Interaction between synaptic inhibition and glial-potassium dynamics leads to diverse seizure transition modes in biophysical models of human focal seizures

2.1 Model description and parameter choices

Network structure and connectivity

The structure of our network of model neurons is based on the arrangement of neurons in minicolumns in the neocortical area (Buxhoeveden and Casanova 2002). In this network model, each minicolumn consists of 16 single-compartment model neurons, of which 12 are excitatory pyramidal cells and 4 are inhibitory interneurons. The neurons are arranged into four layers within each minicolumn (Fig. 1). Each layer consists of four neurons in a square arrangement, with each vertex of the square being the location of one neuron. Pyramidal cells occupy the top three layer and the bottom layer is for the four inhibitory interneurons. (We do not attempt here to replicate the actual 6-layer neocortical structure). The entire cortical network consists of 9 ×9 such minicolumns for a total of 1296 model neurons. Details of the structure configuration of the cortical network model and various distance parameters between model neurons are given in Fig. 1 and Table 1. Unless otherwise stated, we chose a global random connectivity to avoid boundary effects associated with the small size of the simulated cortical network.

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Table 1

Network geometry parameters. Please refer to Fig. 1 for the description of parameters

Parameter	Units	Values
Δd e e1	cm	5×10−4
Δd e e2	cm	5×10−4
Δd i e2	cm	30×10−4
Δd i i1	cm	5×10−4
Δd m m	cm	15×10−4

2.2 Implementing the simulations

The code for the model cortical network simulation was written in C++ (gcc, version 4.4.7, http://gcc.gnu.org). We used open MPI (version 1.8, http://www.open-mpi.org) for parallel processing of each simulation. Each typical 1296-neuron simulation used eight CPUs. A 10-hour simulation on the Brown University computer cluster (https://www.ccv.brown.edu/) usually yielded ∼ 2-3 minutes of data. Parameter scanning was automated via the use of custom Perl (version 5.18.4) and Linux shell scripts. Post simulation analysis of data was carried out using Matlab (version R2013b, http://www.mathworks.com), Octave (version 3.8.1, http://www.gnu.org/software/octave/) and customized Perl scripts. Single neuron simulations (Fig. 2) were performed using xppaut (Ermentrout 2002). Graphical rendering of data was performed by gnuplot (version 4.6.3, http://www.gnuplot.info), xfig (version 3.2) and gimp (version 2.8.10, http://www.gimp.org). The parula palette of the time-frequency plots is from https://github.com/Gnuplotting/gnuplot-palettes. The C++ code is available in the NEURON ModelDB database (https://senselab.med.yale.edu/ModelDB/showModel.cshtml?model=190306). The ModelDB accession number for the network model reported in this paper is 190306.

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3.1 Firing properties of the single-neuron models under physiological and abnormal glia uptake conditions

We used four single pyramidal cell model simulations to show the effects of the potassium pump and glial dynamics on neuronal excitability (Fig. 2). In each of these four simulations, I _{e x t} =0.36, [K ⁺] _{o(e q)}=3, [K ⁺] _{o(t h)}=15, I _{K m a x} =7, g _{K C a} =5. In this section, we define the “physiological” value 𝜃=−1.15, and the “pathological” value as 𝜃=−0.05 (equation set (11)). Table 3 specifies the units of these parameters. Our aim is to show that the neural system tends to settle into a single steady state for a wide range of initial conditions when the increase in the forward binding rate k _f is smooth with increasing [K ⁺] _o (as in the physiological case, effected by a larger absolute value of 𝜃 – see equation set (11)).

However, when there is a sharper increase in k _f with increasing [K ⁺] _o (as in the pathological case, effected by a decreased absolute value of 𝜃), some of the initial conditions can lead the neural system into a state where virtually all of the glial buffers are free (i.e. [K B]→0). In this case, the system’s ability to return potassium to the extracellular space and maintain a target value of extracellular potassium level is impaired. The impairment occurs because Eq. (10) is no longer reversible when [K B]→0.

