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Fractal patterns are seemingly everywhere. They can be analyzed through Fourier and power analyses, and other methods. Cutting, DeLong, and Nothelfer (2010) analyzed as time-series data the fluctuations of shot durations in 150 popular movies released over 70 years. They found that these patterns had become increasingly fractal-like and concluded that they might be linked to those found in the results of psychological tasks involving attention. To explore this possibility further, we began by analyzing the shot patterns of almost twice as many movies released over a century. The increasing fractal-like nature of shot patterns is affirmed, as determined by both a slope measure and a long-range dependence measure, neither of which is sensitive to the vector lengths of their inputs within the ranges explored here. But the main reason for increased long-range dependence is related to, but not caused by, the increasing vector length of the shot-series samples. It appears that, in generating increasingly fractal-like patterns, filmmakers have systematically explored dimensions that are important for holding our attention-shot durations, scene durations, motion, and sound amplitude-and have crafted fluctuations in them like those of our endogenous attention patterns. Other dimensions-luminance, clutter, and shot scale-are important to film style but their variations seem not to be important to holding viewers' moment-to-moment attention and have not changed in their fractional dimension over time.
Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic response to the complexity of exact fractals by manipulating fractal dimensionality, symmetry, recursion, and the number of segments in the generator. Across two studies, a variety of fractal patterns were visually presented to human participants to determine the typical response to exact fractals. In the first study, we found that preference ratings for exact midpoint displacement fractals can be described by a linear trend with preference increasing as fractal dimension increases. For the majority of individuals, preference increased with dimension. We replicated these results for other exact fractal patterns in a second study. In the second study, we also tested the effects of symmetry and recursion by presenting asymmetric dragon fractals, symmetric dragon fractals, and Sierpinski carpets and Koch snowflakes, which have radial and mirror symmetry. We found a strong interaction among recursion, symmetry and fractal dimension. Specifically, at low levels of recursion, the presence of symmetry was enough to drive high preference ratings for patterns with moderate to high levels of fractal dimension. Most individuals required a much higher level of recursion to recover this level of preference in a pattern that lacked mirror or radial symmetry, while others were less discriminating. This suggests that exact fractals are processed differently than their statistical counterparts. We propose a set of four factors that influence complexity and preference judgments in fractals that may extend to other patterns: fractal dimension, recursion, symmetry and the number of segments in a pattern. Conceptualizations such as Berlyne's and Redies' theories of aesthetics also provide a suitable framework for interpretation of our data with respect to the individual differences that we detect. Future studies that incorporate physiological methods to measure the human aesthetic response to exact fractal patterns would further elucidate our responses to such timeless patterns.
The ability to understand and generate hierarchical structures is a crucial component of human cognition, available in language, music, mathematics and problem solving. Recursion is a particularly useful mechanism for generating complex hierarchies by means of self-embedding rules. In the visual domain, fractals are recursive structures in which simple transformation rules generate hierarchies of infinite depth. Research on how children acquire these rules can provide valuable insight into the cognitive requirements and learning constraints of recursion. Here, we used fractals to investigate the acquisition of recursion in the visual domain, and probed for correlations with grammar comprehension and general intelligence. We compared second (n=26) and fourth graders (n=26) in their ability to represent two types of rules for generating hierarchical structures: Recursive rules, on the one hand, which generate new hierarchical levels; and iterative rules, on the other hand, which merely insert items within hierarchies without generating new levels. We found that the majority of fourth graders, but not second graders, were able to represent both recursive and iterative rules. This difference was partially accounted by second graders' impairment in detecting hierarchical mistakes, and correlated with between-grade differences in grammar comprehension tasks. Empirically, recursion and iteration also differed in at least one crucial aspect: While the ability to learn recursive rules seemed to depend on the previous acquisition of simple iterative representations, the opposite was not true, i.e., children were able to acquire iterative rules before they acquired recursive representations. These results suggest that the acquisition of recursion in vision follows learning constraints similar to the acquisition of recursion in language, and that both domains share cognitive resources involved in hierarchical processing.
