Ethics statement

The experimental data were recorded previously [25] and reanalyzed for the purposes of the present study. All experimental procedures were approved by the Czech Academy of Sciences Ethics Committee and the Ethics Committees of the Second Faculty of Medicine (Project Licence No. MSMT-31765/2019-4).

Experimental design

Brain slices were obtained from four male Wistar rats weighing approximately 200g. The animals were kept in an enriched environment with 12/12 hours of light and darkness and regularly checked for health status prior to the experiment. The animals were anesthetized (80 mg/kg ketamine, 25 mg/kg xylazine) and decapitated. The brains were extracted from the skull and placed into an ice-cold, oxygenated protective solution–sucrose-based artificial cerebrospinal fluid–which contained (in mM) 189 mM sucrose, 10 glucose, 2.5 KCl, 0.1 CaCl₂, 5 MgCl₂, 26 NaHCO₃, and 1.25 NaH₂PO₄*H₂0. The brain was then cut into sagittal slices of 350 m thickness using a vibratome (Campden Instruments, Loughborough, UK). The CA3 region was then cut off by a scalpel, as commonly done in in vitro hippocampal epilepsy models to test hypotheses about the origins and spread of epileptiform activity (see [25] for details and a comparison of slice preparations with and without CA3). Then, the hippocampal slices were moved to a holding chamber at room temperature filled with normal artificial cerebrospinal fluid (ACSF) and continuously aerated with a humidified mixture of 95% O₂ and 5% CO₂. The ACSF contained (in mM) 125 NaCl, 26 NaHCO₃, 3 KCl, 2 CaCl₂, 1 MgCl₂, 1.25 NaH₂PO₄, and 10 glucose. After at least 60 minutes, a slice was transferred to a recording interface chamber at 34 1^∘C containing normal ACSF. They were left at rest for 10 minutes before experiments started. Local field potentials were recorded using a glass micropipette (diameter 10 - 15 m) filled with ACSF and inserted in the stratum pyramidale of the CA1 subregion. The signal was amplified using Model 3000 AC/DC Differential Amplifier (A-M Systems, Inc., Carlsborg, Washington, USA), digitized at 10 kHz sampling frequency using Power1401 (CED, Cambridge, UK) and recorded using Spike2 software (CED, Cambridge, UK). To induce epileptic activity, we added KCl solution at first in large steps (1–3 mM) until a total concentration of 6.5–7 mM was reached. After that, we gradually increased the concentration (0.2–0.5 mM at a time) until spontaneous seizure-like episodes appeared. The experimental protocol included perturbations of the CA1 network with regularly delivered stimuli to Schaffer collaterals (input to CA1) with a frequency of 1 Hz. The stimuli were electrical pulses of 200 s duration using a bipolar silver wire electrode and isolated constant current stimulator (Digitimer Ltd., Welwyn, UK). The stimulation was initiated manually, immediately after the end of a seizure-like event, at the beginning of the inter-seizure interval (ISI), as determined by visual inspection of the LFP recording. A stimulated ISI was typically followed by an ISI without stimulation and vice versa. In the analysis, each stimulated ISI was matched to the preceding control ISI. The stimulation duration ranged from a few seconds to nearly two minutes. Table 1 summarizes basic data characteristics. The signals were subsequently analyzed using custom-written Matlab scripts.

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Table 1. Data characteristics of each slice.

K_bath: final potassium concentration per slice, which was sufficient to elicit seizure-like events; n_seiz baseline: number of spontaneous seizures before the stimulation experiment was started; ISI baseline: inter-seizure interval length (mean standard deviation) before the stimulation experiment; n_seiz stim: number of seizure trials with stimulation (where pre-stim seizure trials without stimulation served as a matched control for assessing for relative ISI changes); ISI stim: ISI with stimulation, ISI pre-stim: ISI without stimulation preceding an ISI with stimulation, used as control.

https://doi.org/10.1371/journal.pcbi.1013838.t001

Statistical analysis

To demonstrate the seizure-delaying effect due to LFES in the experimental data, we computed the Spearman correlation between the length of stimulation and the seizure delay for each slice. Seizure delay was calculated as percentage change of the stimulated ISI with respect to the control ISI (which was the unstimulated ISI immediately preceding each respective stimulated ISI, i.e. “pre-stim ISI” in Table 1). We report the correlation coefficient r and the associated p-values. The analysis of the model simulations did not require any statistical analysis.

