We consider N domains Ωin⊂ℝd (d = 1, 2, 3) for n = 1, …, N representing disjoint intracellular regions (physiological cells, e.g., neurons) and an extracellular region Ω_e, and let the complete domain Ω=Ωi1∪⋯∪ΩiN∪Ωe with boundary ∂Ω. See Figure 1 (Right) for an illustration of a sample domain configuration. We denote the cell membrane associated with cell iⁿ, i.e., the boundary of the physiological cell Ωin, by Γ_n. We assume that Γ_n ∩ Γ_m = ∅ for all n≠m and that Γ_n ∩ ∂Ω = ∅ (It follows that ∂Ωin∩∂Ωe=∅ for all n = 1, …, N.). For simplicity and clarity, we present the mathematical model for one intracellular region Ωi1=Ωi with membrane Γ below. The extension to multiple intracellular regions is immediate (but notationally cumbersome).

We will here derive a system of coupled, time-dependent, non-linear partial differential equations to describe ionic electrodiffusion in this domain. We consider a set of ion species K. Typically K will include sodium Na⁺, potassium K⁺, and chloride Cl^-. For each ion species k ∈ K and each region r ∈ {i, e}, we model the ion concentrations [k]_r : Ω_r × (0, T] → ℝ (mol/m³) and the electrical potentials ϕ_r : Ω_r × (0, T] → ℝ (V). Conservation of ions for the bulk of each region Ω_r stipulates that

for t ∈ (0, T]. Here, Jrk:Ωr×(0,T]→ℝd is the regional ion flux density [mol/(m²s)] of ion k. To proceed, we invoke the KNP assumption of bulk electroneutrality. In this case, the ion flux densities Jrk satisfy:

where z^k is the valence of ion species k and F is Faraday's constant. The assumption (2) states that the total net flow of ions (weighted by the respective valences) out of any infinitesimal representative bulk volume is zero. Furthermore, we assume that the each regional ion flux density can be expressed by a Nernst-Planck equation as follows:

Here, Drk denotes the effective diffusion coefficient (m²/s) of ion species k in the region r. The constant ψ = RTF⁻¹ combines Faraday's constant F, the absolute temperature T, and the gas constant R. The ion flux density, i.e., the flow rate of ions per unit area, is thus modeled as the sum of two terms: (i) the diffusive movement of ions due to ionic gradients -Drk∇[k]r and (ii) the ion concentrations that are transported via electrical potential gradients, i.e., the ion migration -Drkzkψ-1[k]r∇ϕr where Drkψ-1 is the electrochemical mobility. This model ignores convective effects, and thus assumes that the underlying material (typically fluid) is at rest. As the potential ϕ_e is only determined up to a constant in Equations (1–3)„ an additional constraint is required, e.g.,

By inserting (3) into (2) we recognize (from volume conductor theory) the following expression for the bulk conductivity σ_r:

Notably, the bulk conductivity σ_r depends on the ion concentrations [k]_r and the diffusion coefficients Drk. Electrodiffusive models without explicit modeling of the ion concentrations typically set the bulk conductivity as a independent parameter (e.g., Krassowska and Neu, 1994; Bokil et al., 2001; Tveito et al., 2017a).

Inserting (3) into (1) and into (2), we thus obtain a system of N|K|+N equations (N|K| parabolic, N elliptic) for the N|K|+N unknown scalar fields. The system remains to be closed by appropriate initial conditions, boundary conditions, and importantly interface conditions.

We next turn to modeling the cell membrane currents and membrane potential across the interface Γ. We denote the membrane potential ϕ_M as the jump in the electrical potential over the membrane:

We introduce the total ionic current density I_M : Γ × (0, T) → ℝ [C/(m²s)] across the interface Γ. By definition and by conservation of total charge, we have that

where n_r denotes the boundary normal pointing out of Ω_r for r ∈ {i, e}. Next, we assume that I_M consists of two components: (i) a total channel current I_ch and (ii) a capacitive current Icap:

We further assume that the total channel current I_ch is the sum of the ion specific channel currents Ichk:

The channel currents Ichk are subject to modeling. Typical models for Ichk notably includes an synaptic input current Isyn, leaky passive neuron, Hodgkin-Huxley etc., and will be detailed further below in section 2.1.4. On the other hand, the capacitive current Icap is defined over to be the capacitance C_M times the rate of change of the voltage (Sterratt et al., 2011), hence:

Inserting (10) into (8) and rearranging gives the following relation for the membrane potential ϕ_M:

It remains to specify a set of interface conditions for the specific ion fluxes Jrk·nr for r ∈ {i, e}. Here, we propose a heuristic approach via ion specific capacitive current modeling. An alternative approach is presented in Mori and Peskin (2009). As for the total current, we assume that the capacitive current can be represented as a sum of ion-associated currents:

Without loss of generality, we let the ion specific capacitive current Icap, r^k in region Ω_r at the interface Γ be some fraction αrk of the total capacitive current Icap:

Specifically, we assume that:

and note that ∑k∈Kαrk=1 for r ∈ {i, e}. By definition of the ion currents and the expression for the capacitive current given by (8), we let the intracellular and extracellular ion fluxes across the membrane be given by:

for k ∈ K.

