pH during non-synaptic epileptiform activity—computational simulations

Our model was based on that proposed by Almeida et al (2008) and Lopes et al (2013), designed to study mechanisms underlying the nonsynaptic epileptiform activity in the granule cell layer of the hippocampal dentate gyrus.

In the present work, new elements were introduced into the model and its basic original characteristics were confirmed. As reported in the literature, recordings of epileptiform activity in hippocampal slices of adult naïve rats are not affected by bumetanide, a specific blocker of the cation–chloride cotransporter NKCC (Margineanu and Klitgaard 2006). As a result, the blockage of the NKCC is expected to decrease the neuronal chloride concentration. This finding indicates that the NKCC expressed in brain slices of naïve rats is almost inactive. As NEA does not seem to depend on NKCC, in this current version of the model the onset of the nonsynpatic epileptiform events was not based on NKCC activity.

It has been reported that the KCC cotransporter operates very close to a thermodynamic equilibrium, meaning that it may revert its flux depending on K⁺ and Cl⁻ chemical gradients (Payne et al 2003). The influx of Cl⁻ induced by extracellular potassium increases is mediated by the KCC. As a result, KCC affinities to Cl⁻ and K⁺ in the intracellular surface of the neuronal membrane were readjusted.

To accomplish the main objective of the present work, that is, to determine possible mechanisms related pH changes during NEA, the ionic dynamics of H⁺ and ${{\rm{HCO}}}_{3}^{-}$ were incorporated into the model. Mechanisms associated with the control of H⁺ and ${{\rm{HCO}}}_{3}^{-}$ concentrations were investigated analyzing correspondent simulations.

All changes performed in the model implicate additional adjustment in parameters. As shown in the supplementary information (SI), these can be divided in steady state parameters or investigative parameters (Almeida et al 2008).

A schematic drawing of the model is shown in figure 1. A tri-dimensional network of functional units represents the dentate gyrus granule cell layer. Functional units were composed of three interconnected compartments: neuronal cell body, glial processes and extracellular space. The communication between each subunit was based on the active and passive ionic transports between the extracellular space, neurons and glial cells. An electrodiffusion process governed the ionic movements through the extracellular space. In this version of the NEA model, the effects of the intracellular electrodiffusion on the intracellular control of the ionic levels have been included.

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A brief description of the subcellular mechanisms used to represent the neuronal electrochemistry activity is present below. Details of the equations, parameters, variables and a pseudocode showing how the model was implemented from a computational standpoint are provided in SI.

The Goldman–Hodgkin–Katz equation for current was used to calculate the ionic density fluxes through ionic channels and gap-junctions. The following ionic channels were represented in granule cells: fast gating Na⁺ channels, persistent Na⁺ channels, K⁺ rectifier channels, A-type K⁺ channels, voltage-dependent Cl⁻ channels and Cl⁻ channels with constant permeability. Ionic channels included in the representation of the glial cells processes were: Na⁺, K⁺ and Cl⁻ constant permeability channels and K⁺ rectifier channels. Permeability changes of the Na⁺ and K⁺ voltage dependent channels were based on the Hodgkin–Huxley formalism (Hodgkin and Huxley 1952).

The inward movement of two potassium ions and the extrusion of three sodium ions by the Na/K pump, which is the source of its electrogenic effect, was represented by a fictitious ion A⁺ flowing according to the electrochemical gradient. The intra and extracellular ionic concentrations of this ion were determined by equations representing the Albers–Post model (Rodrigues et al 2008, 2009).

The representation of the cotransporters was based on the differential equations deduced to represent the chemical reactions of the ionic bindings and the transport through the membrane.

In the previous versions of the model (Almeida et al 2008, Lopes et al 2013), Na⁺/H⁺ (NHE) and ${{\rm{HCO}}}_{3}^{-}$/Cl⁻ (HCE) exchangers were only represented as contributing to control the Na⁺ and Cl⁻ homeostasis. In the present study, these were included to represent the effects of exchangers on concentration changes of H⁺ and ${{\rm{HCO}}}_{3}^{-},$ determining pH changes during the NEA. In addition to these exchangers, the H⁺ pump and the CO₂ hydration reaction, which is catalyzed by carbonic anhydrase, were also incorporated. The mathematical representation and the mechanisms involved in pH regulation are described in the results section.

In the absence of the synaptic connections, mechanisms involved in neuronal couplings, the electric field effect and extracellular ionic fluctuations are dependent on gap-junctions (Almeida et al 2008, Volman et al 2011, Lopes et al 2013). As already mentioned, the gap-junction effect was estimated by calculating ionic fluxes using the Goldman–Hodgkin–Katz current equation. The field effect was determined based on the field produced by transmembrane currents assuming a quasi-stationary process. The extracellular resistivity was estimated based on extracellular volume changes, which were also estimated in terms of osmolarity changes. These resistivity changes modulate the ephaptic effect of the extracellular currents on neighboring cells.

As for neuronal coupling mediated by extracellular ionic fluctuations, extracellular ionic changes promoted by a group of cells modified the ionic concentrations of the neighboring regions to modulate activity of these cells.

The equation used to estimate the ionic changes associated with extracellular electrodiffusion was based on the Nernst–Planck equation, in which the ionic movement is caused by the mutual effect of extracellular diffusion and electric field. The electric field had two components. The first was calculated based on the ionic distribution through the extracellular space and constitutes the slow component of the potential, usually named dc shift (Almeida et al 2004). The second is generated by the transmembrane ionic currents (Almeida et al 2008, Lopes et al 2013).

Osmolarity changes generated by the ionic exchange between intra and extracellular spaces promote tissue volume changes. During the ictal period, the intracellular volume increases due to a higher intracellular osmolarity. This is associated with a decrease in extracellular volume (Lopes et al 2013). Cellular volume changes were determined as a function of the osmolarity gradient between intra and extracellular spaces.

2.2.1. H⁺ and $HC{O}_{3}^{-}$ extracellular electrodiffusion

The inclusion of H⁺ and ${{\rm{HCO}}}_{3}^{-}$ was performed considering exclusively electrodiffusion in the extracellular space and the effect of extracellular volume changes on their concentrations:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[C]}_{{\rm{n}}}}{\partial t}={r}_{{\rm{vol}},{\rm{n}}}^{C},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[C]}_{{\rm{g}}}}{\partial t}={r}_{{\rm{vol}},{\rm{g}}}^{C},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[C]}_{{\rm{o}}}}{\partial t}={r}_{{\rm{vol}},{\rm{o}}}^{C}+{r}_{{\rm{ED}},{\rm{o}}}^{C},\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&{r}_{{\rm{vol}},{\rm{M}}}^{C}=-\displaystyle \frac{1}{{{\rm{Vol}}}_{{\rm{M}}}}{[C]}_{{\rm{M}}}\displaystyle \frac{\partial {{\rm{Vol}}}_{{\rm{M}}}}{\partial t},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{r}_{{\rm{ED}},{\rm{o}}}^{C}={(\displaystyle \frac{\partial {[C]}_{{\rm{o}}}}{\partial t})}_{{\rm{Electrodiffusion}}},\end{eqnarray} \tag{ 5 }$$

and ${(\frac{\partial {[C]}_{{\rm{o}}}}{\partial t})}_{{\rm{Electrodiffusion}}}$ is calculated according to equation (A.31), C represents H⁺ or ${{\rm{HCO}}}_{3}^{-},$ Vol_M the volume to the medium M (NEURON—n, GLIA—g or EXTRA—o).

