Entropy of fractal systems

Zmeškal, Oldřich; Dzik, Petr; Veselý, Michal

doi:10.1016/j.camwa.2013.01.017

Cited by 85 publications

(70 citation statements)

References 8 publications

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“…While fractal dimension is usually though to encode complexity in terms of self-similarity rather than entropy directly, there is a connection between the two values: fractal dimension is related to Renyi entropy, which is itself a generalization of the classical measure of Shannon entropy (Renyi, 1961). Computational models have shown that as the fractal dimension of a shape rises, so does the associated Renyi entropy (Zmeskal et al, 2013). Another measure, the information dimension, relates the fractal dimension to the information content of a fractal at different scales (Renyi, 1959;Wei et al, 2014).…”

mentioning

confidence: 99%

Serotonergic Psychedelics LSD & Psilocybin Increase the Fractal Dimension of Cortical Brain Activity in Spatial and Temporal Domains

Varley

Carhart‐Harris

Roseman

et al. 2019

Preprint

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Psychedelic drugs, such as psilocybin and LSD, represent unique tools for researchers investigating the neural origins of consciousness. Currently, the most compelling theories of how psychedelics exert their effects is by increasing the complexity of brain activity and moving the system towards a critical point between order and disorder, creating more dynamic and complex patterns of neural activity. While the concept of criticality is of central importance to this theory, few of the published studies on psychedelics investigate it directly, testing instead related measures such as algorithmic complexity or Shannon entropy. We propose using the fractal dimension of functional activity in the brain as a measure of complexity since findings from physics suggest that as a system organizes towards criticality, it tends to take on a fractal structure. We tested two different measures of fractal dimension, one spatial and one temporal, using fMRI data from volunteers under the influence of both LSD and psilocybin. The first was the fractal dimension of cortical functional connectivity networks and the second was the fractal dimension of BOLD time-series. We were able to show that both psychedelic drugs significantly increased the fractal dimension of functional connectivity networks, and that LSD significantly increased the fractal dimension of BOLD signals, with psilocybin showing a non-significant trend in the same direction. With both LSD and psilocybin, we were able to localize changes in the fractal dimension of BOLD signals to brain areas assigned to the dorsal-attentional network.These results show that psychedelic drugs increase the fractal character of activity in the brain and we see this as an indicator that the changes in consciousness triggered by psychedelics are associated with evolution towards a critical zone.

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confidence: 99%

Serotonergic Psychedelics LSD & Psilocybin Increase the Fractal Dimension of Cortical Brain Activity in Spatial and Temporal Domains

Varley

Carhart‐Harris

Roseman

et al. 2019

Preprint

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“…The colour was measured on a white light‐stable support, always on five identical places on the sample using a Konica Minolta CM‐700d instrument in the CIELAB colour space. The overall colour difference after printing and after artificial ageing was then calculated from the measured values (Equation )

Δ E_{ab}^{*} = \sqrt{{({normalL}_{2}^{*} - {normalL}_{1}^{*})}^{2} + {({normala}_{2}^{*} - {normala}_{1}^{*})}^{2} + {({normalb}_{2}^{*} - {normalb}_{1}^{*})}^{2}}

where L* is the lightness, a* is the green–red colour axis, and b* is the blue–yellow colour axis.…”

Section: Methodsmentioning

confidence: 99%

“…The colour changes observable by the naked eye can be more simply expressed verbally in a number of ways corresponding to a specific range of overall colour difference values. One of the means of verbal expression is given in Table . Values

Δ E_{ab}^{*}

greater than 1 correspond to colour changes visible to the naked eye.…”

Section: Methodsmentioning

confidence: 99%

Long‐term stability of invisible X‐ray fluorescence identification tags

et al. 2019

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The traditional labelling of archive documents (e.g., hand stamping) entails several limitations. It could affect the aesthetic value of the archive document and is easy to find and remove in case of the theft of the document. The use of a label that would not be easily visible by the naked eye would resolve both of these problems. One possibility in designing such label is the use of the elements that have a good response in X-ray fluorescence (XRF) spectrum, are not disturbing in colour, and are not usually present in the archive materials themselves. This study focuses on such type of labelling, specifically, the tags based on the oxides of lanthanum, dysprosium, samarium, gadolinium, and niobium. For the material to be taken into account by the community, it has to fulfil several important characteristics, such as harmlessness to the archival documents, non-toxicity, or long-term stability. In the study, the long-term stability of these tags was studied on model samples of lignocellulose materials subjected to several types of artificial ageing. The colour changes and changes in the intensity of the responses of the individual elements in the XRF spectra were evaluated. To simulate the practical use, the tags of minimal visibility to the naked eye were printed on samples of real archive materials, and their potential for detection using a handheld XRF spectrometer was studied.Both of the research parts should help bring this type of labelling closer to the field application.

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“…The q-order Renyi entropy is proposed based on the definition of K entropy [31,32]. It is expressed as…”

Section: Two-order Renyi Entropymentioning

confidence: 99%