Investigating the role of gap junctions in seizure wave propagation

Abstract

The effect of gap junctions as well as the biological mechanisms behind seizure wave propagation is not completely understood. In this work, we use a simple neural field model to study the possible influence of gap junctions specifically on cortical wave propagation that has been observed in vivo preceding seizure termination. We consider a voltage-based neural field model consisting of an excitatory and an inhibitory population as well as both chemical and gap junction-like synapses. We are able to approximate important properties of cortical wave propagation previously observed in vivo before seizure termination. This model adds support to existing evidence from models and clinical data suggesting a key role of gap junctions in seizure wave propagation. In particular, we found that in this model gap junction-like connectivity determines the propagation of one-bump or two-bump traveling wave solutions with features consistent with the clinical data. For sufficiently increased gap junction connectivity, wave solutions cease to exist. Moreover, gap junction connectivity needs to be sufficiently low or moderate to permit the existence of linearly stable solutions of interest.

Appendices

A Appendix: Model motivation

A biophysically based and widely used model for describing the somatic membrane potential $V_i(t)$ of a single neuron is Ermentrout (1998); Bressloff (2012):

$$\begin{aligned} C\frac{\mathrm{d}V_i}{\mathrm{d}t}=-I_{\text {con}}+I_{\text {syn}}+I_{\text {gap}}+I_{\text {ext}} \end{aligned}$$

(17)

where C is the cell capacitance, $I_{\text {con}}$ is the membrane current, $I_{\text {syn}}$ denotes the (chemical) synaptic input currents entering the cell, $I_{\text {gap}}$ denotes gap junction input currents and $I_{\text {ext}}$ are any external currents. For the moment, we ignore the external currents.

The membrane current is modeled by Ohm’s Law and determined by the term $I_{\text {con}}$=$\sum _{k} g_k(V-~E_{k})$, where each k determines a specific ion diffusing through channels in the cell membrane, and $g_k$ and $E_k$ determine the gating dynamics and reversal potential of the kth channel, respectively.

We assume that the net chemical synaptic input into neuron i from a population of neurons is determined by $I_{\text {syn}}= \sum _j \sum _m g_{j}(t-T^m_j)(V_{\text {syn}}-V(t))$, where $g_{j}$ represents the synaptic dynamics determined by presynaptic neuron j, $V_{\text {syn}}$ is the synaptic reversal potential, and $T^m_j$ determines a distribution of firing times of neuron j. Here, we consider no synaptic depression and ignore dendritic architecture. Also, we assume that the chemical synaptic inputs sum linearly.

Gap junctions allow direct diffusion of ions and small molecules between adjacent cells (Goodenough and Paul 2009). Therefore, we assume that the gap junctions are also modeled by Ohm’s Law in a diffusive manner $I_{\text {gap}}=\sum _j \frac{1}{R} (V_j-V_i)$ where R is the resistance.

Therefore, we obtain from (17):

$$\begin{aligned} \begin{aligned} C\frac{\mathrm{d}V_i}{\mathrm{d}t}&=-I_{\text {con}}+\sum _j \sum _m g_{j}(t-T^m_j)(V_{\text {syn}}-V(t))\\&\quad +\sum _j \frac{1}{R}(V_j-V_i) \end{aligned} \end{aligned}$$

(18)

We observe that the term $I_{\text {gap}}$ resembles the discretization of a one-dimensional second spatial derivative (similar to Steyn-Ross et al. 2007). That is, considering three aligned neurons $j-1$, j and $j+1$, the input from gap junction coupling on cell j is

$$\begin{aligned} \begin{aligned}&\frac{1}{R}\left( \left( V_{j-1}-V_j\right) +\left( V_{j+1}-V_j\right) \right) \\&\quad = \frac{1}{R}\left( V_{j-1}-2V_j+V_{j+1}\right) \\&\quad =\frac{(\varDelta x)^2}{R}\frac{\left( V_{j-1}-2V_j+V_{j+1}\right) }{(\varDelta x)^2}\\&\quad \approx D_j^2 \frac{\partial ^2 V_j}{\partial x^2} \end{aligned} \end{aligned}$$

