A dynamics model of neuron-astrocyte network accounting for febrile seizures

Abstract

Febrile seizure (FS) is a full-body convulsion caused by a high body temperature that affect young kids, however, how these most common of human seizures are generated by fever has not been known. One common observation is that cortical neurons become overexcited with abnormal running of sodium and potassium ions cross membrane in raised body temperature condition, Considering that astrocyte Kir4.1 channel play a critical role in maintaining extracellular homeostasis of ionic concentrations and electrochemical potentials of neurons by fast depletion of extracellular potassium ions, we examined here the potential role of temperature-dependent Kir4.1 channel in astrocytes in causing FS. We first built up a temperature-dependent computational model of the Kir4.1 channel in astrocytes and validated with experiments. We have then built up a neuron-astrocyte network and examine the role of the Kir4.1 channel in modulating neuronal firing dynamics as temperature increase. The numerical experiment demonstrated that the Kir4.1 channel function optimally in the body temperature around 37 °C in cleaning ‘excessive’ extracellular potassium ions during neuronal firing process, however, higher temperature deteriorates its cleaning function, while lower temperature slows down its cleaning efficiency. With the increase of temperature, neurons go through different stages of spiking dynamics from spontaneous slow oscillations, to tonic spiking, fast bursting oscillations, and eventually epileptic bursting. Thus, our study may provide a potential new mechanism that febrile seizures may be happened due to temperature-dependent functional disorders of Kir4.1 channel in astrocytes.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-021-09706-w.

Introduction

Febrile seizure (FS), one of the most common type of seizures in infants and young children, affecting 3 ~ 5% of the pediatric population, is generally induced by hyperthermia (Dube et al. 2009a; Kim and Connors 2012a; Reid et al. 2009). Raised temperature in fever condition could alter numerous ionic, synaptic and cellular functions, increasing neural excitability and leading to FS. Most of recent studies on the mechanism of FS focus on the overactivation of temperature-sensitive sodium (Thomas et al. 2009) and potassium channels (Kohling and Wolfart 2016; Leo et al. 2015), GABA-mediated network dynamics (Todd et al. 2014), enhanced neuronal excitability by hyperthermia-induced secretion of endogenous fever mediators including interleukin-1b (Dube et al. 2009b; Fukuda et al. 2014). A recent study by Ye et al. showed that neuronal hyperexcitability due to the Na_V1.2 channel over-activation in high temperature may promote seizure onset (Ye et al. 2018a). High temperature could also alter the magnitude of neuronal inward rectifying potassium channel currents (Zhao et al. 2016). Mutations in a Kv2.1-channel modifier (Kv8.2) were found to be associated with a mild form of febrile seizures or a severe form of encephalopathy with seizures, respectively (Jorge et al. 2011). In addition, astrocytes are active neuronal partners, endowed with machinery ion channels, transmitter receptors, and transporters to actively supporting normal neuronal activity. Recent studies suggest that astrocytes may play a crucial role in epilepsy (Coulter and Steinhauser 2015; Siva 2005; Steinhauser and Boison 2012).. Astrocytes have voltage-gated calcium and potassium channels which are also temperature sensitive because of Q10 effect (Bender and Norenberg 1994; Niko et al. 2015; Tigges et al. 1990; Yang et al. 2009). Astrocytes in the cortex also contain high-affinity glutamate transporters that are important for lowering extracellular glutamate concentration after releasing at excitatory synapses. Astrocyte glutamate transporters for uptaking extracellular glutamate are also highly temperature sensitive and may function well at around 36 °C to support normal synaptic transmission of neurons (Bergles and Jahr 1998). A function disorder of Kir4.1 channels that located in astrocytes, could be associated with disordered astrocytic spatial buffering and consecutive activity-dependent pathological accumulation of K⁺ (Dube et al. 2009b; Ohno 2018). This impaired astrocytic spatial buffering may induce large neuronal excitability and lead to epileptic activity (Jabs, Seifert et al. 2008; Du et al., 2018).

The Kir4.1 channels were observed exclusively in glial cells (Higashi et al. 2001). In epilepsy patients, mutations of the astrocytic Kir4.1 channel have been widely reported (Ferraro et al. 2004; Olsen and Sontheimer 2008). Although previous investigators discussed the potential mechanism of Kir4.1 channel in epileptic activity, its potential role in FS is not clear at all. Indeed, the effect of increased temperature on astrocyte Kir4.1 channel should not be ignored (Bender and Norenberg 1994; Ehrengruber et al. 2003; Tigges et al. 1990). Notably, the early Kir4.1 channel current models were built solely on the concentration of extracellular potassium (Cressman et al. 2009; Wei et al. 2014). Since 2013, Alexandra and Jérémie et al. provided mathematical models of the Kir4.1 channel current in astrocytes by experimental data fitting (Biermans et al. 1987; Ransom and Sontheimer 1995), which is dependent on the extracellular space, intra-astrocytic potassium concentration and astrocyte membrane potential. Remarkably, in 2014, Jérémie found a fascinating phenomenon in which a long-term inward current was measured in the Kir4.1 channel that had a similar dynamic characterization as that in sodium and potassium channels in the classical Huxley-Hodgkin neuron model (Jérémie et al. 2014). In addition, the voltage gated characteristics of Kir4.1 channel has been discussed in other biological experiment (Maria et al. 2003). Therefore, a voltage and concentration gated Kir4.1 channel current model will more accurately depict the dynamic characteristic of the Kir4.1 channel in astrocytes and explain the potential impact of Kir4.1 channel on febrile seizures.

