The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus
This is the third essay advocating the use the (non-integer) fractional calculus (FC) to capture the dynamics of complex networks in the twilight of the Newtonian era. Herein, the focus is on drawing a distinction between networks described by monfractal time series extensively discussed in the preq...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-04-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/4/225 |
| _version_ | 1797504780065570816 |
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| author | Bruce J. West |
| author_facet | Bruce J. West |
| author_sort | Bruce J. West |
| collection | DOAJ |
| description | This is the third essay advocating the use the (non-integer) fractional calculus (FC) to capture the dynamics of complex networks in the twilight of the Newtonian era. Herein, the focus is on drawing a distinction between networks described by monfractal time series extensively discussed in the prequels and how they differ in function from multifractal time series, using physiological phenomena as exemplars. In prequel II, the network effect was introduced to explain how the collective dynamics of a complex network can transform a many-body non-linear dynamical system modeled using the integer calculus (IC) into a single-body fractional stochastic rate equation. Note that these essays are about biomedical phenomena that have historically been improperly modeled using the IC and how fractional calculus (FC) models better explain experimental results. This essay presents the biomedical entailment of the FC, but it is not a mathematical discussion in the sense that we are not concerned with the formal infrastucture, which is cited, but we are concerned with what that infrastructure entails. For example, the health of a physiologic network is characterized by the width of the multifractal spectrum associated with its time series, and which becomes narrower with the onset of certain pathologies. Physiologic time series that have explicitly related pathology to a narrowing of multifractal time series include but are not limited to heart rate variability (HRV), stride rate variability (SRV) and breath rate variability (BRV). The efficiency of the transfer of information due to the interaction between two such complex networks is determined by their relative spectral width, with information being transferred from the network with the broader to that with the narrower width. A fractional-order differential equation, whose order is random, is shown to generate a multifractal time series, thereby providing a FC model of the information exchange between complex networks. This equivalence between random fractional derivatives and multifractality has not received the recognition in the bioapplications literature we believe it warrants. |
| first_indexed | 2024-03-10T04:09:13Z |
| format | Article |
| id | doaj.art-73866c56d07a4df6a6a99d3fe8aada0a |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T04:09:13Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-73866c56d07a4df6a6a99d3fe8aada0a2023-11-23T08:15:46ZengMDPI AGFractal and Fractional2504-31102022-04-016422510.3390/fractalfract6040225The Fractal Tapestry of Life: III Multifractals Entail the Fractional CalculusBruce J. West0Office of Research and Innovation, North Carolina State University, Rayleigh, NC 27695, USAThis is the third essay advocating the use the (non-integer) fractional calculus (FC) to capture the dynamics of complex networks in the twilight of the Newtonian era. Herein, the focus is on drawing a distinction between networks described by monfractal time series extensively discussed in the prequels and how they differ in function from multifractal time series, using physiological phenomena as exemplars. In prequel II, the network effect was introduced to explain how the collective dynamics of a complex network can transform a many-body non-linear dynamical system modeled using the integer calculus (IC) into a single-body fractional stochastic rate equation. Note that these essays are about biomedical phenomena that have historically been improperly modeled using the IC and how fractional calculus (FC) models better explain experimental results. This essay presents the biomedical entailment of the FC, but it is not a mathematical discussion in the sense that we are not concerned with the formal infrastucture, which is cited, but we are concerned with what that infrastructure entails. For example, the health of a physiologic network is characterized by the width of the multifractal spectrum associated with its time series, and which becomes narrower with the onset of certain pathologies. Physiologic time series that have explicitly related pathology to a narrowing of multifractal time series include but are not limited to heart rate variability (HRV), stride rate variability (SRV) and breath rate variability (BRV). The efficiency of the transfer of information due to the interaction between two such complex networks is determined by their relative spectral width, with information being transferred from the network with the broader to that with the narrower width. A fractional-order differential equation, whose order is random, is shown to generate a multifractal time series, thereby providing a FC model of the information exchange between complex networks. This equivalence between random fractional derivatives and multifractality has not received the recognition in the bioapplications literature we believe it warrants.https://www.mdpi.com/2504-3110/6/4/225fractal physiologymultifractalsfractional calculus |
| spellingShingle | Bruce J. West The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus Fractal and Fractional fractal physiology multifractals fractional calculus |
| title | The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus |
| title_full | The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus |
| title_fullStr | The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus |
| title_full_unstemmed | The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus |
| title_short | The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus |
| title_sort | fractal tapestry of life iii multifractals entail the fractional calculus |
| topic | fractal physiology multifractals fractional calculus |
| url | https://www.mdpi.com/2504-3110/6/4/225 |
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