Introduction
Epilepsy is characterized by recurrent, spontaneous brain seizures and is well known to be triggered by an imbalance between excitation and inhibition (Engel, 1996; McCormick and Contreras, 2001; Ziburkus et al., 2006; Ben-Ari et al., 2007; Jiruska et al., 2013; Grone and Baraban, 2015; Staley, 2015; Wenzel et al., 2023), but detailed mechanisms are still unclear. Each seizure episode is marked by abnormal neuronal discharge patterns (Buchin et al., 2018; Wenzel et al., 2019) and interneurons play an important yet controversial role in its induction (Yekhlef et al., 2015; Neumann et al., 2017). On the one hand, interneurons release gamma-aminobutyric acid (GABA) to excitatory neurons through GABAergic synapses, allowing influx of chloride and inhibiting the neuron's firing activity, thus reducing the likelihood of seizure (Trevelyan et al., 2007; Cammarota et al., 2013; Liou et al., 2018; Parrish et al., 2019). On the other hand, GABAergic stimulation can also trigger seizure generation (Perreault and Avoli, 1988; Staley et al., 1995; Kaila et al., 1997; Voipio and Kaila, 2000; Cohen et al., 2002; Gulledge and Stuart, 2003; Sipilä et al., 2005; Gnatkovsky et al., 2008; Fujiwara-Tsukamoto et al., 2010; Jedlicka et al., 2011; Sessolo et al., 2015; Librizzi et al., 2017; Elahian et al., 2018; Burman et al., 2019; Desroches et al., 2019; Rich et al., 2020; Lemaire et al., 2021; van Hugte et al., 2023; Weiss, 2023). However, the functional role of GABA in triggering SLE is complicated and not well understood. Indeed, substantial intracellular chloride accumulation has been observed in conjunction with epileptiform activity (Kaila and Voipio, 1987; Payne et al., 2003; Blaesse et al., 2009; Raimondo et al., 2012, 2015; Accardi, 2015; Doyon et al., 2016). Therefore, it is crucial to investigate the functional role of intracellular chloride concentration in regulating epileptic activity at a single neuron level comprehensively.
GABA receptor activation allows both Cl−
influx and efflux of bicarbonate ion (HCO3−)
(Kaila, 1994; Kaila et al., 1997; Farrant and Kaila, 2007; Kim et al., 2009). HCO3−
is produced by carbonic anhydrases using carbon dioxide (CO2) to regenerate intracellular HCO3−
(Lambert and Grover, 1995; Voipio, 1998; Staley and Proctor, 1999). Its production is regulated by pH value (Kaila and Voipio, 1987; Voipio and Ballanyi, 1997), glial cells (Halassa and Haydon, 2010; Matos et al., 2018; Erhardt et al., 2020; Untiet et al., 2023), and the blood–brain barrier (Hladky and Barrand, 2016). The combined dynamics of chloride influx and HCO3−
efflux upon GABA receptor activation, together with concurrent increase of extracellular potassium concentrations (McBain, 1994; Shin et al., 2010; de Curtis et al., 2018; González et al., 2019), in regulating seizure activity has not been fully explored.
While seizures have traditionally been recognized as network events (Traub and Wong, 1982; Fröhlich et al., 2006, 2010; Lytton, 2008; Liou et al., 2020; Otárula and Schuele, 2020), biophysical models have shown that contributing mechanisms are inherent to single neurons (Kager et al., 2000; Jiang et al., 2005; Ullah et al., 2009; Ullah and Schiff, 2010; Barreto and Cressman, 2011; Krishnan and Bazhenov, 2011; Volman et al., 2012; Wei et al., 2014; Călin et al., 2021; Lemaire et al., 2021; Currin and Raimondo, 2022). Previous theoretical studies assumed that factors such as ion concentrations (Chizhov et al., 2019; Lombardi et al., 2021; Gentiletti et al., 2022), oxygen levels (Bradbury and Davson, 1965), and dynamics of potassium (Traynelis and Dingledine, 1988; Durand et al., 2010; Contreras et al., 2021) exert influence over the voltage system, resulting in epileptic behaviors. Yet how the neuron firing activities are regulated by chloride dynamics and the relationship between chloride and potassium interplay in regulating epileptic activity have not been fully explored. Previous work (Krishnan and Bazhenov, 2011) included chloride dynamic without considering the contribution from HCO3−
and focused more on the relationship between Na+ and K+ during seizure induction and termination, while less attention was paid to the relationship between the intracellular chloride dynamics and neural firing property or the interplay between chloride and potassium in regulating neuron seizure-like events (SLEs).