In light of these findings, we first built up a computational model of temperature-sensitive Kir4.1 channel that is also gated by potassium concentration based on recent experimental findings(Biermans et al. 1987; Sibille et al. 2015; Ransom and Sontheimer1995; Sakmann and Trube 1984), and then we built up an astrocyte-neuron coupled network model consisting of a single compartment neuron connected to a surrounding astrocyte with Kir4.1 channels and Na⁺/K⁺-ATPase pumps. Extracellular potassium ions diffused in and out of the space between the neuron and astrocyte. We have validated the model simulation results by experimental data (Ballanyi et al. 1987; Sibille et al. 2015). Then we have carried out a set of simulation experiments to investigate the effect of temperature on the Kir4.1 channel dynamics which may be involved in regulating different neuronal firing patterns. Particularly, we show that with certain parameter range increased temperature over 40 °C will drive neurons generate epileptiform discharges which are similar to that in febrile seizure condition.

Model

The neuron model is described by the following HH-type equations (Du et al.2018;Ullah G et al. 2009; Wang et al.2016):

$$\begin{array}{cc}\hfill C\frac{d{V}_N}{\mathit{dt}}=& -I_{\mathit{Na}}-I_K-I_L+I_{\mathit{ext}}\hfill \\ \hfill I_{\mathit{Na}}=& -g_{\mathit{Na}}{m}^{3}h(V_N-V_{\mathit{Na}})\hfill \\ \hfill I_K=& -g_K{n}^4(V_N-V_K)\hfill \\ \hfill I_L=& -g_{\mathit{NaL}}(V_N-V_{\mathit{Na}})-g_{\mathit{KL}}(V_N-V_K)-g_{\mathit{Cl}}(V_N-V_{\mathit{Cl}})\hfill \\ \hfill \end{array}$$ 1

where C is the membrane capacitance of the neuron. The specific values of the above conductance parameters g_Na, g_K, g_NaL and g_KL are shown in Table 1.V_Na,V_K and V_Cl denote the Na⁺ and K⁺ channel reversal potentials, respectively. n, m and h are gating variables for Na⁺ and K⁺ currents. The equations for the gating variables in Eq. (1) are as follows:

$$\frac{\mathit{dq}}{\mathit{dt}}=\phi [{\alpha }_q(V_N)(1-q)-{\beta }_q(V_N)q],q=m,n,h$$ 2

$\begin{array}{cc}\hfill {\alpha }_m=& \varphi \left(0.1(V_N+30)/[1-exp(-0.1(V_N+30))]\right)\hfill \\ \hfill {\beta }_m=& \varphi \left(4exp[-(V_N+55)/18]\right)\hfill \\ \hfill {\alpha }_n=& \varphi \left(0.01(V_N+34)/[1-exp(-0.1(V_N+34))]\right)\hfill \\ \hfill {\beta }_n=& \varphi \left(0.125exp(-(V_N+44)/80)\right)\hfill \\ \hfill {\alpha }_h=& \varphi \left(0.07exp(-(V_N+44)/20)\right)\hfill \\ \hfill {\beta }_h=& \varphi \left(1/[1+exp(-0.1(V_N+14))]\right)\hfill \\ \hfill \end{array}$

Table 1.

Model Parameters

Parameter	Value and units	Description
CN	$1\mu F/c{m}^2$	Membrane capacitance of the neuron
gNa	$100mS/m^2$	Conductance of the persistent sodium current
gK	$40\text{mS}/{\text{m}}^2$	Conductance of the potassium current
gKL	$0.05mS/m^2$	Conductance of the potassium leak current
gNaL	$0.05mS/m^2$	Conductance of the sodium leak current
gCl	$0.05mS/m^2$	Conductance of the chloride leak current
$v_1$	7.0	The volume rate of extracellular space and neuron
$v_2$	3.0	The volume rate of extracellular space and astrocyte
CA	15pF	Membrane capacitance of astrocyte
VCl	− 81.93 mV	The reversal potentials of chloride
gAL	5.0 mM−1 ms−1	Astrocytic leak conductance(Sibille et al. 2015)

The ϕ = Q₁₀^{( T−23)} is the temperature term, Q₁₀ = 2.3 (Yu et al. 2012). The reversal potentials of Na⁺ and K⁺ and Cl⁻ are calculated by

$$V_j=26.64ln\left(\frac{{[j]}_{\text{Ni}}}{{[j]}_{\text{o}}}\right),j={\text{K}}^{+},{\text{Na}}^{+},{\text{Cl}}^{-}$$ 3

where [j]_Ni and [j]_o denote ionic concentrations (K⁺, Na⁺, and Cl⁻) in the intraneuronal and extracellular spaces, respectively.

Dynamics of ion fluxes in astrocyte-neuron network model

The [K⁺]_o value is continuously updated by the K⁺ currents across the neuronal membrane, K⁺ spatial diffusion (Cressman et al. 2009; Ullah G et al. 2009), neuron and astrocyte Na⁺/K⁺-ATPase pumps, and astrocyte Kir4.1 channels (Djulic et al. 2007; Du et al. 2018; Sibille et al. 2015). The electric current through the cell membrane can cause changes in the ionic concentrations inside and outside of the cell. An electrical current I across a membrane is equal to ion flow per unit of time. Hence, the K⁺ concentration dynamics for the neuron and astrocyte and the extracellular space are described as follows:

$$\frac{d{[K^{+}]}_o}{\mathit{dt}}=J_{\mathit{IK}}-2J_{pump,N}-2J_{pump,A}+J_{\mathit{Kir}}-J_{\mathit{diff}}$$ 4

