A Space-Time Finite Element Method for Neural Field Equations with Transmission Delays

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Abstract

We present and analyze a new space-time nite element method for the solution
of neural eld equations with transmission delays. The numerical treatment of these systems is rare in the literature and currently has several restrictions on the spatial domain and the functionsinvolved, such as connectivity and delay functions. The use of a space-time discretization, with basis functions that are discontinuous in time and continuous in space (dGcG-FEM), is a natural way to deal with space-dependent delays, which is important for many neural eld applications. In
this paper we provide a detailed description of a space-time dGcG-FEM algorithm for neural delay equations, including an a priori error analysis. We demonstrate the application of the dGcG-FEM algorithm on several neural eld models, including problems with an inhomogeneous kernel.
Original languageEnglish
Pages (from-to)B797–B818
JournalSIAM journal on scientific computing
Volume39
Issue number5
DOIs
Publication statusPublished - 21 Dec 2017

Fingerprint

Space-time Finite Elements
Space-time
Finite element method
Delay Equations
Delay-dependent
Error Analysis
Error analysis
Basis Functions
Connectivity
Discretization
kernel
Restriction
Demonstrate
Model

Keywords

  • Neural fields
  • transmission delays
  • Discontinuous Galerkin finite element methods
  • space-time methods

Cite this

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title = "A Space-Time Finite Element Method for Neural Field Equations with Transmission Delays",
abstract = "We present and analyze a new space-time nite element method for the solutionof neural eld equations with transmission delays. The numerical treatment of these systems is rare in the literature and currently has several restrictions on the spatial domain and the functionsinvolved, such as connectivity and delay functions. The use of a space-time discretization, with basis functions that are discontinuous in time and continuous in space (dGcG-FEM), is a natural way to deal with space-dependent delays, which is important for many neural eld applications. Inthis paper we provide a detailed description of a space-time dGcG-FEM algorithm for neural delay equations, including an a priori error analysis. We demonstrate the application of the dGcG-FEM algorithm on several neural eld models, including problems with an inhomogeneous kernel.",
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A Space-Time Finite Element Method for Neural Field Equations with Transmission Delays. / Polner, Monika; van der Vegt, J.J.W.; van Gils, S.A.

In: SIAM journal on scientific computing, Vol. 39, No. 5, 21.12.2017, p. B797–B818.

Research output: Contribution to journalArticleAcademicpeer-review

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AB - We present and analyze a new space-time nite element method for the solutionof neural eld equations with transmission delays. The numerical treatment of these systems is rare in the literature and currently has several restrictions on the spatial domain and the functionsinvolved, such as connectivity and delay functions. The use of a space-time discretization, with basis functions that are discontinuous in time and continuous in space (dGcG-FEM), is a natural way to deal with space-dependent delays, which is important for many neural eld applications. Inthis paper we provide a detailed description of a space-time dGcG-FEM algorithm for neural delay equations, including an a priori error analysis. We demonstrate the application of the dGcG-FEM algorithm on several neural eld models, including problems with an inhomogeneous kernel.

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