Ionic Dynamics Mediate Spontaneous Termination of Seizures and Postictal Depression State

State transitions mediated by ionic dynamics in a single-cell model

To elaborate response properties of a single model neuron, direct current (DC) input was applied to an isolated cell to elicit a brief period of high-frequency spiking activity (Fig. 1A). This resulted in a flow of K⁺ ions to the extracellular space and a gradual increase in the [K⁺]_o during the stimulation period (Fig. 1B). After the termination of DC input, the neuron exhibited a period of bursting activity (with transient depolarization block during each burst) and then tonic spiking activity, before returning to the resting state (Fig. 1A). These findings are similar to our previous results with simpler model that only included [K⁺]_o and [Ca²⁺]_i dynamics (Bazhenov et al., 2004; Fröhlich et al., 2006) and show that this new model including variable [Na⁺]_i/o, [K⁺]_i/o, [Cl⁻]_i, and [Ca²⁺]_i is capable of generating self-sustained transient bursting and tonic activity.

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Figure 1.

Transient seizure-like activity in the isolated pyramidal neuron. A, After brief (5 s) DC stimulation (green line at the top), the neuron displayed self-sustained paroxysmal activity including bursting (indicated by blue line) and tonic spiking (red line) episodes. B, Dynamics of ion concentrations over time. [K⁺]_o and [Na⁺]_i increased during DC stimulation (green bar) and then progressively decreased while [Cl⁻]_i continued to build up until the end of paroxysmal state. The inset shows decrease of [K⁺]_o below baseline immediately after termination of seizure. C, Three-dimensional projections of the state trajectory during transient seizure-like activity indicate coupled dynamics of [K⁺]_o, [Na⁺]_i, [Cl⁻]_i, [Ca²⁺]_i, and membrane voltage. The black, blue, and red portions of the trajectory correspond to the color bars in A.

The [K⁺]_o progressively decreased after termination of DC input, reaching values slightly below baseline (Fig. 1B). The [Na⁺]_i increased during DC stimulation and decreased after its termination. In contrast to [K⁺]_o, however, [Na⁺]_i returned slower to the baseline; since sodium and potassium ion concentrations are coupled in our model through the Na⁺/K⁺ exchange pump, that led to decrease of the [K⁺]_o below baseline after termination of activity. The intracellular calcium ([Ca²⁺]_i) increased during the stimulus and showed oscillatory activity during the bursting period (Fig. 1B). Finally, the [Cl⁻]_i increased during the stimulation period reaching ∼7.5 mm. In normal physiological conditions, KCC2 pump uses the difference in K⁺ across the membrane to transport Cl⁻ and K⁺ ions to maintain a low level of [Cl⁻]_i at ∼5 mm (DeFazio et al., 2000). Experimentally, increase of [K⁺]_o to 9 mm reduces efficiency of this pump and can lead to increase in [Cl⁻]_i to ∼10 mm (Payne et al., 2003). Therefore, increase in [Cl⁻]_i in our model was attributable to the increase in Cl⁻ leak currents and inactivation of KCC2 pump by elevated [K⁺]_o. Approximately 1–2 s after termination of activity, the Cl⁻ pump mediated [Cl⁻]_i return to the baseline.

The three-dimensional plots in Figure 1C show the evolution of voltage and ion concentrations during DC input and self-sustained activity from various projections of the state space (state space is defined as a high dimensional space with all the variables of the model as individual dimensions). Notably, within these plots are distinct regions in the state space that correspond to different states of activity of the neuron. Such segregation of regions, when plotted against the ion concentrations, suggests a strong coupling between different ion concentrations in our model. Indeed, the change in one ion concentration resulted in change of neuronal activity, which in turn resulted in changes in other ion concentrations, leading to a specific trajectory in the state space. For example, increase in [K⁺]_o led to (1) corresponding increase in [Na⁺]_i (Fig. 1C, left) because of increased firing of the neuron, and (2) nonlinear increase in [Cl⁻]_i and [Ca²⁺]_i (Fig. 1C, middle and right).

