The authors have declared that no competing interests exist.
Loss of function mutations of
The voltage-gated sodium channel NaV1.1 is a major target of human mutations implicated in different pathologies. In particular, mutations identified in certain types of epilepsy cause loss of function of the channel, whereas mutations identified in certain types of migraine (in which spreading depolarizations of the cortical circuits of the brain are involved) cause instead gain of function. Here, we study dysfunctions induced by these differential effects in a two-neuron (GABAergic and pyramidal) conductance-based model with dynamic ion concentrations. We obtain results that can be related to experimental findings in both situations. Namely, extracellular potassium accumulation induced by the activity of the GABAergic neuron in the case of CSD, and higher propensity of the GABAergic neuron to depolarization block in the epileptogenic scenario, without significant modifications of its firing frequency prior to it. Both scenarios can induce hyperexcitability of the pyramidal neuron, leading in the migraine condition to depolarization block of both the GABAergic and the pyramidal neuron. Our results are successfully confronted to experimental data and suggest that modification of firing frequency is not the only key mechanism in these pathologies of neuronal excitability.
The experimental data is available in Zenodo at
NaV1.1 is a voltage-gated sodium channel mainly expressed in GABAergic neurons and it is crucial for their excitability. Mutations of
FHM is a rare but severe subtype of migraine with aura, which typically includes hemiparesis, i.e. weakness of one side of the body. Three responsible genes for FHM are currently known.
On the other hand, mutations of the same gene have been found in patients with epileptic disorders. This is the case of Dravet syndrome [
The present study does not aim to model a full blown CSD or a seizure. Instead, we focused on their initiation and on the conditions which can lead to it. We used a modeling approach, building upon previous work. In [
Experiments with mice were carried out according to the European directive 2010/63/UE and approved by institutional and ethical committees (PEA216-04551.02, France; 711/2016-PR, Italy). All efforts were made to minimize the number of animals used and their suffering. Animals were group housed (5 mice per cage, or 1 male and 2 females per cage for breeding) on a 12 h light/dark cycle, with water and food ad libitum.
We developed a conductance-based model formed by a pair of neurons: a GABAergic interneuron and a glutamatergic pyramidal neuron. This model takes into account the dynamics of ion concentrations. It is an essential feature here, since ion gradients, and hence reversal potentials, are modified during migraine and epilepsy attacks. The two neurons are thus coupled through variations of extracellular ion concentrations, in addition to synaptic connections. We implemented several ion transport proteins, such as voltage-gated channels, cotransporters, pumps and synaptic channels, which are sketched in
Consider a pair of interconnected neurons in a closed volume. We modeled a GABAergic synapse from the GABAergic neuron to the pyramidal one, a glutamatergic synapse from the pyramidal neuron to the GABAergic one and a glutamatergic autapse from the pyramidal neuron to itself. The ion transport mechanisms represented here generate transmembrane ionic currents, which modify the membrane potentials of the neurons and the ion concentrations in the different compartments. The diffusion of extracellular potassium takes into account both passive diffusion and glial buffering. We modeled external stimuli, which reflect the activity of the surrounding network or mimic experimental depolarizations, with glutamate inputs on the glutamatergic receptors. The implementation of NaV1.1’s genetic mutations affects only the GABAergic neuron.
