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Research ArticleResearch Article: Methods/New Tools, Novel Tools and Methods

Validating a Computational Framework for Ionic Electrodiffusion with Cortical Spreading Depression as a Case Study

Ada J. Ellingsrud, Didrik B. Dukefoss, Rune Enger, Geir Halnes, Klas Pettersen and Marie E. Rognes
eNeuro 1 April 2022, 9 (2) ENEURO.0408-21.2022; DOI: https://doi.org/10.1523/ENEURO.0408-21.2022
Ada J. Ellingsrud
1Department for Numerical Analysis and Scientific Computing, Simula Research Laboratory, Oslo 0164, Norway
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Didrik B. Dukefoss
2Letten Centre, Division of Anatomy, Department of Molecular Medicine, Institute of Basic Medical Sciences, University of Oslo, Oslo 0317, Norway
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Rune Enger
2Letten Centre, Division of Anatomy, Department of Molecular Medicine, Institute of Basic Medical Sciences, University of Oslo, Oslo 0317, Norway
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Geir Halnes
3CINPLA, Faculty of Mathematics and Natural Sciences, University of Oslo, Oslo 0316, Norway
4Institute of Physics, Faculty of Science and Technology, Norwegian University of Life Sciences, Ås 1432, Norway
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Klas Pettersen
5NORA, Faculty of Mathematics and Natural Sciences, University of Oslo, Oslo 0316, Norway
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Marie E. Rognes
1Department for Numerical Analysis and Scientific Computing, Simula Research Laboratory, Oslo 0164, Norway
6Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Bergen, Bergen 5020, Norway
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Figures

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  • Figure 1.
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    Figure 1.

    Model overview. The tissue is represented as a 1D domain of length 10 mm including neurons, ECS, and glial cells. Within each compartment, the model describes the dynamics of the volume fraction (α), the Na+, K+, Cl– and glutamate concentrations ([Na+],[K+],[Cl−],[Glu] ), and the potential (ϕ). Communication between the compartments occur via ionic and/or water membrane fluxes.

  • Figure 2.
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    Figure 2.

    Simulated CSD wave in the WT model (A) triggered by excitatory fluxes. The upper panels display a snapshot in time (plot of the respective field vs spatial coordinate x) of ECS glutamate (A), ECS ion concentrations (B), potentials (C), and change in volume fractions (D) at 60 s. The lower panels display time evolution of ECS glutamate (E), ECS ion concentrations (F), potentials (G), and change in volume fractions (H) at x = 2.0 mm.

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    Figure 3.

    Comparison of Model A (solid) and Model B (stippled) CSD wave. The upper panels display snapshots (plot of field vs spatial coordinate x) of ECS glutamate (A), ECS ion concentrations (B), potentials (C), and change in volume fractions (D) at 60 s. The lower panels display time evolution of ECS glutamate (E), ECS ion concentrations (F), potentials (G) and change in volume fractions (H) evaluated at x = 2.0 mm.

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    Figure 4.

    Comparison of electrical potentials during simulated CSD where the glial gap junction factor χg is reduced by 0% (Model A), 25%, 50%, 75%, and 100% (Model B). The panels display neuronal potentials (A), glial potentials (B), and ECS potentials (C) evaluated at x = 2.0 mm.

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    Figure 5.

    Comparison of Model A (solid) and Model C (stippled) CSD wave. The upper panels display snapshots (plot of field vs spatial coordinate x) of ECS glutamate (A), ECS ion concentrations (B), potentials (C), and change in volume fractions (D) at 60 s. The lower panels display time evolution of ECS glutamate (E), ECS ion concentrations (F), potentials (G), and change in volume fractions (H) evaluated at x = 2.0 mm.

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    Figure 6.

    Comparison of simulated CSD where the neuronal and glia membrane area-to-volume factors γne and γge are reduced by 0% (Model A), 25%, 50%, and 75% (Model C). The panels display ECS glutamate concentrations (A), ECS potassium concentrations (B), ECS potentials (C), change in ECS volume fractions (D), neuronal potentials (E), glial potentials (F), change in neuronal volume fractions (G), and change in glial volume fractions (H) evaluated at x = 2.0 mm.

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    Figure 7.

    Comparison of intervals (boxes) and min/max-ranges (whiskers) for experimentally and computationally measured values for relevant quantities in WT mice. Red lines indicate median values, and open circles denote outliers. The panels display the mean wave propagation speed (A), amplitude (B), and duration (C) of the DC shift, amplitude of elevated extracellular potassium (D), duration of elevated extracellular glutamate concentration (E), amplitude (F), and duration (G) of neuronal swelling, and amplitude of extracellular shrinkage (H).

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    Figure 8.

    Comparison of changes in volume fractions during simulated CSD where the glial water permeability ηge is reduced by 0% (Model A), 25%, 50%, 75%, and 100% (Model D). The panel displays the largest (over time) change in neuronal, glial, and ECS volume fractions relative to baseline at x = 2 mm for each of the reductions.

  • Figure 9.
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    Figure 9.

    Comparison of Model A (solid) and Model E (stippled) CSD wave. The upper panels display snapshots in time of ECS glutamate (A), ECS ion concentrations (B), potentials (C), and change in volume fractions (D) at 60 s. The lower panels display time evolution of ECS glutamate (E), ECS ion concentrations (F), potentials (G), and change in volume fractions (H) evaluated at x = 2.0 mm.

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    Figure 10.

    Comparison of glial dynamics without stimuli (A) and during induced CSD (B, C) where the Kir 4.1 expression is reduced by 0% (Model A), 10%, 20%, and 30% (Model E), 40%, 50%, 60%, 70%, and 80%. The panels display the resting glial potential (A), the glial potential during simulated CSD (B), and the change in glial volume fractions during simulated CSD (C) evaluated at x = 2.0 mm. Asterisks (*) indicate that CSD could not be induced successfully in the computational models.