The ISI was defined as the time interval in seconds between the end of one seizure and the beginning of the next. In the data, seizure offsets were automatically identified by finding the minimum in the LFP (using Matlab R2021a’s islocalmin() function with ‘MinProminence’ set to half the signal span and a ‘MinSeparation’ 20 seconds, applied to the smoothed signal; smooth() with window size of one second). Seizure onsets were identified by finding the steep increase of LFP (using findchangepts() function with ‘Statistic’ set to ‘linear’) in the time window of up to 30 seconds before each identified seizure offset. In the model, seizures are detected using an equivalent procedure as applied in the data analysis.

Model design

In this study, we aimed to investigate a minimal biophysical model of a neuronal population that: 1) exhibits epileptiform activity similar to our data, 2) incorporates potassium concentration dynamics, and 3) can produce seizure delay in response to low-frequency electrical stimulation. We would like to point out the fact that computational models as well as experimental models focus on mimicking particularly the electrophysiological signatures, rather than behavioral phenomena, and thus cannot be considered seizures in the strict clinical sense and should be interpreted with caution. Here, however, we refer to seizure-like events as seizures for simplicity.

We chose a modified version of the Epileptor-2 model [31]:

(1)

where V is the average membrane potential of pyramidal cells, x_D is the synaptic resource, and and are the extracellular potassium and intracellular sodium concentrations, respectively.

The membrane potential V is driven by the potassium concentration and synaptic input, which is concisely expressed by

(2)

where the first term is the potassium depolarizing current (with being the relative potassium conductance), and I_exc and I_inh are the excitatory and inhibitory synaptic inputs to the excitatory population. The excitatory synaptic input I_exc is defined as:

(3)

with g_exc being the conductance of excitatory synapses, and being the firing rate of the excitatory population. We note here that the excitatory synaptic input is current-based, since the reversal potential for excitatory synapses is so large that its difference from the membrane potential can be considered roughly constant [32]. The inhibitory synaptic input I_inh is defined as [33]:

(4)

with g_inh being the conductance of inhibitory synapses, and their reversal potential.

The firing rate function is chosen to be a sigmoid function with threshold and maximum firing rate :

(5)

The excitatory population is also subjected to low-frequency stimulation, I_stim(t). Each stimulus is represented by a Dirac delta function, and the sequence of stimuli thus reads , where t_n represents stimulation times (once per second in our protocol), and A is the stimulation amplitude. The parameter R_stim was added to ensure consistency of units.

Lastly, the activity of the Na-K pump is computed as follows [34]:

(6)

We made two modifications to the original Epileptor-2 model [31], because the original Epileptor-2 was not sufficient to reproduce the seizure delay effect. First, we changed the firing rate function to a simple sigmoid function, which allows for non-zero firing rates even at small values of the membrane potential. The second modification concerns the implementation of inhibition. The original Epileptor-2 model features a non-depressing inhibitory neuronal population whose activity is proportional to that of the excitatory neuronal population, thus providing a tight balance between the inhibitory and excitatory population. Here, however, we chose an inhibitory population that exhibits a constant firing rate and acts on the excitatory population via conductance-based synapses. Since the reversal potential is close to the resting potential at low concentrations, inhibition doesn’t produce strong hyperpolarizing synaptic potentials. Rather, it has a shunting effect that is roughly proportional to the depolarization of the membrane. Experimental and computational evidence suggests that shunting inhibition is prevalent in the hippocampus, and that it plays a crucial role in the robustness of -oscillations [35,36].