Variable | Description | Unit |
---|---|---|
Time | ms | |
Membrane potential | mV | |
Sodium activating gating variable | ||
Sodium inactivating gating variable | ||
Potassium activating gating variable | ||
[K+]e | Intracellular potassium concentration | mM |
[Na+]e | Intracellular sodium concentration | mM |
[Ca−]e | Intracellular chloride concentration | mM |
[Ca2+]e | Intracellular calcium concentration | mM |
Synaptic variable | ||
Membrane potential | mV | |
Sodium inactivating gating variable | ||
Potassium activating gating variable | ||
[K+]i | Intracellular potassium concentration | mM |
[Na+]i | Intracellular sodium concentration | mM |
Synaptic variable | ||
[K+]o | Extracellular potassium concentration | mM |
[Na+]o | Extracellular sodium concentration | mM |
[Cl−]o | Extracellular chloride concentration | mM |
Parameter | Description | Value | Unit | Source |
---|---|---|---|---|
Membrane capacitance per area unit | 1 | μF ⋅ cm−2 | [ | |
Ratio GABAergic over pyramidal neuron volume |
| - | ||
Vole | Pyramidal neuron volume | 1.4368 ⋅ 10−9 | cm3 | [ |
Temperature | 309.15 | K | [ | |
Pump maximal rate at −70 mV | 30 | μA ⋅ cm−2 | - | |
Pump half activation intracellular [Na+] | 7.7 | mM | [ | |
Pump half activation extracellular [K+] | 2 | mM | [ | |
Parameter for the pump voltage dependence | 0.39 | [ | ||
Parameter for the pump voltage dependence | 1.28 | [ | ||
Extracellular K+ diffusion rate | 5 ⋅ 10−4 | ms−1 | - | |
K+ bath concentration | 3.5 | mM | [ | |
Time constant for the decay of | 3 | ms | [ | |
Voltage threshold defining firing time | 0 | mV | [ | |
Fast inactivating Na+ maximal conductance | 100 | mS ⋅ cm−2 | [ | |
Delayed rectifier K+ maximal conductance | 80 | mS ⋅ cm−2 | [ | |
Ca2+-activated K+ maximal conductance | 1 | mS ⋅ cm−2 | - | |
Ca2+-activated K+ half activation [Ca2+]e | 0.001 | mM | [ | |
Na+ leak conductance | 0.015 | mS ⋅ cm−2 | - | |
K+ leak conductance | 0.05 | mS ⋅ cm−2 | [ | |
Cl− leak conductance | 0.015 | mS ⋅ cm−2 | [ | |
KCC2 cotransporter strength | 0.0003 | mM ⋅ ms−1 | [ | |
NKCC1 cotransporter strength | 0.0001 | mM ⋅ ms−1 | [ | |
NKCC1 cotransporter half activation [K+]o | 16 | mM | [ | |
Glutamatergic current maximal conductance | 0.1 | mS ⋅ cm−2 | [ | |
GABAergic current maximal conductance | 2.5 | mS ⋅ cm−2 | - | |
External glutamatergic conductance | 0-0.3 | mS ⋅ cm−2 | - | |
Ca2+ maximal conductance | 1 | mS ⋅ cm−2 | [ | |
Ca2+ reversal potential | 120 | mV | [ | |
Time constant for Ca2+ extrusion and buffering | 80 | ms | [ | |
Time constant for the decay of | 9 | ms | [ | |
Voltage threshold defining firing time | 0 | mV | [ | |
Fast inactivating Na+ maximal conductance | 112.5 | mS ⋅ cm−2 | [ | |
Delayed rectifier K+ maximal conductance | 225 | mS ⋅ cm−2 | [ | |
Na+ leak conductance | 0.012 | mS ⋅ cm−2 | - | |
K+ leak conductance | 0.05 | mS ⋅ cm−2 | - | |
Glutamatergic current maximal conductance | 0.1 | mS ⋅ cm−2 | [ | |
External glutamatergic conductance | 0-0.3 | mS ⋅ cm−2 | - |
This model is based upon previous work [
We modeled NaV1.1’s FHM-3 and epileptogenic mutations, considering their effect on the GABAergic neuron. The implementation of those mutations is detailed in Section 2.2.6.
We propose a more consistent modeling of ion concentration dynamics. In [
In [
We included the activity of the Na+/K+ ATPase for both neurons, not only for the pyramidal one as in [
We identified several first integrals in system (
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 Volo. Let
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
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:
Fast inactivating sodium current:
Delayed rectifier potassium current:
As [
Contrary to [
Leak sodium current:
Leak potassium current:
Leak chloride current:
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 GABAA receptors, was modeled as in [
To model the activity of the Na+/K+ ATPase, we used this expression [
To summarize, the net currents for each ion are:
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:
Leak K+ current:
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:
The conductance of the calcium-activated potassium current
As in [
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.