Our study focuses on the critical role of chloride dynamics in regulating epileptic activity and investigates the basic mechanisms of how a neuron's firing properties are shaped by the competition between opposing effects of chloride upon GABA stimuli and application of extracellular potassium bath.
Results
We present our results in three parts, investigating models of increasing complexity. In the first part, we illustrate how neuronal intrinsic firing activities are modulated by different levels of intracellular chloride [Cl−]i
.
Bifurcation analysis of neuronal firing activity properties dependence on [Cl−]i
In contrast to potassium, which has been extensively studied (Bazhenov et al., 2004; Cressman et al., 2009; Wei et al., 2014; Depannemaecker et al., 2022), the relationship between chloride dynamics and epileptic seizures, particularly the accumulation of [Cl−]i
during GABA stimulation of a neuron, has not received much attention in computational models. We first demonstrate how the dynamic property of neuron firing activity is regulated by intracellular chloride [Cl−]i
, before considering external GABA input.
Neurons evolve to distinct discharge behaviors with different initial concentrations of [Cl−]i
(Fig. 1A, blue dots). For instance, a neuron starting with a low value of [Cl−]i=5.97mM
from resting state of −70.74 mV is stable at this resting state (Fig. 1A, black curve; Fig. 1D), while a neuron with much higher level of [Cl−]i
at 12.27 mM eventually reached a state of depolarization block (DB; Fig. 1A, red curve; Fig. 1I). In between, neurons with increasing elevation of [Cl−]i
exhibit periodic bursting (Fig. 1E), intensive spiking (Fig. 1F), periodic bursting interrupted with DB (Fig. 1G), and the duration of DB increases with higher [Cl−]i
(Fig. 1H).
Figure 1. Different initial [Cl−]i
leads to varying firing activities with saddle–node (SN) bifurcation and Hopf bifurcation (HB) which characterizes the neuronal firing dynamics. A, Dynamic traces in the phase space with different initial conditions of [Cl−]i
. Solid black curve, dynamic traces to resting states; solid red curve, dynamic trace to depolarization block (DB); the blue dots, different initial conditions of [Cl−]i
. B, SN and HB bifurcations and fixed-point solutions of our model. Solid black curve below the saddle–node (SN) point: the stable resting state; solid red curve located above the HB point: stable DB states. Black dashed curve, fixed points with real eigenvalues; red dashed curve, fixed points with imaginary eigenvalues. C, Fixed-point solutions projected onto the subspace of [K+]o
and voltage (C1) and onto the subspace of [Cl−]i
and voltage (C2). D–I, Six examples of neuronal firing behaviors with increasing [Cl−]i
as initial conditions (with [Na+]i=20.06mM
, [K+]o=2.99mM
unchanged). Within these panels, red, green, and blue curves correspond to the reversal potentials of potassium, sodium, and chloride, respectively. D, Initial value of [Cl−]i
is smaller than 9.87 mM, the system relaxes to a resting state with [Cl−]i=7.85mM
. E, Initial value of [Cl−]i
is set to 9.88 mM; it leads neuron to periodic bursting. F, At an initial level of [Cl−]i
at 10.23 mM, neuron shows intensive firing activities. G, An initial elevation of [Cl−]i
to 10.93 mM leads to periodic bursting with DB. H, With an initial elevation of [Cl−]i
to 11.99 mM, the period of periodic bursting with DB is elongated. I, Neuron evolves to DB state with an extremely higher value of [Cl−]i
of >12.27 mM. E–I share the same legend as D.
To better understand how intracellular chloride concentration of [Cl−]i
shapes neuronal firing patterns, we found fixed-point solutions of the coupled differential equations in high-dimensional space by the XPPAUT software (Ermentrout, 2002) and analyzed their stability by calculating the eigenvalues of the Jacobian's matrix at each fixed-point solution. The distribution of fixed points in our dynamical system in the subspace of ion concentrations is depicted in Figure 1B. Our results show that an SN and an HB occur at [Cl−]i=7.89mM
and [Cl−]i=17.66mM
, respectively. Furthermore, the SN bifurcation projected into the subspace of [Cl−]i
and membrane potential is shown in Figure 1C2. Neurons initiating with values lower than SN point of 7.89 mM evolve toward a resting state, whereas neurons with [Cl−]i
higher than the HB point of 17.66 mM are attracted to stable DB states. Neurons with values between the SN and HB points exhibit a variety of dynamic behaviors including tonic spiking and periodic bursting, corresponding to a set of limit cycles (Fig. 1A, light blue curves) in the phase space; referred to as “loop” structures (Barreto and Cressman, 2011). These “loop” structures show strong cyclic variation of [Cl−]i
and of ECl
(Fig. 1F,G). The SN bifurcation was also observed in the subspace of potassium and membrane potential (Fig. 1C1), consistent with previous findings (Bazhenov et al., 2004; Cressman et al., 2009; Wei et al., 2014; Depannemaecker et al., 2022).