In Fig. 2a and b, we show two single pyramidal neuron model simulations with the same set of “physiological” glial and potassium pump parameters but with different [K ⁺] _o and [B] initial conditions. In Fig. 2a, we set [K ⁺] _o (t=0)=1 mM and [KB] (t=0)=3 mM, and in Fig. 2b, [K ⁺] _o (t=0)=10 mM and [KB] (t=0)=10 mM. It is clear that, despite the difference in initial conditions between the two simulations, the model neuron and the glial buffering system in both simulations converge to the same steady state. The steady state is characterized by a dynamic equilibrium established between the glial buffer’s binding (k _f ) and unbinding (k _b ) of the extracellular potassium. This dynamic equilibrium is evidenced by the small but non-vanishing [K B] in the steady state. Moreover, all the steady state variables (including firing rates, [K ⁺] _o and [K B]) are identical between the two simulations, indicating that the steady states of the two simulations are the same. The glial buffering system enforces an upper ceiling for the extracellular potassium level in both simulations.

In Fig. 2c and d, we show the same two simulations with “pathological” glial parameters. In contrast to Fig. 2a and b, we see that the same set of different initial conditions results in distinct steady states for the two simulations. In Fig. 2c, the model neuron and the glial buffer approach a state where there is a lower [K ⁺] _o and a vanishing value of [K B], while in Fig. 2d, the final [K ⁺] _o and [K B] values are higher and the neuron is excitable with a higher firing rate. Thus, bi- or multi-stable dynamics at the single neuron level are more easily obtained with a smaller absolute value of 𝜃, resulting into multiple fixed points (steady states) having either a lower [K ⁺] _o or higher [K ⁺] _o value.

The main factor behind this bi-stable dynamics is the parameter 𝜃, which determines the level of bias for the forward binding rate k _f against the backward rate k _b (see equation sets (10) and (11)). With a small absolute value of 𝜃, at a critical [K ⁺] _o level lower than [K ⁺] _{o(t h)}, k _b can be overwhelmingly larger than k _f . The large k _b value rapidly lowers [K B]. Although this backward unbinding action releases K ⁺ back to the extracellular space, the I _{K p u m p} (9) counteracts this effect and therefore [K ⁺] _o remains low. Moreover, the small k _f value (due to the bias factor 𝜃) is not sufficient to replenish [K B] via the forward binding action. As a result, the system ends up in a state where virtually all the buffer is in the free state B with a low level of [K ⁺] _o (as in Fig. 2c).

For the neuron and glial system to exit this state, as we have shown in Fig. 2d, one can either artificially increase the values of [K ⁺] _o or [K B] (for example, by setting up the initial conditions). Increasing [K ⁺] _o to a value closer to [K ⁺] _{o(t h)} forces k _b to be less biased against k _f (equation set (11)), so that [K B] can be more efficiently replenished and thus allows a dynamic equilibrium between [B] and [K B]. Increasing [K B] artificially serves similar purposes by increasing the backward unbinding release of K ⁺ to the extracellular space (thus increasing [K ⁺] _o ). In essence, decreasing the absolute value of 𝜃 reduces the range of [K ⁺] _o over which the glial buffering system can efficiently regulate, and thus makes the single neuron more susceptible to multi-stable behaviour. We show in the next section that this apparent multistability also carries over to the network simulations with abnormal glial parameters. As mentioned above, we will associate the state in which the [K ⁺] _o is low with the normal physiological state, and the elevated [K ⁺] _o level state with the “seizure” state. We also note that, although the above single-neuron analysis was based on pyramidal neurons, the same low and elevated [K ⁺] _o states occur in the case of FS inhibitory neurons. The effect of elevated [K ⁺] _o on pyramidal and FS interneuron spiking will, however, differ in the network simulations presented below.

