2.2.1 Conserved quantities.

We identified several first integrals in system (1), namely conservation of mass for sodium and chloride, and a relationship between the membrane potential of a neuron and its intracellular concentrations. In this section, we will explain their role and effect on the system’s dynamics. First of all, we introduce the conversion factors γ_e, γ_i, β₁ and β₂.

Conversion factors. Let z be the valence of an ion. Then, converts the current of this ion across the membrane of the pyramidal neuron in μA ⋅ cm⁻² to variation of concentration inside the cell in mM ⋅ ms⁻¹. The conversion factor γ_e is defined as where Vol_e and S_e are the volume and surface area of the pyramidal neuron, e = 1.6 ⋅ 10⁻¹⁹ C the elementary charge and N_A = 6.02 * 10²³ mol⁻¹ the Avogadro number. We take Vol_e = 1.4368 * 10⁻⁹ cm³ [26] and we compute S_e assuming the neuron is spherical: We obtain The conversion factor γ_i plays the same role as γ_e but for the GABAergic neuron. We assume that the GABAergic neuron is a sphere of volume Vol_i = β₂Vol_e, with , which gives

To convert variation of intracellular concentration to variation of extracellular concentration, we need to multiply by the intracellular volume of the corresponding neuron and to divide by the extracellular volume Vol_o. Let β₁ be the ratio of total intracellular volume over extracellular volume: [42]. We then have

Conservation of mass. In system (1), the sodium and chloride concentrations are only modified by transmembrane currents. We have thus conservation of mass for those ions. Note that it is not the case for potassium, for which we take into account extracellular diffusion (see Section 3.1.4). The conserved quantities enable us to reduce the number of unknowns. For sodium, Let We fix We can then express the extracellular sodium concentration as a function of the intracellular concentrations: Similarly, for chloride we define and choose

Relation between membrane potential and intracellular ion concentrations. System (1) has two other first integrals, which arise from the fact that we take into account the effect of all ionic currents on the dynamics of the intracellular concentrations and that no other mechanism affects those concentrations. For the pyramidal neuron, we have (2) This allows us to remove an additional unknown. Let We fix We can then express the potassium concentration in the pyramidal neuron as We proceed similarly for the GABAergic neuron: let with The potassium concentration in the GABAergic neuron is given by:

Note that, in such a configuration, we should not model external inputs to the neurons with a constant current appearing only in the equation for the membrane potential. For example, if we add a constant external current J to the right hand side of Eq (1a): Eq (2) becomes Integrating, we see that it causes a drift of the system: which thus cannot have any steady state nor limit cycle. Instead, we modeled external inputs to the neurons using synaptic currents. We assumed a constant glutamate input on AMPA receptors, which generates sodium and potassium currents. Those currents appear both in the equation for the membrane potential and in the ones for the intracellular ion concentrations, preserving the first integrals H₁ and H₂. They are defined in Section (2.2.2).

2.2.2 Transmembrane ion currents.

The reversal potentials, which are used to compute ion currents, are typically assumed to be constant. Here, they vary with the ion concentrations, and their dependence on the corresponding ion gradient is given by the Nerst equation: where R = 8, 314 mJ ⋅ (K ⋅ mol)⁻¹ is the ideal gas constant, T the temperature, F = N_A e the Faraday constant and z_ion the valence.

Pyramidal neuron. For the pyramidal neuron, the currents generated by the sodium and potassium voltage-gated channels were modeled as in [26]:

• Fast inactivating sodium current: ,
• Delayed rectifier potassium current: .

As [26], we took into account a calcium-activated potassium current, which is involved in the afterhyperpolarization (AHP) phase of action potentials. It is defined as The implementation of cotransporters was also directly taken from [26]. Those proteins perform secondary active transport: they use the favorable movement of molecules with their electrochemical gradient as an energy source to move other molecules against their gradient. The potassium-chloride transporter member 5 (KCC2) extrudes potassium and chloride (Fig 1), using the potassium gradient to maintain low intracellular chloride concentration: The Na-K-Cl cotransporter isoform NKCC1 transports sodium, potassium and chloride into the cell, with the stoichiometry 1Na:1K:2Cl (Fig 1):

Contrary to [26], we distinguished the sodium, potassium and chloride components of the leak current:

• Leak sodium current: I_Na,L,e = g_Na,L,e(v_e − E_Na,e),
• Leak potassium current: I_K,L,e = g_K,L,e(v_e − E_K,e),
• Leak chloride current: I_Cl,L,e = g_Cl,L,e(v_e − E_Cl,e).

This allowed us to measure their effect on the different ion concentrations. Similarly, we separated the sodium and potassium currents due to the excitatory autapse, assuming an equal permeability of the glutamatergic receptors to both ions:

• ,
• .