Therefore, neuron firing activity is attracted to different states, depending on the initial states of [Cl−]i
. Next, we will explore how neuron firing activity is modified by GABA stimuli through the dynamics of [Cl−]i
and specify when GABA stimuli can switch its effect from inhibition to excitation and trigger SLEs in the neuron.
Pure GABAergic stimuli inhibit neuron firing activity
We applied two patterns of GABA stimuli to the neuron: a step GABA current (Fig. 2A,B) or spike trains of GABA input with different frequencies (Fig. 2C–F).
Figure 2. Exclusive GABA input leads to inhibition. A, Distribution of the fixed-point solution of the neuron state with different step GABA current strengths: black, GGABA=0mS⋅cm−2
; red, GGABA=0.1mS⋅cm−2
; blue, GGABA=0.4mS⋅cm−2
. B, Projection of the fixed-point solutions with different GABA current (as shown in A) to the axis of V−[Cl−]i
. Dotted black curve, reversal potential of chloride ECl
. C1, A periodic bursting with DB neuron is inhibited by GABA inhibition at 80 Hz starting at t = 200 s (dashed vertical line), and the neuron evolves to its resting state. Blue curve: ECl
. C2, An example of depolarizing effect of GABA with higher [Cl−]i
than the cross point shown in Figure 2B. D1, A resting neuron is hyperpolarized by GABA inhibition at 25 Hz starting at t = 0.5 s. D2, GABA spike train dynamics with 25 Hz. F1, F2 are similar to D1, D2 but with a frequency of 80 Hz. E. Dynamic trajectory in the phase space for D1 (red dashed curve) and F1 (blue dashed curve).
With step GABA current input the neuron is mostly inhibited (Fig. 2A,B). As can be seen in Figure 2A, the corresponding [Cl−]i
at the SN bifurcation point increases with higher tonic GABA input (Fig. 2A, blue and red curves compared with the black curves), indicating the hyperpolarization effect of GABA. We computed under which condition the tonic GABA effect can switch from inhibitory to excitatory. By calculating the distribution of the stationary states of the neuron with GABA current input, the voltage can be plotted as a function of [Cl−]i
and compared with the reversal potential of chloride ECl
for each state (Fig. 2B). The cross section between the two types of curves is near the HB point. This implies that only at very depolarized states the tonic GABA effect becomes depolarizing (Fig. 2C2), for all voltages below −35.72 mV GABA is inhibitory (Fig. 2B).
To demonstrate this principle, we applied GABA spike trains (Fig. 2C–F) to the neuron system. Continuous GABA stimuli of 25 Hz (Fig. 2D) and 80 Hz (Fig. 2F) hyperpolarize the membrane potential. Even starting from a higher initial [Cl−]i
corresponding to periodic bursting with DB state (Fig. 1G), stimulation by 80 Hz GABA input still causes inhibition (Fig. 2C1) and returns the neuron to a resting state. Even though [Cl−]i
was already very high, it is still inhibited by GABA and returns to a low level of [Cl−]i
corresponding to the resting state. The excitatory effect of GABA needs to satisfy the condition of ECl>V
, for example, if the neuron starts with a DB state (Fig. 1I), application of 80 Hz GABA stimuli causes a DB state at an even higher voltage (Fig. 2C2).
Our results demonstrate that GABA stimuli can lead to accumulation of [Cl−]i
(Fig. 2E); however, this does not lead to an excitatory effect because even though the reversal potential of chloride ECl
increases by GABA-induced [Cl−]i
accumulation, it usually remains lower than steady state voltage values. In the next section we add the dynamics of HCO3−
efflux upon GABA stimuli to investigate whether it will change the response to GABA activation.
GABA stimuli with HCO3−
outflux trigger a spectrum of neural firing patterns evolving toward DB
Remarkably, 10 Hz GABA spike train stimulation combined with HCO3−
outflux make the neuron generate a series of intensive firing activities evolving to periodic bursting with DB (Fig. 3A,B), which can be seen as a form of SLE. The dynamic trajectory of the neuron firing activity within the phase space evolves from the SN point to the HB point, encompassing all the intermediate states of limit cycles (Fig. 3A), exhibiting tonic firing (Fig. 3G), periodic bursting with DB (Fig. 3H), and it eventually evolves to a DB state (Fig. 3I).