Animal behaviour exhibits fractal structure in space and time. Fractal properties in animal space-use have been explored extensively under the Lévy flight foraging hypothesis, but studies of behaviour change itself through time are rarer, have typically used shorter sequences generated in the laboratory, and generally lack critical assessment of their results. We thus performed an in-depth analysis of fractal time in binary dive sequences collected via bio-logging from free-ranging little penguins (Eudyptula minor) across full-day foraging trips (2(16) data points; 4 orders of temporal magnitude). Results from 4 fractal methods show that dive sequences are long-range dependent and persistent across ca. 2 orders of magnitude. This fractal structure correlated with trip length and time spent underwater, but individual traits had little effect. Fractal time is a fundamental characteristic of penguin foraging behaviour, and its investigation is thus a promising avenue for research on interactions between animals and their environments.
In the past 15 years it has become possible to determine the fractal dimension (Df) of complex objects, including neurons, by automated image analysis methods. However, there are many unresolved issues that need to be addressed. In this paper we discuss how the Df calculated by different methods may vary and how fractal analysis may be of use for retinal ganglion cell characterization. The goal of this work was to acknowledge inherent sources of variation during measurement and evaluate current fractal analysis methods for describing structure. Our results show that different algorithms and even the same algorithm performed by different computer programs and/or experimenters may give different but consistent numerical values. All described methods demonstrated their suitability for classifying cat retinal ganglion cells into distinct groups. Our results reinforce the idea that comparison of measurements of different profiles using the same measurement method may be useful and valid even if an exact numeric value of the dimension is not realised in practice.
One of the main limitations of the technique surface-enhanced Raman scattering (SERS) for chemical detection relies on the homogeneity, reproducibility and reusability of the substrates. In this work, SERS active platforms based on 3D-fractal microstructures is developed by combining corner lithography and anisotropic wet etching of silicon, to extend the SERS-active area into 3D, with electrostatically driven Au@citrate nanoparticles (NPs) assembly, to ensure homogeneous coating of SERS active NPs over the entire microstructured platforms. Strong SERS intensities are achieved using 3D-fractal structures compared to 2D-planar structures; leading to SERS enhancement factors for R6G superior than those merely predicted by the enlarged area effect. The SERS performance of Au monolayer-over-mirror configuration is demonstrated for the label-free real-time gas phase detection of 1.2 ppmV of dimethyl methylphosphonate (DMMP), a common surrogate of G-nerve agents. Thanks to the hot spot accumulation on the corners and tips of the 3D-fractal microstructures, the main vibrational modes of DMMP are clearly identified underlying the spectral selectivity of the SERS technique. The Raman acquisition conditions for SERS detection in gas phase have to be carefully chosen to avoid photo-thermal effects on the irradiated area.
Fractals are self-similar patterns that repeat at different scales, the complexity of which are expressed as a fractional Euclidean dimension D between 0 (a point) and 2 (a filled plane). The drip paintings of American painter Jackson Pollock (JP) are fractal in nature, and Pollock's most illustrious works are of the high-D (~1.7) category. This would imply that people prefer more complex fractal patterns, but some research has instead suggested people prefer lower-D fractals. Furthermore, research has suggested that parietal and frontal brain activity tracks the complexity of fractal patterns, but previous research has artificially binned fractals depending on fractal dimension, rather than treating fractal dimension as a parametrically varying value. We used white layers extracted from JP artwork as stimuli, and constructed statistically matched 2-dimensional random Cantor sets as control stimuli. We recorded the electroencephalogram (EEG) while participants viewed the JP and matched random Cantor fractal patterns. Participants then rated their subjective preference for each pattern. We used a single-trial analysis to construct within-subject models relating subjective preference to fractal dimension D, as well as relating D and subjective preference to single-trial EEG power spectra. Results indicated that participants preferred higher-D images for both JP and Cantor stimuli. Power spectral analysis showed that, for artistic fractal images, parietal alpha and beta power parametrically tracked complexity of fractal patterns, while for matched mathematical fractals, parietal power tracked complexity of patterns over a range of frequencies, but most prominently in alpha band. Furthermore, parietal alpha power parametrically tracked aesthetic preference for both artistic and matched Cantor patterns. Overall, our results suggest that perception of complexity for artistic and computer-generated fractal images is reflected in parietal-occipital alpha and beta activity, and neural substrates of preference for complex stimuli are reflected in parietal alpha band activity.
