Graph fractal dimension and the structure of fractal networks.
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ABSTRACT:
SUBMITTER: Skums P
PROVIDER: S-EPMC7673317 | biostudies-literature |
REPOSITORIES: biostudies-literature
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Project description:Recent evidence suggests that the quantity and quality of conscious experience may be a function of the complexity of activity in the brain and that consciousness emerges in a critical zone between low and high-entropy states. We propose fractal shapes as a measure of proximity to this critical point, as fractal dimension encodes information about complexity beyond simple entropy or randomness, and fractal structures are known to emerge in systems nearing a critical point. To validate this, we tested several measures of fractal dimension on the brain activity from healthy volunteers and patients with disorders of consciousness of varying severity. We used a Compact Box Burning algorithm to compute the fractal dimension of cortical functional connectivity networks as well as computing the fractal dimension of the associated adjacency matrices using a 2D box-counting algorithm. To test whether brain activity is fractal in time as well as space, we used the Higuchi temporal fractal dimension on BOLD time-series. We found significant decreases in the fractal dimension between healthy volunteers (n = 15), patients in a minimally conscious state (n = 10), and patients in a vegetative state (n = 8), regardless of the mechanism of injury. We also found significant decreases in adjacency matrix fractal dimension and Higuchi temporal fractal dimension, which correlated with decreasing level of consciousness. These results suggest that cortical functional connectivity networks display fractal character and that this is associated with level of consciousness in a clinically relevant population, with higher fractal dimensions (i.e. more complex) networks being associated with higher levels of consciousness. This supports the hypothesis that level of consciousness and system complexity are positively associated, and is consistent with previous EEG, MEG, and fMRI studies.
| S-EPMC7017993 | biostudies-literature
Project description:In this paper a family of weighted fractal networks, in which the weights of edges have been assigned to different values with certain scale, are studied. For the case of the weighted fractal networks the definition of modified box dimension is introduced, and a rigorous proof for its existence is given. Then, the modified box dimension depending on the weighted factor and the number of copies is deduced. Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its nearest neighbors. The weighted time for two adjacency nodes is the weight connecting the two nodes. Then the average weighted receiving time (AWRT) is a corresponding definition. The obtained remarkable result displays that in the large network, when the weight factor is larger than the number of copies, the AWRT grows as a power law function of the network order with the exponent, being the reciprocal of modified box dimension. This result shows that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is.
| S-EPMC4678901 | biostudies-other
Project description:Highlights • Epidemic curves can be approached from a fractal interpolation point of view.• Looking at the Covid-19 epidemic curves from a fractal point of view might benefit the medical community to understand better the dynamics and evolution of the pandemic.• Fractal interpolation of data regarding Covid-19 can help retrieve missing pieces of information due to limited testing.• Analyzing maps of the spreading of Covid-19 in different countries from a fractal dimension point of view can be a useful tool in assessing the complexity of the pandemic on the territory of a country. The present paper proposes a reconstruction of the epidemic curves from the fractal interpolation point of view. Looking at the epidemic curves as fractal structures might be an efficient way to retrieve missing pieces of information due to insufficient testing and predict the evolution of the disease. A fractal approach of the epidemic curve can contribute to the assessment and modeling of other epidemics. On the other hand, we have considered the spread of the epidemic in countries like Romania, Italy, Spain, and Germany and analyzed the spread of the disease in those countries based on their fractal dimension.
| S-EPMC7332948 | biostudies-literature
Project description:Fractal Dimension (FD) has shown to be a promising means to describe the morphology of cortical structures across different neurologic and psychiatric conditions, displaying a good sensitivity in capturing atrophy processes. In this study, we aimed at exploring the morphology of cortical areas by means of FD in 58 female patients with Anorexia Nervosa (AN) (38 currently underweight and 20 fully recovered) and 38 healthy controls (HC). All participants underwent high-resolution MRI. Surface extraction was completed using FreeSurfer, FD was computed using the calcFD toolbox. The whole cortex mean FD value was lower in acute AN patients compared to HC (p < 0.001). Recovered AN patients did not show differences in the global FD when compared to HC. However, some brain areas showed higher FD in patients than controls, while others showed the opposite pattern. Parietal regions showed lower FD in both AN groups. In acute AN patients, the FD correlated with age (p < 0.001), body mass index (p = 0.019) and duration of illness (p = 0.011). FD seems to represent a feasible method to explore cortical complexity in patients with AN since it demonstrated to be sensitive to the effects of both severity and duration of malnutrition.