One of the objectives is to investigate the effect of potassium on the neuronal population. In the full model, the sodium and potassium concentrations are slowly changing variables that can be assumed to be roughly constant on the much faster time scales that govern the dynamics of V and x_D. Therefore, to describe the characteristic dynamics of V and x_D, we will assume that and are time-invariant, leading to the following set of equations:

(7)

This system allows us to obtain the bifurcation structure of the system with respect to the ion concentrations, which can be studied as parameters.

In this analysis, we consider two scenarios: 1) using parameter sets fitted to the experimental data (for slices 1 to 4), and 2) using a parameter set which corresponds to the reference set that is closest to the original Epileptor-2 parameters to explore LFES in greater detail, referred to as slice 0. The parameters that are kept constant throughout the analysis are shown in Table 2.

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Table 2. Fixed parameter set used in section Results.

https://doi.org/10.1371/journal.pcbi.1013838.t002

The parameters specific to the slice 0 and the ones specific to the experimental slices 1-4 are shown in Table 3.

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Table 3. Specific parameter set used to fit the experimental data.

https://doi.org/10.1371/journal.pcbi.1013838.t003

The emergence of spontaneous seizure transitions

We first examine the spontaneous transitions into seizures by analyzing how the model parameters influence the duration of seizures and ISIs. Fig 1A shows four example LFP recordings, obtained from the experimental setup described in the Methods section. It shows the heterogeneity in terms of ISI and seizure duration between slices, which is dependent on the amount of added potassium. Based on this observation, we decided to match the recordings with the model in terms of seizure duration and ISI.

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Fig 1. Fitting model ISI and seizure duration to the data.

(A) Time series of data from different slices. Red double arrows indicate ISIs, and orange double arrows indicate seizures. (B) Heatmaps of ISI and seizure duration with respect to the parameters and V_th, for K_bath = 9.5 and K_bath = 5.3. (C) Model fit to slice 1. (D) Model fit to slice 4. In these panels, red curves refer to concentration, and the blue curves refer to concentration. For the specific parameters for each slice, see Table 3.

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The Epileptor-2 model contains various parameters that can be modified to change the temporal structure of the epileptic activity and thus replicate the experimental observations. In our case, we focus on the firing threshold and the maximum firing rate . It is reasonable to assume that slices differ in specific characteristics [37–40], such as cellular excitability and network connectivity, due to intrinsic biological heterogeneity and variations in preparation protocols. This variability underscores the importance of systematically analyzing the parameters and to account for the diversity observed in the dataset.

The heatmaps in Fig 1B determine the length of the ISI and seizures as and are varied, demonstrated for the K_bath values of slices 1 and 4. We note here that if the ISI and seizure length are equal to zero, the model exhibits constant activity with either a low or high firing rate. For fixed values of , the length of the ISI increases and the seizure length decreases as increases.

Using the heatmaps in Fig 1B, we calibrate the model to match two examples from Fig 1A (slice 1 and 4). In Fig 1C and 1D, we first show a time series of the LFP from the data, then a time series of the membrane potential from the model. One can notice the similarity in structure of the epileptic activity of the model: a slow depolarization that characterizes the ISI, then a fast depolarization that starts the seizure. The seizure itself is characterized by initially further depolarization, and subsequent repolarization until the membrane potential reaches its minimum, which terminates the seizure. It exhibits very similar sodium and potassium concentrations as determined experimentally [41]. We note here that the resting sodium concentration in the model is 10mM, which is representative of the normal physiological state. In addition to characterizing the dynamics of potassium and sodium ions, the model provides access to the firing rate and the activity of the Na-K pumps. Shortly before a seizure, the firing rate shows a modest increase, followed by a rapid upturn at the beginning of the seizure. At the end of the seizure, the firing rate declines abruptly. In parallel, Na-K pump activity gradually increases during the seizure. At the seizure end, pump activity initially decreases rapidly as the membrane potential returns to baseline. After reaching baseline, the activity continues to decline but at a slower steady rate.