We present a new automated system for the detection of brain magnetic resonance images (MRI) affected by Alzheimer's disease (AD). The MRI is analyzed by means of multiscale analysis (MSA) to obtain its fractals at six different scales. The extracted fractals are used as features to differentiate healthy brain MRI from those of AD by a support vector machine (SVM) classifier. The result of classifying 93 brain MRIs consisting of 51 images of healthy brains and 42 of brains affected by AD, using leave-one-out cross-validation method, yielded 99.18% ± 0.01 classification accuracy, 100% sensitivity, and 98.20% ± 0.02 specificity. These results and a processing time of 5.64 seconds indicate that the proposed approach may be an efficient diagnostic aid for radiologists in the screening for AD.
When looking around a visual scene, humans make saccadic eye movements to fixate objects of interest. While the extraocular muscles can execute saccades in any direction, not all saccade directions are equally likely: saccades in horizontal and vertical directions are most prevalent. Here, we asked whether head orientation plays a role in determining saccade direction biases. Study participants (n = 14) viewed natural scenes and abstract fractals (radially symmetric patterns) through a virtual reality headset equipped with eye tracking. Participants' heads were stabilized and tilted at -30°, 0°, or 30° while viewing the images, which could also be tilted by -30°, 0°, and 30° relative to the head. To determine whether the biases in saccade direction changed with head tilt, we calculated polar histograms of saccade directions and cross-correlated pairs of histograms to find the angular displacement resulting in the maximum correlation. During free viewing of fractals, saccade biases largely followed the orientation of the head with an average displacement value of 24° when comparing head upright to head tilt in world-referenced coordinates (t (13) = 17.63, p < 0.001). There was a systematic offset of 2.6° in saccade directions, likely reflecting ocular counter roll (OCR; t (13) = 3.13, p = 0.008). When participants viewed an Earth upright natural scene during head tilt, we found that the orientation of the head still influenced saccade directions (t (13) = 3.7, p = 0.001). These results suggest that nonvisual information about head orientation, such as that acquired by vestibular sensors, likely plays a role in saccade generation.
Breast cancer diagnosis is still done by observation of biopsies under the microscope. The development of automated methods for breast TMA classification would reduce diagnostic time. This paper is a step towards the solution for this problem and shows a complete study of breast TMA classification based on colour models and texture descriptors. The TMA images were divided into four classes: i) benign stromal tissue with cellularity, ii) adipose tissue, iii) benign and benign anomalous structures, and iv) ductal and lobular carcinomas. A relevant set of features was obtained on eight different colour models from first and second order Haralick statistical descriptors obtained from the intensity image, Fourier, Wavelets, Multiresolution Gabor, M-LBP and textons descriptors. Furthermore, four types of classification experiments were performed using six different classifiers: (1) classification per colour model individually, (2) classification by combination of colour models, (3) classification by combination of colour models and descriptors, and (4) classification by combination of colour models and descriptors with a previous feature set reduction. The best result shows an average of 99.05% accuracy and 98.34% positive predictive value. These results have been obtained by means of a bagging tree classifier with combination of six colour models and the use of 1719 non-correlated (correlation threshold of 97%) textural features based on Statistical, M-LBP, Gabor and Spatial textons descriptors.
