Fractal-like Behavior Research Articles

Fractal Resonance: Can Fractal Geometry Be Used to Optimize the Connectivity of Neurons to Artificial Implants?

In parallel to medical applications, exploring how neurons interact with the artificial interface of implants in the human body can be used to learn about their fundamental behavior. For both fundamental and applied research, it is important to determine the conditions that encourage neurons to maintain their natural behavior during these interactions. Whereas previous biocompatibility studies have focused on the material properties of the neuron-implant interface, here we discuss the concept of fractal resonance - the possibility that favorable connectivity properties might emerge by matching the fractal geometry of the implant surface to that of the neurons.To investigate fractal resonance, we first determine the degree to which neurons are fractal and the impact of this fractality on their functionality. By analyzing three-dimensional images of rat hippocampal neurons, we find that the way their dendrites fork and weave through space is important for generating their fractal-like behavior. By modeling variations in neuron connectivity along with the associated energetic and material costs, we highlight how the neurons' fractal dimension optimizes these constraints. To simulate neuron interactions with implant interfaces, we distort the neuron models away from their natural form by modifying the dendrites' fork and weaving patterns. We find that small deviations can induce large changes in fractal dimension, causing the balance between connectivity and cost to deteriorate rapidly. We propose that implant surfaces should be patterned to match the fractal dimension of the neurons, allowing them to maintain their natural functionality as they interact with the implant.

A Fractal and Comparative View of the Memory of Bitcoin and S&P 500 Returns

The majority of previous studies used autocorrelation-based methodologies to explore the dependency structure for Bitcoin, but this paper follows Benoit Mandelbrot in taking a fractal point of view. This perspective showed that Bitcoin and S&P 500 returns exhibit fractal-like behavior. Additional evidence suggested that the infinite variance hypothesis cannot be rejected for either asset supporting Mandelbrot’s (1963) early study on cotton price changes. This result held across non-overlapping subsamples. Following Mandelbrot (2008), Hurst exponents were estimated using rescaled/range analysis. The key findings are that (a) Bitcoin returns exhibit a higher level of persistence than S&P 500 returns across various subsamples, (b) the level of persistence in Bitcoin returns did not change over time, (c) the S&P 500 moved from efficiency in the first subsample to inefficiency in the ex-post June 17, 2018, period, (d) even if it was assumed that the variance of S&P 500 returns was finite, the kurtosis remained statistically undefined. The study concluded that the correlation-based methods used to explore the S&P 500 universe result in misleading answers.

From radial symmetry to fractal behavior of aggregation equilibria for repulsive–attractive potentials

For the interaction energy with repulsive–attractive potentials, we give generic conditions which guarantee the radial symmetry of the local minimizers in the infinite Wasserstein distance. As a consequence, we obtain the uniqueness of local minimizers in this topology for a class of interaction potentials. We introduce a novel notion of concavity of the interaction potential allowing us to show certain fractal-like behavior of the local minimizers. We provide a family of interaction potentials such that the support of the associated local minimizers has no isolated points and any superlevel set has no interior points.

New Framework to Study Electromagnetic Turbulence

Combining a current source involving vortical surface currents in the set of Maxwell’s equations offers a functional framework to address the complex phenomena of electromagnetic turbulence. The field structure equations exhibit fluid behavior with associated electromagnetic viscosity and reveal that the electromagnetic field, as a fluid, shows turbulent properties. This is an entirely new mechanism, investigated for the first time to the best of our knowledge. The fluidic–electromagnetic analogy implies that diffraction is the analog phenomenon of EM turbulence. The method clarifies the role of vortical surface currents in generating electromagnetic turbulence and classical fractal-like behavior in optical devices and suggests norms to design suitable plasmon circuity to control electromagnetic turbulence in stealth technology and propulsion machines.

