Effects of Glia in a Triphasic Continuum Model of Cortical Spreading Depression

Abstract

Cortical spreading depression (SD) is a spreading disruption in brain ionic homeostasis during which neurons experience complete and prolonged depolarizations. SD is generally believed to be the physiological substrate of migraine aura and is associated with many other brain pathologies. Here, we perform simulations with a model of SD treating brain tissue as a triphasic continuum of neurons, glia and the extracellular space. A thermodynamically consistent incorporation of the major biophysical effects, including ionic electrodiffusion and osmotic water flow, allows for the computation of important physiological variables including the extracellular voltage (DC) shift. A systematic parameter study reveals that glia can act as both a disperser and buffer of potassium in SD propagation. Furthermore, we show that the timing of the DC shift with respect to extracellular $\hbox {K}^{+}$ rise is highly dependent on glial parameters, a result with implications for the identification of the propagating mechanism of SD.

Appendix: Ion Channels

For each ion species i, we write the outward flux as the sum of transporters and ion channels, with $h_{i}^k$ representing the combined effect of the $\hbox {Na}^{+}/\hbox {K}^{+}$ ATPase and any cotransporters, and m representing the different ion channels. Each channel is written as the product of the proportion of open channels $\hat{g}_i^{k,m}$ and a flux–voltage–concentration relationship $J_i^{k,m}$:

$$\begin{aligned} j_i^k = h_i^k + \sum _m \hat{g}_i^{k,m}J_i^{k,m}, \; k=\mathrm{n,g}. \end{aligned}$$

For the flux–voltage–concentration relationship in some ion channels, we use the Nernst potential for ion i and cell type k, 

$$\begin{aligned} E_i^k = \frac{RT}{z_i F} \ln \left( \frac{c_i^k}{c_i^\mathrm{e}}\right) . \end{aligned}$$

Persistent sodium current (neuron) Kager et al. (2000), Yao et al. (2011)

$$\begin{aligned} \hat{g}_{\mathrm{Na}}^{\mathrm{n,P}}&= m^2h\\ J_{\mathrm{Na}}^{\mathrm{n,P}}&= P_{\mathrm{Na}}^{\mathrm{n,P}}\frac{F\phi _\mathrm{ne}}{RT}\frac{c_{\mathrm{Na}}^\mathrm{n}\exp \left( \frac{F\phi _\mathrm{ne}}{RT}\right) -c_{\mathrm{Na}}^\mathrm{e}}{\exp \left( \frac{F \phi _\mathrm{ne}}{RT}\right) -1}\\ \frac{\hbox {d}m}{\hbox {d}t}&= \alpha _m(1-m)-\beta _mm\\ \frac{\hbox {d}h}{\hbox {d}t}&= \alpha _h(1-h)-\beta _hh\\ \alpha _m&= \frac{1}{\left( 1+\exp \left( {-}\left( 0.143\phi _\mathrm{ne} +5.67\right) \right) \right) 6}\\ \beta _m&= \frac{1}{\left( 1+\exp \left( 0.143\phi _\mathrm{ne} +5.67\right) \right) 6}\\ \alpha _h&= 5.12\times 10^{-6} \exp \left( {-}\left( 0.056\phi _\mathrm{ne} +2.94\right) \right) \\ \beta _h&= \frac{1.6\times 10^{-4}}{1+\exp \left( {-}\left( 0.2\phi _\mathrm{ne} +8\right) \right) } \end{aligned}$$

Potassium delayed rectifier current (neuron) Kager et al. (2000), Yao et al. (2011)

$$\begin{aligned} \hat{g}_\mathrm{K}^{\mathrm{n,DR}}&= m^2\\ J_\mathrm{K}^{\mathrm{n,DR}}&= P_\mathrm{K}^{\mathrm{n,DR}}\frac{F\phi _\mathrm{ne}}{RT}\frac{c_\mathrm{K}^\mathrm{n}\exp \left( \frac{F\phi _\mathrm{ne}}{RT}\right) -c_\mathrm{K}^\mathrm{e}}{\exp \left( \frac{F \phi _\mathrm{ne}}{RT}\right) -1}\\ \frac{\hbox {d}m}{\hbox {d}t}&= \alpha _m(1-m)-\beta _mm\\ \alpha _m&= \frac{0.016(\phi _\mathrm{ne}+34.9)}{1-\exp \left( {-}0.2\left( \phi _\mathrm{ne}+34.9\right) \right) }\\ \beta _m&= 0.25\exp \left( {-}(0.025\phi _\mathrm{ne}+1.25)\right) \end{aligned}$$

