spin fluctuations. Moreover, Knight shift and 1/TT 1 scales linear with the bulk susceptibility which suggests the antiferromagnetic spin fluctuation in the YCrO3 system.

The impact of model rigid fillers in acid-induced sodium caseinate/xanthan gum cooperative protein gels

This work evaluates the impact of model rigid fillers on the deformation and fracture behavior of acid-set sodium caseinate/xanthan gum (NaCas/XG) mixed biopolymer gels. Glass microspheres of varying size ranges were used as idealized filler particles. The large deformation elastic modulus showed a power law scaling behavior with increasing filler volume fraction (φf) for all fillers. The magnitude of this response was strongly dependent on filler size, as decreasing filler diameter produced a higher scaling exponent. For the smallest fillers tested (8 μm), the presence of amino functional groups on the filler surface produced a further increase in the scaling behavior. The stark increase in elastic modulus indicates the microspheres behaved as active fillers which bound to the electrostatically-induced cooperative gel network. The fracture strain (εr*) of both the neat and amino-coated 8 μm microspheres showed an abrupt decrease at low φf, which roughly followed the theoretically expected response proposed by Nielsen. In contrast, the larger fillers (90 μm and 180 μm) produced a roughly linear decrease in εr* with increasing filler loading. Above a threshold values of φf ~0.15, the fracture stress (σr*) displayed a power law scaling behavior, with the effect of filler size and surface chemistry analogous to that seen for the elastic modulus. Water loss results correlated well with the increase in mechanical strength, and T2 relaxometry suggested the hydrophilic fillers reduce water mobility within the gel network. These results indicate that the effect of filler size and surface chemistry can be used to modulate the large deformation and fracture mechanics of particle-filled food gels. This work adds to the growing body of literature on developing a more fundamental understanding of foods as composite soft materials.

Disordered Chromatin Packing Regulates Ensemble Gene Expression and Phenotypic Plasticity

The adaptive potential of organisms is determined not only by their genetic code, but also by their ability to modulate their transcriptome to create new functional states. Transcription itself is a series of chemical reactions governed by local molecular factors and further modulated by macromolecular crowding, which influences the mobility and kinetics of transcriptional reactants. Within the nucleus the predominant crowder is chromatin, the disordered nucleoprotein polymer which controls packing of the genome. Thus, we hypothesized that the three-dimensional structure of chromatin should have a profound influence on large-scale gene expression patterns. First, we developed a multi-scale computational model which integrates disordered chromatin packing (CP) with local macromolecular crowding (MC) to estimate ensemble gene expression patterns. Microscopy and chromosome conformation capture techniques have demonstrated that, at the level of hundreds of kilobases, chromatin is organized into chromatin packing domains that exhibit fractal, power-law scaling behavior. Our CPMC model considers each of these domains to have their own genomic size, fractal dimension, and average chromatin packing density, which we predict are all physical regulators of gene expression. Next, we experimentally validate the importance of these physical properties of chromatin organization on transcription. Specifically, we utilize a combination of nanoscale-sensitive microscopy techniques to measure chromatin structure, including ChromEM (Chromatin Electron Microscopy) and PWS (Partial WaveSpectroscopic) microscopy, combined with RNA sequencing to measure corresponding gene expression levels. Additionally, we evaluate how chromatin packing influences transcriptional responsiveness of a cell population to an exogenous stressor. Finally, we identify an inverse relationship between cancer patient survival post chemotherapy treatment and phenotypic plasticity of tumor cells as determined by publicly available cancer patient transcriptional data. In summary, we demonstrate the crucial role of supranucleosomal chromatin packing on modulating both large-scale transcription patterns and phenotypic plasticity.

Power-law scaling behavior of A-phase events during sleep: Normal and pathologic conditions