Analysis of three-dimensional chromatin packing domains by chromatin scanning transmission electron microscopy (ChromSTEM)

Chromatin organization over multiple length scales plays a critical role in the regulation of transcription. Deciphering the interplay of these processes requires high-resolution, three-dimensional, quantitative imaging of chromatin structure in vitro. Herein, we introduce ChromSTEM, a method that utilizes high-angle annular dark-field imaging and tomography in scanning transmission electron microscopy combined with DNA-specific staining for electron microscopy. We utilized ChromSTEM for an in-depth quantification of 3D chromatin conformation with high spatial resolution and contrast, allowing for characterization of higher-order chromatin structure almost down to the level of the DNA base pair. Employing mass scaling analysis on ChromSTEM mass density tomograms, we observed that chromatin forms spatially well-defined higher-order domains, around 80 nm in radius. Within domains, chromatin exhibits a polymeric fractal-like behavior and a radially decreasing mass-density from the center to the periphery. Unlike other nanoimaging and analysis techniques, we demonstrate that our unique combination of this high-resolution imaging technique with polymer physics-based analysis enables us to (i) investigate the chromatin conformation within packing domains and (ii) quantify statistical descriptors of chromatin structure that are relevant to transcription. We observe that packing domains have heterogeneous morphological properties even within the same cell line, underlying the potential role of statistical chromatin packing in regulating gene expression within eukaryotic nuclei.

The full phase space dynamics of a magnetically levitated electromagnetic vibration harvester

We consider the motion of an electromagnetic vibrational energy harvester (EMVEH) as function of the initial position and velocity and show that this displays a classical chaotic dynamical behavior. The EMVEH considered consists of three coaxial cylindrical permanent magnets and two coaxial coils. The polarities of the three magnets are chosen in such a way that the central magnet floats, with its lateral motion being prevented by enclosion in a hollow plastic tube. The motion of the floating magnet, caused by e.g. environmental vibrations, induces a current in the coils allowing electrical energy to be harvested. We analyze the behavior of the system using a numerical model employing experimentally verified expressions of the force between the magnets and the damping force between the floating magnet and the coils. We map out the phase space of the motion of the system with and without gravity, and show that this displays a fractal-like behavior and that certain driving frequencies and initial conditions allow a large power to be harvested, and that more stable states than two exists. Finally, we show that at leasts fifth order polynomial approximation is necessary to approximate the magnet-magnet force and correctly predict the system behavior.

How neurons exploit fractal geometry to optimize their network connectivity

We investigate the degree to which neurons are fractal, the origin of this fractality, and its impact on functionality. By analyzing three-dimensional images of rat neurons, we show the way their dendrites fork and weave through space is unexpectedly important for generating fractal-like behavior well-described by an ‘effective’ fractal dimension D. This discovery motivated us to create distorted neuron models by modifying the dendritic patterns, so generating neurons across wide ranges of D extending beyond their natural values. By charting the D-dependent variations in inter-neuron connectivity along with the associated costs, we propose that their D values reflect a network cooperation that optimizes these constraints. We discuss the implications for healthy and pathological neurons, and for connecting neurons to medical implants. Our automated approach also facilitates insights relating form and function, applicable to individual neurons and their networks, providing a crucial tool for addressing massive data collection projects (e.g. connectomes).

Diving into a holographic superconductor

Charged black holes in anti-de Sitter space become unstable to forming charged scalar hair at low temperatures T < T_\text{c}T<Tc. This phenomenon is a holographic realization of superconductivity. We look inside the horizon of these holographic superconductors and find intricate dynamical behavior. The spacetime ends at a spacelike Kasner singularity, and there is no Cauchy horizon. Before reaching the singularity, there are several intermediate regimes which we study both analytically and numerically. These include strong Josephson oscillations in the condensate and possible `Kasner inversions’ in which after many e-folds of expansion, the Einstein-Rosen bridge contracts towards the singularity. Due to the Josephson oscillations, the number of Kasner inversions depends very sensitively on TT, and diverges at a discrete set of temperatures \{T_n\}{Tn} that accumulate at T_cTc. Near these T_nTn, the final Kasner exponent exhibits fractal-like behavior.

Analysis and FPGA of semi-fractal shapes based on complex Gaussian map

Abstract This paper studies the fractal-like behavior exhibited by the complex form of Gaussian chaotic map and the capability of digital architectures to mimic that behavior. Digitally realized chaotic attractors had many applications; hopefully, a digital realization of fractals may achieve the same eventually. The Gauss map is viewed concerning its bifurcation behavior, time waveform plots, Lyapunov exponent, and attractor performance through parameter variation. The Fractal-like entities emerging from the perceived complex map are examined versus different map coefficients for the highest chaotic periods to extract an interpretation for the fractal behavior. FPGA implementation of the fractal behavior is discussed viewing an optimized hardware architecture that eventually displays a fractal entity experimentally.