2.2.2. CO₂ hydration

Assuming that [CO₂]_o is kept constant due to the fact that the extracellular space is an abundant source of CO₂ (Chesler 2003, Tong et al 2006) the following reaction was incorporated in the model to investigate its possible contribution for equilibrating the extracellular pH:

$$\begin{eqnarray}&&{{\rm{H}}}^{+}+{{\rm{HCO}}}_{3}^{-}\leftrightarrow {{\rm{H}}}_{2}{\rm{O}}+{{\rm{CO}}}_{2}.\end{eqnarray} \tag{ 6 }$$

Assuming that CO₂ hydration is induced by carbonic anhydrase during the carbonic acid (H₂CO₃) formation (Chesler 2003), and that [CO₂]_o is constant, the variation rate of [H⁺]_o and [${{\rm{HCO}}}_{3}^{-}$]_o were calculated by:

$$\begin{eqnarray}{(\displaystyle \frac{\partial {[C]}_{{\rm{o}}}}{\partial t})}_{{\rm{CA}}} & = & {v}_{{\rm{CA}}}^{{\rm{o}}}({[{{\rm{H}}}^{+}]}_{{\rm{o}}}{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{o}}}\\ & & -{{\rm{Kd}}}_{{\rm{CA}}}^{{\rm{o}}}{[{{\rm{CO}}}_{2}]}_{{\rm{o}}}),\end{eqnarray} \tag{ 7 }$$

where C represents H⁺ and ${{\rm{HCO}}}_{3}^{-},$ ν^o_CA is a proportionality constant determining the reaction speed and ${{{\rm{Kd}}}_{{\rm{CA}}}}^{{\rm{o}}}$ is the equilibrium reaction constant. The term ${{\rm{Kd}}}_{{\rm{CA}}}^{{\rm{o}}}{[{{\rm{CO}}}_{2}]}^{{\rm{o}}}$ was kept constant (=9.56 × 10⁻⁴ mM²) to keep the pH at 7.4. ν^o_CA was set in 0.12 mM⁻¹ s⁻¹, this value was calculated based on the equilibrium constant of the reaction ${{\rm{H}}}^{+}+{{\rm{HCO}}}_{3}^{-}\leftrightarrow {{\rm{H}}}_{2}{{\rm{CO}}}_{3},$ Kd = 0.27 mM, and the reaction rate of the reaction ${{\rm{H}}}_{2}{{\rm{CO}}}_{3}\leftrightarrow {{\rm{H}}}_{2}{\rm{O}}+{{\rm{CO}}}_{2},$ α = 3.32 × 10⁻² mM⁻¹ s⁻¹ (Tong et al 2006) : ${v}_{{\rm{CA}}}^{{\rm{o}}}=\frac{\alpha }{{\rm{Kd}}}$.

Including equation (7) in the model, equation (3) must then be substituted by :

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[C]}_{{\rm{o}}}}{\partial t}={r}_{{\rm{vol}},{\rm{o}}}^{C}+{r}_{{\rm{ED}},{\rm{o}}}^{C}+{r}_{{\rm{CA}},{\rm{o}}}^{C},\end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray}&&{r}_{{\rm{CA}},{\rm{o}}}^{C}=-{(\displaystyle \frac{\partial {[C]}_{{\rm{o}}}}{\partial t})}_{{\rm{CA}}},\end{eqnarray} \tag{ 9 }$$

where C is H⁺ or ${{{\rm{HCO}}}_{3}}^{-}$.

2.2.3. Glial $HC{O}_{3}^{-}$/Cl⁻ exchanger

The following mechanisms were considered as potentially responsible for the ${{\rm{HCO}}}_{3}^{-}$ regulation (Rufin et al 2014), first assuming it as an exclusive glial mechanism: Na⁺-coupled ${{\rm{HCO}}}_{3}^{-}$ transporters (NBCT); the ${{\rm{HCO}}}_{3}^{-}$/Cl⁻ exchangers (HCE). The first would not be effective in reducing extracellular ${{{\rm{HCO}}}_{3}}^{-},$ since it depends on the [Na⁺]_o. A decrease in [Na⁺]_o during bursts is expected to reduce the action of these transporters. Ionic changes during bursts, on the other hand, may help to account for HCE activity. An increase in [Cl⁻]_i leads to the exchange of intracellular Cl⁻ by the extracellular ${{\rm{HCO}}}_{3}^{-},$ which is expected for an extracellular decrease of [${{\rm{HCO}}}_{3}^{-}]$ to occur. Therefore, considering that reactions of Cl⁻ and ${{\rm{HCO}}}_{3}^{-}$ with the exchanger enzyme are always at an equilibrium and that the HCE exchange can be reversed, depending on the Cl⁻ and ${{\rm{HCO}}}_{3}^{-}$ gradients, the following equation was deduced:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{HCE}}}^{{\rm{GLIA}}}={Q}_{{\rm{g}}}^{{\rm{HCE}}}\{\displaystyle \frac{{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{i}}}{[{{\rm{Cl}}}^{-}]}_{{\rm{o}}}}{({[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{i}}}+{{\rm{Kd}}}_{{{\rm{HCO}}}_{3}^{-}}^{{\rm{i}},{\rm{HCE}}}(1+\displaystyle \frac{{[{{\rm{Cl}}}^{-}]}_{{\rm{i}}}}{{{\rm{Kd}}}_{{\rm{Cl}}}^{{\rm{i}},{\rm{HCE}}}}))({[{{\rm{Cl}}}^{-}]}_{{\rm{o}}}+{{\rm{Kd}}}_{{\rm{Cl}}}^{{\rm{o}},{\rm{HCE}}}(1+\displaystyle \frac{{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{o}}}}{{{\rm{Kd}}}_{{{\rm{HCO}}}_{3}^{-}}^{{\rm{o}},{\rm{HCE}}}}))}\\ &&-\displaystyle \frac{{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{o}}}{[{{\rm{Cl}}}^{-}]}_{{\rm{i}}}}{({[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{o}}}+{{\rm{Kd}}}_{{{\rm{HCO}}}_{3}^{-}}^{{\rm{o}},{\rm{HCE}}}(1+\displaystyle \frac{{[{{\rm{Cl}}}^{-}]}_{{\rm{o}}}}{{{\rm{Kd}}}_{{\rm{Cl}}}^{{\rm{o}},{\rm{HCE}}}}))({[{{\rm{Cl}}}^{-}]}_{{\rm{i}}}+{{\rm{Kd}}}_{{\rm{Cl}}}^{{\rm{i}},{\rm{HCE}}}(1+\displaystyle \frac{{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{i}}}}{{{\rm{Kd}}}_{{\rm{HCO}}}^{{\rm{i}},{\rm{HCE}}}}))}\},\end{eqnarray} \tag{ 10 }$$

where the constant ${Q}_{{\rm{g}}}^{{\rm{HCE}}}$ allows adjusting the intensity of the exchanger flux, ${{\rm{Kd}}}_{{\rm{ion}}}^{{\rm{i}},{\rm{HCE}}}$ and ${{\rm{Kd}}}_{{\rm{ion}}}^{{\rm{o}},{\rm{HCE}}}$ are the ionic dissociation constant (${{\rm{HCO}}}_{3}^{-}$ or Cl⁻) at the intra and extracellular surfaces of the cellular membrane, respectively.