As explained in Section (2.2.1), external inputs to the pyramidal neurons were modeled with a constant glutamate input on those receptors:

• ,
• .

Those currents represent average excitatory network activity or experimental depolarizations. The inhibitory synaptic current, created by the movement of chloride ions through GABA_A receptors, was modeled as in [26]:

To model the activity of the Na⁺/K⁺ ATPase, we used this expression [34, 43]: with half-activation parameters from [35]. Based on Bouret et al. [36], we also introduced a voltage dependence of the pump maximal rate: where

To summarize, the net currents for each ion are:

GABAergic neuron. The sodium and potassium currents through the voltage-gated channels of the GABAergic neurons were modeled as in [33]:

• Fast inactivating Na⁺ current: ,
• Delayed rectifier K⁺ current: .

For more details on the gating dynamics of those channels, see Section (2.2.5).

In the same way as for the pyramidal neuron, we separated the components pertaining to the different ions for the leak currents:

• Leak Na⁺ current: I_Na,L,i = g_Na,L,i(v_i − E_Na,i),
• Leak K⁺ current: I_K,L,i = g_K,L,i(v_i − E_K,i),

for the glutamatergic synaptic currents:

• ,
• ,

and for the glutamatergic synaptic currents which model an average external input to the GABAergic neuron:

• ,
• .

The Na⁺/K⁺ ATPase current is given by the same sigmoidal function as for the pyramidal neuron:

We obtained the following sodium and potassium net currents:

2.2.3 Calcium concentration in the pyramidal neuron.

The conductance of the calcium-activated potassium current I_K,AHP,e is determined by the pyramidal neuron’s intracellular calcium concentration. As in [26], we modeled the dynamics of this concentration with Eq (1h) from Wang [41]: I_Ca,e represents high threshold calcium current and is given by: It is a transmembrane current, but for simplicity we did not take into account its effect on the pyramidal neuron’s membrane potential. Unlike in [26], we used the same conversion factor γ_e as for the other ions, for consistency. The second term models various extrusion and buffering mechanism, with a first order decay process.

2.2.4 Diffusion of extracellular potassium.

As in [26], the following term appears in the equation for the extracellular potassium Eq (1p): It accounts for passive diffusion of extracellular potassium, but also, in a simplistic way, for the buffering of this ion by glial cells. This motivated the use of a large value of the diffusion coefficient ϵ (see Table 2), which is a conservative choice.

2.2.5 Gating variables dynamics.

Gating variables represent the state of activation of voltage-gated channels. In Section 2.2.2 and Section 2.2.3, they scale the maximal conductance of those channels.

Pyramidal neuron. For the pyramidal neuron, the dynamics of the potassium and sodium gating variables is given by Eqs (1b)–(1d). As in [26], we used a simplified version of the Traub-Miles model [44] due to Wei et al. [28]: Eq (1h) describes the dynamics of the intracellular calcium concentration, needed to compute the conductance of the calcium-activated potassium channels. As in [26], we assumed that the state of the calcium channels depends instantaneously on the voltage, according to this expression by Wang [41] with parameter values from Gutkin et al. [40]:

GABAergic neuron. For the GABAergic neuron, we replaced the Wang-Buzsáki model [32] with a more recent one by Golomb et al. [33], which is specific to fast-spiking cortical interneurons: The sodium activating variable is assumed to be at its steady state value m_∞,i. The dynamics of the sodium inactivating variable h_i and of the potassium activating variable n_i is given in Eqs (1k) and (1l).

2.2.6 Mutations of Na_V1.1.

Na_V1.1 is mainly expressed in GABAergic neurons and Na_V1.1 mutations affect mainly these neurons [2]. Thus, we assumed in our simulations that the pyramidal neuron is unaffected by mutations of this channel. As we shall see, to model FHM-3 or epilepsy we only modified two parameters: the maximal conductance g_Na,FI,i of the fast-inactivating sodium current and the maximal conductance g_Na,P,i of the persistent sodium current, which we introduce in the case of migraine.

FHM-3 mutations. Increased persistent sodium current is a common effect of most Na_V1.1 FHM-3 mutations [6–9]. To model them, we partially replaced the GABAergic neuron’s fast inactivating sodium current I_Na,FI,i with a persistent sodium current I_Na,P,i. We kept the sum of their maximal conductances constant: g_Na,FI,i + g_Na,P,i = 112.5 mS · cm⁻². We modeled persistent current with the following expression: Note that it does not include the inactivation mechanism represented by the variable h_i for the fast inactivating current. We also shifted the voltage dependence of its activation to more negative potentials [45, 46]. Let p_Na,P be the percentage of maximal voltage-gated sodium conductance corresponding to persistent current: To model Na_V1.1 FHM-3 mutations, we tested values up to p_Na,P = 20%; see Section (3.1).