Hierarchical structures play a central role in many aspects of human cognition, prominently including both language and music. In this study we addressed hierarchy in the visual domain, using a novel paradigm based on fractal images. Fractals are self-similar patterns generated by repeating the same simple rule at multiple hierarchical levels. Our hypothesis was that the brain uses different resources for processing hierarchies depending on whether it applies a "fractal" or a "non-fractal" cognitive strategy. We analyzed the neural circuits activated by these complex hierarchical patterns in an event-related fMRI study of 40 healthy subjects. Brain activation was compared across three different tasks: a similarity task, and two hierarchical tasks in which subjects were asked to recognize the repetition of a rule operating transformations either within an existing hierarchical level, or generating new hierarchical levels. Similar hierarchical images were generated by both rules and target images were identical. We found that when processing visual hierarchies, engagement in both hierarchical tasks activated the visual dorsal stream (occipito-parietal cortex, intraparietal sulcus and dorsolateral prefrontal cortex). In addition, the level-generating task specifically activated circuits related to the integration of spatial and categorical information, and with the integration of items in contexts (posterior cingulate cortex, retrosplenial cortex, and medial, ventral and anterior regions of temporal cortex). These findings provide interesting new clues about the cognitive mechanisms involved in the generation of new hierarchical levels as required for fractals.
Influential views of systems memory consolidation posit that the hippocampus rapidly forms representations of specific events, while neocortical networks extract regularities across events, forming the basis of schemas and semantic knowledge. Neocortical extraction of schematic memory representations is thought to occur on a protracted timescale of months, especially for information that is unrelated to prior knowledge. However, this theorized evolution of memory representations across extended timescales, and differences in the temporal dynamics of consolidation across brain regions, lack reliable empirical support. To examine the temporal dynamics of memory representations, we repeatedly exposed human participants to structured information via sequences of fractals, while undergoing longitudinal fMRI for three months. Sequence-specific activation patterns emerged in the hippocampus during the first 1-2 weeks of learning, followed one week later by high-level visual cortex, and subsequently the medial prefrontal and parietal cortices. Schematic, sequence-general representations emerged in the prefrontal cortex after 3 weeks of learning, followed by the medial temporal lobe and anterior temporal cortex. Moreover, hippocampal and most neocortical representations showed sustained rather than time-limited dynamics, suggesting that representations tend to persist across learning. These results show that specific hippocampal representations emerge early, followed by both specific and schematic representations at a gradient of timescales across hippocampal-cortical networks as learning unfolds. Thus, memory representations do not exist only in specific brain regions at a given point in time, but are simultaneously present at multiple levels of abstraction across hippocampal-cortical networks.
While passive social information (e.g. pictures of people) routinely draws one's eyes, our willingness to look at live others is more nuanced. People tend not to stare at strangers and will modify their gaze behaviour to avoid sending undesirable social signals; yet they often continue to monitor others covertly "out of the corner of their eyes." What this means for looks that are being made near to live others is unknown. Will the eyes be drawn towards the other person, or pushed away? We evaluate changes in two elements of gaze control: image-independent principles guiding how people look (e.g. biases to make eye movements along the cardinal directions) and image-dependent principles guiding what people look at (e.g. a preference for meaningful content within a scene). Participants were asked to freely view semantically unstructured (fractals) and semantically structured (rotated landscape) images, half of which were located in the space near to a live other. We found that eye movements were horizontally displaced away from a visible other starting at 1032 ms after stimulus onset when fractals but not landscapes were viewed. We suggest that the avoidance of looking towards live others extends to the near space around them, at least in the absence of semantically meaningful gaze targets.