The variation rate of [${{\rm{HCO}}}_{3}^{-}],$ previously calculated with equation (2), for glial cells and (8) for the extracellular space, was modified to include the glial HCE effect:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{g}}}}{\partial t}={r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{HCEg}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{o}}}}{\partial t}={r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{HCEg}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}},\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&{r}_{{\rm{HCEg}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}={{\rm{\Phi }}}_{{\rm{HCE}}}^{{\rm{GLIA}}}\displaystyle \frac{{A}_{{\rm{m}}}^{{\rm{GLIA}}}}{{{\rm{Vol}}}_{{\rm{GLIA}}}},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{r}_{{\rm{HCEg}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}=-{{\rm{\Phi }}}_{{\rm{HCE}}}^{{\rm{GLIA}}}\displaystyle \frac{{A}_{{\rm{m}}}^{{\rm{GLIA}}}}{{{\rm{Vol}}}_{{\rm{EXTRA}}}},\end{eqnarray} \tag{ 14 }$$

${A}_{{\rm{m}}}^{{\rm{GLIA}}}$ is the membrane area of the glial cell and the term ${{\rm{\Phi }}}_{{\rm{Cl}}(\text{INTRA}-\text{EXTRA})}^{{\rm{GLIA}}}$ of equation (A.36), describing the Cl⁻ flux through glial membrane, was modified:

$$\begin{eqnarray}{{\rm{\Phi }}}_{{\rm{Cl}}({\rm{INTRA}}\to {\rm{EXTRA}})}^{{\rm{GLIA}}} & = & {{\rm{\Phi }}}_{{\rm{Cl}}}^{{\rm{GLIA}}}+2.{{\rm{\Phi }}}_{{\rm{NKCC}}}^{{\rm{GLIA}}}\\ & & +{{\rm{\Phi }}}_{\text{Cl},\text{leak}}^{{\rm{GLIA}}}+{{\rm{\Phi }}}_{\text{Cl},\text{rest}}^{{\rm{GLIA}}}+{{\rm{\Phi }}}_{{\rm{KCC}}}^{{\rm{GLIA}}}\\ & & +{{\rm{\Phi }}}_{\text{Cl},\text{EDi}}^{{\rm{GLIA}}}-{{\rm{\Phi }}}_{{\rm{HCE}}}^{{\rm{GLIA}}}.\end{eqnarray} \tag{ 15 }$$

2.2.4. Neuronal $HC{O}_{3}^{-}$/Cl⁻ exchanger

To describe HCE exchangers in neurons, the following equation was implemented to the model:

$$\begin{eqnarray}&&{{\Phi }}_{{\rm{HCE}}}^{{\rm{NEURON}}}={Q}_{{\rm{n}}}^{{\rm{HCE}}}\{\displaystyle \frac{{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{i}}}{[{{\rm{Cl}}}^{-}]}_{{\rm{o}}}}{({[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{i}}}+{{\rm{Kd}}}_{{{\rm{HCO}}}_{3}^{-}}^{{\rm{i}},{\rm{HCE}}}(1+\displaystyle \frac{{[{{\rm{Cl}}}^{-}]}_{{\rm{i}}}}{{{\rm{Kd}}}_{{\rm{Cl}}}^{{\rm{i}},{\rm{HCE}}}}))({[{{\rm{Cl}}}^{-}]}_{{\rm{o}}}+{{\rm{Kd}}}_{{\rm{Cl}}}^{{\rm{o}},{\rm{HCE}}}(1+\displaystyle \frac{{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{o}}}}{{{\rm{Kd}}}_{{{\rm{HCO}}}_{3}^{-}}^{{\rm{o}},{\rm{HCE}}}}))}\\ &&-\displaystyle \frac{{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{o}}}{[{{\rm{Cl}}}^{-}]}_{{\rm{i}}}}{({[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{o}}}+{{\rm{Kd}}}_{{{\rm{HCO}}}_{3}^{-}}^{{\rm{o}},{\rm{HCE}}}(1+\displaystyle \frac{{[{{\rm{Cl}}}^{-}]}_{{\rm{o}}}}{{{\rm{Kd}}}_{{\rm{Cl}}}^{{\rm{o}},{\rm{HCE}}}}))\;({[{{\rm{Cl}}}^{-}]}_{{\rm{i}}}+{{\rm{Kd}}}_{{\rm{Cl}}}^{{\rm{i}},HCE}(1+\displaystyle \frac{{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{i}}}}{{{\rm{Kd}}}_{{\rm{HCO}}}^{{\rm{i}},{\rm{HCE}}}}))}\},\end{eqnarray} \tag{ 16 }$$

where the constant ${Q}_{{\rm{n}}}^{{\rm{HCE}}},$ similarly to the HCE implement in the glial cells, is used to adjust the HCE flux intensity. The dissociation constants ${{\rm{Kd}}}_{{\rm{ion}}}^{{\rm{i}},{\rm{HCE}}}$ and ${{\rm{Kd}}}_{{\rm{ion}}}^{{\rm{o}},{\rm{HCE}}}$ were set with the same values adopted for the HCE exchangers of glial cells.

Assuming that neuronal and extracellular $[{{{\rm{HCO}}}_{3}}^{-}]$ are respectively calculated according to equations (1) and (12) and estimating that the HCE rate may be appraised by equations (16), (17) was then included in the model and equation (12) was replaced by (18), as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{n}}}}{\partial t}={r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{HCEn}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}\displaystyle \frac{\partial {[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{o}}}}{\partial t} & = & {r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\\ & & +{r}_{{\rm{HCEg}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{HCEn}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}},\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}&&{r}_{{\rm{HCEn}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}={{\rm{\Phi }}}_{{\rm{HCE}}}^{{\rm{NEURON}}}\displaystyle \frac{{A}_{{\rm{m}}}^{{\rm{NEURON}}}}{{{\rm{Vol}}}_{{\rm{NEURON}}}},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{r}_{{\rm{HCEn}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}=-{{\rm{\Phi }}}_{{\rm{HCE}}}^{{\rm{NEURON}}}\displaystyle \frac{{A}_{{\rm{m}}}^{{\rm{NEURON}}}}{{{\rm{Vol}}}_{{\rm{EXTRA}}}},\end{eqnarray} \tag{ 20 }$$