$$\frac{d{[K^{+}]}_A}{\mathit{dt}}=(-J_{\mathit{kir}}+2I_{\mathit{pumpA}})v_{rate2}$$ 5

$$\frac{d{[K^{+}]}_N}{\mathit{dt}}=(-J_{\mathit{IK}}+2J_{pump,N})v_{rate1}$$ 6

Similar to the K⁺dynamics, the [Na]_ovalue is continuously updated by the Na⁺ currents across the neuronal membrane and neuron and astrocyte Na⁺/K⁺-ATPase pumps (Cressman et al. 2009; Du et al. 2018; Ullah G et al. 2009). In addition, the Na⁺ concentrations in neuron and astrocyte are added to two constant leak terms, J_NaLA and J_NaLN. Thus, the Na⁺ concentration dynamics equations for neurons and astrocytes and the extracellular space are modeled in the astrocyte-neuron network equations:

$$\frac{d{[N{a}^{+}]}_{o,i}}{\mathit{dt}}=J_{Na,N}+3J_{pump,N}+3J_{pump,A}+J_{NaL,N}+3J_{NaL,A}$$ 7

$$\frac{d{[N{a}^{+}]}_A}{\mathit{dt}}=(-3J_{pump,A}-J_{Na\text{L},A})v_{rate2}$$ 8

$$\frac{d{[Na]}_N}{\mathit{dt}}=(-J_{Na,N}-3J_{pump,N}-J_{NaL,N})v_{rate1}$$ 9

In addition, different terms in the expressions (4–9) are described as follows:

$$\begin{array}{c}{J}_{pump,N}=\rho \left(\frac{1}{1.0+exp(25.0-{[Na]}_{\mathit{Ni}})/3.0}\right)\times \left(\frac{1}{1+exp(8-{[K]}_o)}\right)\\ J_{\mathit{diff}}=\epsilon \left({[K]}_o-K_{\mathit{bath}}\right)\\ J_{pump,Ai}=\left(\frac{1}{3}\right)\rho \left(\frac{1}{1.0+exp(25.0-{[Na]}_{\mathit{Ai}})/3.0}\right)\times \left(\frac{1}{1+exp(8-{[K]}_o)}\right)\\ \end{array}$$ 10

where ρ is the pump strength of Na⁺/K⁺-ATPase, [Na]_Niand [Na]_Ai are the sodium concentrations for neurons and astrocytes, respectively, ε is the spatial diffusion coefficient of K⁺, and K_bath is the K⁺ concentration in the largest nearby reservoir.

Computational model of Kir4.1 channel and membrane potential dynamics in astrocytes

The equation of the astrocyte membrane potential V_A is given by:

$$C_A\frac{d{V}_A}{\mathit{dt}}=-I_{\mathit{kir}}-I{}_{\mathit{AL}}$$ 11

where C_A is the astrocytic capacitance, and the leak current is I_AL = g_AI (V_A-V_AL). g_AL is the astrocytic leak conductance. I_kir is the inward rectifier Kir4.1 channel current in astrocyte.

Previous studies reported that the activation of the Kir4.1 channel in astrocytes not only depends on the concentration of extracellular potassium ions (Biermans et al. 1987; Ransom and Sontheimer 1995) but also shows some voltage dependence (Ransom and Sontheimer 1995; Tse et al. 1992). Voltage-activated K⁺ channels may enhance the local K⁺ spatial buffering capabilities of the astrocyte syncytium when extracellular K⁺ increases during neuronal activity (Tse et al. 1992) according to its gating characteristic(Heukelom 1994) and I-V curve (Ransom and Sontheimer 1995). The second term of Eq. 14 describes that the conductance of inwardly rectifying K⁺ currents (Kir4.1) depend strongly on [K⁺]_o and are approximately proportional to the square-root of [K⁺]_o, increasing conductance with increasing [K⁺]_o (Biermans et al. 1987; Ransom andKucheryavykh Sontheimer 1995; Sakmann and Trube 1984; Witthoft et al. 2013). Furthermore, the I_kir is also closely related to the astrocytic membrane potential (Sibille et al. 2015; et al. 2007; Tse et al. 1992; Witthoft et al. 2013). We present a novel equation of gating-activated inwardly rectifying K⁺ currents (Kir4.1) in astrocytes as follows:

$$I_{\mathit{kir}}=g_{\mathit{kir}}\sqrt{{[K^{+}]}_o}m\left(V_{\mathit{ast}}-v_{\mathit{Kir}}\right)$$ 12

$$v_{\mathit{Kir}}=v_{Kir,1}log\left({[K^{+}]}_o\right)-v_{Kir,2}$$

where g_kir is the conductance of Kir4.1 channels and V_ast denotes the membrane potential of astrocytes. [K⁺]_o in units of mM, which represents the K⁺ concentration in the extracellular space. v_Kir,1 and v_Kir,2 are the Kir4.1 reversal potential to extracellular K⁺ and minimum Kir4.1 reversal potential (at low extracellular K⁺), respectively. The dynamics of the gating variable m is described by

$$\frac{\mathit{dm}}{\mathit{dt}}=\frac{\varphi \left(m_{inf}-m\right)}{t}$$ 13

Moreover, the inwardly rectifying K⁺ currents channel to the steady-state open/close partition function of Kir4.1 channels according to the Boltzmann distribution (Heukelom 1994; Tse et al. 1992), which includes dynamic variations of the potassium Nernst potential during neuronal activity, is embodied in Eq. 14.

$$m_{inf}=1/(1+exp(V_{\mathit{ast}}-v_{\mathit{Kir}}/2+a_{v1})/k))$$ 14

where k is the slope of the Boltzmann fit to the steady-state activation parameter m_inf. k = -4.89 mV. The time constants of the Kir4.1 channel was modeled by

$$\tau =1/(aexp(-V/V_0)+bexp(V/V_0))$$ 15

where V₀ = 37.08 mV, a = 0.061 s⁻¹, and b = 0.00817 s⁻¹.