The framework presented thus far allows for general representations of the ion channel current dynamics. In particular, the framework admits different choices of ion specific channel current models Ichk. An advantage of the geometrically explicit framework is that it allows for different channel currents models for individual cells and e.g., geometrically heterogeneous material properties. We here summarize two examples of ion specific channel currents: a passive membrane model (Sterratt et al., 2011) and the Hodgkin-Huxley model (Hodgkin and Huxley, 1952).

We assume that initial conditions are given for all ion concentrations, both intracellularly and extracellularly:

Furthermore, we assume that these conditions are compatible with the assumption of bulk electroneutrality, i.e., that the initial state of the system satisfies:

In addition, we assume that an initial condition is given for the membrane potential:

Finally, a set of boundary conditions will close the system. We describe specific boundary conditions in the numerical experiments in section 2.3.

In summary, the mathematical framework for electrodiffusion with explicit geometrical representation of the cell membranes is comprised of the bulk equations (1), (2) with (3), the interface conditions (7), (11) with (6) and (9), and (15) with (14), the initial conditions (24) and (26), and additional boundary conditions. We will refer to this set of equations as the KNP-EMI framework.

Multiplying (1) with test functions vrk (for r ∈ {i, e}), integration over the intracellular and extracellular domains Ω_i and Ω_e separately, integration by parts, and inserting (15) for the ion fluxes across the membrane, yields

Similarly, multiplying (2) by test functions w_r for r ∈ {i, e}, integration by parts and inserting (7) for the total membrane current, yields

The zero average constraint (4) is enforced by introducing an additional unknown (a Lagrange multiplier) c_e ∈ ℝ along with a test function d_e ∈ R and letting

Finally, multiplying (11) by a test function q, and integrating over Γ yields

We remark that this is a weak formulation of a set of time-dependent, non-linear equations. In particular, recall that I_ch and Ichk depend on ϕ_M and [k]_r cf. (9) while αrk depends on [k]_r cf. (14).

To solve this system numerically, we consider the following approximations.

Moreover, we evaluate I_ch and the discretization of (32) depending on the choice of ion channel model (cf. section 2.1.4) as follows.

The steps are repeated until global end time t^N is reached.

To numerically solve the PDE part of the governing equations defined on the domain Ω = Ω_i∪Ω_e, we use a mortar finite element method. We discretize each subdomain Ω_r by a conforming mesh Tr for r ∈ {i, e}. We assume that the meshes Ti and Te match at the common interface Γ, and define a (lower-dimensional) mesh TΓ of this interface (cf. Figure 2).

Next, we introduce separate finite element spaces for approximating the unknown fields in the weak formulation (27)–(32), [k]_r : Ω_r → ℝ, ϕ_r : Ω_r → ℝ for r ∈ {i, e}, I_m : Γ → ℝ. We approximate the ion concentrations [k]_r and potentials ϕ_r using continuous piecewise linear polynomials (linear Lagrange finite elements) over the meshes Tr. These fields thus have degrees of freedom defined on the vertices of the extracellular and intracellular meshes. The Lagrange multiplier c_e is approximated using a single real number. Furthermore, the transmembrane current I_M is approximated using continuous piecewise linear polynomials over the facet mesh TΓ. We denote the finite element spaces for approximating [k]_r by Vrk, the spaces for approximating ϕ_r by W_r and the spaces for approximating I_M by Q. Let 〈u,v〉Ω=∫Ωuv dx. For notational simplicity, we denote the approximation of [k]_r by [k]_r, the approximation of ϕ_r by ϕ_r, and the approximation of I_M by I_M below. We here use linear polynomials for concreteness, but the formulation also applies directly for higher order polynomials.

We then solve the PDE step in the two-step splitting scheme described in section 2.2.1 as follows: given [k]rn∈Vrk and ϕMn∈Q at time step tⁿ, and the previously computed Ichk and α^k [cf. (35) and (33)], find the ion concentrations [k]r∈Vrk, the potentials ϕ_r ∈ W_r, the total transmembrane current density I_M ∈ Q at time step tⁿ⁺¹ (and the Lagrange multiplier c_e ∈ ℝ) such that:

for all vik∈Vik, vek∈Vek, w_i ∈ W_i, w_e ∈ W_e, d_e ∈ ℝ and q ∈ Q. The ion flux terms on the right-hand side are replaced by appropriate boundary conditions in the subsequent sections.