${A}_{{\rm{m}}}^{{\rm{NEURON}}}$ is the membrane area of the glial cells and the term ${\phi }_{{\rm{Cl}}(\text{INTRA}-\text{EXGTRA})}^{{\rm{NEURON}}}$ of equation (A.36), describing the Cl⁻ flux through the neuronal membrane, was altered:

$$\begin{eqnarray}{{\rm{\Phi }}}_{{\rm{Cl}}({\rm{INTRA}}\to {\rm{EXTRA}})}^{{\rm{NEURON}}} & = & {{\rm{\Phi }}}_{{\rm{Cl}}}^{{\rm{NEURON}}}+{{\rm{\Phi }}}_{\text{Cl},\text{voltage}}^{{\rm{NEURON}}}\\ & & +2.{{\rm{\Phi }}}_{{\rm{NKCC}}}^{{\rm{NEURON}}}+{{\rm{\Phi }}}_{{\rm{KCC}}}^{{\rm{NEURON}}}\\ & & +{{\rm{\Phi }}}_{\text{Cl},\text{rest}}^{{\rm{NEURON}}}+{{\rm{\Phi }}}_{\text{Cl},\text{leak}}^{{\rm{NEURON}}}\\ & & +{{\rm{\Phi }}}_{\text{Cl},\text{EDi}}^{{\rm{NEURON}}}-{{\rm{\Phi }}}_{{\rm{HCE}}}^{{\rm{NEURON}}}.\end{eqnarray} \tag{ 21 }$$

In the simulations used, ${Q}_{{\rm{n}}}^{{\rm{HCE}}}$ and ${Q}_{{\rm{g}}}^{{\rm{HCE}}}$ were assumed as being equal and adjusted in 6 × 10⁻³ μ mol cm⁻² s⁻¹.

2.2.5. Neuronal NHE

Assuming the instantaneous equilibrium of the reaction of Na⁺ and H⁺ with the exchanger enzyme and the fact that NHE can reverse the direction of the ionic transport (i.e. depending on the Na⁺ and H⁺ gradients), the following equation has been derived and implemented in the model:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{NHE}}}^{{\rm{NEURON}}}={Q}_{{\rm{n}}}^{{\rm{NHE}}}\{\displaystyle \frac{{[{{\rm{H}}}^{+}]}_{{\rm{i}}}{[{{\rm{Na}}}^{+}]}_{{\rm{o}}}}{({[{{\rm{H}}}^{+}]}_{{\rm{i}}}+{{\rm{Kd}}}_{{\rm{H}}}^{{\rm{i}},{\rm{NHE}}}(1+\displaystyle \frac{{[{{\rm{Na}}}^{+}]}_{{\rm{i}}}}{{{\rm{Kd}}}_{{\rm{Na}}}^{{\rm{i}},{\rm{NHE}}}}))\;({[{{\rm{Na}}}^{+}]}_{{\rm{o}}}+{{\rm{Kd}}}_{{\rm{Na}}}^{{\rm{o}},{\rm{NHE}}}(1+\displaystyle \frac{{[{{\rm{H}}}^{+}]}_{{\rm{o}}}}{{{\rm{Kd}}}_{{\rm{H}}}^{{\rm{o}},{\rm{NHE}}}}))}\\ &&-\displaystyle \frac{{[{{\rm{H}}}^{+}]}_{{\rm{o}}}{[{{\rm{Na}}}^{+}]}_{{\rm{i}}}}{({[{{\rm{H}}}^{+}]}_{{\rm{o}}}+{{\rm{Kd}}}_{{\rm{H}}}^{{\rm{o}}}(1+\displaystyle \frac{{[{{\rm{Na}}}^{+}]}_{{\rm{o}}}}{{{\rm{Kd}}}_{{\rm{Na}}}^{{\rm{o}},{\rm{NHE}}}}))\;({[{{\rm{Na}}}^{+}]}_{{\rm{i}}}+{{\rm{Kd}}}_{{\rm{Na}}}^{{\rm{i}}}(1+\displaystyle \frac{{[{{\rm{H}}}^{+}]}_{{\rm{i}}}}{{{\rm{Kd}}}_{{\rm{H}}}^{{\rm{i}},{\rm{NHE}}}}))}\},\end{eqnarray} \tag{ 22 }$$

where ${{Q}_{{\rm{n}}}}^{{\rm{NHE}}}$ is constant and allows the adjustment of the NHE flux intensity and ${{{\rm{Kd}}}_{{\rm{ion}}}}^{{\rm{i}},{\rm{NHE}}}$ and ${{{\rm{Kd}}}_{{\rm{ion}}}}^{{\rm{o}},{\rm{NHE}}}$ are the dissociation constants of the ions (Na⁺ and H⁺) in the intra and extracellular surfaces of the neuronal membrane, respectively. The following values were used for these constants: ${{{\rm{Kd}}}_{{\rm{Na}}}}^{{\rm{i}},{\rm{NHE}}}\ =\ 2.5\; mM;$ ${{{\rm{Kd}}}_{{\rm{Na}}}}^{{\rm{o}},{\rm{NHE}}}\ =\ 72,$ 65 mM; ${{{\rm{Kd}}}_{{\rm{H}}}}^{{\rm{i}},{\rm{NHE}}}$ = 1.7 × 10⁻⁵ mM; ${{{\rm{Kd}}}_{{\rm{H}}}}^{{\rm{o}},{\rm{NHE}}}$ = 3.8 × 10⁻⁵ mM. These values were adjusted to give the best fitting to the experimental data and are in the experimental ranges of the ionic affinities of the corresponding NHE sites described for different tissues (Green et al 1988, Krayer-Pawlowska et al 1991, Orlowski 1993, Orlowski and Grinstein 1997, Szabó et al 2000, Wakabayashi et al 2003).

Since our simulations predicted a neuronal and glial accumulation of ${{\rm{HCO}}}_{3}^{-}$ due to the influx of this anion through the HCE and an increase action of the carbonic anhydrase in the intracellular space (Boron 2004, Svichar et al 2009), the following equation was deduced for the estimation of the changes of [H⁺]_i and [${{\rm{HCO}}}_{3}^{-}$]_i:

$$\begin{eqnarray}{(\displaystyle \frac{\partial {[C]}_{{\rm{M}}}}{\partial t})}_{{\rm{CA}}} & = & {v}_{{\rm{CA}}}^{{\rm{i}}}({[{{\rm{H}}}^{+}]}_{{\rm{i}}}{[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{i}}}\\ & & -{{\rm{Kd}}}_{{\rm{CA}}}^{{\rm{i}}}{[{{\rm{CO}}}_{2}]}_{{\rm{i}}}),\end{eqnarray} \tag{ 23 }$$

where C represents H⁺ and ${{{\rm{HCO}}}_{3}}^{-},$ M indicates the neuronal or glial intracellular space, ${{\nu }^{{\rm{i}}}}_{{\rm{CA}}}$ is a proportionality constant related to the speed of the reaction in the intracellular space, and ${{{\rm{Kd}}}_{{\rm{CA}}}}^{{\rm{i}}}$ is the equilibrium constant of the reaction. The term ${{\rm{Kd}}}_{{\rm{CA}}}^{{\rm{i}}}{[{{\rm{CO}}}_{2}]}^{{\rm{i}}}$ was assumed as being equal to 7.37 × 10⁻⁵ mM² in order to maintain a pH of 7.4 in the resting state in both neurons and in glial cells.