Spontaneous seizure termination in a network model

We previously found that when neuron models incorporating [K⁺]_o and [Ca²⁺]_i dynamics were connected to create a network, it led to continuous transitions between slow bursting and tonic firing mediated by slow [K⁺]_o oscillations (Fröhlich et al., 2006). Thus, this earlier model was able to describe experimental observations of state transitions between tonic and clonic states of seizure (Steriade, 2003), although it failed to explain termination of paroxysmal oscillations.

To explore the effect of synaptic connectivity in the new model incorporating [Na⁺]_i/o, [K⁺]_i/o, [Cl⁻]_i, and [Ca²⁺]_i dynamics, a small network of 10 pyramidal neurons and 2 inhibitory interneurons was stimulated by DC input applied to all pyramidal neurons. In this simple model, all excitatory pyramidal cells were connected with AMPA- and NMDA-type synapses and projected to two inhibitory interneurons, which provided GABA_A-type feedback inhibition to pyramidal cells. Figure 2, A and B, shows the activity of a single representative pyramidal neuron and a raster plot of activities of all pyramidal neurons in the network. After stimulus offset (stimulus duration is marked with green bar in Fig. 2A), the network exhibited a series of self-sustained tonic and bursting periods lasting for ∼100 s total before terminating, with bursting lasting ∼75% of the total seizure duration. Excitatory cells displayed trains of single spikes or spike doublets during tonic phase and periodic bursting with typical PDS waveform during clonic phase of oscillations. These bursts typically included few spikes at the beginning of the burst depolarization, few spikes at the end, with no spiking throughout the middle of the burst. The exact pattern depended on the level of [K⁺]_o (Fröhlich and Bazhenov, 2006). This wave form is typical for paroxysmal depolarization shift observed during cortical epileptic seizures (Timofeev et al., 2002; Timofeev and Steriade, 2004). Inhibitory interneurons fired continuously during periods when pyramidal neurons were bursting but were usually silent during tonic spiking epochs (Fig. 2C). This finding was in agreement with experimental data (Timofeev et al., 2002) and suggested that excitatory drive from pyramidal cells was not sufficiently synchronized during tonic spiking to trigger reliable responses of inhibitory interneurons (Fig. 2D). In contrast, during network bursting, all pyramidal neurons fired synchronously, leading to large amplitude excitatory currents from pyramidal to inhibitory neurons that mediated interneuron activity during bursting epochs.

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Figure 2.

Spontaneous termination of self-sustained seizure-like activity in the network model. A, After transient (green line) DC stimulation, membrane voltage of a single pyramidal neuron from the network of 10 PY and 2 IN cells showed several state transitions between tonic (fast runs) and clonic (bursting) periods before spontaneous termination. The bottom plots in A shows activity of pyramidal neuron for 2 s during bursting and tonic periods. The blue line indicates reference potential. B, Space plot of the membrane voltage of all pyramidal neurons shows highly synchronized firing during bursting episodes (light colors indicate depolarized potentials). C, Inhibitory interneuron was active during bursting epochs and showed subthreshold oscillations during tonic firing of pyramidal cells. D, Conductance mediated by AMPA-type synapse from a pyramidal cell to an inhibitory interneuron.

Similar to results obtained with a single-cell model (Fig. 1), the [K⁺]_o increased during transient network stimulation and then oscillated for the duration of seizure (Fig. 3A). The dynamics of [K⁺]_o was closely tied to the spiking frequency; thus, it decreased during bursting and increased during tonic spiking periods. A transition from tonic spiking to bursting occurred at a higher level of [K⁺]_o compared with a transition from bursting back to tonic mode. This was a result of bistability: tonic spiking and bursting regimens coexisted for a range of [K⁺]_o ϵ [[K⁺]_o⁽¹⁾, [K⁺]_o⁽²⁾] (Fröhlich and Bazhenov, 2006) (for more details, see Fig. 4). The neuron stayed in tonic spiking mode (in this state, the outward flow of K⁺ was higher than K⁺ regulatory mechanisms could accommodate, which led to overall increase of [K⁺]_o) until [K⁺]_o reached the high boundary of bistability region [K⁺]_o = [K⁺]_o⁽²⁾ when the neuron switched to periodic bursting. Since the overall outward flow of K⁺ was lower in this state, [K⁺]_o decreased until it reached the low boundary of bistability [K⁺]_o = [K⁺]_o⁽¹⁾, followed by switching back to tonic spiking; this led to repetitive transitions. However, in this new model, transitions between two active states did not last indefinitely but terminated after only a few cycles. The total duration of the active state was ∼100 s, which is consistent with typical seizure duration studied in vivo (Timofeev and Steriade, 2004). After termination of seizure, [K⁺]_o showed a reduction below baseline (Fig. 3A). Similar reduction in [K⁺]_o below baseline has been observed experimentally (Heinemann et al., 1977; Jensen and Yaari, 1997) (Fig. 3B).