The values of the parameters used in the model are listed in Table 1.

In this work, the 4th-order Runge–Kutta method was used for numerical simulation with a time step of h = 0.01 ms. Additionally, the parameter values used in the numerical simulation are shown in Table 1, assuming no special emphasis. If there's no special explanation, control temperature T is set to 23 °C.

Results

Astrocyte-neuron membrane potential dynamics with input stimulus

An astrocyte-neuron coupled model was built up with a single compartment neuron connected to a surrounding astrocyte containing Kir4.1 channels and Na⁺/K⁺-ATPase pumps (Fig. 1a). Based on evidence from previous studies (Cressman et al. 2009; Wang et al. 2016), several key factors mediate the dynamics of extracellular K⁺ concentration (illustrated in Fig. 1a).

Fig. 1.

a. Sketches of the computational model. b. The steady-state opening of the gates m along with v_A and [K⁺]_o in our model. (c, d and e). Superimposition of astrocytic Kir4.1 channel current (I_Kir4.1), extracellular potassium concentration ([K⁺]_o), and astrocytic membrane potential ([K⁺]_A) time series obtained from the Kir4.1 model in Ref. (Sibille et al. 2015) (black line, g_kir = 380 ps) and our model (red line, g_kir = 60 ps) generated by a single pulse simulation f(t) = δ(t), respectively. (f, h and i). Quantification of astrocytic Kir4.1 channel current, extracellular potassium concentration, and astrocytic membrane potential kinetics extracted from numerical simulations using the Kir4.1 model in Ref. (Sibille et al. 2015) (black) and the presented model (red)

In Fig. 1b, we plot the activation curves (m) along with the astrocyte voltage (V_a) and the extracellular potassium concentration ([K⁺]_o), respectively. Note that m is a sigmoid function with 0 represent hyperpolarization and 1 represents full depolarization of astrocyte potential V_a. From these activation curves, we can verify many experimental results using our Kir4.1 channel model. For example, experimental recordings showed that voltage-activated K⁺ currents in acutely isolated hippocampal astrocytes have a half-activated membrane potential (v_Kir/2) of approximately −50 mV (Jensen and Yaari 1997; Meeks and Mennerick 2010; Traynelis and Dingledine 1988; Tse et al. 1992), The half-activation voltage of astrocytes obtained from our simulation results is −48.6 mV, shown at point 1 in Fig. 1b. In addition, the Kir4.1 channels will be closed as the value of [K⁺]_o is equal to 4 mM (shown at point 2 in Fig. 1b) and open when [K⁺]_o is approximately 8.0 (shown at point 1 in Fig. 1b), this is because, at normal concentrations (approximately 4.0 mM), the normal resting potential of the neuron is maintained. However, at higher concentrations (for example, 8 mM), bursts and seizure-like events occur spontaneously in the nervous system (Jensen and Yaari 1997; Meeks and Mennerick 2010; Traynelis and Dingledine 1988). This transient increase in [K⁺]_o is directly related to the “dumping” of potassium during prolonged neuronal discharge, and spontaneous epileptiform activity can be initiated in vitro by raising [K⁺]_o from 4 mM to 8.5 mM (Rutecki et al. 1985). To further verify the new Kir4.1 channel model, we compared the simulated Kir4.1 current time course in responses to a single pulse stimulation (f(t) = δ(t)) with the previous published model (Sibille et al. 2015), shown in Fig. 1c. Noted here, the value Kir4.1 channel conductance for our model is 380 ps while 60 ps in previous model (Sibille et al. 2015). The expanded view of the astrocytic Kir4.1 ionic current time course (see Fig. 1c) shows that all the models results are matched well from each other. Moreover, we measured time course parameters of the astrocyte Kir4.1 channel current from both models for further quantitative comparision. Previous model (Sibille et al. 2015) has peak I_Kir4.1 value −27.39 pA with rise time of 110 ms and decay time of 77 ms, while our model has peak I_Kir4.1 value −27.5 with rise time of 112.06 ms and decay time of 78.3 ms, as shown in Fig. 1f. According to the aforementioned data, we found that the error of the time series of the Kir4.1 channel current corresponding to the external stimulus obtained from our model and the model in Jérémie’s research (Sibille et al. 2015) is within a very small range.