Epileptogenic mutations. Epileptogenic mutations of Na_V1.1 cause a loss of function of the channel [1, 2]. To model them, we decreased the maximal conductance of the GABAergic neuron’s fast inactivating sodium current g_Na,FI,i. Based on what was experimentally observed in Scn1a^+/- mice [1], we set it to 40% of its default value, reflecting the condition of heterozygosis and the expression of other voltage-gated sodium channels than Na_V1.1 in GABAergic neurons:

3.1.1 Persistent sodium current (I_Na,P,i) in GABAergic neurons amplifies spiking-induced modifications of extracellular ion concentrations, even without modifications of their firing frequency.

We first focused on Na_V1.1 FHM-3 mutations, modeled by the increase of I_Na,P,i for the GABAergic neuron which is a common effect of most of those mutations [6–9], and on their effect on the features of the GABAergic neuron itself. To investigate the latter point, we obtained simulations with a version of the model in which the influence of the pyramidal neuron on the GABAergic neuron, through synaptic connections or through variations of extracellular ion concentrations, was removed.

To study how I_Na,P,i modifies the excitability of the GABAergic neuron, we computed the following input-output relationships: number of action potentials elicited during the application of a depolarizing external current of fixed duration versus the conductance of this current (Fig 2A). We found that an increase of I_Na,P,i reduces the rheobase (i.e. minimal external input necessary to trigger at least one action potential): 0.0004 mS ⋅ cm⁻² when p_Na,P = 20% (migraine condition) instead of 0.0051 mS ⋅ cm⁻² when p_Na,P = 0% (control condition), where p_Na,P is defined in Section 2.2.6. This is consistent with a decrease of the rheobase observed experimentally in neocortical mouse neurons transfected with hNa_V1.1-L1649Q, a pathogenic mutant of the Na_V1.1 channel associated with FHM [7]. However, in contrast to [7], in our model FHM-3 mutations do not substantially increase the GABAergic neuron’s firing frequency. With external inputs g_D,i up to approximately 0.1 mS ⋅ cm⁻², an increase of p_Na,P consistently leads to a moderate increase of the number of action potentials generated, but this is not the case for stronger external inputs. Interestingly, there may be variability among GABAergic neurons, concerning how their firing frequency is affected by FHM-3 mutations. In a novel FHM-3 knock-in mouse model, Freilinger et al. (personal communication; see acknowledgments) observed an increase of frequency in fast-spiking GABAergic neurons, but no significant difference in regular spiking interneurons, although they both express Na_V1.1.

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Fig 2. Effect of FHM-3 mutations on the firing properties of the GABAergic neuron.

For different values of p_Na,P (defined in Section 2.2.6), which we increase to model FHM-3 mutations, we applied to the GABAergic neuron 0.4 s long excitatory external inputs of conductances g_D,i. For each p_Na,P value, we took as initial condition the steady state in the absence of external input, i.e. when g_D,i = 0 mS · cm⁻². A: Number of action potentials, considered as overshooting spikes (i.e. exceeding 0 mV of peak membrane potential). B: Extracellular potassium concentration at the end of the 0.4 s long simulations. C: Extracellular sodium concentration at the end of the 0.4 s long simulations.

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On the other hand, persistent sodium current clearly enhances the accumulation of extracellular potassium (Fig 2B) and the uptake of extracellular sodium (Fig 2C) due to the firing of the GABAergic neuron, even when the number of action potentials is similar or slightly smaller than in the control condition. For example, for an external input of conductance g_D,i = 0.3 mS · cm⁻² and p_Na,P = 20%, with 48 spikes in 0.4 s we obtained an extracellular potassium concentration of 8.6 mM (145% increase) and an extracellular sodium concentration of 147.5 mM (3.7% decrease). However, when p_Na,P = 0%, there were 49 spikes in 0.4 s but only 5.9 mM of extracellular potassium (68% increase) and still 150.7 mM of extracellular sodium (1.7% decrease) at the end of the simulation.