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Figure 3.

Ionic concentrations dynamics over the course of seizure. Data correspond to the simulation shown in the Figure 2. A, [K⁺]_o in the extracellular space of one pyramidal neuron increased during tonic states and decreased during bursting. [K⁺]_o reduced below baseline (dashed line) immediately after seizure termination. B, In vivo [K⁺]_o dynamics during epileptic seizure shows similar oscillations and decay below baseline after termination of seizure (Heinemann et al., 1977). C–G, Dynamics of [Na⁺]_i, [K⁺]_i, [Na⁺]_o, [Cl⁻]_i, and [Ca²⁺]_i during epileptic seizure-like activity. Both [Na⁺]_i and [Cl⁻]_i increased progressively until termination of seizure. [Ca²⁺]_i shows large-amplitude oscillations during bursting and remains relatively low during tonic spiking.

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Figure 4.

Bifurcation analysis and Poincaré cross-sections explain termination of epileptic event. A, Left, Bifurcation diagram as a function of [K⁺]_o (steady-state membrane voltage is plotted against parameter [K⁺]_o). Transitions between stable (solid line) and unstable (dashed line) attractors occurred at the SN and the AH bifurcation points. Middle and right, Two parametric—([K⁺]_o, [Na⁺]_i) in the middle and ([K⁺]_o, [Cl⁻]_i) on the right—bifurcation plots. SN and AH lines indicate bifurcation curves corresponding transitions between spiking and silent states. B, The minimum and the maximum values of membrane voltage plotted as a function of parameter [K⁺]_o. Each plot was obtained under different fixed values for [Na⁺]_i and [Cl⁻]_i. All other ionic concentrations remained fixed. The insets show sample voltage traces for two values of [K⁺]_o (6.1 and 9.5 mm). C, Amplitude of [Ca²⁺]_i oscillations plotted in the plane of [K⁺]_o and [Na⁺]_i. Separate plots correspond to different [Cl⁻]_i values. Each plot was obtained for progressively increasing (left) and then decreasing (right) values of [K⁺]_o. The arrows point the direction of [K⁺]_o change. The asymmetry in the image (more prominent for high [Cl⁻]_i) indicates hysteresis because of bistability in the model. The dark blue color corresponds to the silent state of the neuron. Light blue, Tonic spiking. Purple/yellow, Bursting. Note expansion of the tonic spiking region for higher [Cl⁻]_i values. Tonic spiking region disappeared for sufficiently large values of [Na⁺]_i.

To reveal factors contributing to termination of paroxysmal oscillations, we examined the dynamics of sodium and chloride ion concentrations (Fig. 3C–F). The [Na⁺]_i rapidly increased during stimulation phase and then showed oscillatory pattern similar to [K⁺]_o along with an increment at every transition from bursting to tonic states (Fig. 3C). In contrast to the single-cell model (Fig. 1B), both [K⁺]_o and [Na⁺]_i increased during tonic phase of the network activity. This was a result of the higher frequency of cell firing in the network and explained progressive [Na⁺]_i accumulation during seizure. The [Cl⁻]_i increased primarily at the early stage of seizure within the first 50 s, and later maintained a steady range of values ∼9 mm (Fig. 3F) in agreement with experimental observations, showing elevation of GABA_A reversal potential during the early stage of seizure (Timofeev et al., 2002). The increase of [Cl⁻]_i was attributable to the leak Cl⁻ currents, activation of GABA_A conductances, and reduction of KCC2 pump efficacy in conditions of elevated [K⁺]_o. Since in normal physiological conditions, the extracellular Cl⁻ concentration is much higher than the intracellular one, we assumed in the model an infinite pool of extracellular Cl⁻ with constant concentration. The [Na⁺]_o and [K⁺]_i showed changes opposite to their corresponding concentrations on the other side of the membrane. Progressive increase of the mean level of [Na⁺]_i over the entire course of seizure suggests that this may be a primary factor mediating termination of epileptiform activity in our model. Possible role of [Na⁺]_i dynamics in shaping bursts of action potentials was previously shown using a simplified neuronal model in the study by Cressman et al. (2009). In the next section, we will apply bifurcation analysis to reveal the specific role of [Na⁺]_i in terminating paroxysmal states. Furthermore, we will show that changing [Na⁺]_i dynamics to slow down [Na⁺]_i accumulation during seizure significantly extended seizure duration (see Fig. 6B).

