A multiple time renewal equation for neural assemblies with elapsed time model

Torres, Nicolás; Perthame, Benôıt; Salort, Delphine

doi:10.1088/1361-6544/ac8714

Cited by 6 publications

(3 citation statements)

References 47 publications

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“…For this reason, we propose here an entirely different approach based on an adaptation of the so-called Doeblin-Harris method. The latter has proved very performant in a number of contexts recently, as a growing number of papers have focused on the use of this method for many equations from neuroscience or physics [4,3,10,16,17,27,19]. Following these ideas, we eventually prove exponential convergence to the stationary state for Equation (1.1).…”

Section: Introductionmentioning

confidence: 80%

Convergence towards equilibrium for a model with partial diffusion

Salort,

Smets

2024

Communications in Partial Differential Equations

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Section: Introductionmentioning

confidence: 80%

Convergence towards equilibrium for a model with partial diffusion

Salort,

Smets

2024

Communications in Partial Differential Equations

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“…Harris-type theorems have been successfully used in the study of quantitative convergence to equilibrium for several macroscopic equations arising in mathematical biology, particularly in structured population dynamics. Some important examples are the renewal equation, 13 structured neuron population models, 12,[78][79][80] and the growth fragmentation equation. 11 However, applications of these theorems on the mesoscopic models, so-called kinetic equations, in the context of biology are more recent.…”

Section: B Kinetic Equations Arising In Mathematical Biologymentioning

confidence: 99%

On quantitative hypocoercivity estimates based on Harris-type theorems

Yoldaş

2023

Journal of Mathematical Physics

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This Review concerns recent results on the quantitative study of convergence toward the stationary state for spatially inhomogeneous kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in the L1 (or total variation) setting for a large class of initial data. The convergence rates can be made explicit (for both geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates. We aim to present Harris-type theorems, providing a guideline on how to apply these techniques to kinetic equations at hand. We discuss recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, giving some perspectives on potential extensions to nonlinear equations.

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“…In recent years, several systems of integro-differential equations (SIDEs) have paved the way for interpreting the mathematical incidence of real world phenomena occurring in neural activity in electroencephalography (Robinson et al ., 1997), mathematics (Ali et al ., 2017), electrospinning process (Xu et al ., 2007), and biomathematics, such as neuron population (Torres et al ., 2022), plasmodium vivax transmission (Anwar et al ., 2022), epidemiology (Patra, 2023) and COVID-19 epidemic (Laib et al ., 2022). With the involvement of delay phenomenon or known as a time-lag force in a model, a system can govern the physical behaviour of this model at previous-present time.…”

Section: Introductionmentioning

confidence: 99%

A generic numerical method for treating a system of Volterra integro-differential equations with multiple delays and variable bounds

Kürkçü,

Sezer

2024

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PurposeThis study aims to treat a novel system of Volterra integro-differential equations with multiple delays and variable bounds, constituting a generic numerical method based on the matrix equation and a combinatoric-parametric Charlier polynomials. The proposed method utilizes these polynomials for the matrix relations at the collocation points.Design/methodology/approachThanks to the combinatorial eligibility of the method, the functional terms can be transformed into the generic matrix relations with low dimensions, and their resulting matrix equation. The obtained solutions are tested with regard to the parametric behaviour of the polynomials with $\alpha$, taking into account the condition number of an outcome matrix of the method. Residual error estimation improves those solutions without using any external method. A calculation of the residual error bound is also fulfilled.FindingsAll computations are carried out by a special programming module. The accuracy and productivity of the method are scrutinized via numerical and graphical results. Based on the discussions, one can point out that the method is very proper to solve a system in question.Originality/valueThis paper introduces a generic computational numerical method containing the matrix expansions of the combinatoric Charlier polynomials, in order to treat the system of Volterra integro-differential equations with multiple delays and variable bounds. Thus, the method enables to evaluate stiff differential and integral parts of the system in question. That is, these parts generates two novel components in terms of unknown terms with both differentiated and delay arguments. A rigorous error analysis is deployed via the residual function. Four benchmark problems are solved and interpreted. Their graphical and numerical results validate accuracy and efficiency of the proposed method. In fact, a generic method is, thereby, provided into the literature.

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A multiple time renewal equation for neural assemblies with elapsed time model

Cited by 6 publications

References 47 publications

Convergence towards equilibrium for a model with partial diffusion

Convergence towards equilibrium for a model with partial diffusion

On quantitative hypocoercivity estimates based on Harris-type theorems

A generic numerical method for treating a system of Volterra integro-differential equations with multiple delays and variable bounds

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