Transient potassium current (neuron) Kager et al. (2000), Yao et al. (2011)

$$\begin{aligned} \hat{g}_\mathrm{K}^{\mathrm{n,A}}&= m^2h\\ J_\mathrm{K}^{\mathrm{n,A}}&= P_\mathrm{K}^{\mathrm{n,A}}\frac{F\phi _\mathrm{ne}}{RT}\frac{c_\mathrm{K}^\mathrm{n}\exp \left( \frac{F\phi _\mathrm{ne}}{RT}\right) -c_\mathrm{K}^\mathrm{e}}{\exp \left( \frac{F \phi _\mathrm{ne}}{RT}\right) -1}\\ \frac{\hbox {d}m}{\hbox {d}t}&= \alpha _m(1-m)-\beta _mm\\ \frac{\hbox {d}h}{\hbox {d}t}&= \alpha _h(1-h)-\beta _hh\\ \alpha _m&= \frac{0.02(\phi _\mathrm{ne}+56.9)}{1-\exp \left( {-}0.1(\phi _\mathrm{ne}+56.9)\right) }\\ \beta _m&= \frac{0.0175(\phi _\mathrm{ne}+29.9)}{\exp \left( 0.1(\phi _\mathrm{ne}+29.9)-1\right) }\\ \alpha _h&= 0.016\exp \left( {-}(0.056\phi _\mathrm{ne}+4.61)\right) \\ \beta _h&= \frac{0.5}{\exp \left( {-}(0.2\phi _\mathrm{ne}+11.98)\right) +1} \end{aligned}$$

Potassium inward rectifier current (glia) Newman (1993), Steinberg et al. (2005)

$$\begin{aligned} \hat{g}_\mathrm{K}^{\mathrm{n,IR}}&= P_\mathrm{K}^{\mathrm{g,IR}}\sqrt{\frac{c_\mathrm{K}^\mathrm{e}}{3}}\frac{1+\exp \left( 18.5/42.5\right) }{1+\exp \left( (\phi _\mathrm{ge}-E_\mathrm{K}^\mathrm{g}+18.5)/42.5\right) }\\&\qquad \times \,\frac{1+\exp \left( ({-}118.6-85.2)/44.1\right) }{1+\exp \left( ({-}118.6+\phi _\mathrm{ge})/44.1\right) }\\ J_\mathrm{K}^{\mathrm{n,IR}}&= F\left( \phi _{\mathrm{ge}} - E_\mathrm{K}^g\right) \end{aligned}$$

Leak currents (neuron, glia) Kager et al. (2000), Yao et al. (2011)

$$\begin{aligned} \hat{g}_{i}^{k,\mathrm{L}}&= 1\\ J_{i}^{k,\mathrm{L}}&= P_{i}^{k,\mathrm{L}}z_iF\left( \phi _{k\mathrm{e}} - E_i^k\right) \end{aligned}$$

$\hbox {Na}^{+}{/}\hbox {K}^{+}$ ATPase current (neuron, glia) Yao et al. (2011)

$$\begin{aligned} I_k^\mathrm{ATP}&= \frac{\bar{I}_k}{\left( 1+m_\mathrm{K}/c_\mathrm{K}^\mathrm{e}\right) ^2\left( 1+m_{\mathrm{Na}}/c_{\mathrm{Na}}^{k}\right) ^3}\\ h_{\mathrm{Na}}^{k,\mathrm{ATP}}&= 3I_k^\mathrm{ATP}\\ h_\mathrm{K}^{k,\mathrm{ATP}}&= -2I_k^\mathrm{ATP} \end{aligned}$$

Sodium, potassium, chloride cotransporter current (glia) Bennett et al. (2008)

$$\begin{aligned} I_k^{\mathrm{NaKCl}}&= P^{\mathrm{NaKCl}}\ln \left( \frac{c_{\mathrm{Na}}^\mathrm{g}c_\mathrm{K}^\mathrm{g}\left( c_{\mathrm{Cl}}^\mathrm{g}\right) ^2}{c_{\mathrm{Na}}^\mathrm{e}c_\mathrm{K}^\mathrm{e}\left( c_{\mathrm{Cl}}^\mathrm{e}\right) ^2}\right) \\ h_{\mathrm{Na}}^{k,\mathrm{NaKCl}}&= I_k^{\mathrm{NaKCl}}\\ h_\mathrm{K}^{k,\mathrm{NaKCl}}&= I_k^{\mathrm{NaKCl}}\\ h_{\mathrm{Cl}}^{k,\mathrm{NaKCl}}&= 2I_k^{\mathrm{NaKCl}} \end{aligned}$$