After seizure termination, [Na⁺]_i remained elevated before returning to the baseline several seconds later. Since elevation of [Na⁺]_i above baseline leads to stronger activation of Na⁺/K⁺ exchange pump, this [Na⁺]_i dynamics mediated decrease of [K⁺]_o below baseline after termination of paroxysmal activity (Fig. 3A) and may explain experimental observations of undershooting [K⁺]_o level after seizures (Fig. 3B).

Bifurcation analysis explains mechanisms of seizure termination

To understand how changes in ion concentrations may lead to the termination of seizure, a reduced model of a single pyramidal neuron with all ion concentrations fixed was studied. Since ion concentration changes were much slower in our model than dynamics of membrane voltage or the gating variables of intrinsic currents, we used fast-slow decomposition analysis (Rinzel and Ermentrout, 1989) assuming all the ion concentrations to be fixed model parameters. With this “open-loop” method (Fröhlich et al., 2008), the modulation of ionic concentrations by neural activity was artificially removed. By choosing different values for ion concentrations, the corresponding neuronal activity patterns for each set of ion concentrations can be determined (“open loop” dynamics). Thus, the neuron activity was computed for a given fixed set of values of all ion concentrations, and we explored how neuronal dynamics depend on changes in ion concentrations. The neuron dynamics were categorized into resting, spiking/bursting, and depolarization block state for each set of ion concentrations. The resting state of the neuron corresponded to the stable equilibrium of the model. [If neuron dynamics is described by a set of equations, dX/dt = F(X), where X is a vector representing all dynamical variables of the model, then equilibria (or fixed points) are defined as the values of X where F(X) = 0]. A stable equilibrium represents stable resting state of the neuron if any small perturbations (e.g., small transient current injections) decay in time and neuron returns to the resting state. As the model parameters (e.g., ion concentrations) change, the stable equilibrium may become unstable at so-called bifurcation points (these points represent critical values of the model parameters when model dynamics changes); the latter may correspond to transitions to the spiking activity (Fröhlich and Bazhenov, 2006; Izhikevich, 2006; Cressman et al., 2009). Here, we used bifurcation analysis (or numerical continuation) (Kuznetsov, 1995) to identify transitions between different neuron states as we changed ion concentration parameters.

We first changed [K⁺]_o and kept other ion concentrations fixed: [Na⁺]_i = 20 mm, [Na⁺]_o = 130 mm, [K⁺]_i = 130 mm, and [Cl⁻]_i = 5 mm. Figure 4A shows membrane voltage at equilibrium (resting state of the neuron) for various values of [K⁺]_o. A set of stable equilibria representing stable resting states was identified for [K⁺]_o between 0 and 5.34 mm. The resting state of the neuron lost stability by a saddle-node bifurcation (SN) (Kuznetsov, 1995) at [K⁺]_o = 5.34 mm. This corresponded to transition to the spiking activity. For values of [K⁺]_o > 5.34 mm, the silent state was unstable and the neuron demonstrated various types of spiking firing patterns. Another bifurcation point was detected at [K⁺]_o = 11.42 mm. The silent state of the neuron became stable again for values of [K⁺]_o > 11.42 mm. The bifurcation at [K⁺]_o = 11.42 mm was identified to be Andronov–Hopf (AH) bifurcation (Kuznetsov, 1995). This bifurcation corresponded to transition to another silent state of the model neuron corresponding to depolarization block when membrane voltage was locked at a relatively elevated level (−29.67 mV). This state is commonly observed during epileptic seizures (Ziburkus et al., 2006).