Another mechanism involved in the control of intracellular H⁺ levels was the H⁺ pump, present in all regions of the hippocampus (Murata et al 2002, Chesler 2003). This pump is responsible for the efflux of H⁺, when the pH drops in the intracellular space. As a result, it seems to be indispensable for avoiding the intracellular accumulation of H⁺. The flux promoted by the H⁺ pump was determined assuming the instantaneous equilibrium of H⁺ and ATP with the pump enzyme:

$$\begin{eqnarray}{{\rm{\Phi }}}_{{\rm{Hpump}}}^{{\rm{NEURON}}} & = & {P}_{{\rm{H}}}^{{\rm{n}},{\rm{pump}}}\\ & & \times [(({[{{\rm{H}}}^{+}]}_{{\rm{i}}}{[{\rm{ATP}}]}_{{\rm{i}}})/(({[{{\rm{H}}}^{+}]}_{{\rm{i}}}\\ & & +{{\rm{Kd}}}_{{\rm{H}}}^{{\rm{i}},{\rm{pump}}})({[{\rm{ATP}}]}_{{\rm{i}}}+{{\rm{Kd}}}_{{\rm{ATP}}}^{{\rm{i}},{\rm{pump}}})))],\end{eqnarray} \tag{ 24 }$$

where ${P}_{{\rm{H}}}^{{\rm{n}},{\rm{pump}}}$ is constant and allows to adjust the intensity of the pump flux. ${{{\rm{Kd}}}_{{\rm{H}}}}^{i,\text{pump}}$ and ${{{\rm{Kd}}}_{{\rm{ATP}}}}^{{\rm{i}},{\rm{pump}}}$ are dissociation constants of H⁺ and ATP in the intracellular face of the membrane. ${{{\rm{Kd}}}_{{\rm{H}}}}^{{\rm{i}},{\rm{pump}}}$ was assumed equal to 5.7 × 10⁻⁵ mM, making the pump sensitive to the [H⁺] intracellular. The H⁺ pump affinity to ATP was assumed as being equal to the Na/K pump: ${{{\rm{Kd}}}_{{\rm{ATP}}}}^{{\rm{i}},{\rm{pump}}}\ =\ 0175\; mM$ (Almeida et al 2008).

To include the effect of these mechanisms on the variation rates of [H⁺] and [${{{\rm{HCO}}}_{3}}^{-}]$ in neurons, glial cells and extracellular space, it was necessary to add both equations (25) and (26) and to replace equation (9) for H⁺ by (27) and equations (11) and (12) by (28) and (29), respectively:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[{{\rm{H}}}^{+}]}_{{\rm{n}}}}{\partial t}={r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}+{r}_{{\rm{NHE}},{\rm{n}}}^{{H}^{+}}+{r}_{{\rm{CA}},{\rm{n}}}^{{{\rm{H}}}^{+}}+{r}_{{\rm{Hpump}},{\rm{n}}}^{{{\rm{H}}}^{+}},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[{{\rm{H}}}^{+}]}_{{\rm{n}}}}{\partial t}={r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{H}}}^{+}}+{r}_{{\rm{CA}},{\rm{g}}}^{{{\rm{H}}}^{+}},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}\displaystyle \frac{\partial {[{{\rm{H}}}^{+}]}_{{\rm{o}}}}{\partial t} & = & {r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}+{r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}+{r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\\ & & +{r}_{{\rm{NHE}},{\rm{o}}}^{{{\rm{H}}}^{+}}+{r}_{{\rm{Hpump}},{\rm{o}}}^{{{\rm{H}}}^{+}},\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{n}}}}{\partial t}={r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{HCEn}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{CA}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}},\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {[{{\rm{HCO}}}_{3}^{-}]}_{{\rm{g}}}}{\partial t}={r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{HCEg}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}+{r}_{{\rm{CA}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}},\end{eqnarray} \tag{ 29 }$$

where

$$\begin{eqnarray}&&{r}_{{\rm{NHE}},{\rm{n}}}^{{{\rm{H}}}^{+}}=-{{\rm{\Phi }}}_{{\rm{NHE}}}^{{\rm{NEURON}}}\displaystyle \frac{{A}_{{\rm{m}}}^{{\rm{NEURON}}}}{{{\rm{Vol}}}_{{\rm{NEURON}}}},\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&{r}_{{\rm{NHE}},{\rm{o}}}^{{{\rm{H}}}^{+}}={{\rm{\Phi }}}_{{\rm{NHE}}}^{{\rm{NEURON}}}\displaystyle \frac{{A}_{{\rm{m}}}^{{\rm{NEURON}}}}{{{\rm{Vol}}}_{{\rm{EXTRA}}}},\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{r}_{{\rm{Hpump}},{\rm{n}}}^{{{\rm{H}}}^{+}}=-{{\rm{\Phi }}}_{{\rm{Hpump}}}^{{\rm{NEURON}}}\displaystyle \frac{{A}_{{\rm{m}}}^{{\rm{NEURON}}}}{{{\rm{Vol}}}_{{\rm{NEURON}}}},\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&{r}_{{\rm{Hpump}},{\rm{o}}}^{{{\rm{H}}}^{+}}={{\rm{\Phi }}}_{{\rm{Hpump}}}^{{\rm{NEURON}}}\displaystyle \frac{{A}_{{\rm{m}}}^{{\rm{NEURON}}}}{{{\rm{Vol}}}_{{\rm{EXTRA}}}},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{r}_{{\rm{CA}},{\rm{n}}}^{C}=-{(\displaystyle \frac{\partial {[C]}_{{\rm{i}}}}{\partial t})}_{{\rm{CA}}},\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{r}_{{\rm{CA}},{\rm{g}}}^{C}=-{(\displaystyle \frac{\partial {[C]}_{{\rm{i}}}}{\partial t})}_{{\rm{CA}}},\end{eqnarray} \tag{ 35 }$$

where C represents H⁺ or ${{{\rm{HCO}}}_{3}}^{-}.$ With the inclusion of the NHE, the term ${{\rm{\Phi }}}_{{\rm{Na}}({\rm{INTRA}}\to {\rm{EXTRA}})}^{{\rm{NEURON}}}$ of the equation (A.36), that describes the Na⁺ flux through the neuronal membrane, was modified:

$$\begin{eqnarray}{{\rm{\Phi }}}_{{\rm{Na}}({\rm{INTRA}}\to {\rm{EXTRA}})}^{{\rm{NEURON}}} & = & {{\rm{\Phi }}}_{\text{Na},\text{fast}}^{{\rm{NEURON}}}+{{\rm{\Phi }}}_{\text{Na},\text{presist}}^{{\rm{NEURON}}}\\ & & +{{\rm{\Phi }}}_{{\rm{NKCC}}}^{{\rm{NEURON}}}+3.{{\rm{\Phi }}}_{{\rm{PUMP}}}^{{\rm{NEURON}}}\\ & & +{{\rm{\Phi }}}_{\text{Na},\text{rest}}^{{\rm{NEURON}}}+{{\rm{\Phi }}}_{\text{Na},\text{leak}}^{{\rm{NEURON}}}\\ & & +{{\rm{\Phi }}}_{{\rm{Na}},{\rm{ED}}i}^{{\rm{NEURON}}}+{{\rm{\Phi }}}_{{\rm{NHE}}}^{{\rm{NEURON}}}.\end{eqnarray} \tag{ 36 }$$