Tables

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    Table 1

    Overview of the computational models with parameter values: χg: glial gap junction factor, γne: neuronal membrane area-to-volume, γge: glial membrane area-to-volume, ηge: glial membrane water permeability, gKir4.1 : glial Kir 4.1 resting conductance

    Modelχgγne (m– 1)γge (m– 1)ηge (m4/(mol s))gKir4.1 (S/m2)
    A0.055.38 × 1056.38 × 1055.40 × 10– 101.30
    B0––––
    C–1.35 × 1051.6 × 105––
    D–––0–
    E––––0.91
    • Model A corresponds to the default parameters (given in Extended Data 1, only three significant digits included here). The dash (–) indicates no change from the default values.

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    Table 2

    Summary of computational quantities of interest for different models (A, B, C, D, E)

    Quantity of interestABCDE
    Mean wave propagation speed (mm/min)5.845.523.195.835.27
    DC shift (mV)11.023.8010.1811.0210.16
    DC shift duration (s)3250863236
    Neuronal swelling (%)11.6911.675.6914.9311.81
    Glial swelling (%)7.137.053.7406.73
    ECS shrinkage (%)39.7939.5619.3237.3339.60
    Neuronal swelling duration (s)109111183115100
    Glial swelling duration (s)1031011810148
    ECS shrinkage duration (s)135135185136141
    ECS K+ elevation (mm)76.3976.4654.4075.3976.60
    ECS Glu elevation (mm)1.381.380.201.371.39
    ECS Cl– elevation (mm)21.1220.887.4323.1220.81
    ECS K+ elevation duration (s)2626602627
    ECS Glu elevation duration (s)2019252020
    ECS Cl– elevation duration (s)84841788790
    Neuronal membrane potential (mV)63.3663.3851.5163.3264.53
    Glial membrane potential (mV)55.1455.7746.5254.2448.23
    Neuronal membrane potential duration (s)79791588369
    Glial membrane potential duration (s)45409642160
    • Numerical errors are <1.5%.

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    Table 3

    Summary and overview of experimentally reported propagation speeds, DC shift amplitudes (DC) and their duration (DC dur.), peak in extracellular K+ levels (Peak [K+]e ), duration of increased relative changes in mean fluorescence (ΔFF dur.), alteration in neuronal volume fractions (Δαn) and their duration (Δαn dur.), and alteration in the ECS volume fractions (Δαe) during CSD from a selection of studies in either WT mice [Study (WT)] or AQP4 knock-out mice [Study (AQP4– /–)] measured (M) in vivo (IV) or in slices (S)

    ΔFF dur.
    SpeedDCDC dur.Peak [K+]e ΔαnΔαn dur.Δαe
    (mm/min)(mV)(s)(mm)(s)(%)(min)(%)
    Study (WT)M
    Lauritzen and Hansen (1992)IV3.8 ± 0.923 ± 6
    Theis et al. (2003)S3.8 ± 0.416.8 ± 1.158 ± 1538.6 ± 3
    ––4.4 ± 0.3
    Padmawar et al. (2005)IV4.4 ± 0.5
    Takano et al. (2007)IV66 ± 437.1 ± 0.18 – 10
    Chang et al. (2010)IV18.568
    Zhou et al. (2010)S1.56 ± 0.24†11.0 ± 0.9§5 – 7
    Thrane et al. (2013)IV18.71 ± 2.1175.54 ± 1.86
    Enger et al. (2015)IV3.34 ± 0.166.4 ± 3.818.6 ± 1.7
    Yao et al. (2015)IV3.70 ± 0.122 ± 1.831 ± 2.4*34.3 ± 0.870.6**
    ––48 ± 4.8*
    Enger et al. (2017)IV4.60 ± 0.266.7 ± 10.119.5 ± 1.3
    Kucharz et al. (2017)IV12.44 ± 0.65
    Study (AQP4– /–)M
    Thrane et al. (2013)IV12.80 ± 1.1664.49 ± 1.73
    Yao et al. (2015)IV2.90 ± 0.119.6 ± 1.438 ± 3.6*35.5 ± 0.772.6**
    ––58 ± 2.4*
    Enger et al. (2017)IV4.60 ± 0.286.0 ± 13.315.7 ± 1.2
    • * Duration measured at half-maximum amplitude; † speed reported in experiments with TTX, indicated to be 50% of the CSD wave speed without TTX; § deviation from baseline of high [K+]e perfusion; ** baseline ECS volume fraction differs between WT mice (0.18) and AQP4– /– mice (0.23).

Extended Data

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  • Extended Data 1

    Supplementary material. Download Extended Data 1, TEX file.

  • Extended Data 2

    Code. Download Extended Data 2, ZIP file.

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Validating a Computational Framework for Ionic Electrodiffusion with Cortical Spreading Depression as a Case Study
Ada J. Ellingsrud, Didrik B. Dukefoss, Rune Enger, Geir Halnes, Klas Pettersen, Marie E. Rognes
eNeuro 1 April 2022, 9 (2) ENEURO.0408-21.2022; DOI: 10.1523/ENEURO.0408-21.2022

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Validating a Computational Framework for Ionic Electrodiffusion with Cortical Spreading Depression as a Case Study
Ada J. Ellingsrud, Didrik B. Dukefoss, Rune Enger, Geir Halnes, Klas Pettersen, Marie E. Rognes
eNeuro 1 April 2022, 9 (2) ENEURO.0408-21.2022; DOI: 10.1523/ENEURO.0408-21.2022
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Keywords

  • computational modelling
  • cortical spreading depression
  • ionic electrodiffusion and osmosis
  • validation

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