In the previous analysis, [K⁺]_o was varied as a sole bifurcation parameter (Fig. 4A, left), and we found critical values of [K⁺]_o corresponding to transitions from silent to active spiking states for a fixed combination of other ion concentrations ([Na⁺]_i = 20 mm, [Na⁺]_o = 130 mm, [K⁺]_i = 130 mm, and [Cl⁻]_i = 5 mm). The bifurcation analysis was then extended to include two other ion concentrations ([Na⁺]_i and [Cl⁻]_i). Corresponding two-parametric bifurcation diagrams were plotted in Figure 4A, middle and right panels. In these diagrams, the critical values of [K⁺]_o corresponding transitions between silent and spiking states (SN bifurcation points) and between spiking and depolarization block states (AH bifurcation points) were traced for varying concentrations of [Na⁺]_i (Fig. 4A, middle) and [Cl⁻]_i (Fig. 4A, right). Increasing the [Na⁺]_i moved the SN bifurcation point toward higher values of [K⁺]_o; at around [Na⁺]_i = 23 mm, the SN point overlapped with the AH point (Fig. 4A, middle). For [Na⁺]_i beyond this point, the neuron remained silent (either at rest or in depolarization block) for any values of [K⁺]_o. The shift of the SN bifurcation point to the higher values of [K⁺]_o with increase in [Na⁺]_i corresponded to increase of the minimal [K⁺]_o level at which the neuron transitioned from resting state to spiking activity. In contrast, increase of [Cl⁻]_i resulted in the SN point shifting toward lower values of [K⁺]_o (Fig. 4A, right). This corresponded to earlier transition to spiking.

Together, these results suggest that changing of [Na⁺]_i or [Cl⁻]_i significantly affect minimal [K⁺]_o level that is required for transition to spiking and it can potentially make transition to paroxysmal state impossible.

To further illustrate how ion concentrations may affect neuronal activity during seizure, we analyzed dynamics of a single pyramidal neuron as a function of selected ion concentrations. Figure 4B shows minimum and maximum values of membrane voltage of one pyramidal neuron for different combinations of [K⁺]_o and [Na⁺]_i. In this type of analysis, existence of only one value of membrane voltage indicates nonoscillatory state (either at rest or in depolarization block). Oscillations are represented by two voltage values (maximum and minimum). Finally, tonic spiking and bursting states can be separated by analyzing minima of membrane voltage—more negative minima correspond to bursting. Alternatively, we plotted [Ca²⁺]_i level (Fig. 4C); since accumulation of [Ca²⁺]_i was significantly higher in bursting state versus tonic spiking (Figs. 3G, 4C), we could use it to differentiate between these network states. To reveal bistability (i.e., coexistence of different dynamical states for the same range of [K⁺]_o), we conducted analyses for progressively increasing and decreasing values of [K⁺]_o, always starting from initial conditions from the previous simulation (Fig. 4B, solid and dashed lines; C, left and right half of each panel). We found that transitions between different network states occurred at different values of [K⁺]_o based on direction (increase or decrease) of [K⁺]_o change; this indicates hysteresis arising because of bistability (see, e.g., range of [K⁺]_o between 7.6 and 7.9 mm in Fig. 4B, middle/top). The existence of bistability leading to hysteresis was shown in our previous studies (Fröhlich and Bazhenov, 2006; Fröhlich et al., 2006). However, our new model revealed that (1) an increase in [Na⁺]_i reduced the range of [K⁺]_o where tonic spiking regimen was possible (Fig. 4B, compare top and bottom plots; C, light blue region). It also eliminated bistability, so transitions between silent and bursting states occurred for the same level of [K⁺]_o regardless of the direction of [K⁺]_o change. (2) An increase of [Cl⁻]_i had an opposite effect, expanding tonic spiking range and also increasing region of bistability (range of [K⁺]_o) between tonic spiking and bursting.