In the simulations performed, the constants ${Q}_{{\rm{n}}}^{{\rm{NHE}}},$ ${v}_{{\rm{CA}}}^{{\rm{i}}}$ _and ${P}_{{\rm{H}}}^{{\rm{n}},{\rm{pump}}}$ were assumed equal, respectively, to 5 × 10⁻¹⁰ μ mol cm⁻² s⁻¹, 7.0 × 10⁻³ mM⁻¹ s⁻¹ and 1.2 × 10⁻¹⁰ μ mol cm⁻² s⁻¹.

Computational simulations were performed after describing the equations with finite difference methods and admitting a tolerance error of 10⁻⁴. The model was implement in FORTRAN 90. To perform the simulation with a network of 9 × 29 × 19 functional units, the computational code was parallelized and a high performance computer was used (Cluster SGI UV 2000, 80 Cores—Intel Xeon E5-4650v2 10-core, 2.4 GHz, 25MB Cache, RAM 1024GB DDR3 1866 MHz, 80TB HD, SUSE Linux Enterprise Server 11, SGI Performance Suite, Intel Cluster Studio XE).

As can be observed in figure 2(C), the inclusion of the extracellular electrodiffusion of H⁺ and ${{\rm{HCO}}}_{3}^{-}$ into the model suggest that while [H⁺]_o increases pH decreases. This change is in accordance, at least qualitatively, with the experimental findings from Xiong and Stringer (2000a) (figure 3(A)). However, [H⁺]_i is reduced in simulated neurons, which is not in agreement with the intracellular acidification of the granule cells recorded by Xiong et al (2000) (figure 3(A)). As in the simulated neurons, alkalinization of the cytoplasm has also been observed in simulations of the glial cells (figure 3(B)).

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The simulation allowed us to propose mechanism responsible for changes in [H⁺] and [${{\rm{HCO}}}_{3}^{-}]$ (figure 2(D)). The extracellular volume reduction contributed to increase of [H⁺] and [${{\rm{HCO}}}_{3}^{-}]$ (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0$ and ${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0).$ Increases of extracellular volume, which reduced these ionic concentrations (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0$ and ${r}_{\text{vol},o}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0),$ seem to be responsible for interictal recovery. Extracellular electrodiffusion contributed to reduce the accumulation of these ions during bursts (${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0$ and ${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0)$ and to avoid the excessive reduction of ionic concentration in the interictal period (${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0$ and ${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0).$ The intracellular volume increase reduced [H⁺] and [${{\rm{HCO}}}_{3}^{-}]$ in granule cells and glial cells during the ictal period (${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \lt \ 0,$ ${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0,$ ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{H}}}^{+}}\ \lt \ 0$ and ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0).$ Recovery during the interictal period contributed to increase [H⁺] and [${{{\rm{HCO}}}^{-}}_{3}]$ and to bring the levels of these ions to their corresponding baseline (${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \gt \ 0,$ ${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0,$ ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{H}}}^{+}}\ \gt \ 0$ and ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}$ > 0).

The amplitude of extracellular pH changes (>0.15) is often larger in the simulation (figure 3(B)) than in the experimental condition (∼0.1), as recorded by Xiong and Stringer (2000a). Since in the extracellular environment ${{\rm{HCO}}}_{3}^{-}$ is shown to accumulate (Chesler 2003), CO₂ hydration, a reaction catalyzed by the carbonic anhydrase, may contribute to minimize pH changes. As shown in figure 3(A) (simulation ii), the inclusion of the CO₂ hydration reaction catalyzed by the carbonic anhydrase caused extracellular alkalinization during the ictal period. No changes were revealed for either the field potential or the transmembrane and Nernst potentials. Therefore no concentrations of the other ions remained fairly stable (figures 4(A)–(C)). The only change previewed by the simulation was the reduction of [H⁺] during the ictal period, caused by an increased activity of carbonic anhydrase (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0,$ figure 4(D)). In the interictal period, enzymatic activity is restored and the reestablished [H⁺] (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0).$ The extracellular volume reduction contributed to increase [H⁺]_o and [${{\rm{HCO}}}_{3}^{-}$]_o during the ictal period (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0$ and ${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0,$ figure 4(D)). In the interictal period, when the extracellular volume returns to baseline, this change contributes to [H⁺]_o and [${{{\rm{HCO}}}_{3}}^{-}$]_o reduction (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0$ and ${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0,$ figures 4(D) and (E)). In the dentate granule cell cytoplasm and also in glial cells, the increased cellular volume causes reductions in both [H⁺]_i and [${{\rm{HCO}}}_{3}^{-}$]_i during the ictal period (${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \lt \ 0,$ ${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0,$ ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{H}}}^{+}}\ \lt \ 0$ and ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0).$ In the interictal period, these concentrations are restored to baseline after the recovery of cellular volumes (${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \gt \ 0,$ ${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0,$ ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{H}}}^{+}}\ \gt \ 0$ and ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0).$ Extracellular electrodiffusion contributes to ameliorate changes in [H⁺]_o and [${{\rm{HCO}}}_{3}^{-}$]_o: (i) ictal period ${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0$ and ${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0;$ (ii) interictal period ${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0$ and ${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}$ > 0.

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Previous simulations show that an increase in carbonic anhydrase activity promotes extracellular alkalinization due to the accumulation of ${{\rm{HCO}}}_{3}^{-}$ during bursts. However, experimental findings (figure 3(A)) show an accumulation of H⁺, as indicated by a pH reduction. Therefore, the previous simulations were lacking mechanisms responsible for removing ${{\rm{HCO}}}_{3}^{-}$ from the extracellular space. Since neuronal acidification and glial alkalinization have been observed (Xiong et al 2000, Chesler 2003), the first hypothesis tested was that of a glial influx of ${{\rm{HCO}}}_{3}^{-},$ which in the model was represented with the implementation of the equation (10). In this equation, the competition of Cl⁻ and ${{\rm{HCO}}}_{3}^{-}$ to bind the exchanger enzyme in each surface of the cellular membrane were also taken into account. To the best of our knowledge, values of these dissociation constants in neuronal tissues have not been estimated. Therefore, values estimated in other tissues were adopted during simulations (Cassel et al 1988, Nord et al 1988, Whitcomb and Ermentrout 2004): ${{\rm{Kd}}}_{{{\rm{Cl}}}^{-}}^{{\rm{i}},{\rm{HCE}}}$ = ${{\rm{Kd}}}_{{{\rm{Cl}}}^{-}}^{{\rm{o}},{\rm{HCE}}}\ =\ 10\; mM;$ ${{\rm{Kd}}}_{{{\rm{Cl}}}^{-}}^{{\rm{i}},{{\rm{HCO}}}_{3}^{-}}$ = ${{\rm{Kd}}}_{{{\rm{Cl}}}^{-}}^{{\rm{o}},{{\rm{HCO}}}_{3}^{-}}\ =\ 1\; mM.$ ${Q}_{{\rm{g}}}^{{\rm{HCE}}}$ was incremented progressively in successive simulations aiming to intensify the influx of ${{\rm{HCO}}}_{3}^{-}$ during the ictal period. However, the increment of ${Q}_{{\rm{g}}}^{{\rm{HCE}}}$ was limited by the maximum numerical error accepted in the simulations (1 × 10⁻⁴), which limits the adjusted value of ${Q}_{{\rm{g}}}^{{\rm{HCE}}}$ at 2 × 10⁻² μ mol cm⁻² s⁻¹ (figure 3(A)—simulation (iii)). For this simulation, the extracellular pH exhibits transient reduction. However, during the ictal period, the pH increases again in the extracellular space and also increases in the neuronal and glial cytoplasm (figure 3(B)—simulation (iii)).