Gender comparison in consistency in the basketball free throw by an event-driven approach

This paper compares the free-throw performance of men and women in the National Collegiate Athletic Association (NCAA), taking into account differences in ball size, ball stiffness, and release height. Although some claim that men are more athletic, based on the assumption that athleticism correlates with physical traits, the average free-throw percentages in NCAA Division 1 basketball have been close for decades and across gender. The larger men’s basketball could create a disadvantage for men. On the other hand, the stiffer women’s basketball could create a disadvantage for women. In addition, women typically launch the ball from a lower height above the floor than men do. Therefore, the question of which gender is more consistent was unanswered. To answer this question, we turned to simulations of basketball trajectories. We patched together closed-form trajectory events that terminate at the front rim, back rim, backboard, and a fictitious plane below the ring. This produced what we call speed lines. Using the speed lines and NCAA average free-throw percentages, the consistencies of men- and women-free throws were determined. Examination of the speed lines reveals that they exhibit fractal-like behavior. We analyzed sensitivities with respect to ball size, ball bounce, and release height. We found that the most influential factor in determining free-throw success is the average release height of a gender. Under the stipulated assumptions, this paper found that women are about 3% more consistent than men are.

A Computational Approach to Studying Protein Folding Problems Considering the Crucial Role of the Intracellular Environment.

Intracellular protein folding (PF) is performed in a highly inhomogeneous, crowded, and correlated environment. Due to this inherent complexity, the study and understanding of PF phenomena is a fundamental issue in the field of computational systems biology. In particular, it is important to use a modeled medium that accurately reflects PF in natural systems. In the current study, we present a simulation wherein PF is carried out within an inhomogeneous modeled medium. Simulation resources included a two-dimensional hydrophobic-polar (HP) model, evolutionary algorithms, and the dual site-bond model. The dual site-bond model was used to develop an environment where HP beads could be folded. Our modeled medium included correlation lengths and fractal-like behavior, which were selected according to HP sequence lengths to induce folding in a crowded environment. Analysis of three benchmark HP sequences showed that the modeled inhomogeneous space played an important role in deeper energy folding and obtained better performance and convergence compared with homogeneous environments. Our computational approach also demonstrated that our correlated network provided a better space for PF. Thus, our approach represents a major advancement in PF simulations, not only for folding but also for understanding functional chemical structure and physicochemical properties of proteins in crowded molecular systems, which normally occur in nature.

High Activity Oxygen Evolution Reaction Catalysts from Additive-Controlled Electrodeposited Ni and Nife Films

We use electrodeposition for energy and microelectronics purposes. Electrodeposition of Ni or NiFe films exhibiting fractal-like behavior from plating baths containing an inhibitor, such as 3,5-Diamino-1,2,4-triazole (DAT), is found to yield oxygen evolution reaction (OER) catalysts for alkaline solutions exhibiting high current densities (100 mA/cm2), high mass activity (~ 1200 A/g of catalyst), high stability (> 72 hours) and low overpotentials (~ 300 mV). By changing electrodeposition time, the activity of the catalyst can be tuned, with longer times yielding higher activities. The electrodeposition method works with any conductive substrate yielding unprecedented performance and providing an easy route to high activity catalysts. We also investigate the mechanism of 3-mercapto-1-propanesulfonate (MPS) acceleration of Cu electrodeposition in the presence of Cl- through comparison with the inactive but related model compounds 1-butanesulfonate and 1-butanethiol (BuSH). In situ Shell-Isolated Nanoparticle-Enhanced Raman Spectroscopy (SHINERS) was used to evaluate these additives as a function of potential. MPS adsorbs to the Cu electrode in the presence of Cl- through the thiol functional group. DFT calculations reveal that the adsorption of thiol has a depolarizing effect on the strength of the interaction between adsorbed Cl- and the electrode surface, the consequences of which are observed spectroscopically. The sulfonate moiety does not interact directly with the electrode. However, cyclic voltammetry used to assess the permeability of adsorbed MPS and BuSH adlayers indicates that the sulfonate group has a more direct role in Cu deposition beyond simple disruption of the surface layer.