This study shows that increase of [Na⁺]_i makes neurons stay in resting state at values of [K⁺]_o that would otherwise (for lower values of [Na⁺]_i) result in tonic spiking activity. It suggests that a slow increase of [Na⁺]_i during seizure can lead to the disappearance of bistability required for self-sustained periodic transitions between tonic spiking and bursting and could result in termination of paroxysmal activity. In other words, during late phase of seizure when [Na⁺]_i is sufficiently elevated, a progressive decrease of [K⁺]_o that would normally lead to transition to tonic spiking starting a new cycle of oscillations, would result in transition to silent state and termination of seizure.

To further elucidate the role of [Na⁺]_i on seizure termination, we again allowed all the ion concentrations to evolve freely according to the model equations. This corresponds to “close-loop” dynamics (Fröhlich et al., 2008) with all the ion concentration being updated based on the neuronal activity rather than being fixed as we did in the bifurcation analysis on the Figure 4. The entire episode of seizure was plotted in the state space of the neuron model along with the bifurcation points. Figure 5B shows the state trajectory in the space of [K⁺]_o and [Na⁺]_i corresponding to the membrane voltage evolution during episode of seizure presented in the Figure 5A. The blue lines in Figure 5B indicate SN bifurcation points (critical values of [K⁺]_o for different values of [Na⁺]_i corresponding to transition from silence to tonic spiking) obtained for two different levels of [Cl⁻]_i (9 and 5 mm) as described above. This analysis revealed that seizure was initiated once the initial transient stimulation (Fig. 5, red portion of trajectory) forced the state trajectory to cross the curve of SN points and to move beyond a critical value of [K⁺]_o corresponding to the left boundary of the bistability region between tonic and bursting modes. This initiated self-sustained activity and the state trajectory stayed in the region of [K⁺]_o that corresponded to slow transitions between tonic spiking and bursting (black portion of trajectory). Epileptic activity was accompanied by progressive increase of [Na⁺]_i and [Cl⁻]_i. The [Na⁺]_i increase tended to move state trajectory to the left; the seizure would terminate once trajectory crosses the SN bifurcation boundary. However, increase in [Cl⁻]_i moved the SN boundary to the left, causing the trajectory to stay in the oscillatory state longer. Nevertheless, once the [Na⁺]_i reached a high enough value, the trajectory again crossed the SN bifurcation boundary. At this point, the silent state became stable (green portion of trajectory), which corresponded to seizure termination. This analysis concisely describes the course of seizure and provides additional evidence that [Na⁺]_i increase leads to termination of seizure.

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Figure 5.

Dynamics of the epileptic activity in the model state space. A, Membrane voltage trace of a pyramidal neuron from small network (10 PY–2 IN). Different colors indicate different episodes of seizure. B, Top view of activity of pyramidal neuron in three-dimensional space of [K⁺]_o, [Na⁺]_i, and membrane voltage, along with lines of bifurcation boundaries indicating transition from resting state to spiking. The dashed blue line indicates bifurcation boundary for [Cl⁻]_i = 5 mm, [Na⁺]_o = 130 mm, [K⁺]_i = 130 mm (conditions similar to the resting state). The solid blue line indicates bifurcation boundary for [Cl⁻]_i = 9 mm, [Na⁺]_o = 120 mm, and [K⁺]_i = 128 mm (conditions similar to termination of seizure in the network).

Seizure duration depends on synaptic connectivity and chloride buildup

Our analysis (Figs. 4, 5) suggests that duration of seizure may depend on the speed of [Cl⁻]_i buildup that itself depends on the strength of the cation-chloride channel KCC2 pump. In normal physiological conditions, the KCC2 pump uses the difference in [K⁺] across the membrane to maintain a low level of [Cl⁻]_i at ∼5 mm (Payne et al., 2003). In our model, the [Cl⁻]_i rapidly increased during the early stage of seizure (Fig. 3), similar to results of experimental studies (Timofeev et al., 2002). This increase in [Cl⁻]_i partly resulted from reduction of KCC2 pump efficacy in conditions of elevated [K⁺]_o.