3.2 Sustained gamma epileptiform activity results from an abnormal but balanced glial dynamics and inhibitory synaptic strength

In this section we show how most features of gamma seizures observed in the human focal epilepsy data (Truccolo et al. 2011, 2014) can be replicated by our network model. These gamma seizures in the human data are characterized by sustained narrow band gamma LFP oscillations (∼30−60 Hz). However, few fine temporal synchrony transients exist in the spiking activities of the measured neurons. Neuronal spiking tends to be asynchronous and heterogeneous (Truccolo et al. 2011, 2014). Individual neurons also fire at a lower rate than the gamma frequency of observed LFP oscillations.

Similar neural dynamics is observed in our model simulations (Fig. 3). Critical ingredients for the appearance of sustained gamma band LFP oscillations with asynchronous neuronal firing at a lower rate include high inhibitory conductance values (for both \({g_{syn}^{{i}\rightarrow {i}}}\) and \({g_{syn}^{{i}\rightarrow {e}}}\)) and an axonal conduction time delay (Brunel 2000; Brunel and Wang 2003). We also require a value of [K ⁺] _{o(t h)} that is not too high (see equation set (11)). A moderate [K ⁺] _{o(t h)} value ensures that the model FS inhibitory interneurons have a high excitability in the “epileptic” state, while preventing the interneurons from being overly excited to enter depolarization block. We used \({g_{syn}^{{e}\rightarrow {e}}}=0.0007\), \({g_{syn}^{{e}\rightarrow {i}}}=0.0007\), \({g_{syn}^{{i}\rightarrow {i}}}=0.025\), \({g_{syn}^{{i}\rightarrow {e}}}=0.025\), g _e0=0.01026, σ _e =0.0025, g _i0=0.084, σ _i =0.02 for this gamma seizure simulation (Table 5 introduces and defines these synaptic parameters). As for intrinsic and glial properties of model neurons, we chose 𝜃=−0.15, [K ⁺] _{o(e q)}=3.6, [K ⁺] _{o(t h)}=7.5, I _{K m a x} =1.45, g _{K C a} =5 for pyramidal cells and 𝜃=−0.15, [K ⁺] _{o(e q)}=3, [K ⁺] _{o(t h)}=7.5, I _{K m a x} = 1.9, g _K = 6.8 for interneurons. The value of g _K for interneurons was chosen so that they only enter depolarization block at a relatively high level of [K ⁺] _o (\(\gtrsim 8\) mM).

Figure 3a shows the first 90 seconds of the temporal progression of several biophysical variables in the model and the time-frequency spectrum of the average membrane potentials across all the pyramidal neurons in the network (Fig. 3a, lowest panel). We use this average value as a proxy for the LFP activity.

The model cortical network during the first 40 seconds of the simulation is in the “low activity” state (in which [K ⁺] _o converges to ∼3mM). This “low-activity” state at the network level is similar to the single neuron level in Fig. 2c where the replenishment of [K B] is insufficient for the glial buffer to maintain a targeted level of [K ⁺] _o . At time t=40s, a DC stimulation (2.5 μA/cm ² added to I _{e x t} ) was applied to every neuron in the model network for a short period of 2.5 seconds. This stimulation kick-started the extracellular potassium accumulation process and forced the cortical network to enter into a “high-activity” or “epileptic” state in which the glial buffer attempts to lock into a higher [K ⁺] _o level (Fig. 3a, 4 ^{t h} panel), as determined by the parameter [K ⁺] _{o(t h)}. Clear gamma band (∼40 Hz) oscillations are readily observed in the LFP power spectrum after the DC stimulation (Fig. 3a, lowest panel). Importantly, the gamma activity is sustained, even after DC stimulation is terminated, persisting until the end of the simulation.

Furthermore, at the individual neuron level, both the fast-spiking interneurons and pyramidal cells spike at lower rates of ∼ 3 and 1 spike per second, respectively, than the frequency of the gamma LFP activity (see Fig. 3a, topmost panel). The second panels from the top of Fig. 3a and b show raster plots of 2 of the 81 minicolumns in the simulation. The neuron indices in the raster plots are grouped by neuron types (pyramidal or interneuron).