(19)

Without the inclusion of gap junctions, (18) can be further reduced. This is done by considering a temporal averaging of (18) to obtain a closed system of integral equations to later reduce to Wilson–Cowan or Amari-type equations (Ermentrout 1998; Bressloff 2012). The potential $V_i$ can be considered to be converted into a firing rate by means of a nonlinear function. This can help to establish the potential in a network of neurons as a set of Volterra equations that can be further reduced into the voltage-based model (20). In this reduction, the variable $u_e$ now accounts for a mean variable denoting the activity of a neuronal population. We note that model (20) does not include action potentials. For this reduction to happen, it is necessary to include the assumption of a slowly acting synaptic current. However, we have mentioned that gap junctions are faster-acting than chemical synaptic current, so this assumption is not valid for gap junctions. We suggest that this may be rectified by considering instead an anomalous diffusion. As a step in this direction, we propose a simple model considering an excitatory and inhibitory population, together with both chemical and electrical synapses, where the electrical synapses are modeled by simple diffusion:

$$\begin{aligned} \begin{aligned} \frac{\partial u_e}{\partial t} \left( x,t \right)&= - \alpha _e u_e \left( x,t \right) + \alpha _e g_{ee} \otimes H(u_e (x,t)-k_e) \\&- \alpha _e g_{ie} \otimes H(u_i (x,t)-k_i) + D_\mathrm{e}^2\frac{\partial ^2 u_e}{\partial x^2}\left( x,t \right) \\ \frac{\partial u_i}{\partial t} \left( x,t \right)&= - \alpha _i u_i \left( x,t \right) + \alpha _i g_{ei} \otimes H(u_e (x,t)-k_e)\\&- \alpha _i g_{ii} \otimes H(u_i (x,t)-k_i) + D_i^2\frac{\partial ^2 u_i}{\partial x^2}\left( x,t \right) \\ \end{aligned} \end{aligned}$$

(20)

In this model, there is no dynamic component for the gap junction coupling and architecture of the gap junction distribution is not considered. However, we propose that this model is of interest for determining if wave propagation as observed in vivo preceding seizure termination is possible under a simple scenario. This could provide a useful foundation for developing more realistic models including gap junction coupling that mimic wave features observed in vivo.

B Appendix: Traveling wave solutions

We now provide a sketch of the derivation of the traveling wave solutions (3) and (4). We first consider system (1) in moving frame (z, t) where $z=x-ct$. We look for stationary solutions of this system such that $\frac{\partial u_j}{\partial t}(z,t)=0$, implying $u_j(z,t)=u_j(z)$ where $j=\{e,i\}$. We derive a Green’s function (Stakgold and Holst 2011; Evans 2010) that helps solve the inhomogeneous differential system that arises. The Green’s function that we derive has the following form:

$$\begin{aligned} G(z,s)= {\left\{ \begin{array}{ll} \frac{\alpha _j}{\sqrt{c^2+4\alpha _j D_j^2}} \exp \left( r_1 \left( z-s\right) \right) &{}\text {if } z \le s \\ \frac{\alpha _j}{\sqrt{c^2+4\alpha _j D_j^2}} \exp \left( r_2 \left( z-s\right) \right) &{}\text {if } s < z \end{array}\right. } \end{aligned}$$

(21)

where $r_1=\frac{-c+\sqrt{c^2+4 \alpha _j D_j^2}}{2 D_j^2}$, and $r_2=\frac{-c-\sqrt{c^2+4 \alpha _j D_j^2}}{2 D_j^2}$. This Green’s function solves the inhomogeneous system by considering:

$$\begin{aligned} u_j(z)= & {} \int _{-\infty }^{\infty } G(z,s)P(s)\hbox {d}y \end{aligned}$$

(22)

$$\begin{aligned} P(s)= & {} \int _{-\infty }^{\infty }g_{ej}(s-y)H(u_e(y)-k_e)\nonumber \\&-g_{ij}(s-y)H(u_i(y)-k_i)\hbox {d}y \end{aligned}$$