To evaluate the accuracy of the numerical solutions defined over Ω_r for r ∈ {i, e}, we use the standard L² and H¹ norms denoted by ||·||₀ and ||·||₁, respectively: for u : Ω_r → ℝ,

In addition, for I : Γ → ℝ, we define the broken L²-norm by summing over the L²-norms over the mesh cells of the interface mesh TΓ:

The numerical scheme was implemented using a mixed dimensional framework from the FEniCS finite element library (Alnæs et al., 2015). The linear systems arising in the numerical experiments were solved using a direct (MUMPS) solver. The code is publicly available at: https://zenodo.org/record/3492075#.XahQOhh9g5k.

In the numerical experiments comparing the KNP-EMI and the EMI models, the EMI model is discretized using the mortar finite element formulation as presented in Tveito et al. (2017b).

For Model A, we consider a two-dimensional domain Ω=Ωi∪Ωe=[0,6.0·10-5]×[0,6.0·10-5] m, with one intracellular domain (cell) Ωi=[6.0·10-6,5.6·10-5]×[2.8·10-5,3.4·10-5] m. We mesh this domain by dividing the domain into n × m rectangles, with Δx = 6.0·10⁻⁵/n and Δy = 3.0·10⁻⁵/m, and dividing each rectangle into two triangles by a diagonal, for a series of Δx = Δy = 2.0·10⁻⁶, 1.0·10⁻⁶, 5.0·10⁻⁷, 2.5·10⁻⁷ m. We model I_ch using the passive model, as described in section 2.1.4, and prescribe a synaptic input I_syn of the form

where α is the synaptic time constant, H(x) = {1 for x ∈ Z and 0 elsewhere} for an interval Z. We let Z = [5.0 · 1.0⁻⁵, 1.0 · 10⁻⁵] m, and set t₀ = 0, gsyn=1.25·103 S/m². At the exterior boundary ∂Ω, we apply the boundary condition

describing that no ions can leave or enter the system.

To evaluate the numerical accuracy of the mortar finite element scheme presented in section 2.2, we construct an analytical solution using the method of manufactured solutions (Roache, 1998). In particular, we let the analytical solution to (1)–(15) be given by:

with the passive model I_ch = ϕ_M and with Isyn = 0. We assume that the parameter values all equal one: Cm=Dik=Dek=F=G=R=1, and that K = {Na⁺, K⁺, Cl⁻}. We consider a two-dimensional domain Ω = Ω_i∪Ω_e = [0, 1] × [0, 1], with one intracellular domain Ω_i = [0.25, 0.75] × [0.25, 0.75]. The domain is meshed as for Model A (cf. section 2.3.1) for a series of n = m = 8, 16, 32, 64, 128, 256. In the numerical experiments for this test case, we initially let Δt=164·10-5, and then quarter the timestep in each series. The errors are evaluated at t=264·10-5.

For Model C, we define three different two-dimensional domains: (C1) a domain with one intracellular region (cell), (C2) a domain with two intracellular regions with a distance of 4.0·10⁻⁶m in the y-direction between the cells, and (C3) a domain with two intracellular regions with a distance of 1.0·10⁻⁵m in the y-direction between the cells. More precisely, we let

The ion channel currents Ichk are modeled using the passive model described in section 2.1.4. The synaptic input current model (37) is applied with gsyn=1.25·103 S/m², t₀ = 0, and with Z = [3.5·10⁻⁵, 4.0·10⁻⁵] m for Model C1, Z1=[6.0·10-5,6.5·10-5] m for Model C2, and Z2=[5.5·10-5,6.0·10-5] m for Model C3. At the exterior boundary ∂Ω, we apply the boundary condition (38). In order to compare the KNP-EMI and the EMI framework, we set the bulk conductivity σ_r in the EMI model by (5) with initial values [k]r0 for the ion concentration [k]_r. Note that σ_r will generally change over time.

For Model D, we consider a domain Ω=Ωi1∪⋯∪Ωi9∪Ωe=[0,4.0·10-4]·[0,1.4·10-6]×[0,1.4·10-6] m, where 9 cuboidal cells of size 3.9·10⁻⁴ × 2.0·10⁻⁷ × 2.0·10⁻⁷ m are placed uniformly throughout Ω (cf. Figure 1). The distance between the cells is 1.0·10⁻⁷ m. The domain is meshed as in section 2.3.1 with Δy = Δz = 1.0·10⁻⁷ m and Δx = 1.25·10⁻⁶ m. The ion channel currents are modeled using the Hodgkin-Huxley model as described in section 2.1.4. An action potential is induced every 20 ms throughout the simulations by applying the synaptic input current model (37) with g_syn = 40 S/m², α = 0.002 s, and t₀ = 0, 0.02, 0.04 s. We ran two sets of simulations: (1) stimulating, i.e., applying the synaptic current to the membrane of, the middle axon only (axon A in Figure 1), and (2) stimulating the 8 axons around axon A (axons B,C in Figure 1). At the exterior boundary, we apply the boundary condition (38).