High Activity Oxygen Evolution Reaction Catalysts from Additive-Controlled Electrodeposited Ni and NiFe Films

Electrodeposition of Ni or NiFe films exhibiting fractal-like behavior from plating baths containing an inhibitor, such as 3,5-diamino-1,2,4-triazole (DAT), is found to yield oxygen evolution reaction (OER) catalysts for alkaline solutions exhibiting high current densities (100 mA/cm2), high mass activity (∼1200 A/g of catalyst), high stability (>72 h), and low overpotentials (∼300 mV). By changing electrodeposition time, the activity of the catalyst can be tuned, with longer times yielding higher activities. The electrodeposition method works with any conductive substrate yielding unprecedented performance and providing an easy route to high activity catalysts.

Symmetries of the Feinberg–Zee random hopping matrix

We study the symmetries of the spectrum of the Feinberg–Zee Random Hopping Matrix introduced in [J. Feinberg and A. Zee, Spectral curves of non-Hermitian Hamiltonians, Nucl. Phys. B 552 (1999) 599–623] and studied in various papers thereafter (e.g. [S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, eigenvalue Problem meets Sierpinski triangle: Computing the spectrum of a non-self-adjoint random operator, Oper. Matrices 5 (2011) 633–648; S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators, Oper. Matrices 7 (2013) 739–775; S. N. Chandler-Wilde and E. B. Davies, Spectrum of a Feinberg–Zee sandom hopping matrix, J. Spectral Theory 2 (2012) 147–179; R. Hagger, On the spectrum and numerical range of tridiagonal random operators, preprint (2014), arXiv: 1407.5486; D. E. Holz, H. Orland and A. Zee, On the remarkable spectrum of a non-Hermitian random matrix model, J. Phys. A: Math. Gen. 36 (2003) 3385–3400]). In [J. Spectral Theory 2 (2012) 147–179], Chandler-Wilde and Davies proved that the spectrum of the Feinberg–Zee Random Hopping Matrix is invariant under taking square roots, which implied that the unit disk is contained in the spectrum (a result already obtained slightly earlier in [Oper. Matrices 5 (2011) 633–648]. In a similar approach we show that there is an infinite sequence of symmetries at least in the periodic part of the spectrum (which is conjectured to be dense). Using these symmetries and the result of [J. Spectral Theory 2 (2012) 147–179], we can exploit a considerably larger part of the spectrum than the unit disk. As a further consequence we find an infinite sequence of Julia sets contained in the spectrum. These facts may serve as a part of an explanation of the seemingly fractal-like behavior of the boundary.

Combined heart rate variability and dynamic measures for quantitatively characterizing the cardiac stress status during cycling exercise

In this study, we aimed to seek for different ways of measuring cardiac stress in terms of heart rate variability (HRV) and heart rate (HR) dynamics, and to develop a novel index that can effectively summarize the information reflected by these measures to continuously and quantitatively characterize the cardiac stress status during physical exercise. Standard deviation, spectral measure of HRV as well as a nonlinear detrended fluctuation analysis (DFA) based fractal-like behavior measure of HR dynamics were all evaluated on the RR time series derived from windowed electrocardiogram (ECG) data for the subjects undergoing cycling exercise. We recruited eleven young healthy subjects in our tests. Each subject was asked to maintain a fixed speed under a constant load during the pedaling test. We obtained the running estimates of the standard deviation of the normal-to-normal interval (SDNN), the high-fidelity power spectral density (PSD) of HRV, and the DFA scaling exponent α, respectively. A trend analysis and a multivariate linear regression analysis of these measures were then performed. Numerical experimental results produced by our analyses showed that a decrease in both SDNN and α was seen during the cycling exercise, while there was no significant correlation between the standard lower frequency to higher frequency (LF-to-HF) spectral power ratio of HRV and the exercise intensity. In addition, while the SDNN and α were both negatively correlated with the Borg rating of perceived exertion (RPE) scale value, it seemed that the LF-to-HF power ratio might not have substantial impact on the Borg value, suggesting that the SDNN and α may be further used as features to detect the cardiac stress status during the physical exercise. We further approached this detection problem by applying a linear discriminant analysis (LDA) to both feature candidates for the task of cardiac stress stratification. As a result, a time-varying parameter, referred to as the cardiac stress measure (CSM), is developed for quantitatively on-line measuring and stratifying cardiac stress status.