This is a counterintuitive result that we have better investigated comparing detailed features of action potentials and underlying ionic currents for those parameter values: g_D,i = 0.3 mS · cm⁻² and p_Na,P = 0% or 20% (Fig 3). With increased I_Na,P,i, simulations showed larger action potential half width (0.365 ms for the 25^th action potential with p_Na,P = 0% and 0.565 ms with p_Na,P = 20%, Fig 3C₁). However, modifications of action potential features were relatively small compared to the increase of I_Na,P,i. We reasoned that the I_Na,P,i increase could induce a larger activation of potassium channels during the action potentials, able to limit the modifications of action potential features, but leading to larger potassium currents and consequent spiking-induced extracellular potassium build up. Indeed, the plots of the potassium action currents show a large increase when I_Na,P,i is higher (0.520 μA ⋅ cm⁻² for the 25^th action potential in the discharge with p_Na,P = 0%, 1.311 μA ⋅ cm⁻² with p_Na,P = 20%, Fig 3C₂), leading to a larger accumulation of extracellular potassium at each action potential (Fig 3A₃, 3B₃ and 3C₃). As expected, action sodium currents show a large increase as well, in particular during the repolarization phase (0.478 μA ⋅ cm⁻² for the 25^th action potential in the discharge with p_Na,P = 0%, 1.284 μA ⋅ cm⁻² with p_Na,P = 20%, Fig 3C₄), leading to a consequent larger decrease of extracellular sodium concentration at each action potential (Fig 3A₅, 3B₅ and 3C₅).

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Fig 3. Effect of FHM-3 mutations on the firing and action currents of the GABAergic neuron, and correlation with the dynamics of [K⁺]_o and [Na⁺]_o.

Plots of the voltage (action potentials; upper row), total potassium current (second row), extracellular potassium concentration (third row), total sodium current (fourth row) and extracellular sodium concentration (bottom row) corresponding to simulations displayed in Fig 2, for the condition in which g_D,i = 0.3 mS · cm⁻². A: No persistent sodium current: p_Na,P = 0%. B: p_Na,P = 20%. C: Zoom showing the 25^th action potential of A (in green) and B (in red). The dashed lines are the curves obtained with p_Na,P = 20% shifted to allow a better comparison with the curves obtained with p_Na,P = 0%.

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3.1.2 Dynamics of neuronal firing at CSD initiation.

In this section, we show how FHM-3 mutations in the GABAergic neuron influence the entry of the pyramidal neuron into depolarization block, which we consider as the initiation of a CSD, when the two neurons are coupled. We depolarized them with the same external inputs g_D,e = g_D,i = 0.3 mS · cm⁻², and we compared the control condition (p_Na,P = 0%) with a pathological one (p_Na,P = 15%).

In the first case, tonic spiking of the GABAergic neuron begins immediately while there is a latency of a few seconds before the pyramidal neuron also starts to generate repetitive action potentials (Fig 4A). Those initial subthreshold oscillations demonstrate the inhibitory nature of the GABAergic neuron in a physiological situation. As expected, the firing frequency is larger in the GABAergic neuron than in the pyramidal one (Fig 5A and 5B). With pathological persistent sodium current in the GABAergic neuron, the voltage traces are very different (Fig 4B). As in the control condition, there is a delay before sustained firing of the pyramidal neuron, but both the GABAergic interneuron and the pyramidal neuron show two phases of activity, and during the first phase extracellular potassium rises faster and to a larger extent than in control (Fig 5C). This is coherent with the results obtained on the GABAergic neuron alone in Section 3.1.1. Shortly after the pyramidal neuron begins to spike, we observe an increase of its firing frequency to towards an initial plateau, together with a steeper slope for the increase in extracellular potassium concentration. This is followed by a further increase of the pyramidal neuron’s firing frequency. The second phase leads to the beginning of depolarization block for both neurons, while extracellular potassium continues to grow. Notably, sodium overload during the depolarization block (Fig 5D) can contribute to silencing of firing.

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Fig 4. Effect of the FHM-3 mutation on CSD initiation: Representative time traces.

A-B: Model simulations. Starting from neurons at rest, similarly to Fig 2, we stimulated them with constant excitatory conductances g_D,i = g_D,e = 0.3 mS · cm⁻² for 30 s. A: No persistent sodium current for the GABAergic neuron (p_Na,P = 0%). B: Pathological condition (p_Na,P = 15%). C: Experimental data. Representative dual juxtacellular-loose patch voltage recordings of a GABAergic interneuron (C₁ and C₃) and a pyramidal neuron (C₂ and C₄) showing the dynamics of the firing at the site of CSD initiation; CSD was induced by spatial optogenetic activation of GABAergic neurons and is the slow negative deflection observable in C₁ and C₂. The blue bar in C₁ shows the optogenetic stimulation with blue light.

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Fig 5. Behavior of the firing frequency and of the extracellular ion concentrations before the onset of CSD.