In Fig 2A, we first test different values of K_bath to study their effect on the time series. The potassium concentration in the surrounding bath, K_bath, is increased instantaneously at the vertical dashed line. We note that the addition of potassium allows the appearance of seizures only when a specific concentration threshold is exceeded, then the ISI progressively reduces and the seizure increases in length until the added potassium stabilizes the membrane potential at a high value. This status could correspond to so-called status epilepticus, i.e. an epileptic seizure prolonged for more than 5 minutes or by repeated seizures without regaining consciousness between them [42–44]. We remark here that the alternative scenario of depolarization block is not captured by our model, because the firing rate saturates at high values of the membrane potential, which represents intense firing activity. Instead, the model exhibits a baseline switch which emerges from slow ionic dynamics and sustained recurrent activity. The baseline switch occurs in the absence of spikes because the model represents population firing rates rather than individual spiking activity. Note that the length of ISI and seizure duration depends also on the and , which is fitted individually to each slice in Fig 1C and 1D. For each setting of and the effect of increasing K_bath corresponds to the one shown in Fig 2A. Comparing K_bath concentrations between slices 1 and 4, we found longer ISI and shorter seizures for higher K_bath, while increasing K_bath within one slice will lead to shorter ISI and longer seizures. This discrepancy is resolved when considering that each slice has different characteristics, which corresponds to different parameter combinations placing it in a different point in Fig 1B.

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Fig 2. Highlighting the effect of K_bath on the modified Epileptor-2 model compared to the data.

(A) Time series of the model for different values of K_bath. (B) Superposition of the experimental protocol of sequential addition of potassium performed in the medium containing the slices, with the simulated slow addition protocol. The red vertical dashed lines correspond to the moment when the model is stimulated with a single spike of A = 15mA. (C) Effect of the same experimental protocol: the blue time series corresponds to the membrane potential, and the red one to the moment when the experimentalists stimulated the slice. (D) Effect of the slow addition protocol on the model. For panel (C) and (D), zooms on the time series at the moment of the stimulation effect are provided. For the specific parameters, see slice 0 in Table 3.

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In the next step, we focus on the potassium addition protocol: in Fig 2B, the experimental protocol (blue curve), which corresponds to regular additions in the medium with a delay long enough to allow the potassium to diffuse into the tissue, is superimposed onto our protocol applied to the model (black curve). In the experimental protocol, potassium is added until the first appearance of spontaneous, recurring seizures. However, sometimes the experimenter probed the tissue excitability by stimulating the tissue in order to induce the appearance of seizures, as commonly done in epilepsy research and clinical practice [45]. In order to reproduce the experimental setup, we added a constant slow input of potassium to our model. The continuous addition of potassium in the model, as opposed to the step-wise manner during the experiment, is justified by the slow diffusion process at each step, so we can assume that our process is similar. Furthermore, we tested the induction of seizures by external stimulation with spike trains of 15 mA, which are indicated in Fig 2D by red dashed vertical lines.

We first examine the effect of stimulations in the experimental protocol. Fig 2C shows the recorded LFP from slice 3, overlaid with the different stimulations applied to induce seizures. At low potassium concentrations, the stimulation produced only transient depolarization-repolarization responses, regardless of the number of pulses that were applied. However, at higher potassium concentrations later during the experiments, the stimulation was capable of provoking seizure events if the number of pulses was sufficiently large.

In Fig 2D, the experimental protocol is reproduced in the model, showing the membrane potential of the neuronal population. In these simulations, the stimulation consists of a train of spikes delivered at 1 Hz. We observe that when the extracellular potassium concentration is low, the stimulation only produces artifacts without triggering any pathological activity. However, when the potassium concentration is elevated, it is the first spike of the stimulation train that is sufficient to initiate a seizure.

The anti-seizure effect of LFES

To demonstrate the seizure-delaying effect of LFES, we analyzed the LFP recordings with regards to the difference in length between stimulated ISIs and control ISIs (which are ISIs without stimulation preceding the stimulated ISI, i.e. “pre-stim ISI” in Table 1). The seizure delay was calculated as percentage change of the stimulated ISI with respect to the control ISI.

