Similarity and self-similarity in random walk with fixed, random and shrinking steps

Mitra, Tushar; Hossain, Mohammad Tomal; Banerjee, Santo; Hassan, Md. Kamrul

doi:10.1016/j.chaos.2021.110790

Cited by 6 publications

(5 citation statements)

References 33 publications

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“…Temporal scale-invariance is in practice—as opposed to geometric fractality—always of purely statistical nature, e.g. in phenomena based on random-walk processes ( Mitra et al. 2021 ).…”

Section: Scale-free Systems In General and In Naturementioning

confidence: 99%

The fractal brain: scale-invariance in structure and dynamics

et al. 2022

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The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.

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Section: Scale-free Systems In General and In Naturementioning

confidence: 99%

The fractal brain: scale-invariance in structure and dynamics

et al. 2022

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“…It means that the numerical values of the dimensional quantities are different in different snapshots but the corresponding dimensionless quantities remain the same. The litmus test of this property is the dynamic scaling [29][30][31] We can characterize each block by their area of the blocks in the lattice and investigate the nature of their distribution. To this end, we define an observable quantity C(a, t)da as the number of blocks whose area lies in the range a and a+da at time t. To calculate the distribution function we collect data for different times by finding the frequency of number of blocks using δa as an interval size.…”

Section: Area Size Distribution Function and Dynamic Scaling In The S...mentioning

confidence: 99%

Multi-multifractality and dynamic scaling in stochastic porous lattice

Mitra,

Hassan

2022

Preprint

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In this article, we extend the idea of stochastic dyadic Cantor set to weighted planar stochastic lattice that leads to a stochastic porous lattice. The process starts with an initiator which we choose to be a square of unit area for convenience. We then define a generator that divides the initiator or one of the blocks, picked preferentially with respect to their areas, to divide it either horizontally or vertically into two rectangles of which one of them is removed with probability q = 1 − p. We find that the remaining number of blocks and their mass varies with time as t p and t −q respectively. Analytical solution shows that the dynamics of this process is governed by infinitely many hidden conserved quantities each of which is a multifractal measure with porous structure as it contains missing blocks of various different sizes. The support where these measures are distributed is fractal with fractal dimension 2p provided 0 < p < 1. We find that if the remaining blocks are characterized by their respective area then the corresponding block size distribution function obeys dynamic scaling.

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Section: We Assume That μmentioning

confidence: 99%

Multi-multifractality and dynamic scaling in stochastic porous lattice

Mitra

Hassan

2021

Eur. Phys. J. Spec. Top.

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In this article, we extend the idea of stochastic dyadic Cantor set to weighted planar stochastic lattice that leads to a stochastic porous lattice. The process starts with an initiator which we choose to be a square of unit area for convenience. We then define a generator that divides the initiator or one of the blocks, picked preferentially with respect to their areas, to divide it either horizontally or vertically into two rectangles of which one of them is removed with probability q = 1 − p. We find that the remaining number of blocks and their mass varies with time as t p and t −q , respectively. Analytical solution shows that the dynamics of this process is governed by infinitely many hidden conserved quantities each of which is a multifractal measure with porous structure as it contains missing blocks of various different sizes. The support where these measures are distributed is fractal with fractal dimension 2p provided 0 < p < 1. We find that if the remaining blocks are characterized by their respective area, then the corresponding block size distribution function obeys dynamic scaling.

show abstract

Similarity and self-similarity in random walk with fixed, random and shrinking steps

Cited by 6 publications

References 33 publications

The fractal brain: scale-invariance in structure and dynamics

The fractal brain: scale-invariance in structure and dynamics

Multi-multifractality and dynamic scaling in stochastic porous lattice

Multi-multifractality and dynamic scaling in stochastic porous lattice

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