Overexpression of Adenylyl Cyclase 8 (AC8) in Mice Increases Intrinsic Heart Rate (IHR) and Reduces Heart Rate Variability (HRV), and Detaches HR and HRV from Autonomic Modulation

While changes in HRV have been ascribed to changes in neural autonomic input to the heart, recent studies have shown that beating rate variability also exists in isolated sinoatrial (SA) node tissue and SA node cells (SANC). Adenylate cyclases are constitutively expressed in SANC, where they ensure a high basal level of phosphorylation of pacemaking proteins even in the absence of autonomic receptor stimulation. AC8 is a Ca+2/calmodulin-activated AC which, when overexpressed in mice, leads to an increased rate and amount of Ca+2 cycling from the sarcoplasmic reticulum. We hypothesized that AC8 overexpression would affect IHR and HRV in vivo.ECGs were recorded from wild type (WT) C57 and transgenic AC8 mice under light anesthesia before and after dual autonomic blockade with atropine and propranolol. Time- and frequency-domain parameters of HRV were measured. AC8 mice displayed significantly higher basal and intrinsic HRs, and significantly lower basal and intrinsic measures of HRV (total power 13% of WT basal; 21% of WT intrinsic). Dual autonomic blockade caused reductions in most HRV parameters in WT mice, but had no effect in AC8 mice. A shift in β coefficients from log-log plots of FFT-derived power spectra (basal: −1.82 in WT to −1.17 in AC8; intrinsic: −1.62 to −1.10) indicated more fractal-like behavior in the HR of AC8 mice.In summary, genetic manipulation of a cAMP-generating mechanism intrinsic to SANC alters not only basal and intrinsic HR and HRV, but also dissociates HR and HRV from autonomic control. Thus mechanisms of intrinsic SANC automaticity, in isolation of and in concert with autonomic neural impulses, can regulate HR and HRV in vivo.

The fractal nature of blood glucose fluctuations

AimsFluctuations of blood glucose are generated by multiple external and internal factors continuously modifying glucose concentrations through complex feedback loops. This equilibrium may be perturbed during physiological or pathological conditions. The traditional theory suggests that physiological systems achieve homeostasis when disturbed and restore equilibrium through linear feedback loops. Complex systems on the other hand, may function nonlinearly with feedback loops that operate at different time scales, exhibiting chaotic or fractal behavior. We hypothesized that blood glucose fluctuations recorded for prolonged time periods show chaotic, fractal-like behavior that may be altered in diabetes. MethodsWe applied nonlinear analytical methods such as detrended fluctuation analysis to glucose data derived from continuous glucose monitoring devices for prolonged time periods in healthy volunteers, diabetes type 1 and pregnant diabetes type 1 patients. ResultsGlucose fluctuations extracted for prolonged time periods show fractal-like behavior and power law behavior of the system. ConclusionsHidden features underlying glucose fluctuations in health and in disease were revealed by using dynamic nonlinear analyses methods to discrete glucose readings extracted from continuous glucose monitoring devices. By using such methods we can enhance our understanding of the dynamics of blood glucose fluctuations in health and disease.