We find that stimulated ISIs were significantly longer than the control ISIs in each of the slices, as assessed via Spearman correlation (see Fig 3B and Table 1; see also S1 Fig). This indicates that the 1 Hz stimulation had indeed a seizure delaying, or anti-seizure, effect. Fig 3A shows two examples of control and subsequent stimulated ISI (their beginnings and ends marked by red and green vertical lines, respectively). The longer stimulation in example 2 leads to a substantially longer stimulated ISI as compared to control ISI.

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Fig 3. LFES-induced delay phenomena observed from the data and from the model.

(A) Time series from experimental data. The beginning (end) of the ISI is marked by a red (green) vertical line. (B) Seizure delay correlation with stimulation length from experimental data, as assessed via Spearman correlation. Seizure delay is measured as the percent change of stimulated ISI relative to control ISI. Each dashed line is a linear regression fit. For slice 4, two scenarios are proposed: one with the whole data (yellow dashed line), and one without outlier (red dashed line). The outlier is circled in red. (C) Time series of the simulated model with standard parameters (“slice 0”) for different stimulation lengths. (D) Seizure delay correlation with stimulation length for the model fitted to slices 1 to 4. For the specific parameters for each slice, see Table 3.

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In Fig 3C and 3D, we apply the LFES protocol to the model. In all the simulations, the stimulation is applied right after the seizure. In Fig 3C, we can observe that the stimulation pulse does not have an immediate effect for approximately the first 10 to 15s. Only after this initial period, we observe a tangible effect of the stimulation, which is a brief increase in the depolarization rate, and subsequently the convergence of the system onto a new forced equilibrium until the stimulation ends. Thus there exists an optimal time window in which the stimulation is effective, which we investigate further below.

Fig 3D shows the effect of stimulation on the ISI in the same way as panel B. For values up to t_stim = 10s, the stimulation does not have an effect on the ISI, in line with the (descriptive) experimental observations in Fig 3B that for short stimulation lengths LFES is ineffective. For stimulation periods greater than t_stim = 10s, in the case of slices 1 to 3, the ISI increases linearly with the stimulation period. The model reproduces the same order of slope as observed in the experimental data (Data: Slope0.830, Slope0.933, Slope0.937; Model: Slope1.027, Slope1.183, Slope1.268). This demonstrates its ability to predict the individual effectiveness of the anti-seizure protocol. However, in the case of slice 4, the ISI decreases and then stabilizes, as LFES induces an early seizure. This phenomenon arises from the choice of parameters optimized to fit the experimental data. The impact of LFES on seizure delay as a function of and will be examined in the S2 Fig.

Model predictions regarding optimal parameters and the mechanism of the anti-seizure effect

In order to understand the mechanism behind the anti-seizure effect, we first investigate the optimal stimulation parameters—timing and amplitude—that maximize it. If the stimulation causes a new seizure, then this is considered the end of the ISI, even as the stimulation continues.

We portray the changes to the ISI in a heatmap in Fig 4. In panel A, we plot the effect of amplitude and onset time while keeping the stimulation duration fixed at 50 s. When the amplitude is low, e.g., A = 0.5 mA, the ISI does not change significantly. When the amplitude increases from 2.5 mA to 12.5 mA, the stimulation reduces the ISI. However, when the amplitude exceeds 15 mA and the onset time is 16s or less, the stimulation increases the ISI by 40% to 60% compared to the unperturbed ISI. If the stimulation onset occurs too late (17s or more), then the stimulation creates a new seizure immediately. We can speak here of a critical point for the stimulation amplitude and stimulation onset. In Fig 4B, we plot some corresponding time series that are around this critical point. In panels B1, B2 and B4, the LFES causes premature seizures. In Fig 4B3, stimulation effectively delays the next seizure and pushes the system onto a new forced equilibrium during stimulation. One can also observe that in panel B4, the LFES does indeed induce a premature seizure, however unlike panels B1 and B2 where the stimulation amplitude is not high enough to delay subsequent seizures, what follows the first premature seizure is the beginning of a delay process. This implies that even if the stimulation starts too late and triggers an earlier seizure it can delay the next seizure if the amplitude is high enough.