NaV1.1 is mainly expressed in GABAergic neurons and NaV1.1 mutations affect mainly these neurons [
Whole-cell patch clamp recordings were performed with the F1 generation of crosses between heterozygous
Brain slices of the somatosensory cortex were prepared as previously described [
Patch-clamp recordings were performed with a Multiclamp 700B amplifier, Digidata 1440a digitizer and pClamp 10.2 software (Axon Instruments, USA); signals were filtered at 10 kHz and acquired at 50 kHz. Whole-cell recordings of neuronal firing were done at 28°C in current-clamp mode applying the bridge balance compensation; the external recording solution was ACSF (see above) and the internal solution contained (mM): K-gluconate, 120; KCl, 15; MgCl2, 2; EGTA, 0.2; Hepes, 10; Na2ATP, 2; Na2GTP 0.2; leupeptine, 0.1; P-creatine 20, pH 7.25 with KOH. Patch pipettes were pulled from borosilicate glass capillaries; they had resistance of 2.5-3.0 MΩ and access resistance of 5-10 MΩ. We held the resting potential at −70 mV by injecting the appropriate holding current, and neuronal firing was induced injecting depolarizing current pulses of increasing amplitude. Neurons with unstable resting potential and/or unstable firing were discarded from the analysis. Juxtacellular-loose patch recordings of neuronal firing were performed in voltage-clamp mode perfusing slices with modified mACSF at 34°C (which contained (in mM): 125 NaCl, 3.5 KCl, 1 CaCl2, 0.5 MgCl2, 1.25 NaH2PO4, 25 NaHCO3 and 25 glucose, bubbled with 95% O2—5% CO2) and using the same pipettes used for whole cell experiments, but filled with ACSF. Recordings were performed from GABAergic neurons of Layer 2-3, identified by their fluorescence and morphology. Fast-spiking neurons were selected for the analysis of whole cell recordings, identified by their firing properties (short, < 1 ms, action potentials with pronounced after-hyperpolarization, non-adapting discharges reaching several hundred Hz of maximal firing frequency).
For the statistical analysis, the reported
The original raw data is available at
We first focused on NaV1.1 FHM-3 mutations, modeled by the increase of
To study how
For different values of
On the other hand, persistent sodium current clearly enhances the accumulation of extracellular potassium (
This is a counterintuitive result that we have better investigated comparing detailed features of action potentials and underlying ionic currents for those parameter values:
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
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
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 (
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 [
More generally, we studied whether those findings are robust to modifications of the value of parameter
Very close to the threshold that is approximated in
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 (
When NaV1.1 carries a FHM-3 mutation, modeled by
Similarly to
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 (
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 NaV1.1, 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
To better understand the enhanced transition from the firing regime to depolarization block of the GABAergic neuron when NaV1.1 carries an epileptogenic loss of function mutation, which we observed in
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
We saw in the previous section that NaV1.1’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 (
When both neurons are coupled, taking as initial condition the steady-state when there is no external input, we applied external inputs
Plots of the extracellular potassium concentration (upper row), GABAergic neuron’s inhibitory current on the pyramidal neuron (second row) and pyramidal neuron’s firing frequency (bottom row) corresponding to the simulations displayed in Figs
We have developed a two-neuron model with one pyramidal neuron and one GABAergic neuron, building upon our previous modeling framework [
Interestingly, our model did not display a clear-cut increase in firing frequency of the GABAergic neuron implementing a common effect of NaV1.1 FHM-3 mutations, although these mutations cause a clear gain of function of the channel. Notably, in a study where FHM-3 mutations were implemented in an extended Hodgkin–Huxley model with dynamic ion concentrations [
Experimentally, the effect of FHM-3 mutations on firing features is not completely clear yet. An increase of firing frequency has been observed in GABAergic neurons transfected with the FHM-3 mutant L1649Q [
Overall, a noteworthy outcome of the present work is that, in our model, an increase of the firing frequency of the GABAergic neuron is not necessary for FHM-3 mutations to promote network hyperexcitability that leads to CSD initiation. We observed an alternative mechanism in the simulations: although in our model the FHM-3 condition induced just small modifications of the GABAergic neuron’s firing frequency, the ion fluxes at each action potential were increased, leading to a build-up of extracellular potassium. This is possible because each action potential generates larger and more sustained sodium currents that induce increased activation of potassium currents, causing higher net translocation of ions, including potassium, across the membrane, which is consistent with modeling results from Barbieri et al. [