Synchronization of sinoatrial node pacemaker cell clocks and its autonomic modulation impart complexity to heart beating intervals

A reduction of complexity of heart beating interval variability that is associated with an increased morbidity and mortality in cardiovascular disease states is thought to derive from the balance of sympathetic and parasympathetic neural impulses to the heart. However, rhythmic clocklike behavior intrinsic to pacemaker cells in the sinoatrial node (SAN) drives their beating, even in the absence of autonomic neural input. To test how this rhythmic clocklike behavior intrinsic to pacemaker cells interacts with autonomic impulses to the heart beating interval variability in vivo. We analyzed beating interval variability in time and frequency domains and by fractal and entropy analyses: (1) in vivo, when the brain input to the SAN is intact; (2) during autonomic denervation in vivo; (3) in isolated SAN tissue (ie, in which the autonomic neural input is completely absent); (4) in single pacemaker cells isolated from the SAN; and (5) after autonomic receptor stimulation of these cells. Spontaneous beating intervals of pacemaker cells residing in the isolated SAN tissue exhibit fractal-like behavior and have lower approximate entropy compared with those in the intact heart. Isolation of pacemaker cells from SAN tissue, however, leads to a loss in the beating interval order and fractal-like behavior. β-Adrenergic receptor stimulation of isolated pacemaker cells increases intrinsic clock synchronization, decreases their action potential period, and increases system complexity. Both the average beating interval in vivo and beating interval complexity are conferred by the combined effects of clock periodicity intrinsic to pacemaker cells and their response to autonomic neural input.

Biorelevant dissolution of poorly soluble weak acids studied by UV imaging reveals ranges of fractal-like kinetics

Much pharmaceutical research has been invested into drug dissolution testing and its mathematical modeling. Even today, there is no complete understanding of the dissolution process but novel imaging tools have been introduced into pharmaceutics that may spur further scientific advancement. We used UV imaging to study the intrinsic dissolution of various poorly soluble acidic model drugs to understand the effects of heterogeneity on early intrinsic drug dissolution using a biorelevant medium: celecoxib, ketoprofen, naproxen, and sulfathiazole. All compounds were characterized using X-ray powder diffraction and thermal analysis. Raman spectroscopy and scanning electron microscopy were employed before and after the initial dissolution phase. As a result, ranges of fractal-like dissolution behavior were found with all model compounds. Intrinsic dissolution rate exhibited a power law mainly at early time points. Subsequently, after several minutes, pseudo-equilibrium was reached with a nearly constant dissolution rate. Further research should investigate whether compounds other than acids demonstrate similar early dissolution kinetics in biorelevant media. The observed fractal-like intrinsic dissolution behavior has several pharmaceutical implications. This study primarily helps us to better understand in vitro dissolution testing, particularly on a miniaturized scale. This improved understanding of early dissolution events may advance future correlations with in vivo data. Therefore, fractal-like dissolution should be considered during biopharmaceutical modeling.

Fractal-Like Behavior of the Heart-Beat Intervals is Encoded within Intrinsic Complexity of Pacemaker Cells Residing in the Sinoatrial Node and Modulated by Autonomic Input to the Heart

The heart rate and rhythm are controlled by complex chaotic neural, chemical and hormonal networks which are not strictly regular, but exhibit fluctuations across multiple-time scales. Therefore, it is not surprising that decoding of the ECG in mammals, even under resting conditions, reveals scale-invariant dynamics and beat-interval variability (BIV). Moreover, fractal-like behavior of heart beat intervals, which reveals itself in a power-law dependence of the frequency distribution of its intrinsic regimes, contributes to the complexity of the mammalian heart's rhythm. The traditional explanation for BIV and fractal-like behavior of heart beat intervals is that this result from the balance of sympathetic and parasympathetic autonomic input to the heart. But whether the sinoatrial node (SAN), the heart's primary pacemaker, or pacemaker cells isolated from the SAN exhibit fractal-like behavior of beating intervals in the absence of autonomic input is unknown.We analyzed beating rhythms: (i) in vivo, when the brain input to the SAN is intact; ii) during autonomic denervation in vivo; iii) in intact isolated (denervated) SAN; and iv) in single pacemaker cells isolated from the SAN. By segregating each component of the brain-pacemaker cascade, we discovered that fractal-like beating interval exhibit in SAN tissue and although the beat interval of single pacemaker cells isolated from the SAN is rhythmic it does not exhibit fractal-like behavior. Therefore, cell-to-cell communication among pacemaker cells within the SAN tissue is required to impart their fractal-like beating interval behavior within the SAN. Autonomic input from the brain to the heart in vivo modulates both the rate and rhythm at which pacemaker cells beat and its fractal-like behavior, but it is not required for isolated SAN fractal-like complexity.