Figure 3. Neuron's intensive firing activity triggered by GABA spike stimuli with HCO3−
outflux. A, Dynamic trajectory of the neural firing activity in the phase space by 10 Hz GABA spike input and HCO3−
outflux shows spinning up toward the higher [Cl−]i
direction. B, Dynamics of neuron membrane potential (black curve) by 10 Hz GABA stimuli starting at t = 100 s (vertical dashed line) and reversal potential for each ion (ENa
, green; EK
, red; ECl
, blue), corresponding to dynamic trajectory shown in A. C, Dynamics of the ion concentrations. D, Reversal potentials of chloride ECl
and GABA EGABA
. F–I, The voltage dynamics from B at different time points, showing the details of the voltage dynamics. F–I share the same legends as in B.
For the initial period of GABA input, the GABA effect on the membrane potential is hyperpolarizing (Fig. 3E). At approximately t = 130 s, the GABA's effect turns to depolarizing but does not trigger action potentials (Fig. 3B); at approximately t = 356.7 s, the neuron begins to generate action potentials (Fig. 3F), indicating the GABA effect turns to excitatory. At approximately t = 360 s, neuron starts to exhibit pseudoperiodic bursting activities interrupted by DB (Fig. 3A,B,G) and with continuing input of GABA and HCO3−
outflux, neuron finally evolves to a stable DB state at approximately t = 590 s (Fig. 3I). With the HCO3−
outflux upon GABA receptor activation, the system reversal potential of GABA EGABA
is systematically higher than the reversal potential of chloride ECl
(Fig. 3D).
Our simulation results demonstrate that evolution to SLE can result from HCO3−
outflux upon GABA receptor activation. The basic mechanism is the high reversal potential of HCO3−
(EHCO3=−13mV)
driving the neuron's membrane potential and pushing it toward higher [Cl−]i
. These findings suggest a potential way in which GABA input can trigger epileptic seizures.
However, in mature animals, only in disease states GABA receptor activation will induce epileptic firing activities. This indicates that most of the time, HCO3−
's effect is compensated by other mechanisms, one of which may be extracellular potassium homeostasis. Experimental evidence showed that when the extracellular potassium concentration is kept at a low level ([K+]bath=3mM)
, neurons stay at resting state (Kaila et al., 1997; Kofuji and Newman, 2004; de Curtis et al., 2018; González et al., 2018). In the next sections, we investigate the effect of applying an external potassium bath with [K+]bath=3mM
on neuron firing activity with simultaneous GABA stimuli with outflux of HCO3−
.
Synergistic interplay between intracellular chloride and extracellular potassium results in different firing activities
Previous theoretical work (Wei et al., 2014) suggested that [K+]bath
determines the neuronal steady state. We next investigate how [K+]bath
affects the dynamic changes in [Cl−]i
during GABA stimuli and the resulting neuronal firing activity.
First, we show that connecting a neuron in DB state with a high [Cl−]i
of 18.75 mM to a potassium bath with [K+]bath=3.0mM
successfully drives the neuron toward its resting state. During this transition process, the neuron exhibits a seizure-like event (Fig. 4A), meanwhile the dynamic trajectory spins down to a low [Cl−]i
(Fig. 4E, black dashed curve). Second, modulating the rate of K+ exchange with the bath (diffusion rate εk
) strongly affects the neuron firing activity during its journey to the resting state (Fig. 4E,F). A slower exchange rate with εk=0.025s−1
will cause long-lasting SLE before returning to the resting state (Fig. 4E,F1, red curves), while the neuron goes to resting state much faster with εk=0.25s−1
(Fig. 4E,F2, blue curve), without experiencing periodic bursting with DB state. Third, we show that neurons starting with different initial levels of [Cl−]i
all evolve to the same resting state with [K+]bath=3.0mM
but exhibit different firing activities (Fig. 4G,H). Our findings suggest that [Cl−]i
and [K+]o
interact synergistically in regulating neuron firing activities, shedding light on the basic mechanism of interplay between potassium and chloride dynamics in regulating SLEs.
Figure 4. Extracellular potassium exchange with a low [K+]bath=3mM
drives the system to a resting state with low [Cl−]i
. A, Voltage dynamics from a DB state by adding [K+]bath=3mM
at t = 0 s. B, Dynamics of ion concentrations in the process of A. C1, C2, D1, D2 show more detailed plots of the typical firing patterns and their transitions in A. E, Dynamic trajectories in the phase space showing the voltage evolution from DB state to resting state for different exchange rates: black dashed curve with εk=0.0025s−1
(A), red curve with εk=0.025s−1
(F1), blue curve with εk=0.25s−1
(F2). G, Dynamic trajectories showing that neuron with initial high level of [Cl−]i
(black curve, H3), middle level (red curve, H2), and low level (blue curve, H1) of initial [Cl−]i
are all driven to the resting state by [K+]bath=3mM
.