A-D: Model simulations. For the first 5 s of the simulations presented in Fig 4, we compared the pathological situation (p_Na,P = 15%) with the physiological one (p_Na,P = 0%). A: Instantaneous firing frequency, i.e. inverse of the interspike interval, for the GABAergic neuron. B: Instantaneous firing frequency for the pyramidal neuron. C: Extracellular potassium concentration. D: Extracellular sodium concentration. E-F: Experimental data. E: Instantaneous firing frequency of the loose patch recording of the GABAergic neuron shown in Fig 4C₁ and 4C₃. F: Instantaneous firing frequency of the loose patch recording of the pyramidal neuron shown in Fig 4C₂ and 4C₄. CSD was induced by the optogenetic stimulation of the GABAergic neuron; the two phases in the firing dynamics, with lower and higher frequency respectively, are evident and similar to those observed in the simulation. Although the first phase was longer for GABAergic neurons (55.3 ± 4.5 s, n = 7) than for pyramidal neurons (1.6 ± 0.2 s, n = 5; p = 0.006 Mann-Whitney test), the second phase had similar duration (2.0 ± 0.2 s for GABAergic neurons; 1.7 ± 0.3 s for pyramidal neurons; p = 0.28, Mann-Whitney test) and firing frequency was not significantly different (mean instantaneous firing frequency in the first phase was 48.4 ± 6.8 Hz for GABAergic neurons and 32.4 ± 8.5 Hz for pyramidal neurons, p = 0.19 Mann-Whitney test; in the second phase it was 349 ± 36 Hz for GABAergic neurons and 214 ± 57 Hz for pyramidal neurons, p = 0.06 Mann-Whitney test).

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In order to study experimentally the dynamics of the firing of both GABAergic interneurons and pyramidal neurons at the site of CSD initiation, we performed pairs of juxtacellular-loose patch voltage recordings, inducing CSD by spatial optogenetic activation of GABAergic neurons as in [29]. Although this method of induction does not completely reproduce the condition of the simulation, it mimics hyperexcitability of the GABAergic neurons and allows recordings at the pre-determined site of CSD initiation, which cannot be performed with other experimental models of CSD. Notably, the firing dynamics was similar to that obtained with the simulation. We found that GABAergic neurons began to fire at the beginning of the illumination, whereas pyramidal neurons began to fire later during the illumination, just for few seconds before CSD initiation (Figs 4C, 5E and 5F). As in the simulation, the firing dynamics of both GABAergic and pyramidal neurons showed two phases, a first phase with lower firing frequency (which was on average 34.6-fold longer for the GABAergic neurons) and a second phase of similar duration (few seconds) for the two types of neurons, in which firing frequency increased on average 8.3±1.8-fold for GABAergic neurons and 8.1±2.5-fold for pyramidal neurons, leading to depolarization block. Although few GABAergic neurons fired at very high frequency, instantaneous frequency was on average not different between GABAergic and pyramidal neurons.

3.1.3 Persistent sodium current in GABAergic neurons reduces the threshold for CSD initiation.

More generally, we studied whether those findings are robust to modifications of the value of parameter p_Na,P in the pathological case. This is important since we do not know precisely which level of persistent sodium current best models the effects of FHM-3 mutations. Migraine is an episodic disorder, which means that patients exhibit symptoms only during attacks: a trigger factor causes the shift from a physiological state to a pathological state [12]. In our model, FHM-3 mutations reduce the threshold for this transition. Indeed, the more persistent sodium current in the GABAergic neuron, the smaller the minimal external input necessary to initiate CSD (Fig 6A). This is consistent with our recent experimental work where it was shown that bath application of Hm1a, a toxin which mimics the effects of FHM-3 mutations by increasing persistent sodium current, can lead to spontaneous CSD initiation in mouse brain slices [29]. Moreover, we found that, for a given large external input, persistent sodium current decreases the latency to CSD (Fig 6B). Here also, our simulations agree qualitatively with experimental data. Indeed, optogenetic activation of GABAergic neurons induces CSD earlier in slices perfused with Hm1a than in control slices [29]. Those two points are also supported by experiments on the knock-in mouse model of the L1649Q FHM-3 mutation: the threshold for eliciting CSD by electrical stimulation was substantially lowered in heterozygous mice, and the latency from KCL application to the onset of CSD was significantly reduced (Freilinger et al., personal communication; see acknowledgments).

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Fig 6. Effect of FHM-3 mutations on CSD initiation.

A: We show here how increasing the GABAergic neuron’s persistent sodium current influences the minimal external input necessary to induce a depolarization block in the pyramidal neuron, varying p_Na,P from 0 to 20%. We used as external inputs constant excitatory conductances, equal for both neurons, taking values up to g_D,i = g_D,e = 0.3 mS · cm⁻². We considered that beyond these values, inputs would be unrealistically strong, leading to a response of the model that would not be interpretable. Again, we chose as initial condition the steady state when the neurons are not stimulated. We defined a depolarization block by the absence of oscillations of amplitude larger than 5 mV of the voltage for at least half a second, with the additional condition that it must be between −55 mV and −20 mV at the end of this interval. For each p_Na,P, we estimated the smaller external input for which this criterion is met for the pyramidal neuron with a bisection method. B: Starting from neurons at rest, for given large external inputs g_D,e = g_D,i = 0.3 mS · cm⁻², we computed the time it takes for a CSD to be initiated, if it is initiated at all.