Increase of [Cl⁻]_i was crucial for sustaining paroxysmal activity. The range of [K⁺]_o in which the neuron was able to maintain hysteresis between tonic spiking and bursting was extended in conditions of elevated [Cl⁻]_i, which significantly extended duration of seizure (Figs. 4B, 5). To directly test the role of KCC2 pump efficacy on seizure duration, we reduced the time constant of KCC2 pump (Fig. 6A). As a result, pump performance was less affected by extracellular potassium concentration, leading to slower [Cl⁻]_i buildup; this decreased seizure duration and could even prevent initiation of seizure for the same strength of external DC stimulation. When KCC2 pump time constant was small, paroxysmal activity terminated almost immediately after offset of DC stimulation, even though [K⁺]_o still reached high levels (Fig. 6A). In contrast, increasing KCC2 pump time constant allowed [Cl⁻]_i to rise higher and led to longer seizures.

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Figure 6.

Effect of KCC2 pump and rate of [Na⁺]_i buildup on seizure duration. A, Plot shows the seizure duration as a function of the time constant of KCC2 pump, which inversely relates to its strength. The plots on the right show the activity of a single neuron and [K⁺]_o and [Na⁺]_i traces. B, Plot shows the seizure duration as a function of the rate of [Na⁺]_i buildup. The last was varied by changing the contribution to [Na⁺]_i from activity of the voltage-gated fast Na⁺ channels mediating the action potentials. Hundred percent mark corresponds to the [Na⁺]_i change (per action potential) used in the rest of the simulations. C, The image shows seizure duration (left) and the ratio of total bursting to tonic spiking duration (right) for various levels of AMPA and GABA_A synaptic conductance strength. The plots on the bottom show the activity of a single neuron and corresponding [K⁺]_o and [Na⁺]_i traces. Stimulus duration was identical in all experiments as indicated by the green bar.

According to our hypothesis, progressive increase of [Na⁺]_i over the course of seizure leads to its termination. If correct, this would suggest that changing the rate of [Na⁺]_i accumulation would affect seizure duration. Below, we tested this prediction. Since action potentials are one of the main sources of [Na⁺]_i increase during active states in our model, we artificially slowed down accumulation of [Na⁺]_i by decreasing the amount of Na⁺ ions entering the cell during each action potential. This was done without changing voltage-gated Na⁺ conductance itself and therefore without affecting spiking dynamics. Indeed, we found that slower accumulation of [Na⁺]_i significantly extended seizure duration (Fig. 6B). When [Na⁺]_i remained low, the paroxysmal state never ended for even very long simulations (Fig. 6B, left, three leftmost data points). In contrast, when Na⁺ contribution from action potentials was high, we could not initiate self-sustained seizure and network activity terminated almost immediately after stimulus offset.

Duration of seizure also depended on the strength of excitatory and inhibitory connections. In the model of 10 pyramidal cells and 2 inhibitory interneurons, self-sustained seizure was absent for low values of PY–PY (AMPA) and IN–PY (GABA_A) connections (Fig. 6C). In this case, the network model behaved similar to an isolated neuron. There was, however, one important difference. Even with all synaptic connections blocked, the neurons within a network could communicate by means of ion diffusion between extracellular compartments associated with individual neurons. As a result, when strength of initial DC stimulation was increased, it led to short episodes of seizure-like activity (data not shown). The seizure duration depended on both excitatory (PY–PY) and inhibitory (IN–PY) connections; removing either excitation or inhibition reduced seizure duration (Fig. 6C, left). For high values of PY–PY and IN–PY connection strength, seizure-like activity lasted continuously. The relative strength of PY–PY and IN–PY connections also determined the dominant state in which neurons stayed during seizure. Increase in both PY–PY and IN–PY connection strength led to longer burst duration (Fig. 6C, right). Since [Na⁺]_i level remained relatively low in bursting mode, it was not sufficient to promote termination of seizure when the model burst continuously, which explains continuous oscillations for high synaptic strength.