As can be seen in Equation 6c, the concentration variation of intracellular chloride depends on the cotransporters’ contribution (Eqs. 7a, 7b). As [K+]o
decreases, the contribution of ρnkcc1
decreases, and that of ρKCC2
increases, both of which decrease [Cl−]i
. Meanwhile, when [K+]o
decreases, the term of γ*(gClL(V−ECl))
also tends to decrease; therefore, all three factors contribute to decreasing [Cl−]i
. Figure 5, A–C, shows the detailed contribution of cotransporters and leak chloride current to chloride decrease in Figure 4C1, where εk=0.0025s−1
. The increase of KCC2 current (Fig. 5B, red curve) and decrease of NKCC1 (Fig. 5B, yellow curve) current both diminish [Cl−]i
; meanwhile, the leak current (Fig. 5B, blue curve) also decreases. Similarly, Figure 5, D–F, shows the details of how the [K+]o
drives the [Cl−]i
decrease with much larger εk=0.25s−1
, corresponding to Figure 4F1.
Figure 5. Detailed dynamic process showing how chloride concentration decreases during potassium bath application with [K+]bath=3mM
. A–C show detailed dynamics from 0 to 4 s in Figure 4C1; D–F show detailed dynamics in Figure 4F1 from 0 to 0.4 s. The red curves in B and E represent the contribution from KCC2, while the yellow curves in B and E represent the contribution from NKCC1 and the blue curves in B and E represent the leak current of chloride. The black curves in B and E represent the summation of these three currents. C, F, Change of the same three currents relative compared with the original values at t = 0.
Our initial investigation of the effect of [K+]bath=3.0mM
had no GABA input. Next, we examine how neuron responds to both application of [K+]bath=3.0mM
and GABA input with HCO3−
outflux.
Competitive and synergistic interplay between GABA stimuli and potassium exchange regulates the neuron firing activity
When the K+ exchange rate is high at εk=0.25s−1
the neuronal response depends on the frequency of GABA spike stimulation (Fig. 6A1–3). In Figure 6A1, GABA spike input with 10 Hz did not cause the neuron to fire, indicating that the effect from [K+]bath
to decrease [Cl−]i
is stronger than that of GABA stimuli to increase [Cl−]i
. Therefore, the neuron goes to resting state with low [Cl−]i
(Fig. 6A4, magenta curve). Increasing GABA frequency to 40 Hz causes action potentials, showing that the accumulation of [Cl−]i
by GABA stimuli overcomes the effect of [K+]bath
(Fig. 6A4, cyan curve with limit cycle). Further increasing the GABA frequency to 80 Hz causes more intensive firing activities as shown in Figure 6A3, demonstrating that the [Cl−]i
increase from GABA input is now much stronger, as can be seen from the dynamic trajectories in Figure 6A4 with light blue curve.
Figure 6. Neuron firing activities by applying different frequency of GABA spike train stimuli with different diffusion rate εk
to [K+]bath
. A1–3, Neuron firing activities by stimuli of GABA input with 10, 40, and 80 Hz, respectively. A4, The corresponding dynamic trajectories with 10 Hz (magenta), 40 Hz (cyan), and 80 Hz (light blue), with εk=0.25s−1
. B1–4, Same as A1–4 but with εk=0.025s−1
. C1–4, Same as A1–4 but with even lower εk=0.0025s−1
. A2,3, B1–3, C1–3 share the same legend as A1; and B4 and C4 share the same legend as A4.
With a lower K+ exchange rate (εk=0.025s−1)
, GABA stimulation with frequencies of 10, 40, and 80 Hz all trigger firing activities (Fig. 6B1–3), with earlier onset of firing and higher action potential frequencies. With 10 Hz stimulation, the neuron exhibits intensive firing activities (Fig. 6B1; Fig. 6B4, magenta curve). Increasing the frequency to 40 Hz causes periodic bursting interrupted with DB state (Fig. 6B2; Fig. 6B4, cyan curve); and with 80 Hz stimuli, the neuron immediately turns to the DB state after a few action potentials (Fig. 6B3; Fig. 6B4, light blue curve), indicating the overwhelming strength of GABA input over [K+]bath
.
Finally, with a very small exchange rate εk=0.0025s−1
, all GABA stimuli frequencies eventually lead to a DB state with different trajectories (Fig. 6C1–4).
The mechanism affecting [Cl−]i
dynamics when GABA spike input is combined with [K+]o
exchange is shown in detail in Figure 7 by taking the example of Figure 6A1 (Fig. 7A–C) and Figure 6A3 (Fig. 7D–F). In Figure 7B, the contribution from KCC2 (red curve) increases, the contribution from NKCC1 (yellow curve) did not change, while the leak current (blue curve) decreases, all of which contribute to the [K+]o
exchange-induced decrease of [Cl−]i
(black curve); however, as shown in Figure 7C, the GABA input-induced [Cl−]i
current increases to a much larger degree than the black curve in Figure 7B; therefore, the GABA input-induced increase of [Cl−]i
wins over the [K+]o
exchange-induced decrease of [Cl−]i
and [Cl−]i
increases to a higher level (Fig. 7A2).