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Very close to the threshold that is approximated in Fig 6A, the response oscillates between non-CSD and CSD solutions, when gradually increasing the external input. This shows great sensitivity of the model at the transition between physiological and pathological behaviors. We cannot assert whether this is a numerical effect or a property inherent to the model. In any case, such a phenomenon is not surprising given the dimensionality of the model and its multiple time scales. Similar issues already arise in much simpler conductance-based models, e.g. FitzHugh-Nagumo, Morris-Lecar or a 2D reduction of Hodgkin-Huxley, which are type-2 neuron models for which the firing threshold may not be well defined [52]. The approximation of the threshold for CSD initiation that we have obtained is nevertheless sufficient to study the roles of persistent sodium current and extracellular ion concentrations. Computing the exact threshold, if it exists, is challenging, and we do not consider it relevant for the purpose of the present study. This is an interesting question for future work, where we will focus on reducing our current model while keeping its most salient features.

3.1.4 The accumulation of extracellular potassium is crucial for CSD initiation.

In our model, the two neurons are coupled through synaptic connections and through variations in extracellular ion concentrations. The synaptic connection from the GABAergic neuron to the pyramidal one is inhibitory in all the tested conditions (see Section 3.3). Modifications of extracellular ion concentrations can thus be the cause of the large increase of firing frequency that precedes the depolarization block observed in the pathological case (Fig 5B). In the first seconds of the simulation, both extracellular potassium and sodium concentrations are substantially modified (Fig 5C and 5D), but the relative change is much more important for potassium, consistent with the simulations with the isolated GABAergic neuron (Fig 2). A rise in extracellular potassium increases the reversal potential for this ion, which is a depolarizing effect, while a decay of the extracellular sodium has the opposite effect. This suggests that potassium plays a major role in promoting CSD initiation. To confirm it, we implemented unrealistically strong buffering of either extracellular potassium or sodium, to keep their concentration constant (Fig 7A and 7B). In the same pathological setting as in Fig 5B, we applied strong external inputs g_D,i = g_D,e = 0.3 mS · cm⁻² to the neurons. We observe that only the strong buffering of extracellular potassium prevents CSD initiation (Fig 7B). More generally, similarly to (Fig 6), we computed the thresholds for CSD initiation and the latencies to CSD for strong external inputs (Fig 8A). Remarkably, keeping the extracellular potassium concentration low completely prevents CSD initiation over the examined ranges of persistent current and external inputs. This is not the case with sodium; in fact the strong buffering of its extracellular concentration even facilitates CSD initiation, and leads to a larger increase of extracellular potassium concentration.

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Fig 7. The roles of extracellular ions in CSD initiation: Representative time traces.

When Na_V1.1 carries a FHM-3 mutation, modeled by p_Na,P = 15%, we tested the effect of keeping either extracellular potassium or sodium constant using unrealistically high diffusion rates. We depolarized the neurons at rest with constant excitatory conductances g_D,i = g_D,e = 0.3 mS · cm⁻² for 30 s. A: Voltage traces for the default parameters. B: Voltage traces when ε = 0.1 ms⁻¹ to simulate a strong buffering of extracellular potassium. C: Voltage traces when there is a strong buffering of extracellular sodium. To implement it, we introduced the diffusion term I_diff,Na = ε_Na([Na⁺]_o − Na_bath) in the equation for extracellular sodium, with ε_Na = 0.1 ms⁻¹ and Na_bath equal to the steady-state extracellular sodium in the absence of external input. D: [K⁺]_o traces in the cases described in A-C. E: [Na⁺]_o traces in the cases described in A-C.

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Fig 8. The roles of extracellular ions in CSD initiation.

Similarly to Fig 6, we show in the left panel the minimal external input necessary to induce a depolarization block in the pyramidal neuron, and in the right panel the latency to CSD. A: We tested the effect of keeping either extracellular potassium or sodium approximately constant using the same method as for Fig 7. B: We tested the effect of reducing the extracellular potassium diffusion rate. The default value of this parameter is ε = 5 ⋅ 10⁻⁴ ms⁻¹.

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We also tested smaller values of the extracellular potassium diffusion rate. This can model, in a simplistic way, the contribution of other GABAergic neurons to the accumulation of extracellular potassium, or a less efficient buffering by the glial network. As expected, it reduces the threshold for CSD initiation and CSD is ignited earlier for a given stimulus (Fig 8B).

3.2.1 Reduced sodium current in GABAergic neurons makes them more susceptible to depolarization block.

As with the FHM-3 mutations (Section 3.1.1), we first investigated the effects of epileptogenic mutations on the GABAergic neuron itself, when it does not interact with the pyramidal neuron. To model epileptogenic mutations of Na_V1.1, we reduced the GABAergic neuron’s fast-inactivating sodium maximal conductance, as explained in Section 2.2.6.