Below baseline reduction of the extracellular potassium mediates postictal depression state

The network model described in the previous sections included only 10 excitatory pyramidal neurons and 2 inhibitory interneurons. To test whether our results can be generalized to larger networks, we simulated a model including 100 PY and 20 IN cells. To make simulations more realistic, we also included background noise that was modeled as Poisson-distributed spike trains delivered through AMPA-type synapses to all neurons in the network. To account for homeostatic changes found experimentally in the neurons driven by external noise, we increased slightly conductance of the [K⁺] leak current, as seen in the study by van Welie et al. (2006), reduced conductances for high threshold [Ca²⁺] (O'Leary et al., 2010) and fast sodium currents (Grubb and Burrone, 2010), and increased conductance for Cl⁻ leak current (see Materials and Methods). With these changes, the network was stable and displayed spontaneous random spiking activity at a mean rate of 0.5 Hz (Fig. 7).

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Figure 7.

Seizure termination in a large network with noise and postictal depression. A, Space–time plot of activity of 100 pyramidal neurons from a network over time. The red bar indicates transient increase of the external Poisson rate. B, Activity of a single pyramidal neuron includes episodes of baseline low-frequency spiking before stimulation (a), self-sustained seizure-like oscillations including slow bursting and tonic firing (see expanded plots) (b), and period of postictal depression after termination of paroxysmal activity (c). Inset, Distribution of membrane potential from all cells before the onset of seizure and in postictal state. C, Left, Peaks of cross-correlation computed between all pairs of pyramidal neurons for bursting and spiking periods. Right, Delays corresponding to the peak of cross-correlation for all pairs of pyramidal neurons.

The mean frequency of the external Poisson input was transiently increased from 140 to 400 Hz for 2 s (Fig. 7B, red bar). This initiated seizure-like oscillations, which included sequences of bursting and tonic spiking episodes lasting for ∼200 s total. Similar to smaller networks (Fig. 2), this larger network displayed a high level of synchrony across the entire population of neurons during bursting episodes and reduced synchrony during tonic spiking (Fig. 7A). Bursting episodes were primarily found during earlier phases of epileptiform activity and were interrupted by epochs of fast runs during later phases of the seizure (Fig. 7B). To quantify the change in synchronization, cross-correlation was computed between all pairs of pyramidal neurons separately for bursting and spiking periods. Bursting periods were identified as the periods when membrane potential was consecutively above −20 mV for >200 ms; spiking periods were identified as consecutive 500 ms periods with firing rate >10 Hz when there was no bursting. Cross-correlation was higher during bursting periods compared with spiking periods (Fig. 7C), with the nearby cells showing highest cross-correlation as seen by increase in the values near the diagonal in Figure 7C (left). Distribution of delays to the main peak of the cross-correlation function suggests formation of clusters during bursting periods while spiking periods showed random pattern of delays (Fig. 7C, right).

To initiate self-sustained paroxysmal oscillations, transient increase of the Poisson input had to exceed certain minimal level suggesting existence of the threshold for seizure initiation as we previously reported (Fröhlich et al., 2010). Indeed, small perturbations only led to short transient firing rate increase followed by return to the baseline firing, therefore indicating stability of the physiological (random spiking) state in our model including dynamics of different ion concentrations.

Concentration of sodium, [Na⁺]_i, increased progressively during seizure, reaching it maximum just before activity termination in agreement with our results from smaller networks (Fig. 8A). Interestingly, [K⁺]_o displayed complex nonlinear dynamics, reaching its peak soon after seizure onset and remaining relatively constant (except for low amplitude oscillations associated with tonic-clonic transitions) during a later phase. Seizure termination was followed by a period of postictal depression. The last appeared as a relatively small hyperpolarization of membrane voltage after termination of paroxysmal rhythm (Fig. 7B, inset). Postictal state lasted for several minutes. Neuronal excitability was reduced during postictal state that was sufficient to prevent spontaneous spiking in the model. As we previously described, this effect was mediated by residual elevation of intracellular sodium concentration that activated Na⁺/K⁺ exchange pump forcing [K⁺]_o to remain below baseline (Figs. 3A, 8B).

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Figure 8.

Ion concentration dynamics in a large network. A, Raster plots of the [K⁺]_o, [Na⁺]_i, and [Cl⁻]_i dynamics over time. Data correspond to the simulation shown in the Figure 7. B, Time course of [K⁺]_o, [Na⁺]_i, and [Cl⁻]_i for a single pyramidal neuron (no. 25) from the network shown in the Figure 7.