ObjectiveA-phases are short-time cortical events during sleep that repeatedly disrupt the basal fluctuations of electrical brain activity. They are the basic units of the cyclic alternating pattern (CAP), which is related to the instability and consolidation of the sleep process. The main purpose of this study is to evaluate the temporal occurrence of A-phases (TOAP) under normal and pathologic sleep conditions. MethodsTo derive a quantitative description of TOAP, we have applied detrended fluctuation analysis (DFA), to unveil scale-free behavior from non-stationary time series. Data from sleep recordings of 15 healthy subjects (H), 37 with nocturnal frontal lobe epilepsy (NFLE), 9 with periodic leg movement (PLM), and 22 with REM Sleep Behavior Disorder (RBD) from the Physionet database were used. TOAP was computed from binary time series constructed from A-phase annotations, where symbols 1 and −1, represent the presence or absence of an A-phase, respectively. These time series were analyzed through DFA and characterized by the scaling exponent. ResultsIn all cases, numerical results show evidence that a statistical fractal structure is embedded in TOAP, where persistent scaling behavior is observed with scaling exponent close to 1. ConclusionThe sleep process maintains a similar structure from the A-phase perspective despite of internal or external factors that could affect the sleep process, suggesting that sleep may be a resilient process. SignificanceThe scaling exponent of the A-phase occurrence provides new information about the sleep process that may have clinical relevance and complementary to standard clinical indices such as the CAP-rate.

Controllability limit of edge dynamics in complex networks.

Edge dynamics is relevant to various real-world systems with complex network topological features. An edge dynamical system is controllable if it can be driven from any initial state to any desired state in finite time with appropriate control inputs. Here a framework is proposed to study the impact of correlation between in- and out-degrees on controlling the edge dynamics in complex networks. We use the maximum matching and direct acquisition methods to determine the controllability limit, i.e., the limit of acceptable change of the edge controllability by adjusting the degree correlation only. Applying the framework to plenty complex networks, we find that the controllability limits are ubiquitous in model and real networks. Arbitrary edge controllability in between the limits can be achieved by properly adjusting the degree correlation. Moreover, a nonsmooth phenomenon occurs in the upper limits, and exponential and power-law scaling behaviors are widespread in the approach or separation speed between the upper and lower limits.

Collapsing Complexity: Quantifying Multiscale Properties of Reef Topography

AbstractSeafloor topography affects a wide range of physical and biological processes; therefore, collapsing the three‐dimensional structure of the bottom to roughness metrics is a common challenge in studies of marine systems. Here we assessed the properties captured by metrics previously proposed for the seafloor, as well as metrics developed to characterize other types of rough surfaces. We considered three classes of metrics: properties of the bottom elevation distribution (e.g., standard deviation), length scale ratios (e.g., rugosity), and metrics that describe how topography varies with spatial scale (e.g., Hölder exponents). The metrics were assessed using idealized topography and natural seafloor topography data from airborne lidar measurements of a coral reef. We illustrate that common roughness metrics (e.g., rugosity) can have the same value for topographies that are geometrically very different, limiting their utility. Application of the wavelet leaders technique to the reef data set demonstrates that the topography has a power law scaling behavior, but it is multifractal so a distribution of Hölder exponents is needed to describe its scaling behavior. Using principal component analysis, we identify three dominant modes of topographic variability, or ways metrics covary, among and within reef zones. Collectively, the results presented here show that coral reef topography is both multiscale and multifractal. While individual metrics that capture specific topography properties relevant to a given process may be suitable for some studies, many applications will require a set of metrics that includes statistics that capture how topography varies with spatial scale.

Scale-free amplitude modulation of low-frequency fluctuations in episodic migraine.

Arrhythmic fluctuations in neural activity occur at many levels of the nervous system. Such activity does not have a characteristic temporal periodicity but can exhibit statistical similarities, most commonly power-law scaling behavior, which is indicative of scale-free dynamics. The recurrence of scaling laws across many different systems and its manifestation in behavior has prompted a search for unifying principles in human brain function. With this in mind, a focused search for abnormities in scale-free dynamics is of considerable clinical relevance to migraine and other clinical pain disorders. Here, we examined the scale-free properties of the resting-state functional magnetic resonance imaging (fMRI) signal in the broadband frequency range known to be related to spontaneous neural activity (0.01-0.1 Hz). In a large cohort of episodic migraine patients (N = 40), we observed that the strength of long-range temporal correlations in the fMRI signal (captured by the scaling exponent α) was significantly higher in the sensorimotor network compared with healthy controls. Increases in the scaling exponent were positively correlated with fMRI signal variance and negatively correlated with the patient's self-reported headache intensity. These changes in the fMRI signal suggest that the temporal structure of amplitude fluctuations carries valuable information about the dynamic state of the underlying neuronal networks and ensuing sensory impairments in migraine. The demonstrated scaling laws pose a novel quantitative approach for examining clinically relevant interindividual variability in migraine and other pain disorders.