3.3 Transition into spike-and-wave complex (SWC) seizures: role of synaptic inhibition and depolarization block

In this section we show how variations in synaptic inhibition and depolarization block in the examined neuronal network model can reproduce the dynamics of the observed SWC seizures. In this type of seizures, neural activity transitions into high amplitude rhythmic 2−3 Hz LFP discharges. Each SWC event in the LFP lasts for about 300 ms and is characterized by an initial fast “spike” followed by a slow wave potential. Neuronal spiking tends to occur during the LFP “spike” phase and is highly suppressed during the LFP wave phase (e.g. Truccolo et al. 2014). Preliminary examination of neuronal spiking characteristics has indicated that although both putative principal and FS inhibitory interneurons tend to increase their firing rates in the initial stages of spike-wave seizures (Ahmed et al. 2014), putative FS interneurons tend to shut down later, just before the emergence of full spike-wave discharges. Ahmed et al. (2014) observed that this termination of activity is preceded by a progressive decreasing of action potential amplitudes in FS interneurons – an indication that FS interneurons enter depolarization block, resulting eventually in the cessation of their activity.

Figure 4 shows simulation results for the case where the overall inhibitory synaptic strength is kept at a low level during the entire simulation, leading to SWC seizures. Because the aim is to have FS interneurons to eventually enter depolarization block, we set a higher [K ⁺] _{o(t h)} and a lower g _K value for the interneurons than in the gamma seizure simulation (Fig. 3). Higher [K ⁺] _{o(t h)} values increase FS interneuron excitability, while lower g _K values lead FS interneurons to enter depolarization block at a lower [K ⁺] _o level. Parameters for model neurons and glial properties of model neurons corresponded to: 𝜃=−0.15, [K ⁺] _{o(e q)}=3.6, [K ⁺] _{o(t h)}=8.5, I _{K m a x} =1.8, g _{K C a} =12 for pyramidal cells and 𝜃=−0.15, [K ⁺] _{o(e q)}=3, [K ⁺] _{o(t h)}=7.75, I _{K m a x} =1.9, g _K =3.5 for interneurons. Unitary inhibitory conductance values \({g_{syn}^{{i}\rightarrow {i}}}\) and \({g_{syn}^{{i}\rightarrow {e}}}\) were set here to only one-tenth of that in the gamma seizure simulation (Fig. 3). Synaptic parameters corresponded to: \({g_{syn}^{{e}\rightarrow {e}}}=0.0007\), \({g_{syn}^{{e}\rightarrow {i}}}=0.0007\), \({g_{syn}^{{i}\rightarrow {i}}}=0.0025\), \({g_{syn}^{{i}\rightarrow {e}}}=0.0025\), g _e0=0.01016, σ _e =0.0025, g _i0=0.084, σ _i =0.02.

As before, the simulation starts at a “low-activity” state (t=0−40 seconds) and a DC stimulation (1.6 μA/cm ²) is delivered between t=40 and 42.5 seconds, “kicking” the system into a high-activity “epileptic” state. Since the overall inhibitory synaptic strength is low, but [K ⁺] _{o(t h)} is higher than that in the gamma seizure case, FS inhibitory interneurons fire at a higher frequency after the DC stimulation than in Fig. 3, creating a positive feedback which further increases [K ⁺] _o . When [K ⁺] _o reaches a critical value, inhibitory interneurons enter depolarization block, which eventually leads to cessation of inhibition (at around t=46 seconds in Fig. 4). We note that, in contrast to the gamma seizures, the attainment of the critical value for [K ⁺] _o is possible here because [K ⁺] _{o(t h)} is set to a higher level, such that the glial potassium ceiling is higher.

After cessation of synaptic inhibition, the network dynamics is dominated by the interaction amongst pyramidal cells and evolve into rhythmic SWC discharges. Throughout, abnormal glial buffering maintains the high [K ⁺] _o required for keeping the interneurons in the depolarization block regime, thus creating a long-lasting “seizure” state. In the model simulations, I _{K C a} adaptation currents in the pyramidal cells and sufficiently strong excitatory-excitatory (\({g_{syn}^{{e}\rightarrow {e}}}\)) couplings (Van Vreeswijk and Hansel 2001; Dur-e-Ahmad et al. 2012; Nicola and Campbell 2013b; Ferguson et al. 2015) are critical for both the “spike” and “wave” firing suppression phases of the the SWC discharges.