Figure 7. Competition between GABA input-induced increase of [Cl−]i
and exchange of [K+]o
-induced decrease of [Cl−]i
from Figure 6A1,3 for the first 200 s. A1, The voltage dynamics; A2, the dynamics of [Cl−]i
; B, the contribution to the decrease of [Cl−]i,
where the red curve is contribution from KCC2; yellow curve is contribution from NKCC1, the blue curve is the contribution from chloride leak current, and the black curve is the summation of all the three contributions. C, The contribution to the increase of [Cl−]i
. D–F are similar to A–C but with 80 Hz stimuli as shown in Figure 6A3. Figure 7E shares the same legend as Figure 7B.
A similar phenomenon is shown in Figure 7D–F for the first 200 s from Figure 6A3. The GABA input-induced contribution of [Cl−]i
increase (Fig. 7F) is much larger than the [K+]bath
application-induced decrease of [Cl−]i
(Fig. 7E, black curve); therefore, the [Cl−]i
increases as shown in Figure 7D2.
Our results show that the presence of SLE depends on the competition between two effects: one drives [Cl−]i
to lower levels because of the potassium exchange with [K+]bath
, and the counterbalancing one tends to elevate [Cl−]i
through GABA stimuli with HCO3−
outflux. Smaller values of εK
decrease [Cl−]i
slowly, while larger values of εK
decrease [Cl−]i
fast, effectively preventing epileptic seizures at the single neuron level.
To quantitatively plot the parameter space within which the effect of GABA input wins out that from potassium exchange, or vice versa, we performed an analytical calculation of the steady-state distribution of the voltage with GABA step current input and potassium bath application with [K+]bath=3.0mM
. Our results show that the system exhibits SN and HB bifurcations (Fig. 8A) and that the different potassium exchange rates give to different steady states (Fig. 8A, red, black, blue curves). For example, by comparing Figure 8B1 (εk=0.025s−1)
with Figure 8B2 (εk=0.0025s−1)
, it is clear that with the higher εk
the neuron requires larger GABA stimuli to escape from the resting state and generate spiking than the condition with lower εk
value, and the same applies for HB point with the transition to DB state.
Figure 8. Neuron discharge state determined by competition between two parameters: step current input with constant GGABA
and potassium exchange rate εk
. A, Steady-state distribution of voltage with step current input of GABA and application of potassium diffusion with [K+]bath=3mM
. Black, εk=0.25s−1
; red, εk=0.025s−1
; blue, εk=0.0025s−1
. B, The projection of steady-state voltage onto the axis of strength of GABA step current, red, εk=0.025s−1
; blue, εk=0.0025s−1
. C, The phase diagram for the SN and HB point with different εk
and GGABA
. D, Neuron firing patterns by different strength of GABA current input (εk=0.25s−1)
, blue, periodic firing; green, intensive firing; red, subthreshold periodic oscillation. E, Dynamic trajectory in the phase space corresponding to the firing patterns shown in D (colored curves). F, Examples of dynamic trajectories for neuron firing activity shown in G (GGABA=0.24mS⋅cm−2)
, H (GGABA=0.33mS⋅cm−2)
, I (GGABA=2.07mS⋅cm−2)
corresponding to blue, green, and red curves, respectively. G1 and H2 show the firing activity at a smaller time scale. G1, H1 share the same legend as I.
The dependence of the bifurcation points delineating the different firing regimes on GGABA
and exchange rate εk
is plotted in Figure 8C. The HB (red curve) point leading to DB state is much more sensitive to the exchange rate εk
than the SN (black curve).
In Figure 8D–I, we show examples of the neuron responses in the case of a large exchange rate of potassium with the bath [K+]bath=3mM
, εk=0.25s−1
. The relationship between the membrane potential and GGABA
is plotted in Figure 8D, where the black curve indicates that the neuron requires at least GGABA=0.20mS⋅cm−2
to generate action potentials, the blue curve (Fig. 8D,E) shows the periodic bursting activities (example in Fig. 8G; GGABA=0.24mS⋅cm−2
) and the green curve (Fig. 8D,E) the intensive firing of the neuron (example in Fig. 8H; GGABA=0.33mS⋅cm−2
). The envelop shrinks with increasing GGABA
, and with very large GGABA
∼1.8 mS cm−2, the neuron shows a subcellular periodic oscillation (example in Fig. 8I, periodic little bump; GGABA=2.07mS⋅cm−2
). Finally, the neuron enters the DB state with GGABA=2.15mS⋅cm−2
. The dynamic trajectories for Figure 8G–I are plotted in Figure 8F by blue, green, and red curves, respectively, showing that the stronger GABA input, the higher [Cl−]i
becomes, leading to a more depolarized state.