| S-EPMC7141241 | biostudies-literature
Project description:Brain activity is complex; a reflection of its structural and functional organization. Among other measures of complexity, the fractal dimension is emerging as being sensitive to neuronal damage secondary to neurological and psychiatric diseases. Here, we calculated Higuchi's fractal dimension (HFD) in resting-state eyes-closed electroencephalography (EEG) recordings from 41 healthy controls (age: 20-89 years) and 67 Alzheimer's Disease (AD) patients (age: 50-88 years), to investigate whether HFD is sensitive to brain activity changes typical in healthy aging and in AD. Additionally, we considered whether AD-accelerating effects of the copper fraction not bound to ceruloplasmin (also called "free" copper) are reflected in HFD fluctuations. The HFD measure showed an inverted U-shaped relationship with age in healthy people (R2 = .575, p < .001). Onset of HFD decline appeared around the age of 60, and was most evident in central-parietal regions. In this region, HFD decreased with aging stronger in the right than in the left hemisphere (p = .006). AD patients demonstrated reduced HFD compared to age- and education-matched healthy controls, especially in temporal-occipital regions. This was associated with decreasing cognitive status as assessed by mini-mental state examination, and with higher levels of non-ceruloplasmin copper. Taken together, our findings show that resting-state EEG complexity increases from youth to maturity and declines in healthy, aging individuals. In AD, brain activity complexity is further reduced in correlation with cognitive impairment. In addition, elevated levels of non-ceruloplasmin copper appear to accelerate the reduction of neural activity complexity. Overall, HDF appears to be a proper indicator for monitoring EEG-derived brain activity complexity in healthy and pathological aging.
| S-EPMC4752290 | biostudies-literature
Project description:A failure of adaptive inference-misinterpreting available sensory information for appropriate perception and action-is at the heart of clinical manifestations of schizophrenia, implicating key subcortical structures in the brain including the hippocampus. We used high-resolution, three-dimensional (3D) fractal geometry analysis to study subtle and potentially biologically relevant structural alterations (in the geometry of protrusions, gyri and indentations, sulci) in subcortical gray matter (GM) in patients with schizophrenia relative to healthy individuals. In particular, we focus on utilizing Fractal Dimension (FD), a compact shape descriptor that can be computed using inputs with irregular (i.e., not necessarily smooth) surfaces in order to quantify complexity (of geometrical properties and configurations of structures across spatial scales) of subcortical GM in this disorder. Probabilistic (entropy-based) information FD was computed based on the box-counting approach for each of the seven subcortical structures, bilaterally, as well as the brainstem from high-resolution magnetic resonance (MR) images in chronic patients with schizophrenia (n = 19) and age-matched healthy controls (n = 19) (age ranges: patients, 22.7-54.3 and healthy controls, 24.9-51.6 years old). We found a significant reduction of FD in the left hippocampus (median: 2.1460, range: 2.07-2.18 vs. median: 2.1730, range: 2.15-2.23, p<0.001; Cohen's effect size, U3 = 0.8158 (95% Confidence Intervals, CIs: 0.6316, 1.0)), the right hippocampus (median: 2.1430, range: 2.05-2.19 vs. median: 2.1760, range: 2.12-2.21, p = 0.004; U3 = 0.8421 (CIs: 0.5263, 1)), as well as left thalamus (median: 2.4230, range: 2.40-2.44, p = 0.005; U3 = 0.7895 (CIs: 0.5789, 0.9473)) in schizophrenia patients, relative to healthy individuals. Our findings provide in-vivo quantitative evidence for reduced surface complexity of hippocampus, with reduced FD indicating a less complex, less regular GM surface detected in schizophrenia.
| S-EPMC4866699 | biostudies-literature
Project description:Settlement centers of various types, including cities, produce basins of attraction whose shape can be regular or complexly irregular (from the point of view of geometry). This complexity depends in part on properties of the space surrounding a settlement. This paper demonstrates that by introducing a dynamic approach to space and by including an equation of motion and space resistance, a dramatic change in the stylized static CPT (Central Place Theory) image occurs. As a result of the interplay of gravitational forces, basins of attraction arise around cities, whose boundaries appear to be fractals. This study provides a wealth of spatial fractal complex images which may change the traditional understanding of CPT.
| S-EPMC9911407 | biostudies-literature
Project description:BackgroundFractal geometry measures the morphology of the brain and detects CNS damage. We aimed to assess the longitudinal changes on brain's fractal geometry and its predictive value for disease worsening in patients with Multiple Sclerosis (MS).MethodsWe prospectively analyzed 146 consecutive patients with relapsing-remitting MS with up to 5 years of clinical and brain MRI (3 T) assessments. The fractal dimension and lacunarity were calculated for brain regions using box-counting methods. Longitudinal changes were analyzed in mixed-effect models and the risk of disability accumulation were assessed using Cox Proportional Hazard regression analysis.ResultsThere was a significant decrease in the fractal dimension and increases of lacunarity in different brain regions over the 5-year follow-up. Lower cortical fractal dimension increased the risk of disability accumulation for the Expanded Disability Status Scale [HR 0.9734, CI 0.8420-0.9125; Harrell C 0.59; Wald p 0.038], 9-hole peg test [HR 0.9734, CI 0.8420-0.9125; Harrell C 0.59; Wald p 0.0083], 2.5% low contrast vision [HR 0.4311, CI 0.2035-0.9133; Harrell C 0.58; Wald p 0.0403], symbol digit modality test [HR 2.215, CI 1.043-4.705; Harrell C 0.65; Wald p 0.0384] and MS Functional Composite-4 [HR 0.55, CI 0.317-0.955; Harrell C 0.59; Wald p 0.0029].ConclusionsFractal geometry analysis of brain MRI identified patients at risk of increasing their disability in the next five years.
