Emergence of Fractional Kinetics in Spiny Dendrites

Vitali, Silvia; Mainardi, Francesco; Castellani, Gastone

doi:10.3390/fractalfract2010006

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…In [7] the authors show that time fractional generalization of the cable equation arises naturally in the continuous time random walk, by considering a superposition of Markovian processes and in a ggBm-like construction of the random variable. This model is also used to describe tranmembrane potentital in spiny dendrites.…”

Section: Fractional Methods In Bio-medical Areasmentioning

confidence: 99%

Fractional Dynamics

Cattani

Spigler

2018

Fractal Fract

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Modelling, simulation, and applications of Fractional Calculus have recently become increasingly popular subjects, with impressive growth concerning applications. The founding and limited ideas on fractional derivatives have achieved an incredibly valuable status. The variety of applications in mathematics, physics, engineering, economics, biology, and medicine, have opened new, challenging fields of research. For instance, in soil mechanics, a suitable definition of the fractional operator has shed some light on viscoelasticity, explaining memory effects on materials. Needless to say, these applications require the development of practical mathematical tools in order to extract quantitative information from models, newly reformulated in terms of fractional differential equations. Even confining ourselves to the field of ordinary differential equations, the well-known Bagley-Torvik model showed that fractional derivatives may actually arise naturally within certain physical models, and are not merely fanciful mathematical generalizations. This Special Issue focuses on the most recent advances in fractional calculus, applied to dynamic problems, linear and nonlinear fractional ordinary and partial differential equations, integral fractional differential equations, and stochastic integral problems arising in all fields of science, engineering, and other applied fields. In this issue, we have collected several significant papers devoted to applications of fractional methods with a focus on dynamical aspects. The applications range from theoretical mathematical-numerical aspects [1,2] to bio-medical subjects [3-7]. Applications to complex materials are investigated in [8], aiming at proposing a generalized definition of fractional operators. Special diffusion models are studied in [9-11].

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Section: Fractional Methods In Bio-medical Areasmentioning

confidence: 99%

Fractional Dynamics

Cattani

Spigler

2018

Fractal Fract

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“…The Wright function arises in the theory of the spacetime fractional diffusion equation with the temporal Caputo derivative [1] and the fractional cable equation used in the modeling of apical dendrites of neurons [2]. The eponymous function provides a unified treatment of several classes of special functions, notably the Bessel functions, the probability integral function erf, the Airy function Ai , and the Whittaker function, among others.…”

Section: Introductionmentioning

confidence: 99%

The Wright function – hypergeometric representation and symbolical evaluation

Prodanov

2023

2023 International Conference on Fractional Differentiation and Its Applications (ICFDA)

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The Wright function, which arises in the theory of the space-time fractional diffusion equation, is an interesting mathematical object which has diverse connections with other special and elementary functions. The Wright function provides a unified treatment of several classes of special functions, such as the Gaussian, Airy, Bessel, and Error functions, etc. The manuscript demonstrates an algorithm for symbolical representation in terms of finite sums of hypergeometric (HG) functions and polynomials. The HG functions are then represented by known elementary or other special functions, wherever possible. The algorithm is programmed in the open-source computer algebra system Maxima and can be used to for testing numerical algorithms for the evaluation of the Wright function.

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Fractional Diffusion and Medium Heterogeneity: The Case of the Continuous Time Random Walk

Sposini

Vitali

Paradisi

et al. 2021

SEMA SIMAI Springer Series

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In this contribution we show that fractional diffusion emerges from a simple Markovian Gaussian random walk when the medium displays a power-law heterogeneity. Within the framework of the continuous time random walk, the heterogeneity of the medium is represented by the selection, at any jump, of a different time-scale for an exponential survival probability. The resulting process is a non-Markovian non-Gaussian random walk. In particular, for a power-law distribution of the time-scales, the resulting random walk corresponds to a time-fractional diffusion process. We relates the power-law of the medium heterogeneity to the fractional order of the diffusion. This relation provides an interpretation and an estimation of the fractional order of derivation in terms of environment heterogeneity. The results are supported by simulations.

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Emergence of Fractional Kinetics in Spiny Dendrites

Cited by 3 publications

References 21 publications

Fractional Dynamics

Fractional Dynamics

The Wright function – hypergeometric representation and symbolical evaluation

Fractional Diffusion and Medium Heterogeneity: The Case of the Continuous Time Random Walk

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