In addition, we investigated the extracellular potassium concentration ([K⁺]_o) and astrocytic membrane potential ([K⁺]_A) dynamics in response to single pulse stimulation. This pulse stimulus induced an action potential, resulting in an ~ 0.173 mM increase in [K⁺]_o (mV) within 326 ms, which slowly decayed back to baseline levels over 10 s, as shown in Fig. 1d. The kinetics of the extracellular potassium concentration obtained using our Kir4.1 model are comparable to those obtained by the Kir4.1 model in Ref. (Sibille et al. 2015) during single stimulation (peak [K⁺]_o: 4.1738 mM, rise time: 326.2 ms, and decay time: 2.3554 s for numerical simulation obtained by the Kir4.1 channel model from Ref. (Sibille et al. 2015); peak [K⁺]_o: 4.1731 mM, rise time: 326.15 ms, and decay time: 2. 3551 s for numerical simulation obtained by our model), shown in Fig. 1h. After validating the extracellular potassium responses of the astrocyte-neuron coupled model to pulse stimulation, we also examined the impact of the above pulse stimulus on the dynamics of the astrocyte membrane potential. The kinetics of the astrocyte membrane potential obtained with the numerical simulations in the present work are comparable to the results obtained with the simulation recordings in Ref. (Sibille et al. 2015) (peak V_A: -78.79 mV, rise time: 34.8 ms, and decay time: 1.8372 s for numerical simulation obtained by the Kir4.1 channel model from Ref. (Sibille et al. 2015); peak V_A: -78.8 mV, rise time: 33.6 ms, and decay time: 1.8516 s for numerical simulation obtained using our model), shown in Fig. 1i. These data show that the dynamics of the extracellular potassium concentration and astrocyte membrane potential obtained by our Kir4.1 channel model and the Kir4.1 channel model in Ref. (Sibille et al. 2015) are the same. Thus, these data suggest that our model captures the key players sufficient to mimic the evoked extracellular potassium concentration and astrocyte membrane potential dynamics observed in experimentals in different conditions.

Kir4.1 channel contribution to extracellular K⁺ levels

To further validate the regulation dynamics of the extracellular K⁺ concentration by our Kir4.1 channel model, we simulated the effect of the Kir4.1 channel blocker Ba²⁺ (Ballanyi et al. 1987) by setting the Kir4.1 channel conductance g_kirA to 0.1 ps (see Eq. 14). in order to verify our numerrical results in comparison with the experimental recording in study(Ballanyi et al. 1987), here, the sine stimulus amplitude and frequency are 1μA/cm² and 10 Hz, respectively. The numerical simulations showed that inhibition of astrocyte Kir4.1 channels led to a higher transient peak increase in extracellular space K⁺ concentration ([K⁺]_o) during long-lasting sine stimulation (10 s) compared to control conditions (Fig. 2b). In addition, the maximum amplitude of the extracellular K⁺ concentration for g_kirA = 0.1 ps was higher than that for g_kirA = 360.0 ps during a 10 s external stimulation input. In response to a stimulus under the normal conditions (black curves), the astrocyte responds with a quick rise in the intracellular K⁺ concentration (Fig. 2b). In contrast, in the presence of a Kir4.1 blocker (gray curves), the astrocyte K⁺ concentration baseline is higher and rises more slowly to a lower peak concentration. These computational results are all in good qualitative agreement with previously recorded experimental results (Ballanyi et al. 1987), shown in Fig. 2a, b), for comparison. Here, the baseline K⁺ concentration in astrocytes in the blocking condition is higher and rises more slowly to a lower peak value after an external stimulus train compared to that in the control condition. The high baseline K⁺ concentration in astrocytes is due to the normal function of the Na⁺/K⁺-ATPase pump. Altogether, these results show that astrocyte Kir4.1 channels are prominently involved in extracellular K⁺ buffering during neuron firing and thereby have a significant effect on the neuron resting membrane potential to control firing during an external stimulus sequence.

Fig. 2.

a. Adapted experimental data of the K⁺ concentration in the extracellular space ([K⁺]_o) and astrocyte ([K⁺]_A) in Kir4.1 channel control and blocked states during and after a 10 s stimulus, interpolated from Fig. 7 in Ballanyi et al. (Ballanyi et al. 1987). b The simulation results of [K⁺]_o and [K⁺]_A corresponding to the experimental results using our model

We also examined the role of threshold value of g_kir in the generation of spontaneous epileptic seizures of neurons in the absence of an external stimulus in our model. Model simulation results suggested that spontaneous periodic epileptic activities can be induced when g_kir <  = 50 ps and without external stimulation input (Fig. S1b), but increasing g_kir beyond the threshold transforms the pathological periodic epileptic discharge pattern into normal spontaneously rapid firing, as illustrated in the left trace of Fig. S1a. These oscillations are remarkably similar to the experimental results reported by several investigators, for example, Figs. 1 and 6 in Ref. (Jensen and Yaari 1997), in which the authors use a high-potassium in vitro preparation, and Fig. 2 of Ref. (Jokubas et al. 2006). In addition, Fig. S1c and d exhibit the neuron action potential during action potential generation for g_kir equal to 380.0 ps (Fig. S1c) and 0.1 ps ((Fig. S1d), respectively. Note that the action potentials become shorter in duration and smaller in amplitude when g_kir decreases from 380.0 ps to 0.1 ps.