| S-EPMC8045041 | biostudies-literature
Project description:PurposeTo compare the fractal dimensions of the peripapillary microvasculature as obtained by optical coherence tomography angiography (OCTA) between primary open-angle glaucoma (POAG) and controls.MethodsOptic nerve head and peripapillary area images were taken using the 20° × 20°-scan of Spectralis OCTA (Heidelberg Engineering) in 97 subjects (64 POAG patients, 33 control patients). The optic nerve head microvasculature was evaluated according to predefined slabs: the superficial vascular complex (SVC) and the avascular complex (AVC). The en face image of each slab was processed by ImageJ software (National Institutes of Health) in order to calculate the vessel density and the fractal dimension using the box-counting method. For comparison, the peripapillary retinal nerve fiber layer (RNFL) thickness was obtained from Spectralis OCT circle scans. The utilities of the parameters for discriminating between the POAG and control groups were assessed using areas under the receiver operating characteristic curves (AUCs).ResultsThe SVC fractal dimension was lower in the POAG than in the control group (p < 0.001), while AVC showed no intergroup difference (p = 0.563). The fractal dimension showed a good correlation with the vessel density in both SVC and AVC (both p < 0.001). In a multivariable regression analysis, the SVC fractal dimension was negatively correlated with age (p < 0.001) and axial length (p < 0.001) and positively correlated with average RNFL thickness (p < 0.001), while the AVC fractal dimension was positively correlated with the Bruch's membrane opening size (p = 0.013). In terms of diagnostic utility, the AUC was significantly larger for the average RNFL thickness (AUC, 0.889) than for the SVC fractal dimension (AUC, 0.772; p = 0.008).ConclusionsThe fractal dimension of SVC was associated with the average RNFL thickness and was reduced in POAG patients. Fractal dimension analysis could be used in evaluating peripapillary vascularity by OCTA.
| S-EPMC9745347 | biostudies-literature
Project description:BackgroundChronic obstructive pulmonary disease (COPD) is characterized by airway remodeling. Characterization of airway changes on computed tomography has been challenging due to the complexity of the recurring branching patterns, and this can be better measured using fractal dimensions.MethodsWe analyzed segmented airway trees of 8,135 participants enrolled in the COPDGene cohort. The fractal complexity of the segmented airway tree was measured by the Airway Fractal Dimension (AFD) using the Minkowski-Bougliand box-counting dimension. We examined associations between AFD and lung function and respiratory morbidity using multivariable regression analyses. We further estimated the extent of peribronchial emphysema (%) within 5 mm of the airway tree, as this is likely to affect AFD. We classified participants into 4 groups based on median AFD, percentage of peribronchial emphysema, and estimated survival.ResultsAFD was significantly associated with forced expiratory volume in one second (FEV1; P < 0.001) and FEV1/forced vital capacity (FEV1/FVC; P < 0.001) after adjusting for age, race, sex, smoking status, pack-years of smoking, BMI, CT emphysema, air trapping, airway thickness, and CT scanner type. On multivariable analysis, AFD was also associated with respiratory quality of life and 6-minute walk distance, as well as exacerbations, lung function decline, and mortality on longitudinal follow-up. We identified a subset of participants with AFD below the median and peribronchial emphysema above the median who had worse survival compared with participants with high AFD and low peribronchial emphysema (adjusted hazards ratio [HR]: 2.72; 95% CI: 2.20-3.35; P < 0.001), a substantial number of whom were not identified by traditional spirometry severity grades.ConclusionAirway fractal dimension as a measure of airway branching complexity and remodeling in smokers is associated with respiratory morbidity and lung function change, offers prognostic information additional to traditional CT measures of airway wall thickness, and can be used to estimate mortality risk.Trial registrationClinicalTrials.gov identifier: NCT00608764.FundingThis study was supported by NIH K23 HL133438 (SPB) and the COPDGene study (NIH Grant Numbers R01 HL089897 and R01 HL089856). The COPDGene project is also supported by the COPD Foundation through contributions made to an Industry Advisory Board comprised of AstraZeneca, Boehringer Ingelheim, Novartis, Pfizer, Siemens, Sunovion and GlaxoSmithKline.
| S-EPMC6264725 | biostudies-literature