# The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus

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## Abstract

**:**

## 1. Introduction

…it has occurred to a number of the more philosophically attuned contemporary scientists that we are now at another point of transition, where the implications of complexity, memory, and uncertainty have revealed themselves to be barriers to our future understanding of our technological society. The fractional calculus (FC) has emerged from the shadows as a way of taming these three disrupters with a methodology capable of analytically smoothing their singular natures.

## 2. Fractality

#### 2.1. Fractal Time Series

**Scaling PDF**: The hallmarks of fractal statistics are spatial (x) inhomogeneity, temporal (t) intermittency and the phase space trajectory ($x;t$) exchanged for the dynamic variable $X\left(t\right)$. In phase space, the scaling of the dynamic variable is replaced by a scaling of the PDF $P(x;t)$:

#### 2.2. Multifractal Time Series

**Multifractal signal processing:**Mandelbrot was the first to recognize that signals that are singular at almost every point, fractal signals, are typical of physiological datasets. So how do we interpret a multifractal signal, or a fractal signal for that matter? A long-standing strategy for interpreting a signal is to construct the Fourier transform applied to an experimental time series. Although the method is mathematically unassailable, the utility of the various derived quantities has been questioned. The basis for these queries is the mutually exclusive treatment of time and frequency in the specification of the signal, that is, the time series is assumed to be infinitely long and each frequency is defined for a monochromatic infinitely long wave train. However, all time series in medicine are of finite duration and dominant frequencies change over time. The recognition of this limitation of the (time, frequency) representation of Fourier signals lead to the development of the wavelet transform method for representing one-dimensional signals as a function of time and frequency; see, e.g., [86].

#### 2.2.1. Ergodicity Breaking

#### 2.2.2. Information Transfer

## 3. Cross-Correlation Cube

- (1)
- A complex network belonging to region 2 cannot exert any asymptotic influence on a complex network belonging to region 1. This is the square denoted II on the CCC and is where LRT supposedly died.
- (2)
- A complex network belonging to region 2 exerts varying degrees of influence on a complex network belonging to region 2. This follows from PCM and is indicated by IV on the CCC.
- (3)
- A complex network belonging to region 1 exerts varying degrees of influence on a complex network belonging to region 1. This follows from PCM and is indicated by I on the CCC.
- (4)
- (5)
- When the two IPL indices are equal to 2 there is an abrupt jump up from zero (square II) to one (square III), or down from one to zero, depending on the values of the IPL indices just before they converge on 2. This is a singular point where the spectra of the two networks display exact $1/f$-noise fluctuations.

## 4. Fractional Calculus

We are all deeply conscious today that the enthusiasm of our fore bears for the marvelous achievements of Newtonian mechanics lead them to make generalizations in this area of predictability which, indeed, we may have generally tended to believe before 1960, but which we now recognize were false. We collectively wish to apologize for having misled the general educated public by spreading ideas about determinism of systems satisfying Newton’s laws of motion that, after 1960, were to be proved incorrect…

#### 4.1. Nexus with Multifractality

#### 4.1.1. Fractional Linear Langevin Equation (FLLE)

#### 4.1.2. Stochastic Fractional Index

**Multifractals**: As reviewed in [65], the interval $1/2\ge H>0$ has in the past been interpreted in terms of an anti-persistent random walk. An anti-persistent explanation of time series was made by Peng et al. [10] for the differences in time intervals between heart beats, now called HRV. They interpreted their time series, as did a number of subsequent investigators, in terms of random walks with $H<1/2$. In this model, the anti-persistent behavior lead to an avoidance of the extremes, so that the time intervals became neither too large nor too small. However, from these results, it is clear that the SFLE as a model for the dynamics provides an equivalent description of the underlying dynamics. The scaling behavior alone cannot distinguish between these two models. What is needed is a complete statistical distribution and not just the time dependence (scaling behavior) of the central moments.