Hyperthermia induces febrile seizure activities

Temperature has multiple effects on cells in the central nervous system (CNS) (Moser et al., 1993; Andersen and Moser, 2010; Kim and Connors, 2012b). Numerous experiments have observed that small changes in temperature can seriously affect neuron firing activity (Jerison 1976; Kim and Connors 2012b). For example, with temperature increasing, the firing amplitude of neurons decreases, but firing frequency increases (Ye et al. 20108a). In addition, the neuron firing pattern switches from burst spiking to regular firing during an increase in temperature from room temperature (RT, 20 °C ~ 25 °C) to physiological temperature (PT, 37 °C) (Jack W, 2011). Specially, pathologically high temperatures can have dramatic effects on brain function and even lead to very serious clinical consequences. For instance, hyperthermia up to 40 °C can induce febrile seizure activities of hippocampal pyramidal neurons (Kim and Connors 2012a; Ye et al. 2018b). Here, we studied the effects of acute hyperthermia on neuron firing activities using a hippocampus astrocyte-neuron coupled model. Figure 3 presents numerical simulations showing that an increased temperature can induce spontaneous neuron firing pattern changes in the absence of external stimulus input. For example, neurons exhibit a slow periodic bursting firing state for the low temperature range [21 °C ~ 29 °C] (Fig. 3a). However, neurons switch to regular spiking pattern for temperature around 37.5 °C (Fig. 3b), which is consistent with experimental findings (Jack W 2011). A further increase temperature to the range of [37.6 °C ~ 39.8 °C] (Fig. 3c) drives neurons to fast burst pattern, while temperature above 40 °C can induce spontaneous epileptic firing patterns (Fig. 3d) similar to what was reported in hippocampal pyramidal neurons(Kim and Connors 2012a). This febrile seizure activity induced by hyperthermia (40 °C), which is characterized by a depolarization block phenomenon in pyramidal cells in rat hippocampal slices (Fig. 1d (Bikson et al. 2003)). In addition, the simulation results show the total frequencies of the neuron firing in 80 s (Fig. 3e) and the neuron firing frequencies in every periodic firing period (Fig. 3f) as the temperature increases. Specially, in Fig. 3f, because of neuron exhibits regular firing states in a temperature range (30 °C ~ 37 °C), the neuron firing frequencies in every periodic firing period is replaced by the total firing frequencies of 80 s. we can find that the total frequencies of the neuron firing in 80 s increase gradually, but the neuron firing frequencies in every periodic firing period decrease gradually in a temperature range (22 °C ~ 29 °C). this This inconsistency of discharge frequency is due to neuron discharges showed from sparse periodic bursting discharges to dense bursting discharges. Moreover, no matter which method is used to calculate the firing frequency of neurons, after 30 °C, the discharge frequency of neurons increases gradually with increasing temperature, and there is a particularly large jump in discharge frequency from 39 °C to 40 °C, as shown in Fig. 3f. The action potential demonstrates a large and prolonged afterhyperpolarization (AHP) for low temperature (e.g., T = 22 °C) and a smaller and shorter duration AHP for higher temperatures (e.g., T = 38 °C and 40 °C, respectively), as shown in Fig. 3h.

Fig. 3.

Spontaneous neuronal firing patterns as a function of temperature in the absence of external stimulus input. Time trains of the neural membrane potential V (mV) (black lines), astrocyte membrane potential V_A (mV) (red lines) and extracellular K⁺ concentration ([K⁺]_o) at temperatures (a), 37 °C (b), 38.0 °C (c) and 40 °C (d), respectively, found from the model equations presented in the model section. E and F. Neuron firing frequencies in the total 80 s and every periodic firing period as temperature increases. H. The action potential demonstrates a large and prolonged afterhyperpolarization (AHP) for low temperature (e.g., T = 22 °C and 37 °C, respectively) and a smaller and shorter duration AHP for higher temperatures (e.g., T = 38 and 40 °C, respectively)

In addition, Fig. 4 shows time series of neuron membrane potential and the corresponding Na⁺ current (I_Na) and K⁺ current (I_K) of neuron, Kir4.1 channel current (I_Kir) and extracellular potassium concentration ([K⁺]_o) at different temperature respectively( 22 °C (Fig. 4a), 37 °C (Fig. 4b), and 38 °C (Fig. 4c), 40 °C (Fig. 4d)). We found that the amplitude of neuronal action potentials will gradually decrease with temperature increasing. Moreover, Moreover, elevated temperature results in a marked decrease in the duration of action potentials as well as a significant decrease in excess K⁺ entry owing largely to the increased of the inactivation rate of Na⁺ channels (Fig. 4). In Fig. 4a, for a single neuron action potential at 22.0 °C, there is a overlap between the Na⁺ and Na⁺/K⁺ currents, which lead to a large offset effect in the total current. At 37.0 °C, the overlap of the Na⁺ and K⁺ currents is greatly reduced, and the half-width of the action potential also decreases. Interestingly, when temperature increases to 40.0 °C, the Na⁺ and K⁺ currents show very little overlap, which resulting in a very large I_c peak (similar to the time course of the I_K amplitude). Also, the maximum peak of dV/dt is similar to the time course of Na⁺ current at 40 °C. In short, when the temperature rises, the offset effect of the Na⁺ and K⁺ currents corresponding to a action potential gradually decreases, the half-width and energy consumption of the action potential are also reduced. In addition, it is noted that the amplitude of the Kir4.1 channel current reaches its maximum amplitude at the physiological temperature of 37 °C, and the current in Kir4.1 channel will decrease when the temperature continues to increase, especially there is a particularly large jump in discharge frequency from 37 to 40 °C, as shown in Fig. 4b, c and d). The results indicated that astrocytes Kir4.1 channel has the strongest ability to absorb extracellular potassium ions at physiological temperature (37 °C), while lower or higher temperature will weaken the uptake of extracellular potassium ions by the Kir4.1 channel. This conclusion also better explains the phenomenon that the lowest extracellular potassium concentration ([K⁺]_o) oscillation amplitude at 37.0 °C when the temperature increases in Fig. 4. From Fig. 4, we can also verify the time-lag mechanism of astrocyte buffering extracellular potassium that astrocytic Kir4.1 channel and neuron potassium channel open almost simultaneously, and then, the termination of neuronal potassium channel current corresponding to the maximum value of extracellular potassium concentration and astrocytic Kir4.1 channel amplitude.

Fig.4.