3.4 Transition into seizures consisting of initial low-voltage fast LFP oscillations followed by spike-and-wave discharges

Next, we demonstrate the generation of the second type of SWC seizures. In this case, the emergence of rhythmic SWC discharges is preceded by low voltage higher frequency (∼10−20 Hz) LFP oscillations. Neuronal spiking activity during these fast oscillations tends to be asynchronous as in the previously examined gamma seizures. In other words, while there is a clear fast frequency oscillation at the LFP level, individual neurons spike at a lower rate without obvious synchrony with the global (i.e. LFP) oscillation patterns. The similarity of the firing dynamics of individual neurons between these transient oscillations and gamma seizures leads us to speculate that both phenomena might share a common synaptic and glial mechanism. As shown in Fig. 5, the seizure begins with low voltage fast (∼20 Hz) LFP oscillations which eventually evolve into low-frequency rhythmic SWC discharges. Similarly to the gamma seizures (Fig. 3), the initial fast oscillations in the model require a high overall inhibitory synaptic strength, while the evolution and maintenance of spike-wave discharges require the opposite.

To reproduce this type of seizure dynamics, the following sequence of events was implemented in the model: a substantial increase in inhibitory synaptic strength at seizure transition (leading to low voltage fast oscillations) is followed by a rapid decrease, which is hypothesized here to occur via GABA depletion (Zhang et al. 2012). Once synaptic inhibition is below normal levels, the same path as in the first type of SWC seizures follows: inhibitory interneurons enter depolarization block and the network evolves into rhythmic SWC discharges supported mostly by the population of pyramidal neurons. Specifically, the synaptic parameters \({g_{syn}^{{i}\rightarrow {i}}}\) and \({g_{syn}^{{i}\rightarrow {e}}}\) were set to constant values during the entire simulation.

The increased s _{m a x} value for interneurons was then held constant during the fast oscillations phase (from 42.5 to 49.5 seconds). After that it was decreased to simulate the loss of inhibition. We set s _{m a x} →0.1×s _{m a x} for individual interneuron after each spike of the same interneuron during this “depletion” phase. (The start of this depletion phase is marked by a red triangle in Fig. 5.) Eventually the inhibition approaches zero, the “depleted” state.

In summary, Fig. 5 shows the emergence of clear low voltage fast (∼ 20 Hz) LFP oscillations right after the onset of the DC stimulation. These fast oscillations disappear shortly after the start of the inhibition “depletion” phase (marked by a red triangle in Fig. 5). The strong build up of inhibitory synaptic strength (up to 70 times the initial strength) during the DC stimulation is necessary because it slows the rise of [K ⁺] _o during DC stimulation and keeps the interneurons from entering depolarization block. Moreover, this very high level of synaptic inhibition keeps the population frequency in the ∼10−20 Hz range. A lower, but still large inhibitory strength would have led to the emergence of gamma oscillations, as in Fig. 3.

The initial decrease in [K ⁺] _o during the transient fast oscillations (Fig. 5a, fourth plot from top) is due to the strong inhibition which limits the firing of inhibitory interneurons. As inhibition begins to wear off at t=49.5 seconds, the trajectory of [K ⁺] _o reverses its course because both pyramidal cells and interneurons fire more as a result of decreased inhibitory synaptic strength. Finally, when [K ⁺] _o reaches a sufficiently high level (∼7 mM for this simulation), the interneurons enter depolarization block and inhibition ceases, as clearly shown by the complete termination of interneuron firing (Fig. 5b, raster plot at around t=63 seconds). After cessation of inhibition, SWC discharges emerge supported only by the activity of interacting pyramidal neurons. With the exception of the time varying nature of s _max for interneurons and a DC stimulation here set to 2.5 instead of 1.6 μA/cm², the numerical values of the synaptic (including \({g_{syn}^{{i}\rightarrow {i}}}\) and \({g_{syn}^{{i}\rightarrow {e}}}\)), intrinsic and glial parameters were the same as those for the first type of SWC seizures (Fig. 4).