Rebound behavior after termination of GABA input
We complete the analysis of the neuronal response to GABA input by demonstrating the rebound behavior after the termination of a GABA spike train stimuli, as reported experimentally (Perkel and Mulloney, 1974; Kuwada and Batra, 1999; Felix et al., 2011; Adhikari et al., 2012; Chang et al., 2018). We set [K+]bath=3.0mM
during the whole simulation process and apply a spike train of GABA stimuli during 100–600 s (Fig. 9). As can be seen in Figure 9D, with 10 Hz stimulation the neuron slowly depolarizes and then starts firing intensively (black curve), and the dynamic trajectory is evolving upward with increasing [Cl−]i
showing a limit cycle in the phase space (Fig. 9A, black curve). When the GABA stimulation terminates at 600 s, the neuron shows rebound activity (Fig. 9A,D, red curve) and the trajectory evolves downward to lower [Cl−]i
and ends up with the resting state.
Figure 9. Rebound firing activity upon termination of GABA stimuli. [K+]bath=3.0mM,εk=0.025s−1
. A–C, Dynamic trajectory of neuron firing activity by GABA stimuli during 100–600 s with 10, 40, and 80 Hz, respectively. Black arrow, the direction of the dynamic trace during GABA stimulation; red arrow, dynamic trace after ending the GABA stimuli. D–F, Neuron firing activity with dynamics of voltage by GABA stimuli of 10, 40, and 80 Hz, respectively. The first dashed vertical line shows the onset of GABA input at t = 100 s; the second vertical dashed line indicates the end of stimulation at t = 600 s. Black curve, neuronal firing activities during GABA stimulation; red curve, neuronal rebound behavior after end of stimulation. A and D, B and E, and C and F share the same legends, respectively.
With higher frequency of GABA stimuli (40 Hz), the neuron exhibits intensive firing and then periodic bursting with DB activities (Fig. 9B,E, black curves). Similar to the 10 Hz case, the neuron generates rebound firing activities with larger limit cycles after withdrawn GABA spike input; and it evolves toward the lower [Cl−]i
direction and a resting state (Fig. 9B,E, red curves). At 80 Hz stimulation, the neuron immediately exhibits the firing activity followed by a stable DB state (Fig. 9C,F, black curves), the rebound activity (Fig. 9C,F, red curves) shows intensive firing but leads to lower [Cl−]i
and eventually neuron goes to resting state. Our results demonstrate the basic mechanism of opposing effects on [Cl−]i
: by withdrawing GABA input, the drive to increase [Cl−]i
suddenly disappears, and the potassium exchange-induced decrease of [Cl−]i
dominates; thus, the neuron evolves into the resting state.
Discussion
In summary, we describe how the influence of increasing [Cl−]i
on intrinsic neuronal properties can be characterized by SN and HB bifurcations, similar to the voltage-dependent dynamics (Rinzel and Ermentrout, 1998). Our results reveal that neuron firing activity is regulated by the interplay between two opposing effects: upregulation of [Cl−]i
by GABA input with HCO3−
efflux and downregulation of [Cl−]i
by potassium exchange to a low [K+]bath
. Our results elucidate the crucial role of [Cl−]i
in enhancing epileptic SLEs within the framework of a single-compartment neuron model.