The inclusion of the HCE in glial cells changed the duration of the interictal period (reduction from 227 s to 205 s), but not the ictal period or dc shift (figure 5(A)). In addition, no remarkable changes in cellular volumes, amplitude of the transmembrane or ions Nernst potential were previewed by simulation, as shown in figure 5(B).

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In respect to changes in [H⁺] or [${{\rm{HCO}}}_{3}^{-}],$ two main aspects must the highlighted: (i) during the ictal period, [H⁺]_o had a transient positive change followed by a progressive reduction. Simultaneously, after an initial drop, [${{\rm{HCO}}}_{3}^{-}$]_o tends to increase. In neurons, [H⁺] and [${{\rm{HCO}}}_{3}^{-}]$ decrease. In glia, while a decrease is observed in [H⁺], [${{\rm{HCO}}}_{3}^{-}]$ is increased; (iii) in the interictal period, [H⁺]_o and [${{\rm{HCO}}}_{3}^{-}$]_o return to baseline. However, [H⁺]_o had an initial overshoot and [${{\rm{HCO}}}_{3}^{-}$]_o an undershoot. While [H⁺]_i and [${{\rm{HCO}}}_{3}^{-}$]_i went back to their respective baseline, an initial undershoot was observed in glial cells.

While investigating the mechanisms responsible for changes in [H⁺] and [${{\rm{HCO}}}_{3}^{-}],$ as shown in figures 5(D) and (E), relevant observations could be extracted from simulations concerning the ictal and interictal periods. During the ictal period, similar to the simulations described in the item 3.2, volume changes caused both H⁺ and ${{\rm{HCO}}}_{3}^{-}$ accumulation in the extracellular space (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0$ and ${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0)$ and a cytoplasmic reduction in neurons and glial cells (${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \lt \ 0,$ ${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0,$ ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{H}}}^{+}}\ \lt \ 0$ and ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0).$ The extracellular electrodiffusion contributed for the [H⁺]_o increase (${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0)$ and [${{\rm{HCO}}}_{3}^{-}$]_o decrease (${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0).$ The carbonic anhydrase caused an initial [H⁺]_o increase (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0,$ at the beginning of the ictal period) and later promoted its reduction (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0,$ in almost all of the ictal period). Carbonic anhydrase had no significant effect on the extracellular concentration of ${{\rm{HCO}}}_{3}^{-}$ (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \approx \ 0)$ when compared with the changes caused by the other mechanisms. However, the HCE exchanger was responsible for the ${{\rm{HCO}}}_{3}^{-}$ influx in glial cells (${r}_{{\rm{HCEg}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0$ and ${r}_{{\rm{HCEg}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}$ < 0).

During the interictal period, volume changes reduced [H⁺]_o and [${{\rm{HCO}}}_{3}^{-}$]_o (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0$ and ${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0)$ and contributed to the cytoplasmic increase of ionic concentrations in neurons and glial cells (${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \gt \ 0,$ ${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0,$ ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{H}}}^{+}}\ \gt \ 0$ and ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0).$ The extracellular electrodiffusion favored the [H⁺]_o increase (${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0).$ Even during the interictal period, the HCE exchanger was still responsible for the ${{\rm{HCO}}}_{3}^{-}$ influx into glial cells (${r}_{{\rm{HCEg}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0$ and ${r}_{{\rm{HCEg}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0),$ however, with less intensity.

Due to the action of the carbonic anhydrase, [${{\rm{HCO}}}_{3}^{-}$]_o seems to be necessary for [H⁺]_o increase (extracellular pH reduction). As shown in figure 5(C), ${{\rm{HCO}}}_{3}^{-}$ influx into glial cells was not enough to reduce [${{\rm{HCO}}}_{3}^{-}$]_o during the whole ictal period. Despite the initial increase due to glial [Cl⁻]_i, HCE activity was later decreased following a [Cl⁻]_o reduction. This [Cl⁻]_o reduction was promoted by the Cl⁻ influx into neurons and glial cells. Therefore, in neurons, simulation also showed Cl⁻ accumulation (figure 5(C)).

Considering that the HCE exchangers are also expressed in neurons (Rufin et al 2014), the neuronal ${{\rm{HCO}}}_{3}^{-}$ influx can also occur and would also contribute to the reduction of ${{\rm{HCO}}}_{3}^{-}$ concentration in the extracellular space as well as the regulation of pH. The inclusion of this exchanger in the neurons allows the reproductions of the pH excursion similar to those recorded in the extracellular space (figure 3(A)—simulation (iv)). In addition, it did not affect the typical excursion of the field potential, the transmembrane potential or Nernst potentials when compared with previous simulations (figures 6(A) and (B)). However, the simulation shows a decrease in dc shift as well as the duration of ictal and interictal periods. In contrast, no remarkable changes were observed in the intra and extracellular concentrations of Na⁺ and K⁺, as well as in intra and extracellular volumes (figure 6(C)). The inclusion of HCE exchangers affected the extracellular Cl⁻, resulting in the slow accumulation of this ion during the ictal period (due to the exchange of extracellular ${{\rm{HCO}}}_{3}^{-}$ by intracellular Cl⁻; figure 6(E) -${r}_{{\rm{HCE}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0).$ In the extracellular space, an increase in [H⁺] (figure 6(C)) has also contributed to a pH decrease (figure 3(B)). In the interictal period, [H⁺] is re-established to baseline levels. In the neuronal and glial cytoplasm, [H⁺]_i is reduced during the ictal period and only returns to baseline values during the interictal period. Reductions in [${{\rm{HCO}}}_{3}^{-}$]_o during the ictal period are also observed with this ion returning to baseline levels during the interictal phase following an initial overshoot. In neurons and glial cells, [${{\rm{HCO}}}_{3}^{-}$]_i increases during the ictal period and recovers during the interictal phase.

