The simplified Kirchhoff network model (SKNM): a cell-based reaction–diffusion model of excitable tissue

Figures

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  Figure 1. Illustration of the Kirchhoff network model (KNM). The model consists of a network of connected cells, each with a surrounding extracellular compartment. The model is derived by applying Kirchhoff’s current law for each cell and extracellular compartment.

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  Figure 2. Traveling wave solutions of KNM and SKNM for the default model set-up for a collection of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$40\times 40$$\end{document}40×40 hiPSC-CMs. The figure displays snapshots of the membrane potential solution at three points in time during the simulation. In this case, the SKNM assumption (21) holds, and the solutions of the two models appears to be identical.

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  Figure 3. Comparison of the conduction velocity (CV) computed using KNM and SKNM for a collection of hiPSC-CMs with an increasing cell length to width ratio, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α (see the “Parameter variations” section). We consider the cases of 50%, 20%, 10%, and 2% extracellular space as indicated in the panel titles.

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  Figure 4. Comparison of the conduction velocity (CV) computed using KNM and SKNM for a collection of hiPSC-CMs with an increasing gap junction conductance variation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}γ (see the “Parameter variations” section). We consider the cases of 50%, 20%, 10%, and 2% extracellular space as indicated in the panel titles.

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  Figure 5. Sources of difference between KNM and SKNM. The left panel illustrates that the first factor of the SKNM approximation (40) increases as the gap junction conductance variation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}γ, increases, while the right panel illustrates that the second factor of (40) decreases as the extracellular volume fraction, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _e$$\end{document}δe increases, explaining the SKNM and KNM differences observed in Fig. 4. In the left panel, the extracellular space is 20% (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta…

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  Figure 6. Traveling wave solutions of BD and MD for the default model set-up for a collection of hiPSC-CMs. The figure displays snapshots of the membrane potential solution at three points in time during the simulation. In this case, the MD assumption (21) holds, and the solutions of the two models appears to be identical.

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  Figure 7. Comparison of the conduction velocity (CV) computed using BD and MD for a collection of hiPSC-CMs with an increasing cell length to width ratio, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α (see the “Parameter variations” section). We consider the cases of 50%, 20%, 10%, and 2% extracellular space as indicated in the panel titles.

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  Figure 8. Comparison of the conduction velocity (CV) computed using BD and MD for a collection of hiPSC-CMs with an increasing gap junction conductance variation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}γ (see the “Parameter variations” section). We consider the cases of 50%, 20%, 10%, and 2% extracellular space as indicated in the panel titles.

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  Figure 9. Sources of difference between BD and MD. The left panel illustrates that the first factor of the MD approximation (43) increases as the gap junction conductance variation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}γ, increases, while the right panel illustrates that the second factor of (43) decreases as the extracellular volume fraction, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _e$$\end{document}δe increases, explaining the MD and BD differences observed in Fig. 8. In the left panel, the extracellular space is 20% (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _e=0.2$$\end{document}δe=0.2) and…

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  Figure 10. Snapshots of the membrane potential, v, computed using KNM and SKNM for the default pancreatic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β cell collection set-up.

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  Figure 11. Snapshots of the membrane potential, v, computed using KNM and SKNM for a collection of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β cells with a variable gap junction coupling, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =1$$\end{document}γ=1 (see the “Parameter variations” section), and 2% extracellular volume.

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  Figure 12. Comparison of the conduction velocity (CV) computed using KNM and SKNM for a collection of pancreatic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β cells with an increasing gap junction conductance variation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}γ (see the “Parameter variations” section). We consider the cases of 50%, 20%, 10%, and 2% extracellular space as indicated in the panel titles.