Neuron action potential and the corresponding Na⁺ and K⁺ currents of neuron, Kir4.1 channel current and [K⁺]_o during action potential generation for temperature at 22 °C (a), 37 °C (b), and 38 °C (c), 40 °C (d), respectively. Note that the overlap of Na⁺ and K⁺ currents during action potential generation is also reduced when temperature increases. The action potential demonstrates a large and prolonged afterhyperpolarization (AHP) for low temperature (e.g., T = 22 °C and 37 °C, respectively) and a smaller and shorter duration AHP for higher temperatures (e.g., T = 38 and 40 °C, respectively)

Moreover, Fig. 5 showed neuron action potential and the corresponding Na⁺ and K⁺ currents of neuron, Kir4.1 channel current and extracellular potassium concentration ([K⁺]_o) during action potential generation at 40 °C at three time points (the first action potential in a periodic epileptic activity (Fig. 5a), the single action potential when a periodic epileptic activity enters 2 s (Fig. 5b and the last action potential before ‘Depolarization block’ in a periodic epileptic activity (Fig. 5c), The amplitude of the action potential, the overlap of inward Na⁺ and outward K⁺ currents and the Kir4.1 current decrease gradually from the first action potential to the last action potential before ‘Depolarization block’ in a periodic epileptic activity, but neuronal firing frequency and the corresponding potassium concentrations increase significantly. In Fig. 5, at the beginning of neurons inter periodic epileptic firing after a period of resting state, we found that, the extracellular potassium concentration is not very high because neurons are just starting to enter the cluster discharge state. At this time, although high temperature (40 °C) reduced the enzyme activity of the Kir4.1 channel, extracellular potassium ions can be also absorbed normally in a very short period. As time goes on, high temperature further affects the enzyme activities of neuronal sodium and potassium channels, which significantly reduced the oscillations amplitude of neuronal membrane potential. More importantly, high temperature also greatly reduces the activity of Kir4.1 channel enzyme, leading to the failure of rapid clearance of excess extracellular potassium ions, which changes the reversal potential of neurons, and further causes neurons to produce higher frequency and lower amplitude neuron discharge. Eventually, the accumulation of extracellular potassium ions over a certain amplitude (17.28 mM), which leads to 'Depolarization block' epileptic discharges.

Fig. 5.

Neuron action potential and the corresponding Na⁺ and K⁺ currents of neuron, Kir4.1 channel current and [K⁺]_o during action potential generation for temperature at 40 °C during different period

Discussion

Increased temperature increases nervous system excitability and even induce febrile seizures. There have been many studies about the effects of temperature changes on neuronal action potentials (Petracchi et al. 1994; Yu et al. 2012), but the phenomenon that elevated temperature changes neuron firing patterns in neuron-astrocyte coupled system has not been well-studied. Astrocytic Kir4.1 channels play a crucial role in regulating neuronal discharges by maintaining extracellular K⁺ concentration balance (Jensen and Yaari 1997; Jokubas et al. 2006; Larsen and Macaulay 2014). This paper constructed a Hodgin-Huley-type model for voltage-gated astrocyte Kir4.1 channel current based on it’ s voltage gated characteristic published in previous studies (Biermans et al. 1987; Heukelom 1994; Ransom and Sontheimer 1995; Sakmann and Trube 1984; Tse et al. 1992), and the model were validated by comparison with published results (Ballanyi et al. 1987; Sibile et al. 2015; Jokubas et al. 2006). Moreover, this work verified the experimental observations that spontaneous febrile seizures can be induced directly by hyperthermia in the absence of stimuli with our neuron-astrocyte coupled model, and also revealed the critical role of Kir4.1 channel in the generation of febrile seizures.

Firstly. we presented a voltage-gated Kir4.1 channel current model based on previous studies(Heukelom 1994; Tse et al. 1992) and examined its validity. We found that the activation curves of Kir4.1 channel along with astrocyte voltage and extracellular potassium concentration are both incremental sigmoid functions. Note that the data points of the activation curves are consistent with some results of previous experiments (Jensen and Yaari 1997; Meeks and Mennerick 2010; Rutecki et al. 1985; Traynelis and Dingledine 1988). In addition, we performed a statistical analysis comparing the astrocyte Kir4.1 channel current dynamics obtained by our model to the simulation recordings in Ref. [Fig. 2 in (Sibille et al. 2015)] performed in individual astrocyte during single pulse stimulation. Considering the experimental accuracy and other environmental factors, our Kir4.1 channel current model presented in this work may well describe the actual Kir4.1 channel current in reality. Furthermore, we simulated the effects of the Kir4.1 channel dysfunction on the extracellular K⁺ concentration balance and the resultant neuronal discharges at a certain temperature (23 °C). We verified that extracellular potassium accumulation occurs when the Kir4.1 channel conductance is less than a certain threshold using our computational model. During the Kir4.1channel is blocked, there is a higher transient peak increase in the extracellular K⁺ concentration, but the astrocyte K⁺ concentration baseline is higher and rises more slowly to a lower peak concentration than in the control condition. Here, the high baseline K⁺ concentration in astrocytes is due to the normal function of the Na⁺/K⁺-ATPase pump. In addition, we further demonstrated epileptiform activities in vitro that downregulation of astrocytic Kir4.1 channels is closely related to spontaneous seizure activities (Ohno et al. 2014; Witthoft et al. 2013). This result reveals that the central idea of extracellular potassium accumulation regulating neuronal discharges is that increased extracellular K⁺ concentration enhances neuronal excitability, which in turn further increases extracellular K⁺ concentration in a positive feedback manner.