## 5. Fractional Probability Calculus

#### 5.1. Fractal Diffusion

**Lévy statistics:**The theory of the influence of long-time memory on stochastic phenomena was developed to explain diffusion in which the second moment of the diffusion variable does not increase linearly in time, that is, the diffusion is anomalous. Following West et al. [130], this long-time memory is captured by assuming an IPL form for the autocorrelation function:

**Fractional in time and space:**Memory can be introduced into the FPE by means of a subordination process whereby two times are introduced into the process and one is subordinated to the other. The concept of using different clocks to measure different aspects of interacting complex dynamic networks dates back to the middle of the 19th century. It was then proposed that the two clocks defined subjective and objective times and were used to justify the empirical Weber–Fechner law [140]. Due to the present-day availability of time resolved datasets, life science investigators have begun adopting the notion of multiple clocks to distinguish between cell-specific and organ-specific clocks in biology, which is analogous to person-specific and group-specific clocks in sociology. While the global activity of an organ, such as the brain or the heart, might be characterized by quite regular behaviors, the activity of single neurons or pacemaker cells demonstrate statistical intermittency resulting in global 1/f variability.

#### 5.2. Fractal Random Walks

## 6. Discussion and Conclusions

- (1).
- The simple analytic functions of the IC have been found to be insufficient to describe the time dependence of most physiology networks. The notion of fractality was introduced to capture the true complexity of such biomedical network time series through fractal geometry, fractal statistics and fractal dynamics.
- (2).
- A fractal function diverges when an integer-order derivative is taken, so that such a fractal function cannot be the solution to a Newtonian equation of motion. However, when a fractional-order derivative of a fractal function is taken, it results in a new fractal function. Consequently, a time-dependent fractal process can have an equation of motion that is a FDE.
- (3).
- The network effect is the influence exerted by a complex dynamic network on each member of the network. When the network dynamics is a member of the Ising universality class, the interconnected set of IDEs for the probability of an individual being in one of two states during its non-linear interaction with the other members of the network can be replaced by an equivalent linear FDE and solved using the FC.
- (4).
- Even the simplest FDEs has a built-in memory resulting from the hidden interaction of the observable with its environment, which is manifest in the non-integer order of the time derivative, as in the network effect.
- (5).
- The solution to a linear FRE is a MLF for $\alpha <1$ and becomes an exponential function for $\alpha =1$. The MLF is the workhorse of the FC just as the exponential is for the IC.
- (6).
- A truly complex stochastic dynamic process can have more than one fractal dimension. A multifractal process is characterized by a uni-modal spectrum $f\left(h\right)$ peaked at the value of the Hurst exponent $h=H$.

- (7).
- The flow of information due to interaction of two complex networks each generating a multifractal time series is from the network with the broader to that with the narrower multifractal spectrum. This is summarized in the interpretation of the efficiency of information transfer using the CCC.
- (8).
- FREs with random fractional derivatives are shown to generate multifractal processes and therefore can be used to model the dynamics of both healthy and pathological physiologic networks.
- (9).
- Multifractality emerges from three distinct sources: (1) the introduction of random fractional derivatives into the dynamics of complex networks; (2) a FKT developed to define the evolution of PDF over fractional trajectories; (3) fractional random walks with diverging central moments.
- (10).
- A simple FDE that has a built-in non-locality in space is the FSDE. The solution to this fractional diffusion equation in space is a Lévy PDF, whose index is given by the order of the spatial fractional derivative. Yet another fractional diffusion equation differs in having a built-in memory and is the FTDE. The solution to this fractional diffusion equation in time is expressed in terms of the inverse Fourier transform of a MLF.
- (11).
- The health of a physiologic network is manifest by the width of the multifractal spectrum of the time series generated by that network. Experiments include but are not limited to CBF, HRV, BRV and SRV, which also show that pathologies in each of the underlying networks narrow the approprate multifractal spectrum.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