In our previous work [
In our model, loss of function of NaV1.1, typical of mutations causing epilepsy (including the developmental and epileptic encephalopathy Dravet syndrome), makes GABAergic neurons more susceptible to depolarization block. The action potential frequency during repetitive firing appears unchanged prior to the depolarization block. Simultaneous to the suppression of spike generation by the GABAergic neuron, we observed the transition to a phase of hyperactivity of the pyramidal neuron. This firing pattern cannot be considered as a seizure-like epileptiform activity, but can be interpreted as an earlier stage of hyperexcitability. A limitation of our model is that it only takes into account two neurons, without including any network dynamics. This allowed us to keep its size manageable, but network effects may be necessary for observing seizure-like activity in simulations. Nevertheless, our work suggests the potentially important role of the depolarization block of GABAergic neurons in epilepsies caused by NaV1.1 loss of function. In particular, our model could reproduce conditions of the pre-epileptic period identified in mouse models, in which there is network hyperexcitability but not spontaneous seizures [
There is experimental evidence in favor of facilitated depolarization block of GABAergic neurons as a mechanism of pro-epileptic network hyperexcitability, for both NaV1.1-related and other models.
Our experimental data show that depolarization block is induced by smaller injected currents in fast spiking GABAergic neurons from cortical brain slices of
In an experimental model in which seizure-like events were induced in rat hippocampal slices from wild type mice using the potassium channel blocker 4-aminopyridine together with decreased magnesium, a sequence of events similar to what we obtained in our simulations was reported: seizure generation correlated with long-lasting depolarization blocks in GABAergic neurons and the simultaneous increase of firing frequency in pyramidal cells [
Interestingly, a neuronal mass computational model of Dravet syndrome generated, when abnormal depolarizing GABAA currents were implemented (which would make GABAergic synaptic connections excitatory), seizure-like activity that was similar to some EEG patterns observed in Dravet syndrome patients [
Overall, our results suggest that depolarization block can be involved in the mechanism of both gain of function migraine mutations and loss of function epilepsy mutations of NaV1.1, but with different features. In the migraine condition spiking-induced increased extracellular potassium leads to depolarization block of both GABAergic and glutamatergic neurons, whereas in the epilepsy condition depolarization block of GABAergic neurons leads to hyperexcitability of glutamatergic neurons. Notably, modifications of firing frequency of the GABAergic neurons are not necessary for inducing these effects.
Our results that disclose different pathological mechanisms leading to CSD and epileptic activity are consistent with the finding that often epileptic networks are resistant to CSD induction: In several models, the propensity to CSD generation seems to decline during the course of epileptogenesis, whereas the propensity to spontaneous epileptic seizures increases. For instance, the threshold for high potassium-induced CSD was increased in neocortical slices both from patients who had undergone surgery for intractable epilepsy and from chronic epilepsy rats following pilocarpine-induced status epilepticus, whereas brain slices from age-matched healthy control rats that showed a lower threshold [
A further future investigation would be to develop a reduced model more amenable to theoretical analysis while retaining the salient features of the present model. Bifurcation theory is indeed a very powerful tool to dissect the spectrum of activity regimes that a model can produce, as well as offer a cartography of these regimes in parameter space. Furthermore, the obvious presence of multiple timescales brings a strategy to reduce the model. We initiated this approach in the present work by studying a particular fast subsystem of the full model in
We thank Tobias Freilinger (Department of Neurology and Epileptology, Hertie Institute for Clinical Brain Research, University of Tübingen, Tübingen, Germany) for sharing unpublished results as personal communication. Our laboratories are members of the Interdisciplinary Institute for Modeling in Neuroscience and Cognition (NeuroMod) of the Université Côte d’Azur (France) and of the “Fédération Hospitalo-Universitaire” InovPain (FHU-InovPain, France).