Nernst-Planck-Gaussian modelling of electrodiffusional recovery from ephaptic excitation between mammalian cardiomyocytes

Abstract

Introduction: In addition to gap junction conduction, recent reports implicate possible ephaptic coupling contributions to action potential (AP) propagation between successive adjacent cardiomyocytes. Here, AP generation in an active cell, withdraws Na⁺ from, creating a negative potential within, ephaptic spaces between the participating membranes, activating the initially quiescent neighbouring cardiomyocyte. However, sustainable ephaptic transmission requires subsequent complete recovery of the ephaptic charge difference. We explore physical contributions of passive electrodiffusive ion exchange with the remaining extracellular space to this recovery for the first time. Materials and Methods: Computational, finite element, analysis examined limiting, temporal and spatial, ephaptic [Na⁺], [Cl^-], and the consequent Gaussian charge differences and membrane potential recovery patterns following a ΔV∼130 mV AP upstroke at physiological (37°C) temperatures. This incorporated Nernst-Planck formalisms into equations for the time-dependent spatial concentration gradient profiles. Results: Mammalian atrial, ventricular and purkinje cardiomyocyte ephaptic junctions were modelled by closely apposed circularly symmetric membranes, specific capacitance 1 μF cm^-2, experimentally reported radii a = 8,000, 12,000 and 40,000 nm respectively and ephaptic axial distance w = 20 nm. This enclosed an ephaptic space containing principal ions initially at normal extracellular [Na⁺] = 153.1 mM and [Cl^-] = 145.8 mM, respective diffusion coefficients D _Na = 1.3 $\times$ 10⁹ and D _Cl = 2 $\times$ 10⁹ nm²s^-1. Stable, concordant computational solutions were confirmed exploring ≤1,600 nm mesh sizes and Δt≤0.08 ms stepsize intervals. The corresponding membrane voltage profile changes across the initially quiescent membrane were obtainable from computed, graphically represented a and w-dependent ionic concentration differences adapting Gauss's flux theorem. Further simulations explored biological variations in ephaptic dimensions, membrane anatomy, and diffusion restrictions within the ephaptic space. Atrial, ventricular and Purkinje cardiomyocytes gave 40, 180 and 2000 ms 99.9% recovery times, with 720 or 360 ms high limits from doubling ventricular radius or halving diffusion coefficient. Varying a, and D _Na and D _Cl markedly affected recovery time-courses with logarithmic and double-logarithmic relationships, Varying w exerted minimal effects. Conclusion: We thereby characterise the properties of, and through comparing atrial, ventricular and purkinje recovery times with interspecies in vivo background cardiac cycle duration data, (blue whale ∼2000, human∼90, Etruscan shrew, ∼40 ms) can determine physical limits to, electrodiffusive contributions to ephaptic recovery.

Keywords: action potential propagation; cardiomyocytes; electrodiffusion; ephaptic conduction; sodium channels.

FIGURE 1

Background geometrical parameters describing the model. (A) The intercalated disc (a) and ephaptic junction (b) with three-dimensional arrangement of the intercalated disc (c) gap-junctions hold the membranes of two adjacent cells in close apposition with Na_v 1.5 β-subunits restricting the axial distance in other regions Reproduced from Salvage et al. (2020a), licensed under CC-BY 4.0. (B) Simplified physical scheme describing recovery processes at the ephaptic junction axial distance, w, radius, a containing Na⁺ and Cl⁻ with diffusion coefficients D _Na, D _Cl, following initial Na⁺ withdrawal consequent upon action potential generation (a). This lists electrodiffusive (b) and other, Na⁺-K⁺-ATPase (c) and Kir2.1 (d) mediated membrane transport contributions to ionic composition and consequent membrane potential in the ephaptic space.

FIGURE 2

Midline slice heat maps of concentration and voltage profiles close to the end of the action potential upstroke. (A–C) Heat maps of [Cl⁻] (A) and [Na⁺] (B) and resulting ion concentration and membrane potential differences (C) at different axial (vertical axis) and radial positions (horizontal axis) in the atrial ephaptic space. Slices obtained at end of a 0.7 ms period of sodium extrusion representing the cardiac action potential upstroke. Note the very small axial gradients generated. Given the size of these gradients and the time-period of recovery to allow dissipation of these gradients we have assumed a uniform concentration profile for the initial conditions. Computational mesh size 400 nm. Stepsize interval 0.01 ms. (D, E) Series of midline slice heat maps at successive time points of recovery (left axis) over the first millisecond of recovery beginning from the initial conditions derived from the Na⁺ transfer established in (A–C). (D) [Na⁺] and (E) ionic concentration differences. Scale bars on right: concentration differences (D, E) and membrane potential changes (E) at the passive membrane. Mesh size 400 nm; stepsize interval 0.001 ms.

FIGURE 3

Heat maps of atrial ephaptic recovery. (A) Radial distributions of [Na⁺] (a, b) and of charged ion concentration differences (c) at the start (a) and end (b, c) of the recovery period. (B, C) Midline slice heat maps of [Na⁺] (B) and of charged ion concentration differences (C) at critical time points (1.25 ms, 10 ms, and 80 ms) in the atrial ephaptic space during recovery.

FIGURE 4

Time and spatial dependences of the atrial ephaptic recovery process. (A) Midline slice heat maps at successive time points of recovery representing (a) [Na⁺] and (b) ionic concentration differences and membrane potential changes. (B) Quantification of the temporal and spatial recovery. (a) [Na⁺] spatial profiles with time; Recovery timecourses of (b) [Na⁺] and membrane potential and (c) ionic concentration differences at the ephaptic rim, ephaptic centre and half-way between the two.

FIGURE 5

Temporal and spatial [Cl ⁻ ] gradients through atrial ephaptic recovery. (A, B) Radial gradients at the outset (A) and 80 ms into (B) the recovery period. (C–E) [Cl⁻] through midline slices taken at successive timepoints (left axis) at the outset (C), 10 ms (D) and 80 ms (E) into the recovery period. (F, G) Sequence of midline slices at successive time points through the recovery period (left axis): (F) uses the same colour scale as (A–E). (G) uses an expanded colour range to permit demonstration of small changes in [Cl⁻] distribution.

FIGURE 6

Time and spatial dependences of the ventricular ephaptic recovery process. (A) Midline slice heat maps at successive time points of recovery representing (a) [Na⁺] and (b) ionic concentration differences and membrane potential changes. (B) Quantification of the temporal and spatial recovery: (a) [Na⁺] spatial profiles with time and (b, c) recovery timecourses of (b) [Na⁺] and membrane potential, and (c) ionic concentration differences at the ephaptic rim, ephaptic centre and half-way between the two, respectively.

FIGURE 7

Quantification of temporal and spatial ephaptic recovery at varying a and constant w, D _Na and D _Cl . (A) [Na⁺] spatial profiles with time; (B) Recovery timecourses of [Na⁺] and membrane potential and ionic concentration differences at the ephaptic rim, ephaptic centre and half-way between the two. Values of a varied through a = 4000 (a), 8000 (b), 12000 (c), 24000 (d) and 40000 nm (e) respectively.

FIGURE 8

Dependences of recovery half times upon a, w and D _Na and D _Cl . Relationships between recovery half times and: (A, B) ephaptic radius, a (A), and a ² (B), (C) atrial (black lines) and ventricular (red lines) axial distance, w, and (D) in double logarithmic plots, relative diffusion coefficients. (E, F) Small trends in atrial (E) and ventricular recovery half times (F) with axial distance, w, following magnification of the ordinate.