Fractional integration toolbox

Marinov, Toma; Ramirez, Nelson; Santamarı́a, Fidel

doi:10.2478/s13540-013-0042-7

Cited by 26 publications

(18 citation statements)

References 17 publications

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“…We recently introduced the fractional leaky integrate-and-fire model (LIF) [ 14 ], which we have used to replicate the firing rate activity of adapting cortical neurons. We have also developed tools to efficiently integrate such equations [ 19 ]. Other groups have used the fractional derivative of the voltage to study the Hodgkin-Huxley [ 15 – 17 ] model or to model the power-law firing rate adaptation observed in cortical and brain stem neurons [ 12 , 13 ].…”

Section: Introductionmentioning

confidence: 99%

Power-Law Dynamics of Membrane Conductances Increase Spiking Diversity in a Hodgkin-Huxley Model

2016

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We studied the effects of non-Markovian power-law voltage dependent conductances on the generation of action potentials and spiking patterns in a Hodgkin-Huxley model. To implement slow-adapting power-law dynamics of the gating variables of the potassium, n, and sodium, m and h, conductances we used fractional derivatives of order η≤1. The fractional derivatives were used to solve the kinetic equations of each gate. We systematically classified the properties of each gate as a function of η. We then tested if the full model could generate action potentials with the different power-law behaving gates. Finally, we studied the patterns of action potential that emerged in each case. Our results show the model produces a wide range of action potential shapes and spiking patterns in response to constant current stimulation as a function of η. In comparison with the classical model, the action potential shapes for power-law behaving potassium conductance (n gate) showed a longer peak and shallow hyperpolarization; for power-law activation of the sodium conductance (m gate), the action potentials had a sharp rise time; and for power-law inactivation of the sodium conductance (h gate) the spikes had wider peak that for low values of η replicated pituitary- and cardiac-type action potentials. With all physiological parameters fixed a wide range of spiking patterns emerged as a function of the value of the constant input current and η, such as square wave bursting, mixed mode oscillations, and pseudo-plateau potentials. Our analyses show that the intrinsic memory trace of the fractional derivative provides a negative feedback mechanism between the voltage trace and the activity of the power-law behaving gate variable. As a consequence, power-law behaving conductances result in an increase in the number of spiking patterns a neuron can generate and, we propose, expand the computational capacity of the neuron.

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Section: Introductionmentioning

confidence: 99%

Power-Law Dynamics of Membrane Conductances Increase Spiking Diversity in a Hodgkin-Huxley Model

2016

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“…Such a time-fractional derivative naturally arises in many context, including geophysics [14], neurology [18,36] (see also [29] and the references therein) and viscoelasticity [5], and can be seen as a natural consequence of classical models of diffusion in highly ramified media such as combs [4]. In addition, from the mathematical point of view, equations involving the Caputo derivatives can be framed into the broad line of research devoted to Volterra type integrodifferential operators, see [28,40].…”

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confidence: 99%

Decay Estimates in Time for Classical and Anomalous Diffusion

2020

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We present a series of results focused on the decay in time of solutions of classical and anomalous diffusive equations in a bounded domain. The size of the solution is measured in a Lebesgue space, and the setting comprises time-fractional and space-fractional equations and operators of nonlinear type. We also discuss how fractional operators may affect long-time asymptotics. Decay estimates, methods, results and perspectivesIn this note we present some results, recently obtained in [2,19], focused on the long-time behavior of solutions of evolution equations which may exhibit anomalous diffusion, caused by either time-fractional or space-fractional effects (or both). The case of several nonlinear operators will be also taken into account (and indeed some of the results that we present are new also for classical diffusion run by nonlinear operators).The results that we establish give quantitative bounds on the decay in time of smooth solutions, confined in a smooth bounded set with Dirichlet data. The size of the solution will be measured in classical Lebesgue spaces, and we will detect different types of decays according to the different cases that we take into consid-

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“…FIT is the Fractional Integration Toolbox developed by Santamaria Laboratory at the University of Texas at San Antonio, [52]. It is for the numerical computation of fractional integration and differentiation of the Riemann-Liouville (R-L) type, and is designed for large data size, which allows parallel computing of multiple fractional integration/differentiation on GPUs (graphical processing units).…”

Section: Fitmentioning

confidence: 99%

A review and evaluation of numerical tools for fractional calculus and fractional order controls

Liu

Dehghan

et al. 2016

International Journal of Control

145

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In recent years, as fractional calculus becomes more and more broadly used in research across different academic disciplines, there are increasing demands for the numerical tools for the computation of fractional integration/differentiation, and the simulation of fractional order systems. Time to time, being asked about which tool is suitable for a specific application, the authors decide to carry out this survey to present recapitulative information of the available tools in the literature, in hope of benefiting researchers with different academic backgrounds. With this motivation, the present article collects the scattered tools into a dashboard view, briefly introduces their usage and algorithms, evaluates the accuracy, compares the performance, and provides informative comments for selection.

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Fractional integration toolbox

Cited by 26 publications

References 17 publications

Power-Law Dynamics of Membrane Conductances Increase Spiking Diversity in a Hodgkin-Huxley Model

Power-Law Dynamics of Membrane Conductances Increase Spiking Diversity in a Hodgkin-Huxley Model

Decay Estimates in Time for Classical and Anomalous Diffusion

A review and evaluation of numerical tools for fractional calculus and fractional order controls

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