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Fig 4. Relative seizure delay as a function of two variables: the amplitude of the stimulation (mA), and the stimulation onset after the end of the first seizure (s).

(A) Relative seizure delay plotted against amplitude and stimulation onset, with the stimulation length fixed at 30s. (B) Time series numbered from 1 to 4 illustrate model behavior around the critical point in (A). For the specific parameters, see slice 0 in Table 3.

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Note that our experimental dataset was not sufficient to validate the model predictions regarding optimal parameters. Stimulation amplitude information was not available. Moreover, the stimulation onset time with regard to last seizure offset time varied only between –4.8 and 7.3 seconds with a median of 1.2 seconds (see S3 Fig).

Having identified the critical thresholds for each stimulation parameter, we next aim to elucidate the biological processes underlying the effects of LFES. To achieve this, we analyze the model time series by mapping them onto the Na-K phase plane. To interpret this representation, we highlight the stimulation threshold and employ a bifurcation diagram to delineate the distinct dynamical regimes—namely, ictal and interictal states. This approach allows us to assess whether LFES effectively drives the system away from the ictal into the interictal region.

Fig 5A shows one time series obtained from the model, namely the membrane potential, Na/K concentrations, firing rate, and Na-K pump activity, for the case where the LFES is active for 150 s. The stimulation sends the system into a regime where the Na-K pump is active (Na: 25.4–25.8 mM, K: 3.4–3.7 mM). A stimulation pulse causes an instantaneous increase of both Na and K concentration. Then, the active Na-K pump slowly lowers both again. This requires sufficiently strong Na-K pump activity, otherwise the mechanism is likely to fail.

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Fig 5. Deciphering the biological mechanisms underlying LFES.

(A) Time series demonstrating seizure delay in response to LFES. Red and green vertical bar indicate the beginning and end of the stimulated ISI. (B) 3D bifurcation diagram obtained from the fast-subsystem (7), overlaid with a non-stimulated solution obtained from (1) (black arrowed curve). For the superposition to be in perfect adequacy, we impose parameter in front of the equation of [K⁺] and [Na⁺], in such a way, they are slower and follow the bifurcation structure. The stable states are the green and purple surface, which refer respectively to the inter-ictal region and the ictal region. The grey surface represents the unstable states. HB denotes Hopf bifurcation, and LP, fold bifurcation point. (C) Effect of our protocol on (1) represented on a {[Na]⁺, [K]⁺} phase plane respectively for and 30 mA by a stimulated trajectories (arrowed curves), and colormaps of the activity of Na-K pumps, and the firing rate. The start of each stimulation is represented by a vertical red arrow, and last 150s. Trajectories with LFES are plotted in red, trajectories without LFES are plotted in black. The trajectories highlighted in orange or purple indicates where LFES respectively induces an earlier seizure or delays its onset. The blue cross represents the critical point up to which a stimulation causes a delay, as obtained in Fig 4. For the specific parameters, see slice 0 in Table 3.

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This explains why a stimulation pulse that occurs too soon after the previous seizure is ineffective in delaying or suppressing the next seizure: the K concentration is still too low and the Na-K pump does not respond to the stimulation. Note that increasing the stimulation amplitude does not change the dynamics of the Na-K pump. Only later stimulation pulses, when the external K concentration increased sufficiently (above mM), are effective in producing seizure delay. This is concurrent with a marked increase in the Na-K pump activity. Thus, during prolonged stimulation only later stimulation pulses contribute to the observed seizure-delaying effect of LFES.

We now characterize the effects produced by LFES in greater detail. We first focus on the bifurcation diagram obtained from (7) (S-shaped, colored surface in Fig 5B), which is superimposed with the trajectory of a solution of the model (1) in the absence of stimulation (arrowed black curve). Superimposing these two representations allows us to break down the different regions of the bifurcation diagram: when the black trajectory is in the region with low values of membrane potential V, i.e. the red part of the surface, then it is the interictal region. When the curve approaches the LP₁ bifurcation point, then the trajectory jumps to the purple surface where the values of the membrane potential are high, which represents the ictal phase. Then the curve jumps back down at the HB₁ bifurcation point. This abrupt jump is due to a periodic branch which emerges from HB₁, forming a direct homoclinic connection to the S-shaped surface. For clarity, we have omitted the periodic branch in Fig 5B, which is further elaborated on in S4 Fig. LP₁ can be regarded as the potassium threshold for which the model produces a seizure if it is exceeded. The grey colored surface corresponds to the unstable states of the system that separate the basins of attraction of the ictal and interictal states.