In addition, we first simulated the specific dynamic characteristics of hyperthermia inducing spontaneous epileptic activities in the absence of external stimulus input using an astrocyte-neuron coupled model that consists of extracellular potassium dynamics. Mammals and other warmed-blood animals have a warm body temperature of approximately 37 °C, unlike ectotherms, whose body temperature fluctuates with the environment (Jerison, 1976). Previous experiments have found that neuron firing patterns switch from burst spiking to regular firing during changes in temperature from room temperature to physiological temperature (Waters, 2011)(Jack W 2011). We not only verified these results using the proposed computational model but also found that there is an abundant transformation of neural firing patterns as temperature changes. For example, neurons present burst firing when the temperature parameter is in the parameter region [21 °C ~ 29 °C]; an increase in the temperature into the parameter region [29.1 °C ~ 37.5 °C] (Fig. 3(B)) causes neurons to switch from periodic burst firing to regular firing states. Increasing temperature to the parameter region [37.6 °C ~ 39.8 °C], the neuron firing pattern switches to burst spiking again from regular firing. Unlike physiological temperature, neurons exhibit more dense bursting discharge activities in the temperature range of [37.6 °C ~ 39.8 °C]. Interestingly, our simulation results also demonstrated the experimental observation that hyperthermia up to 40 °C can induce febrile seizure activities in hippocampal pyramidal neurons (Kim and Connors, 2012a, b). Notably, there was a particularly large jump in firing frequencies from 39 °C to 40 °C during the fast-wave period (Fig. 4). This phenomenon is probably because the temperature-sensitive ionic currents open, causing the neuron membrane potential to remain suprathreshold level while neurons begin seizure-like firing at 40 °C. In addition, we found that excess Na⁺ entry decreases significantly with increasing temperature, resulting in a marked reduction in the overlap of the inward Na⁺ and outward K⁺ currents and a shortening of action potential duration. Furthermore, we can come to a consistent conclusion that the changes of firing activity of neurons with slightly lower potassium in the bath solution where no seizures occur at room temperature but the cell begin to seize at higher temperatures. For example, Fig. S4 showed time trains of the neural membrane potential, astrocyte membrane potential and potassium concentration when extracellular K⁺ solution is 6.0 mM (A), 7.0 mM (B) and 8.0 mM (C), respectively. Moreover, this results also found a phenomenon that the lower potassium in the bath solution corresponds to sparse bursting discharges at the same temperature.

Furthermore, our simulation results better verified the physiological experiments that the key role of the Kir4.1 channel in normal discharges and febrile seizures. Such as, we found that the amplitude of astrocytic Kir4.1 channel current is the largest value at 37 °C (shown in Fig. 4). However, the amplitude of the Kir4.1 channel current would decrease as temperature continue to increasing, particularly, the Kir4.1 channel have the smallest amplitude in 40 °C. In addition, corresponding to the action potential, ions channels currents in neuron and astrocyte, decreases gradually with temperature rising from 37 °C to 40 °C. However, the oscillation amplitude of extracellular potassium concentration increase markedly with temperature rising from 37 °C to 40 °C, shown in Fig. 3 and Fig. 4. In Fig. 4, the oscillation amplitude of the extracellular potassium concentration in 22 °C is larger than 37 °C, this may be due to reduced Na⁺/K⁺-ATPase pump activity at low temperatures. Specially, in Fig. 5, form the beginning of neurons inter periodic epileptic firing after a period of resting state to the last action potential before the ‘Depolarization block’ at 40 °C, the ions channels in neuron and astrocyte decrease rapidly, the kir4.1 channel current not only decreases rapidly but also delays the removal of extracellular potassium ions. However, the concentration of extracellular potassium ions increases significantly, even exceeding the threshold that induces neuron inter the ‘Depolarization blocking’ period of epileptic discharge. In addition, we also test neuron discharges and ions currents in each channel of the neuron with temperature increasing in the absence of the kir4.1 channel (shown in Fig. S2 and Fig. S3). we only found a reduction in the overlap of the inward Na⁺ and outward K⁺ currents, a shortening of action potential duration, and an increase of discharge frequency as the temperature elevates. However, neurons do not show changes in discharge pattern and febrile seizures due to temperature rise. These results may explain a key role of astrocytic Kir4.1 channel in the induction of febrile seizures: elevated temperature causes an increase in the frequency of action potential of neuron, in the meantime, the corresponding frequency of astrocytic Kir4.1 channel current also increases but its amplitude decreases gradually, specially, at 40 °C, significantly reduced current amplitude of the Kir4.1 channel in a short time can not remove extracellular "excessive" potassium ions in time. This delayed clearance of potassium ions could lead to rapid accumulation of extracellular potassium ions, and then, febrile seizures can be induced when the concentration of extracellular potassium ions exceeds a certain threshold. Hence, we can also predict that febrile seizures is a high potassium -induced seizures at febrile temperature. This indicated that potassium continued release during neuronal “depolarization block” requires both open potassium channels and a driving force for potassium efflux. Considering the amplitude reduction of the Kir4.1 channel buffering extracellular potassium in 40 °C. Persistent inward potassium currents in astrocyte would still depolarize the neuron membrane by providing a driving force and helping maintain open potassium channels. These results predicted the possibility that reductions in febrile seizures activity might be resulted from the development of depolarization block, during which potassium release would continue.

Hyperthermia can induce febrile seizures in the central nervous system. This study aimed to provide an in-depth understanding of how temperature regulate febrile seizures by facilitating construction of more accurate dynamic models of neuron-astrocyte networks to improve recognition, forecasting and control of febrile seizures.

Supplementary Information

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Declarations

Conflict of interest

The authors declare no competing financial interests.