Epileptic seizures are intricate pathological phenomena influenced by many factors (Timofeev and Steriade, 2004; Moshé et al., 2015; Gentiletti et al., 2022; Wenzel et al., 2023). Our single-neuron model demonstrates that GABA input with HCO3−
efflux can increase the intracellular chloride concentration and lead to a spectrum of distinct firing activities, comparable with SLE, and consistent with experimental observations of elevated intracellular chloride concentrations by external stimuli (Timofeev et al., 2002; Somjen, 2004; Lillis et al., 2012; Staley, 2015; Magloire et al., 2019). However, the conclusions derived from various experiments regarding the role of inhibitory neurons in epilepsy showed inconsistent results (de Curtis and Avoli, 2016; Weiss, 2023; Wenzel et al., 2023). It is well known that inhibitory neurons have the capacity to mitigate epileptic seizures by hyperpolarizing excitatory neurons and numerous experimental and modeling studies have underscored that disruption of this delicate excitatory–inhibitory balance can precipitate epileptic seizures (McCormick and Contreras, 2001; Liu et al., 2020; van Hugte et al., 2023). Additionally, prolonged GABA input can result in the accumulation of [Cl−]i
, thereby facilitating epileptic seizures (Trevelyan et al., 2006, 2007; Ben-Ari et al., 2007; Burman et al., 2019). Our findings can explain this double role of GABA input. Upon GABA receptor activation, if there is only chloride influx, then the effect of GABA will be inhibition provided that the reversal potential of chloride remains below the membrane potential. If there is also HCO3−
efflux upon GABA receptor activation, then it will turn the inhibition of GABA into excitation by depolarization, caused by the elevated reversal potential of HCO3−
. The permeability of HCO3−
is between 0.18 and 0.44 of the chloride permeability (Bormann et al., 1987; Kaila and Voipio, 1987; Fatima-Shad and Barry, 1993; Kaila et al., 1997, 2014), and we assumed a constant HCO3−
reversal potential EHCO3
of −13 mV. This value may change during neural activity as the dynamics of EHCO3
depend on the detailed chemical reaction and transport of HCO3−
in neuronal systems. It is possible that the concentration of HCO3−
varies, allowing for dynamic modulation of the depolarizing effect of HCO3−
(Farrant and Kaila, 2007). Possible mechanisms might include secondary active HCO3−
uptake via electroneutral and electrogenic Na+/HCO3−
symporters (Hübner and Holthoff, 2013) and intracellular pH changes by the carbonic anhydrases (Sinning and Hüebner, 2013). Elucidating HCO3−
dynamics in the brain is an important question for future investigations.
In addition, the potassium homeostatic mechanism serves as a regulator of [Cl−]i
during spontaneous discharge behaviors. In our work, we have demonstrated that it is the interplay between potassium homeostasis and GABA stimuli with HCO3−
efflux that shapes neuron firing activities. Nevertheless, the precise dynamical properties of the potassium homeostatic mechanism remain elusive, with potentially important role of astrocytes (Bradbury and Davson, 1965; Volman et al., 2012; Chizhov et al., 2019; Palabas et al., 2022). Our study demonstrates that a larger potassium exchange rate (denoted as εK
) can mitigate epileptic discharges by shortening the maximum seizure duration and drive the neuron firing activities to a state with lower [Cl−]i
. The strength and timing of the potassium homeostasis will be critically important for the preventive treatment of epilepsy.
Existing reports indicate that potassium concentration [K+]o
is maintained at ∼3 mM in vivo (Raimondo et al., 2015; Staley, 2015), while the corresponding value of εK
remains largely unknown. The physiological significance of the diffusion coefficient εK
covers neuron–glia interactions and neuron–capillary interactions. Consequently, the means to expediently reduce extracellular potassium concentrations, whether through blood vessels or alternative mechanisms (Somjen, 2004; de Curtis et al., 2018), seem critically important. Our work underscores the importance of precise measurements of the strength and speed of the potassium homeostatic mechanism in physiological environments to better understand seizure events.
We examine the role of the K-Cl cotransporter NKCC1 and KCC2 in regulating SLE and [Cl−]i
dynamics in Figures 5 and 7. KCC2 transport one chloride ion and one potassium ion into the extracellular space, making it a potentially effective means of reducing [Cl−]i
(Payne et al., 2003; Gamba, 2005; Kahle et al., 2008; Blaesse et al., 2009; Kaila et al., 2014), especially with exchange process with [K+]bath
. Some studies have indicated at the possibility of pathological alterations in KCC2 contributing to certain behaviors in neural networks (Kaila et al., 2014; Moore et al., 2017; González et al., 2018). We found that the relative strength of NKCC1 and KCC2 contributes to the change of chloride concentration and that the Cl−
leak current is also quite important. How the cotransporters of NKCC1 and KCC2 coherently can regulate the dynamics of chloride and mitigate epileptic seizures, and how their roles change in various physiological situations, still need further investigation.
During epileptic seizures, changes in ion concentrations lead to alterations in intracellular and extracellular osmotic pressures, resulting in dynamic fluctuations in the ratio of neuronal intracellular to extracellular volume (McBain et al., 1990; Østby et al., 2009; Hrabetova et al., 2018). The volume factor in the model indicates how strongly intracellular ion concentration variations affect the reversal potential. In physiologically realistic systems, the volume factor changes with neuronal firing properties, which involves an additional level of complexity that is challenging to simulate and analyze accurately. In our study we assumed the volume factor is fixed at β=7
. We also tested our results with different volume factors, the fixed-point solution of the neuron system and the SN and HB points still exist but the distribution of them at each level of chloride concentration varies with
β. The neuron exhibits qualitatively similar results yet distinct discharge patterns based on the SN and HB points in the system (data not shown).