Having established the different regions, in Fig 5C we study the solutions of the model subjected to LFES in the phase plane , indicating the threshold values obtained in Fig 5B (HB₁, LP₁, LP₂) by horizontal lines for better readability. The half-planes below LP₂ and above LP₁ in Fig 5C belong to the interictal and the ictal region, respectively. There is a hysteresis effect due to the coexistence of the ictal and interictal regime between LP₁ and LP₂. If a trajectory approaches this interval from the interictal regime (below LP₂), then it remains in the interictal regime; and conversely, if it approaches this interval from the ictal regime (above LP₁), then it will remain in the ictal regime. In the top panels of Fig 5C, we plot several trajectories with different stimulation onsets (indicated by the dots of different colors). The blue cross indicates the critical stimulation onset established in Fig 4. For any stimulation before this critical point, the LFES is effective, otherwise, it causes a premature seizure. For the case A = 10 mA, the stimulation pushes the system onto a smaller limit cycle (compared to the unperturbed system) that passes through the thresholds. Thus, for any stimulation below the critical stimulation amplitude, seizures are shorter with smaller amplitudes, and shorter ISIs. When A = 15 mA, for any stimulation arriving before the critical point of stimulation onset, the system stabilizes on a smaller limit cycle. This limit cycle is also observable in the case A = 30 mA. Since these limit cycles no longer pass through the LP₁ point, they are no longer associated with seizure behavior, but instead, keep the system locked in a subpart of the interictal domain. These results corroborate our observations in Fig 4, and further suggest that the stronger the stimulation (above A = 15 mA), the faster is the stabilization at this new stationary cycle.

In Fig 4, we predicted that if we choose the right amplitude to induce a delay but stimulate too late, then we induce a single transient seizure before converging to the delay process. In Fig 5C we observe this behavior in the phase plane for A = 15 mA and A = 30 mA. The trajectory passes through the seizure domain once, before converging to the new attractor. This implies that the onset threshold relative to the last seizure only acts as a threshold for the initial conditions of the LFES. In fact, for any LFES above 15 mA, the delay appears with or without a single transient seizure.

The second and third row of panels in Fig 5C show the trajectories obtained from the same simulations as the first row, but with color coding of the current induced by the Na-K pump I_pump, and the firing rate , respectively. We can observe the activation and deactivation of the Na-K pump during a seizure: initially, the intracellular sodium concentration is low, and the pumps have very low activity. During the seizure, at the peak of activity, the pump reactivates due to the accumulated intracellular sodium, and decreases the extracellular potassium concentrations, which brings the neuronal population back to the baseline. When stimulating with A = 10 mA, this process also occurs during the appearance of the premature seizure. When the amplitude is large enough to delay seizures, the system converges onto a new attractor where the Na-K pump becomes increasingly active compared to the resting state without stimulation. The firing rate provides an indication of the effect of the stimulation protocol on the model. During the interictal phase, the firing rate is close to zero, whereas in the ictal phase it is close to the maximum . When LFES is applied to the system, the firing rate shows a burst-like response to the stimulation pulses. However, the amplitude of these bursts only reaches its maximum when the stimulation is effective in delaying the subsequent seizure. The bursts of firing activity increase K and Na concentrations, which is counteracted by the Na-K pumps, thus establishing a new attractor state.

In summary, this analysis highlights the critical role of the Na-K pump in the processes induced by LFES, stabilizing ionic concentrations at relatively low values. We demonstrate that the observed phenomenon is not associated with epileptic activity. Instead, it represents an interictal dynamic that maintains the system at a safe distance from the seizure transition.