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Potential mechanisms responsible for [H⁺] and $[{{\rm{HCO}}}_{3}^{-}]$ changes during ictal and interictal periods are shown in figures 6(D) and (E). During the ictal phase, carbonic anhydrase contributed to the initial transient increase of the [H⁺]_o (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0)$ and its subsequent reduction (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0)$ in the extracellular space. As in previous simulations, the extracellular volume reduction favored H⁺ accumulation (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0),$ an effect that is countered by electrodiffusion (${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0).$ In the neuronal and glial intracellular spaces, the volume increase reduced [H⁺]. Analyzing [${{\rm{HCO}}}_{3}^{-}$]_o, the simulation does not show expressive activity of the carbonic anhydrase, when compared with the changes caused by the other mechanisms (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}$ ≪ ${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}},$ ${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}},$ ${r}_{{\rm{HCEg}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}$ and ${r}_{{\rm{HCEn}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}).$ The volume changes made the [${{\rm{HCO}}}_{3}^{-}$]_o reduction less intense (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0)$ and also reduced anionic accumulation in neurons and glial cells (${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0$ and ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0).$ The neuronal and glial HCE exchangers contributed to the [HCO₃]_o reduction (${r}_{{\rm{HCEg}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0$ and ${r}_{{\rm{HCEn}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0)$ and the anionic accumulation in neurons (${r}_{{\rm{HCEn}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0)$ and glial cells (${r}_{{\rm{HCEg}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}$ > 0).

During the interictal period, simulation shows the carbonic anhydrase promoting an initial transient reduction of [H⁺]_o (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0)$ followed by increase (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0).$ The extracellular volume recovery contributed to reduce [H⁺]_o (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0).$ The extracellular electrodiffusion was initially responsible for the [H⁺]_o reduction (${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0).$ Subsequently, the electrodiffusion countered the undershoot of [H⁺]_o (${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0).$ In neuronal and glial intracellular spaces, the reinstated volume contributed to the return of [H⁺]_i to baseline levels (${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \gt \ 0,$ ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{H}}}^{+}}\ \gt \ 0).$ On the other hand, the volume change slowed [${{\rm{HCO}}}_{3}^{-}]$ recovery in the extracellular space, as well as in neurons and glial cells (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0,$ ${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0$ and ${r}_{{\rm{vol}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0).$ [${{\rm{HCO}}}_{3}^{-}]$ recovery was mediated by neuronal and glial HCE exchangers (${r}_{{\rm{HCEg}},{\rm{g}}}^{{{\rm{HCO}}}_{3}^{-}}\ \lt \ 0,$ ${r}_{{\rm{HCEg}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0,$ ${r}_{{\rm{HCEn}},{\rm{o}}}^{{{\rm{HCO}}}_{3}^{-}}\ \gt \ 0$ and ${r}_{{\rm{HCEn}},{\rm{n}}}^{{{\rm{HCO}}}_{3}^{-}}$ < 0).

Comparing the simulated pH changes with the experimental data from Xiong and Stringer (2000a), it can be seen that changes in the extracellular space are more intense in the simulation (ΔpH ∼ 0,15, figure 3(B), simulation (iv)) than in the recordings (ΔpH ∼ 0,1). Moreover, as shown in figure 3(B) (simulation (iv)), the simulation shows neuronal intracellular alkalinization, whereas the experimental recording reveal acidification (figure 3(A)). Therefore, it is appropriate to suspect that mechanisms able to promote H⁺ influx in neurons are required. According to Chesler (2003), the Na⁺/H⁺ exchanger (NHE) is vastly expressed in the brain. This mechanism is normally described as an acid extruder, due to the elevated level of extracellular Na⁺ that contributes to the outflow of H⁺ in normal conditions (Chesler 2003, Ruffin et al 2014). However, under specific circumstances that effectively reduce Na⁺ this exchanger can cause H⁺ influx (Cha et al 2009). As shown in figure 3(A) (simulation (v)), the inclusion of NHE in the model allow to simulate the dynamic of the pH excursion of the intra and extracellular spaces, as observed experimentally (Xiong and Stringer 2000). The amplitude of changes was also reproduced in the same range. In glial cells, the model shows alkalinization associated with the intracellular ${{\rm{HCO}}}_{3}^{-}$ accumulation. Our simulation suggested that the inclusion of the NHE and H⁺ pump and of the effects of carbonic anhydrase on the extracellular space in the model did not alter the dynamic of the field potential (figure 7(A)), the transmembrane potential, the Nernst potentials (figure 7(B)), ionic concentrations in intra and extracellular spaces and cellular volume changes (figure 7(C)). Remarkable changes were related to the [H⁺] in the extracellular space and also in the neuronal and glial intracellular spaces. These changes are not enough to alter the concentration of other ions, since the mechanism responsible for [H⁺] changes are less intense than the activities of the mechanisms that affect their concentrations. Comparing the effects of mechanisms that changed [H⁺] and [${{\rm{HCO}}}_{3}^{-}]$ with the previous simulations (figures 6(D) and (E)), there were no significant changes in [${{\rm{HCO}}}_{3}^{-}].$ The changes were only relevant for [H⁺] (figure 7(E)).

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The simulation suggests that, in the ictal period, the extracellular volume change, the activity of the carbonic anhydrase, and the H⁺ pump are influenced by H⁺ accumulation (${r}_{{\rm{vol}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0,$ ${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0,$ ${r}_{{\rm{pump}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0).$ The NHE, which transports H⁺ to the neuronal intracellular space (${r}_{{\rm{NHE}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \lt \ 0),$ is counteracted, which reduces H⁺ accumulation. Compared to these mechanisms, electrodiffusion did not have a significant effect (${r}_{{\rm{ED}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \approx \ 0).$ In neurons, the NHE causes [H⁺] to increase (${r}_{{\rm{NHE}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \gt \ 0),$ which is restricted by the effects of the neuronal volume increase and the activities of the carbonic anhydrase and H⁺ pump (${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \lt \ 0,$ ${r}_{{\rm{CA}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \lt \ 0,$ ${r}_{{\rm{pump}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \lt \ 0).$ In glial cells, the volume increase and the carbonic anhydrase activity reduce the [H⁺].

During the interictal period, the simulation proposes that the carbonic anhydrase and the H⁺ pump continue contributing for the [H⁺]_o increase with a lesser intensity (${r}_{{\rm{CA}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0,$ ${r}_{{\rm{pump}},{\rm{o}}}^{{{\rm{H}}}^{+}}\ \gt \ 0).$ The re-establishment of the extracellular volume and of NHE activity restricted [H⁺] increase in the extracellular space. In neurons, the NHE and the cell volume reduction delayed the [H⁺] decrease (${r}_{{\rm{NHE}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \gt \ 0,$ ${r}_{{\rm{vol}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \gt \ 0).$ The H⁺ pump and the carbonic anhydrase are responsible for the neuronal [H⁺] recovering (${r}_{{\rm{CA}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \lt \ 0,$ ${r}_{{\rm{pump}},{\rm{n}}}^{{{\rm{H}}}^{+}}\ \lt \ 0).$ In glial cells, the [H⁺] recovery is due to the carbonic anhydrase activity and the cell volume reduction.