(23)

where $g_{jk}=\frac{1}{2\sigma _{jk}}\exp {\left( -\frac{\mid x \mid }{\sigma _{jk}}\right) }$ and H is the Heaviside function. Assuming $u_j(y)>k_j$ for $w_{j0}<y<w_{jf}$ we obtain:

$$\begin{aligned} P(s)=\int _{s-{w_{ef}}}^{s-w{e0}}g_{ej}(y)\hbox {d}y -\int _{s-w{if}}^{s-w{i0}}g_{ij}(y)\hbox {d}y \end{aligned}$$

(24)

Assuming $w_{i0}\le w_{e0} \le w_{if} \le w_{ef}$ and substituting (21) and (24) into (2223) we obtain the traveling wave solutions (3) and (4):

$$\begin{aligned} u_j^{\star } (z)= {\left\{ \begin{array}{ll} u_{j1} &{}\text {if } z \le w_{i0} \\ u_{j2} &{}\text {if } w_{i0}< z \le w_{e0} \\ u_{j3} &{}\text {if } w_{e0}< z \le w_{if} \\ u_{j4} &{}\text {if } w_{if} < z \le w_{ef} \\ u_{j5} &{}\text {if } z > w_{ef} \end{array}\right. } \end{aligned}$$

(25)

where

$$\begin{aligned} u_{j1}(z)= & {} \biggl ( \frac{\alpha _j}{2 \sqrt{c^2+4 \alpha _j D_j^2}}\biggr )\biggl [\exp \biggl (\frac{z-w_{e0}}{\sigma _{ej}}\biggr )\nonumber \\&\times \biggl ( \frac{\sigma _{ej}}{1-\sigma _{ej}r_2}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_1} \biggr )\nonumber \\&+ \exp \biggl (\frac{z-w_{ef}}{\sigma _{ej}}\biggr )\biggl ( \frac{\sigma _{ej}}{1-\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_2} \biggr ) \nonumber \\&+\exp \biggl (\frac{z-w_{if}}{\sigma _{ij}}\biggr )\biggl ( \frac{\sigma _{ij}}{1-\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1-\sigma _{ij}r_1} \biggr ) \nonumber \\&+\exp (\frac{z-w_{i0}}{\sigma _{ij}})\biggl ( \frac{\sigma _{ij}}{1-\sigma _{ij}r_1}-\frac{\sigma _{ij}}{1-\sigma _{ij}r_2} \biggr )\nonumber \\&+\exp \biggl (r_1(z-w_{e0})\biggr )\biggl (\frac{\sigma _{ej}}{1-\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1+\sigma _{ej}r_1} +\frac{2}{r_1} \biggr )\nonumber \\&+\exp \biggl (r_1(z-w_{ef})\biggr )\biggl (\frac{\sigma _{ej}}{1+\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_1} -\frac{2}{r_1} \biggr ) \nonumber \\&+ \exp \biggl (r_1(z-w_{i0})\biggr )\biggl (\frac{\sigma _{ij}}{1+\sigma _{ij}r_1}-\frac{\sigma _{ij}}{1-\sigma _{ij}r_1} -\frac{2}{r_1} \biggr ) \nonumber \\&+\exp \biggl (r_1(z-w_{if})\biggr )\biggl (\frac{\sigma _{ij}}{1-\sigma _{ij}r_1}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_1} +\frac{2}{r_1} \biggr )\biggr ] \nonumber \\ \end{aligned}$$

(26)