4.1 Abnormal glial buffering and its relationship to seizure

In this work we have established a computational model of seizure through the occurrence of bi-stability at the single neuron level. This bi-stability is the result of a hypothesized pathological glial potassium dynamics in which there is an imbalance between potassium buffers in their bound and unbound forms. Bi- (or multi)stability, either at the single neuron or at the network level, has been implicated in numerous physiological and pathological conditions in the brain (Van Ooyen et al. 1992; Sasaki et al. 2007; Freyer et al. 2009; Fröhlich et al. 2010; Anderson et al. 2012; Ho et al. 2012; He 2014; Ho et al. 2014; Hübel et al. 2014). In the particular case of epileptic seizures, Fröhlich et al. (2010) have demonstrated in computational models that it is possible for network bistability to exist even when the glial potassium dynamics are within a physiological range (through a higher absolute value of 𝜃, the parameter determining the bias favouring forward versus backward binding rate). In this scenario, Fröhlich et al. (2010) associate higher [K ⁺] _o states with epileptic seizures. In the models used here, we did not observe bi-stability either at the single neuron or network level when 𝜃 was set to physiological values. We only observed bi-stability with pathological (i.e. smaller in absolute value) values of 𝜃. We have singled out (via single neuron simulations) the differential in transition rates between [B] and [K B] as the important factor underlying such bi-stability. The fact that Fröhlich et al. (2010) used two-compartment neuron models, while we used single-compartment, might explain this difference between the two studies. Moreover, different synaptic connection parameters were used by Fröhlich et al. (2010). Different synaptic connectivity might affect the length and strength of perturbation required to elicit the bi-stable behaviour, if at all possible. A more thorough examination of how glial potassium buffering may affect neuronal dynamics should involve detailed bifurcation analyses of the combined neuron-glial system (Touboul 2008; Nicola and Campbell 2013a; Kim and Nykamp 2014; Nicola and Campbell 2014). Such analyses could determine the intrinsic and glial parameters that allow multistability. From the existence and stability of these states, one would expect to gain some insight into how resilient the network is to seizure initiation and propagation. Although we have focused on their role on seizure propagation, potassium and glial buffering may also play an important role in seizure termination (Kramer et al. 2012; González-Ramírez et al. 2015).

4.2 Changes in inhibitory and excitatory conductances

Our focus on changes in inhibition was motivated by several previous (in vivo/vitro) animal models (Trevelyan et al. 2006; Zhang et al. 2012; Grasse et al. 2013; žiburkus et al. 2013; Uva et al. 2015) and in vitro studies of human epileptic cortical tissue (e.g. D’Antuono et al. 2004), where inhibitory interneuron activity appears to change significantly during preictal and initial periods of the seizure, before major changes in principal cells are detected. Recent work by žiburkus et al. (2013) has also examined in detail how changes in both inhibitory and excitatory conductances may lead to imbalances in excitation/inhibition preceding and during seizure like events in hippocampal slices under the potassium channel blocker 4-aminopyridine and reduced extracellular magnesium. In that respect, we emphasize that the changes in inhibition as implemented in our simulations did not only affect inhibitory conductances. Changes in inhibition in our model were implemented in part by systematically varying the levels of unitary synaptic inhibition, which in turn affected several other network properties such as the number of neurons generating spikes at any given time, and thus firing rates in both populations of pyramidal neurons and FS interneurons. In this way, by varying unitary synaptic inhibition, the E-I balance in the simulated networks also varied dynamically, with both excitatory and inhibitory conductances being altered. In addition, after transitions into spike-wave discharges in the simulated SWC seizures (Figs. 4 and 5), these discharges were supported only by the synaptic interactions among pyramidal neurons.