The simulations show that in this condition the rheobase is slightly increased, action potential amplitude is decreased and depolarization block is induced by smaller depolarizing external inputs (Figs 9A, 9B and 10A). Notably, action potential firing frequency shows just very small modifications in the input-output relationship (Fig 10A₁), in which a slight increase is surprisingly observed for the epileptic condition between g_D,i = 0.2 and g_D,i = 0.4 mS · cm⁻². We observed similar modifications in experimental traces, comparing cortical layer 2-3 fast spiking GABAergic neurons recorded in heterozygous Scn1a knock-out mice (Scn1a^+/-), model of the developmental and epileptic encephalopathy Dravet syndrome [1, 2], and wild type littermates as control (Figs 9C, 9D and 10B). Notably, the comparison of single representative neurons with similar input-output relationships showed that, although firing frequency could be slightly larger for the Scn1a^+/- neuron, depolarization block was induced by a smaller depolarizing current (Figs 9C, 9D and Fig 10B₁) and action potential amplitude was reduced (Fig 10B₂). The comparison of average features did not disclose modifications of firing frequency before the induction of depolarization block in Scn1a^+/- (Fig 10B₃), but depolarization block was consistently induced with smaller depolarizing currents. Of note, for obtaining these average input-output curves, we excluded for each cell the traces in which there was depolarization block, to avoid the influence of depolarization block in the evaluation of firing frequency with larger depolarizing currents. To statistically compare action potential amplitude, we quantified the mean amplitude of the first suprathreshold action potential, which was reduced in Scn1a^+/- mice (Fig 10B₄).

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Fig 9. Effect of Na_V1.1 epileptogenic mutations on the firing of GABAergic neurons: Representative traces.

A-B: Model simulations. We depolarized the GABAergic neuron at rest with constant excitatory conductances g_D,i = 0.0075 (A₁, B₁), g_D,i = 0.05 (A₂, B₂) or g_D,i = 0.5 mS · cm⁻² (A₃, B₃), for the default parameters (A: control condition), or when the sodium fast-inactivating maximal conductance is reduced to 40% of its default value (B: Na_V1.1 epileptogenic mutation). C-D: Experimental data. Action potential discharges elicited in a fast spiking GABAergic layer 2-3 cortical neuron from a wild type mouse (C) or a Scn1a^+/- mouse (D), injecting depolarizing current steps of 10 pA (C₁,D₁), 100 pA (C₂,D₂) or 260 pA (C₃,D₃).

https://doi.org/10.1371/journal.pcbi.1009239.g009

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Fig 10. Effect of Na_V1.1 epileptogenic mutations on the firing of GABAergic neurons: Input-output relationship and action potential amplitude.

A: Model simulations. We applied to the GABAergic neuron at rest 2.5 s long excitatory external inputs of conductances g_D,i, for the default parameters (wild-type), or for a reduced sodium fast-inactivating maximal conductance g_Na,FI,i (epileptogenic mutations). A₁: Number of action potentials. A₂: Representative action potentials, for g_D,i = 0.3 mS · cm⁻². B: Experimental data. B₁: Representative fast spiking neurons recorded in cortical slices from Scn1a^+/- mice and wild type littermates, selected for having similar input-output curves with lower injected current, showing that the Scn1a^+/- neuron was more prone to depolarization block (sharp decrease of action potential number). B₂: Comparison of the first suprathreshold action potential recorded in the neurons selected for panel B1. B₃: Mean input-output relationships for fast spiking neurons recorded from Scn1a^+/- (n = 7) and wild type littermates (n = 7), the number of elicited action potentials was not different, but depolarization block was induced with lower depolarizing current (the traces in which there was depolarization block were excluded, to avoid the influence of depolarization block in the evaluation of the number of action potentials elicited with larger depolarizing currents). B₄: Mean peak amplitude (absolute value) of the first suprathreshold action potential for fast spiking neurons recorded from Scn1a^+/- (22.6 ± 1.5 mV, n = 7) and wild type littermates (28.7 ± 2.0 mV, n = 7); p = 0.044 Mann-Whitney test. * p < 0.05.