Amorphous poly(p-vinyl phenol) (PVPh) was added into semicrystalline poly(p-dioxanone) (PPDO) to induce a uniquely novel dendritic/ringed morphology. Polarized-light optical, atomic-force and scanning electron microscopy (POM, AFM, and SEM) techniques were used to observe the crystal arrangement of a uniquely peculiar cactus-like dendritic PPDO spherulite, with periodic ring bands not continuingly circular such as those conventional types reported in the literature, but discrete and detached to self-assemble on each of the branches of the lobs. Correlations and responsible mechanisms for the formation of this peculiar banded-dendritic structure were analyzed. The periodic bands on the top surface and interior of each of the cactus-like lobs were discussed. The banded pattern was composed of feather-like lamellae in random fractals alternately varying their orientations from the radial direction to the tangential one. The tail ends of lamellae at the growth front spawned nucleation cites for new branches; in cycles, the feather-like lamellae self-divided into multiple branches following the Fibonacci sequence to fill the ever-expanding space with the increase of the radius. The branching fractals in the sequence and the periodic ring-banded assembly on each of the segregated lobs of cactus-like dendrites were the key characteristics leading to the formation of this unique dendritic/ringed PPDO spherulite.
Brain connectivity in autism spectrum disorders (ASD) has proven difficult to characterize due to the heterogeneous nature of the spectrum. Connectivity in the brain occurs in a complex, multilevel and multi-temporal manner, driving the fluctuations observed in local oxygen demand. These fluctuations can be characterized as fractals, as they auto-correlate at different time scales. In this study, we propose a model-free complexity analysis based on the fractal dimension of the rs-BOLD signal, acquired with magnetic resonance imaging. The fractal dimension can be interpreted as measure of signal complexity and connectivity. Previous studies have suggested that reduction in signal complexity can be associated with disease. Therefore, we hypothesized that a detectable difference in rs-BOLD signal complexity could be observed between ASD patients and Controls.
The sensitivity of medial superior olive (MSO) neurons to tens of microsecond differences in interaural temporal delay (ITD) derives in part from their membrane electrical characteristics, kinetics and timing of excitatory and inhibitory inputs, and dendrite structure. However, maturation of these physiological and structural characteristics are little studied, especially in relationship to the onset of auditory experience. We showed, using brain slices at physiological temperature, that MSO neurons exhibited sensitivity to simulated temporally delayed (TD) EPSCs (simEPSC), injected through the recording electrode, by the initial phase of hearing onset at P10, and TD sensitivity was reduced by block of low threshold potassium channels. The spike generation mechanism matured between P10 and P16 to support TD sensitivity to adult-like excitatory stimuli (1-4 ms duration) by P14. IPSP duration was shorter at physiological temperature than reported for lower temperatures, was longer than EPSP duration at young ages, but approached the duration of EPSPs by P16, when hearing thresholds neared maturity. Dendrite branching became less complex over a more restricted time frame between P10 and P12. Because many physiological and structural properties approximated mature values between P14 and P16, we studied temporal integration of simEPSCs and IPSPs at P15. Only a narrow range of relative onset times (< 1 ms) yielded responses showing sensitivity to TD. We propose that shaping of excitatory circuitry to mediate TD sensitivity can begin before airborne sound is detectable, and that inhibitory inputs having suboptimal neural delays may then be pruned by cellular mechanisms activated by sensitivity to ITD.
Introduction: Fetal heart rate variability (FHRV) evaluates the fetal neurological state, which is poorly assessed by conventional prenatal surveillance including cardiotocography (CTG). Accurate FHRV on a beat-to-beat basis, assessed by time domain and spectral domain analyses, has shown promising results in the scope of fetal surveillance. However, accepted standards for these techniques are lacking, and the influence of fetal breathing movements and gross movements may be especially challenging. Thus, current standards for equivalent assessments in adults prescribe rest and controlled respiration. The aim of this review is to clarify the importance of fetal movements on FHRV. Methods: A systematic review in accordance with the PRISMA guidelines based on publications in the EMBASE, the MEDLINE, and the Cochrane Library databases was performed. Studies describing the impact of fetal movements on time domain, spectral domain and entropy analyses in healthy human fetuses were reviewed. Only studies based on fetal electrocardiography or fetal magnetocardiography were included. PROSPERO registration number: CRD42018068806. Results: In total, 14 observational studies were included. Fetal movement detection, signal processing, length, and selection of appropriate time series varied across studies. Despite these divergences, all studies showed an increase in overall FHRV in the moving fetus compared to the resting fetus. Especially short-term, vagal mediated indexes showed an increase during fetal breathing movements including an increase in Root Mean Square of the Successive Differences (RMSSD) and High Frequency power (HF) and a decrease in Low Frequency power/High Frequency power (LF/HF). These findings were present even in analyses restricted to one specific fetal behavioral state defined by Nijhuis. On the other hand, fetal body movements seemed to increase parameters supposed to represent the sympathetic response [LF and Standard Deviation of RR-intervals from normal sinus beats (SDNN)] proportionally more than parameters representing the parasympathetic response (RMSSD, HF). Results regarding entropy analyses were inconclusive. Conclusion: Time domain analyses as well as spectral domain analyses are affected by fetal movements. Fetal movements and especially breathing movements should be considered in these analyses of FHRV.