$$\begin{aligned} u_{j2}(z)= & {} \biggl (\frac{\alpha _j}{2 \sqrt{c^2+4 \alpha _j D_j^2}}\biggr )\biggl [\exp \biggl (\frac{z-w_{e0}}{\sigma _{ej}}\biggr )\nonumber \\&\times \biggl ( \frac{\sigma _{ej}}{1-\sigma _{ej}r_2}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_1} \biggr )\nonumber \\&+ \exp \biggl (\frac{z-w_{ef}}{\sigma _{ej}}\biggr )\biggl ( \frac{\sigma _{ej}}{1-\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_2} \biggr )\nonumber \\&+\exp \biggl (\frac{z-w_{if}}{\sigma _{ij}}\biggr )\biggl ( \frac{\sigma _{ij}}{1-\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1-\sigma _{ij}r_1} \biggr ) \nonumber \\&+\exp (\frac{-(z-w_{i0})}{\sigma _{ij}})\biggl ( \frac{\sigma _{ij}}{1+\sigma _{ij}r_1}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_2} \biggr )\nonumber \nonumber \\&+ \exp \biggl (r_1(z-w_{e0})\biggr )\biggl (\frac{\sigma _{ej}}{1-\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1+\sigma _{ej}r_1} +\frac{2}{r_1} \biggr ) \nonumber \\&+\exp \biggl (r_1(z-w_{ef})\biggr )\biggl (\frac{\sigma _{ej}}{1+\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_1} -\frac{2}{r_1} \biggr )\nonumber \\&+ \exp \biggl (r_2(z-w_{i0})\biggr )\biggl (\frac{\sigma _{ij}}{1+\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1-\sigma _{ij}r_2} -\frac{2}{r_2} \biggr )\nonumber \\&+\exp \biggl (r_1(z-w_{if})\biggr )\biggl (\frac{\sigma _{ij}}{1-\sigma _{ij}r_1}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_1} +\frac{2}{r_1} \biggr )\nonumber \\&+\biggl ( \frac{2}{r_2}-\frac{2}{r_1}\biggr )\biggr ]\nonumber \\ u_{j3}(z)= & {} \biggl ( \frac{\alpha _j}{2 \sqrt{c^2+4 \alpha _j D_j^2}}\biggr )\biggl [\exp \biggl (\frac{-(z-w_{e0})}{\sigma _{ej}}\biggr )\nonumber \\&\times \biggl ( \frac{\sigma _{ej}}{1+\sigma _{ej}r_2}-\frac{\sigma _{ej}}{1+\sigma _{ej}r_1} \biggr )\nonumber \\&+ \exp \biggl (\frac{z-w_{ef}}{\sigma _{ej}}\biggr )\biggl ( \frac{\sigma _{ej}}{1-\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_2} \biggr )\nonumber \\&+\exp \biggl (\frac{z-w_{if}}{\sigma _{ij}}\biggr )\biggl ( \frac{\sigma _{ij}}{1-\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1-\sigma _{ij}r_1} \biggr ) \nonumber \\&+\exp (\frac{-(z-w_{i0})}{\sigma _{ij}})\biggl ( \frac{\sigma _{ij}}{1+\sigma _{ij}r_1}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_2} \biggr )\nonumber \\&+ \exp \biggl (r_2(z-w_{e0})\biggr )\biggl (\frac{\sigma _{ej}}{1-\sigma _{ej}r_2}-\frac{\sigma _{ej}}{1+\sigma _{ej}r_2} +\frac{2}{r_2} \biggr ) \nonumber \\&+\exp \biggl (r_1(z-w_{ef})\biggr )\biggl (\frac{\sigma _{ej}}{1+\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_1} -\frac{2}{r_1} \biggr )\nonumber \\&+ \exp \biggl (r_2(z-w_{i0})\biggr )\biggl (\frac{\sigma _{ij}}{1+\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1-\sigma _{ij}r_2} -\frac{2}{r_2} \biggr )\nonumber \\&+\exp \biggl (r_1(z-w_{if})\biggr )\biggl (\frac{\sigma _{ij}}{1-\sigma _{ij}r_1}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_1} +\frac{2}{r_1} \biggr )\biggr ] \nonumber \\ \end{aligned}$$