Furthermore, we note that there are a couple of important differences between the animal model in žiburkus et al. (2006, 2013) and our model simulations. First, apart from temporary depolarization block periods, the activity of inhibitory neurons remained throughout the seizures (i.e. until seizure termination) in žiburkus et al.’s model. In our computational model, it is only during gamma seizures that the spiking activity of FS inhibitory interneurons is preserved throughout the seizure and depolarization block is absent. By contrast, FS interneurons enter depolarization block and remain in that state during the SWC seizures. Second, while in žiburkus et al.’s study transient depolarization block events in pyramidal neurons are also present and play an important role, in our computational model pyramidal neurons tend to be robust to depolarization block both in SWC and gamma seizures. This robustness resulted primarily from the slow calcium-mediated potassium afterhyperpolarization (AHP) currents (I _{K C a} , Eq. (7)) in the model pyramidal neurons, which prevented them from evolving into regimes of high enough firing rates required for depolarization block. These features of the biophysical model examined here seem consistent with the dynamics of propagated seizures (in neocortical patches distal from putative onset areas) observed in our human data. First, both inhibitory and excitatory spiking activity appears to be preserved throughout the gamma seizures (Truccolo et al. 2011). Second, in SWC seizures, spiking in putative FS interneurons appears to terminate before the transition into large amplitude spike wave discharges. This cessation of FS interneuron spiking activity is accompanied by signatures of depolarization block, while no similar signatures were found in putative principal cells (Ahmed et al. 2014).

4.3 Network inhibition and its relationship to seizure propagation

Using biophysical neuronal network models of neocortical patches, we have demonstrated how FS inhibitory interneurons may play an important role in three types of observed human seizures. In particular, we have quantified the role of FS interneurons by varying the inhibitory synaptic strength and controlling how easily interneurons enter depolarization block. For the two types of SWC seizures modeled here (Figs. 4 and 5), ultimately it is the failure of inhibition (when the FS inhibitory interneurons enter depolarization block) that leads the network to transition into SWC activity. We examined two main scenarios that led to the failure of inhibition, resulting in SWC seizures. In the first scenario, FS inhibitory interneurons enter depolarization block when there is an overall low inhibitory synaptic strength. When interneurons are excited during seizure initiation, the lower inhibitory synaptic strength allows interneurons to fire at a higher frequency, thus hastening the [K ⁺] _o accumulation process and leading eventually to depolarization block (Fig. 4). In the second scenario, an initially large inhibitory synaptic strength is followed by GABA depletion (Zhang et al. 2012). Initially, interneurons are mutually inhibited by the large inhibitory synaptic strength and their firing frequencies are too low to trigger [K ⁺] _o accumulation. If such level of inhibition was not perturbed, seizure propagation would fail. The decreased effective inhibitory strength resulting from subsequent GABA depletion allows for higher firing rates in interneurons (and principal cells) to develop, again leading eventually to depolarization block in inhibitory interneurons (Fig. 5).

Some previous studies have emphasized the role of feedforward inhibition in seizure propagation in neocortex (Trevelyan et al. 2006; Schevon et al. 2012; for a review see Paz and Huguenard 2015). The main idea is that the propagation of initially localized ictal activity to more distal cortical sites involves predominantly feedforward inhibition. This feedforward inhibitory drive could be a mechanism for the initial strengthening of inhibition during seizure propagation assumed in our model in two of the three seizure propagation scenarios (gamma and SWC with preceding fast oscillations). Furthermore, Trevelyan and colleagues (Trevelyan et al. 2006; Schevon et al. 2012) have emphasized the role of an “inhibitory restraint” or “veto” mechanism during neocortical propagation of focal seizures. Inhibitory veto of seizure propagation has been shown in low Mg ²⁺ mouse models of epilepsy (Trevelyan et al. 2006) and has been argued to be present also in human focal seizures (Schevon et al. 2012). This inhibitory veto mechanism would lead to “penumbra areas” ahead of the ictal wavefront. The failure of such inhibitory veto would allow the successful propagation of the ictal wavefront into new recruited ictal areas. Otherwise, the inhibition would be strong enough to contain the seizure spread. In this way, one could also argue that the gamma seizures examined here would represent a failure in seizure propagation. This possibility remains an open and important experimental question. Given that these sustained gamma oscillations reflect abnormal dynamics during secondarily generalized focal seizures, we currently consider this activity as propagated seizures. The gamma seizures in our model (Fig. 3) represent a situation in which there is a balance between enhanced inhibition and [K ⁺] _o accumulation. In other words, the seizure propagates while inhibitory interneuronal activity is preserved. In a related experimental study, our group (Lu et al. 2015) has shown that constant, strong enough depolarization driven by optogenetic stimulation can induce sustained (∼50 Hz) gamma oscillations in healthy primate motor cortex.