https://doi.org/10.1371/journal.pcbi.1009239.g010

To better understand the enhanced transition from the firing regime to depolarization block of the GABAergic neuron when Na_V1.1 carries an epileptogenic loss of function mutation, which we observed in Fig 10A₁, we studied it as a dynamic bifurcation phenomenon [53]. Even though system (1) is not explicitly slow-fast, it is clear from its time traces that it has different time scales. We considered the sodium concentration in the GABAergic neuron [Na⁺]_i as a slow variable, and analyzed the corresponding fast subsystem where [Na⁺]_i is a parameter, with or without epileptogenic mutation. In Fig 11A, we show the voltage traces of the complete system when applying an excitatory conductance g_D,i = 0.3 mS · cm⁻² to the GABAergic neurons at rest. In Fig 11B, we projected the same solutions onto the ([Na⁺]_i, v_i) plane, superimposed onto the bifurcation diagram of the fast subsystem with respect to [Na⁺]_i, what is called a slow-fast dissection [54]. In the wild-type case, the solution first follows limit cycles of large amplitude, which corresponds to repetitive firing, while the sodium slowly increases (Fig 11B₁). For [Na⁺]_i approximately 28 mM there is a fold of limit cycle bifurcation, which causes the solution to jump to a stable steady state. This is the beginning of a quiescent phase, where the neuron does not spike while [Na⁺]_i decreases slowly until a Hopf bifurcation is encountered, where the steady state becomes unstable. This causes the solution to return to the large-amplitude limit cycles. In this way, the solution alternates between active and quiescent phases (Fig 11A₁), a neuronal behavior typically referred to as bursting. It is not surprising since bursting is a possible behavior of the model we have used to describe the gating dynamics of the GABAergic neuron’s voltage-gated channels [33]. The subHopf/fold-cycle hysteresis loop which enables this bursting is not preserved for the reduced value of fast-inactivating sodium maximal conductance which we use to model Na_V1.1’s epileptogenic mutations (Fig 11B₂). In this case, the solution of the complete system also first follows large-amplitude limit cycles while [Na⁺]_i slowly increases. However, once it reaches the fold of limit cycle bifurcation and jumps to the stable steady state, it does not leave it anymore: the GABAergic neuron completely stops producing action potentials (Fig 11A₂).

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Fig 11. Effect of Na_V1.1 epileptogenic mutations on the GABAergic neuron bifurcation structure.

We considered the intracellular sodium concentration as a slow variable and we studied the corresponding fast subsystem where this concentration is a parameter. We focused here on the case where g_D,i = 0.3 mS · cm⁻². A: Voltage trace of the full system starting from a neuron at rest, for the default parameter values (A₁) or when the fast-inactivating sodium maximal conductance is reduced to 40% of its default value (A₂). B: Trajectory of A₁ (B₁) or A₂ (B₂), projected onto the ([Na⁺]_i, v_i) plane and bifurcation diagram of the fast subsystem with respect to the intracellular sodium concentration. We represented steady-states with black lines and limit cycles with purple lines.

https://doi.org/10.1371/journal.pcbi.1009239.g011

3.2.2 Pyramidal neurons firing frequency suddenly increases when GABAergic neurons enter depolarization block.

We saw in the previous section that Na_V1.1’s epileptogenic loss of function mutations make GABAergic neurons more susceptible to depolarization block. We now focus on the consequences on the firing of the pyramidal neuron in our computational model, when the two neurons are coupled. We stimulated them with strong depolarizing external inputs (g_D,i = g_D,e = 0.3 mS · cm⁻²), with or without Na_V1.1’s epileptogenic loss of function mutations mutation (Fig 12). In the pathological condition, after firing for about ten seconds (Fig 12B₁ and 12C₁), the GABAergic neuron enters depolarization block. Simultaneously, we observe a sharp increase of the firing frequency of the pyramidal neuron (Fig 12B₂ and 12C₂). This behavior is easily understood: when the GABAergic neuron stops to generate action potentials, the synaptic inhibitory restraint that it exerts on the pyramidal neuron is suddenly removed, allowing the pyramidal neuron to fire at a greater pace. Although this effect cannot be considered epileptiform activity, it may model early hyperexcitability that has been observed in mouse models of Dravet syndrome before the appearance of spontaneous seizures [2, 47].

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Fig 12. Effect of Na_V1.1 epileptogenic mutations on the firing frequency of the pyramidal neuron.

When both neurons are coupled, taking as initial condition the steady-state when there is no external input, we applied external inputs g_D,i = g_D,e = 0.3 mS · cm⁻². We tested both the default parameters, which represent a wild-type GABAergic neuron, and a sodium fast-inactivating maximal conductance g_Na,FI,i reduced to 40% of its default value, to model epileptogenic mutations of Na_V1.1. A: Voltage of the GABAergic (A₁) and pyramidal (A₂) neurons in the wild-type condition. B: Voltage of the GABAergic (B₁) and pyramidal (B₂) neurons with the epileptogenic mutation. C: Instantaneous firing frequency of the GABAergic (C₁) and pyramidal (C₂) neurons in the two conditions.

https://doi.org/10.1371/journal.pcbi.1009239.g012