BRV | breathrate variability |

GLRT | generalized linar response theory |

CBF | cerebral blood flow |

HRV | heartrate variability |

CCC | cross correlation cube |

IC | integer calculus |

CE | crucial event |

IDE | integer differential equation |

CTRW | continuous time random walk |

IPL | inverse power law |

DMM | decision-making model |

LE | Langevin equation |

FBM | fractional Brownian motion |

LRT | linear response theory |

FC | fractional calculus |

MLF | Mittag-Leffler function |

FDE | fractional diffusion equaton |

PCM | principle of compexity management |

FFPE | fractional Fokker–Planck equation |

probability density function | |

FKE | fractional kinetic equaition |

RG | renormalization group |

FKT | fractional kinetic theory |

RHS | right hand side |

FLE | fractional Langevin equation |

RW | random walk |

FLLE | fractional linear Langevin equation |

SFLE | simplest fractional Langevin equation |

FO | fractional order |

SOC | self-organized criticality |

FPE | Fokker–Planck equation |

SOTC | self-organized temporal criticality |

FPC | fractional probability calculus |

SRV | striderate variability |

FRE | fractional rate equation |

WF | Weirstrass function |

FTDE | fractional time diffusion equation |

WRW | Weirstrass random walk |

FSDE | fractional space diffuson equation |

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**Figure 1.**The multifractal spectra $f\left(h\right)$ are plotted for the indicated values of the IPL index $\mu =1.1,1.5,2.0,2.5,3.0,$ and $3.5$ without truncation. When $\mu $ is close to 1, the spectrum is very broad. As $\mu $ is increased, the spectrum narrows and its peak moves to $h=0.5$. From [96] with permission.

**Figure 3.**Walking in synchrony with a multifractal metronome. The multifractal spectra for the participant (black circles) and that for the metronome (white circle) are shown. This figure is derived with permission from the left panel of Figure 3 in [28].

**Figure 4.**Effects of perturbation of network R by network P as captured by the multifractal spectrum. (

**Upper**): the perturber network has ${\mu}_{P}=2.4$ response network has ${\mu}_{R}=3.4$ and is depicted before and after the perturbation; (

**Lower**): the perturber network has ${\mu}_{P}=3.4$ and response network has ${\mu}_{R}=2.4$ and is depicted before and after the perturbation. The spectrum of the perturbed system is strongly affected in the first case in the upper panel, in contrast to the second case in the lower panel.

**Figure 5.**Effects of perturbation on the response network by a perturbing network, as captured by the multifractal spectra. (

**Upper**): perturber having ${\mu}_{P}=1.2$ perturbs network having ${\mu}_{R}=3.4$ shown both before and after perturbation; (

**Lower**): perturber having ${\mu}_{P}=3.4$ perturbs network having ${\mu}_{R}=1.2$ shown both before and after perturbation. The perturbation significantly changes the spectrum only when the IPL complexity index of the perturbed network is larger than that of the perturbing one. From [96] with permission.

**Figure 6.**The cross-correlation cube (CCC) depicts the asymptotic response of the cross-correlation function, graphed as a function of the IPL indices of the responding network R and the stimulating network P. The height of the CCC, that being the vertical axis perpendicular to the (${\mu}_{R},{\mu}_{P})$ plane, is normalized to a maxmum value of one. Adapted from [2] with permission.

**Figure 7.**Non-ergodic complexity management CCC: The asymptotic in time cross-correlation function between the P and R networks as a function of the IPL indices. The details of the numerical analysis are given in [41], as are the analytic calculations given by the red stripes.

**Figure 9.**The landing sites for the WRW are connected by jumps and the islands of clusters discussed in the text are readily observed. From [150] with permission.

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West, B.J.
The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus. *Fractal Fract.* **2022**, *6*, 225.
https://doi.org/10.3390/fractalfract6040225

**AMA Style**

West BJ.
The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus. *Fractal and Fractional*. 2022; 6(4):225.
https://doi.org/10.3390/fractalfract6040225

**Chicago/Turabian Style**

West, Bruce J.
2022. "The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus" *Fractal and Fractional* 6, no. 4: 225.
https://doi.org/10.3390/fractalfract6040225