In view of the strong inhibition in simulations of gamma seizures (Fig. 3), we do not preclude the possibility that some of the “high-activity” states in the simulated gamma seizures may in fact be long-lasting transients as opposed to asymptotic stable states. In other words, the elevated [K ⁺] _o in the “high-activity” states may nevertheless settle back to a much lower value after a long enough time. However, this issue has little effect on our main claims in this paper, given the long time scale (at least in the order of minutes) over which [K ⁺] _o remains elevated and increasing.

4.4 Model limitations and future work

Our analyses and simulations show how different types of seizure transitions observed in intracortical recordings of propagated human focal seizures can be accounted for by variations in synaptic inhibition and extracellular potassium concentration. Several limitations of the model, however, warrant future improvement. First, although the interaction of [K ⁺] _o and inhibitory synaptic dynamics can account for the phenomena observed in our human epilepsy data (Ahmed et al. 2014), several other ionic or synaptic mechanisms are possible and have been studied in different animal models of epilepsy (e.g. Alfonsa et al. 2015).

Second, our model exploration was restricted to the transition into ictal states. In particular, we did not examine how seizures terminate, which is also a complex topic and may involve multiple ionic and synaptic components (such as glutamate depletion–Lado and Moshé 2008; GABA upregulation–Wen et al. 2015; fluctuations in ionic concentrations–Krishnan and Bazhenov 2011; Kramer et al. 2012). A more detailed model, including more complex ionic dynamics, will be required to address this issue. Furthermore, gap junction effects were not included in the model. Although electrical synapses have been suggested as an underlying mechanism for interneuronal bursting (Skinner et al. 1999) and synchronous population gamma oscillations (Traub et al. 2001), neither of these two features appeared to be prominent in our human data where neuronal spiking during gamma seizures is largely asynchronous. Thus, it remains an open question whether gap junction effects are critical to account for transitions to and maintenance of gamma seizures. Nevertheless, gap junctions may still play an important role during SWC discharges as previously hypothesized by Traub and colleagues (e.g. Traub et al. 2005). We hope to address these open issues in the future.

Third, in the hypothesized contributing factors to seizure transitions examined here, we have remained agnostic about what ultimately leads to transient dysfunctions in inhibitory activity and in potassium glial-buffering processes. We note that several other factors involving changes in oxygen, Na+/K+ ATP pumps and cell volume (e.g. Ingram et al. 2014; Wei et al. 2014a), for example, can also contribute to elevations in extracellular potassium. We also emphasize that the goal in this study is to replicate the dynamics observed in propagated seizures, in other words, ictal dynamics in neocortical patches distal to the putative seizure onset zones. As described in Truccolo et al. (2011, 2014) and Wagner et al. (2015), our recorded neocortical patches were close, but distal to the putative seizure onset zones. The dynamics involving seizure initiation in seizure onset/focus areas might involve different mechanisms.

As stated above, we have focused on fast time-scale changes in synaptic inhibition which have been reported in several previous animal studies and in vitro studies of human epileptic cortical tissue. Although these studies have provided some initial evidence for transient changes in inhibitory interneuron activity preceding seizures, it remains a major open question how these dysfunctions in interneuron network activity (and potassium glial buffering) would arise in the longer time scale of epileptogenesis.