The topological structure of the macromolecules of lignins isolated from oat husk and fir wood was studied by means of macromolecular hydrodynamic methods. The macromolecular properties were analyzed by evaluating the intrinsic viscosity and coefficients of the translational diffusion and the sedimentation velocity of the lignins in dilute dimethylformamide solutions. The average molecular weights (MDη) and polydispersity parameters were calculated based on the results of the fractionation, as follows: Mw = 14.6 × 103, Mn = 9.0, and Mw/Mn = 1.62 for lignins from fir wood and Mw = 14.9 Mn = 13.5 and Mw/Mn = 1.1 for lignins from oat husks. The fractal analysis of the lignin macromolecules allowed us to identify the distinctive characteristics of the fractal and topological structures of these lignins. The measurements indicated that the fractal dimension (df) values of the guaiacyl-syringyl lignins from oat husks were between 1.71 and 1.85, while the df of a typical guaiacyl lignin from fir wood was ~2.3. Thus, we determined that the lignin macromolecules of oat husks belong to the diffusion-limited aggregation-type cluster-cluster class of fractals of the Meakin-Kolb type, with a predominance of characteristics common to a linear configuration. The lignins of softwood fir trees exhibited a branched topological structure, and they belong to the diffusion-limited aggregation-type particle-cluster class of fractals of the Witten-Sander type. Lignins from oat husks have the linear topology of macromolecules while the macromolecules of the lignins from fir wood can be characterized as highly branched polymers.
Much is known regarding the structure and logic of genetic regulatory networks. Less understood is the contextual organization of promoter signals used during transcription initiation, the most pivotal stage during gene expression. Here we show that promoter networks organize spontaneously at a dimension between the 1-dimension of the DNA and 3-dimension of the cell. Network methods were used to visualize the global structure of E. coli sigma (sigma) recognition footprints using published promoter sequences (RegulonDB). Footprints were rendered as networks with weighted edges representing bp-sharing between promoters (nodes). Serial thresholding revealed phase transitions at positions predicted by percolation theory, and nuclei denoting short steps through promoter space with geometrically constrained linkages. The network nuclei are fractals, a power-law organization not yet described for promoters. Genome-wide promoter abundance also scaled as a power-law. We propose a general model for the development of a fractal nucleus in a transcriptional grammar.
Blender interstitial volume is a novel method that utilizes 3D modeling techniques to accurately and efficiently quantify the volume of interstitial gaps in marine benthic habitats, as well as the space provided by substrate rugosity. This method builds upon the analog methods routinely used on rocky shores and intertidal habitats, including those that measure rugosity, topography, fractals and volume. The method provides a direct Euclidean measurement and uniquely allows retrospective analysis if historical data on species composition are available. Blender interstitial volume allows users to quickly build and measure a large number of samples at no extra cost. •The program for Blender is free and opensource, and requires no extra equipment.•Once 3D models of species are made, the entire method takes less than ten minutes to complete.•Blender interstitial volume is as accurate as Fractal analysis in determining structural complexity on rocky shores, but is more consistent and precise, and better at discerning differences.
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