The extracellular potential and sodium (Na⁺) concentration of Model A for four different mesh resolutions are shown at t = 10 ms in Figure 3. The system quickly (after 3 ms) reaches a semi-steady state where the membrane potential does not change notably over time, but there is a slow exchange of sodium (Na⁺) and potassium (K⁺) ions due to the leak currents. The extracellular sodium concentration does not appear to change visibly under mesh refinement, and the extracellular potential seems to reach a converged state for the finest mesh resolution. More precisely, the mean relative difference (over time) between the solutions for the finest mesh (Δx = 0.25) and the coarser mesh resolutions Δx = 2.0, Δx = 1.0, and Δx = 0.5 are 4.8·10⁻⁵, 4.6·10⁻⁶, and 1.6·10⁻⁶ for [Na]_e, respectively, and 2.0·10⁻², 2.3·10⁻³, and 4.3·10⁻⁴ for ϕ_e, respectively, at the point (4, 31). We conclude the differences between the solutions are small and decreasing, indicating convergence of the method.

Using Model B, we analyzed the rates of convergence for the approximations of all solution variables under refinement in space and time. Based on properties of the approximation spaces and the time discretization, the optimal theoretical rate of convergence is 1 in the H¹-norm and 2 in the L²-norm and the broken L²-norm. Our numerical findings (Table 2) are in agreement with the theoretically optimal rates. We observe second order convergence in the L²-norm for the approximation of the extracellular and intracellular concentrations and potentials, and first order convergence in the H¹-norm. For the transmembrane ionic current I_M, we observe a convergence rate of 1.5 in the broken L²-norm. The loss of convergence of ~0.5 for I_M is likely due to a lack of smoothness of the interface in the test domain.

To examine whether the size of the extracellular space (ECS) affects the solution near the axon, we consider Model C1 with both the default size of the ECS (120 × 120 μm) and with an extended ECS (240 × 240 μm). The axon is placed in the same position in both cases. The two cases do not differ substantially in the extracellular Na⁺ concentrations or the extracellular potentials near the axon (Figure 4). The maximal difference between the default model and the model with extended extracellular space, measured 2μm above the axon at t = 10 ms, for ϕ_e and [Na]_e are 6.33·10⁻³ mV and 5.72·10⁻⁶ mM, respectively.

To investigate ephaptic coupling, we first apply a synaptic current to stimulate the cell membrane of the middle axon of the axon bundle (axon A, Figure 1, Model D). The synaptic current induces a series of action potentials in axon A and also induces substantial changes in the surrounding extracellular potential (Figures 7A,B). The extracellular potential fluctuations further spread to axon B. However, the ephaptic effect on the membrane potential in axon B is relatively small (1–2 mV), and is not sufficient to reach the threshold for inducing an action potential (Figure 7C).

The ephaptic effect is stronger if we simultaneously stimulate the cell membranes of all eight peripheral axons (axons B–C). Again, we observe a series of action potentials in the eight stimulated axons. Moreover, the combined ephaptic currents have a pronounced excitatory effect on axon A, but again fail to induce an action potential there (Figure 7D).

The difference between the EMI and KNP-EMI simulations are due to the time evolution of the intracellular and extracellular ion concentrations, accounted for by the KNP-EMI model but not by the EMI model. For each action potential fired, the Nernst potential will change due to alterations in the ionic concentrations using the KNP-EMI framework (Figure 8), whereas in the EMI framework the Nernst potential is constant.

Our predictions differ from those made in a similar study by Bokil et al. (2001), who found that a single active neighbor can induce action potentials in all nearby axons. We hypothesize that the main explanation for these differences is that the bulk conductivities differ between the two studies. Here, in the KNP-EMI framework, the bulk conductivities are functions of the ion concentrations [cf. (5)]. Using realistic values for the intra- and extracellular ion concentrations, we obtained bulk conductivities values of σ_i≈2.01μS/μm and σ_e≈1.31 S/m. In contrast, Bokil et al. set the bulk conductivities as free parameters, with σ_i = 1S/m and σ_e = 0.1S/m as the corresponding effective bulk conductivities in the EMI model. Tveito et al. (2017b) found that the ephaptic current was inversely proportional to σ_e, which suggests that the ephaptic current was more than seven times stronger in Bokil et al. (2001) than here.

In light of this, we repeated the simulations of the EMI model using the lower effective bulk conductivity values (σ_i = 1 S/m and σ_e = 0.1S/m). In this case, simultaneous stimulation of the 8 peripheral axons (B–C) induced an action potential in axon A (Figure 7F). Stimulation of axon A alone did not induce an action potential in the 8 peripheral axons (Figure 7E).