Well-posedness and discrete-to-continuum convergence of the Kirchhoff Network model
Abstract
In this work, we study the connection between the bidomain model of cardiac electrophysiology and the Kirchhoff Network model (KNM), that was recently introduced by Jäger and Tveito. In the KNM, each cell of the cardiac tissue is represented as a node in a discrete network and the dynamics of the electric potentials in the heart are described by a system of ordinary differential equations on the nodes. We first study the well-posedness of the KNM and then prove that, when the nodes are situated on the vertices of a regular lattice and the mesh size ε tends to zero, the solutions of the KNM system at cell-size ε converge to the solution of the bidomain model, along appropriate subsequences. Regarding the convergence, we adopt the framework presented by Pennacchio, Savaré, and Colli Franzone in "Multiscale modeling for the electrical activity of the heart", 2005, which is based on (a) the time-discretization of the evolution equations of the bidomain and ε-KNM systems via a semi-implicit Euler scheme, (b) the identification of the time-discrete solutions as extremal points of a Minimizing Movement scheme, and (c) the Γ-convergence of the Minimizing Movements functionals.
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BibTeX
@misc{oyarzun2026wellposedness,
title = {{Well-posedness and discrete-to-continuum convergence of the Kirchhoff Network model}},
author = {Joaquín Oyarzún and Marco Veneroni},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18351},
eprint = {2609.18351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18351},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Well-posedness and discrete-to-continuum convergence of the Kirchhoff Network model
AU - Joaquín Oyarzún
AU - Marco Veneroni
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18351
UR - https://arxiv.org/abs/2609.18351
AB - In this work, we study the connection between the bidomain model of cardiac electrophysiology and the Kirchhoff Network model (KNM), that was recently introduced by Jäger and Tveito. In the KNM, each cell of the cardiac tissue is represented as a node in a discrete network and the dynamics of the electric potentials in the heart are described by a system of ordinary differential equations on the nodes. We first study the well-posedness of the KNM and then prove that, when the nodes are situated on the vertices of a regular lattice and the mesh size ε tends to zero, the solutions of the KNM system at cell-size ε converge to the solution of the bidomain model, along appropriate subsequences. Regarding the convergence, we adopt the framework presented by Pennacchio, Savaré, and Colli Franzone in "Multiscale modeling for the electrical activity of the heart", 2005, which is based on (a) the time-discretization of the evolution equations of the bidomain and ε-KNM systems via a semi-implicit Euler scheme, (b) the identification of the time-discrete solutions as extremal points of a Minimizing Movement scheme, and (c) the Γ-convergence of the Minimizing Movements functionals.
ER -