(27)

$$\begin{aligned} u_{j4}(z)= & {} \biggl ( \frac{\alpha _j}{2 \sqrt{c^2+4 \alpha _j D_j^2}}\biggr )\biggl [\exp \biggl (\frac{-(z-w_{e0})}{\sigma _{ej}}\biggr )\nonumber \\&\times \biggl ( \frac{\sigma _{ej}}{1+\sigma _{ej}r_2}-\frac{\sigma _{ej}}{1+\sigma _{ej}r_1} \biggr )\nonumber \\&+ \exp \biggl (\frac{z-w_{ef}}{\sigma _{ej}}\biggr )\biggl ( \frac{\sigma _{ej}}{1-\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_2} \biggr )\nonumber \\&+\exp \biggl (\frac{-(z-w_{if})}{\sigma _{ij}}\biggr )\biggl ( \frac{\sigma _{ij}}{1+\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_1} \biggr ) \nonumber \\&+\exp (\frac{-(z-w_{i0})}{\sigma _{ij}})\biggl ( \frac{\sigma _{ij}}{1+\sigma _{ij}r_1}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_2} \biggr )\nonumber \\&+ \exp \biggl (r_2(z-w_{e0})\biggr )\biggl (\frac{\sigma _{ej}}{1-\sigma _{ej}r_2}-\frac{\sigma _{ej}}{1+\sigma _{ej}r_2} +\frac{2}{r_2} \biggr ) \nonumber \\&+\exp \biggl (r_1(z-w_{ef})\biggr )\biggl (\frac{\sigma _{ej}}{1+\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_1} -\frac{2}{r_1} \biggr )\nonumber \\&+ \exp \biggl (r_2(z-w_{i0})\biggr )\biggl (\frac{\sigma _{ij}}{1+\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1-\sigma _{ij}r_2} -\frac{2}{r_2} \biggr )\nonumber \\&+\exp \biggl (r_2(z-w_{if})\biggr )\biggl (\frac{\sigma _{ij}}{1-\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_2} +\frac{2}{r_2} \biggr )\nonumber \\&+ \biggl (\frac{2}{r_1}-\frac{2}{r_2}\biggr )\biggr ] \end{aligned}$$

(28)

$$\begin{aligned} u_{j5}(z)= & {} \biggl ( \frac{\alpha _j}{2 \sqrt{c^2+4 \alpha _j D_j^2}}\biggr )\biggl [\exp \biggl (\frac{-(z-w_{e0})}{\sigma _{ej}}\biggr )\nonumber \\&\times \biggl ( \frac{\sigma _{ej}}{1+\sigma _{ej}r_2}-\frac{\sigma _{ej}}{1+\sigma _{ej}r_1} \biggr )\nonumber \\&+ \exp \biggl (\frac{-(z-w_{ef})}{\sigma _{ej}}\biggr )\biggl ( \frac{\sigma _{ej}}{1+\sigma _{ej}r_1}-\frac{\sigma _{ej}}{1+\sigma _{ej}r_2} \biggr )\nonumber \\&+\exp \biggl (\frac{-(z-w_{if})}{\sigma _{ij}}\biggr )\biggl ( \frac{\sigma _{ij}}{1+\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_1} \biggr ) \nonumber \\&+\exp (\frac{-(z-w_{i0})}{\sigma _{ij}})\biggl ( \frac{\sigma _{ij}}{1+\sigma _{ij}r_1}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_2} \biggr )\nonumber \\&+ \exp \biggl (r_2(z-w_{e0})\biggr )\biggl (\frac{\sigma _{ej}}{1-\sigma _{ej}r_2}-\frac{\sigma _{ej}}{1+\sigma _{ej}r_2} +\frac{2}{r_2} \biggr )\nonumber \\&+\exp \biggl (r_2(z-w_{ef})\biggr )\biggl (\frac{\sigma _{ej}}{1+\sigma _{ej}r_2}-\frac{\sigma _{ej}}{1-\sigma _{ej}r_2} -\frac{2}{r_2} \biggr )\nonumber \\&+ \exp \biggl (r_2(z-w_{i0})\biggr )\biggl (\frac{\sigma _{ij}}{1+\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1-\sigma _{ij}r_2} -\frac{2}{r_2} \biggr )\nonumber \\&+\exp \biggl (r_2(z-w_{if})\biggr )\biggl (\frac{\sigma _{ij}}{1-\sigma _{ij}r_2}-\frac{\sigma _{ij}}{1+\sigma _{ij}r_2} +\frac{2}{r_2} \biggr )\biggr ]\nonumber \\ \end{aligned}$$

(29)