Dynamic Mechanism of Epilepsy Generation and Propagation After Ischemic Stroke

Epilepsy is the second largest neurological disease which seriously threatens human life and health. The one important reason of inducing epilepsy is ischemic stroke which causes insucient oxygen supply from blood vessels to neurons. However, few studies focus on the underlying mechanism of the generation and propagation of epilepsy after ischemic stroke by utilizing modeling methods. To explore the mechanism, this paper establishes a coupled network model consisting of neurons and astrocytes, and introduces a blood vessel to simulate the condition of ischemic stroke. First we study the effect of the degree of vascular blockage on the generation of epilepsy. The results demonstrate that the important reason of epilepsy after ischemic stroke is the disruption of ion concentration gradient. Then we study three factors that inuence the epileptic propagation after ischemic stroke: massive glutamate release, excessive receptor activation and high extracellular potassium concentration. The results demonstrate that massive glutamate acting on postsynaptic neurons and the excessive activation of glutamate receptors on postsynaptic neurons promote the epileptic propagation in neuronal population, and massive glutamate acting on astrocytes and excessive activation of metabotropic glutamate receptors on presynaptic neurons inhibit the epileptic propagation, and the potassium uptake by astrocytes suppresses the epileptic propagation. The results are consistent with the experimental phenomena. The simulation results also shed light on the fact that astrocytes have neuroprotective effect. Our results on the generation and propagation of epilepsy after ischemic stroke could offer theoretical guidelines for the treatment of epilepsy after ischemic stroke. to {G}_{glia}=6 and 10, respectively. From the gure 4 (a), we observe the peak value and range of {\left[{K}^{+}\right]}_{o} drops with the increase {G}_{glia}, and the ring state from epileptic-like ring (small gure (1) in gure 4 (a), {G}_{glia}=6) to normal ring (small gure (2) in gure 4 (a), {G}_{glia}=10). The results show that {\left[{K}^{+}\right]}_{o} decreases gradually as the increase of the strength of astrocyte potassium uptake, resulting in the reduction of neuronal excitability and transition of neuron ring state. Figure 4 (b) shows that the neuron ring frequency in single epileptic-like ring event decreases with the increase of {G}_{glia}, which indicates neuronal excitability decreasing as potassium uptake of astrocyte enhanced. results demonstrate that astrocyte could adjusts neuronal excitability and inuences epilepsy dynamics in neuron by taking extracellular potassium which is consistent with the experiment results that astrocyte neuronal epilepsy extracellular vessel coupled network model, we mainly investigated four factors which inuence the generation and propagation of epilepsy after ischemic stroke: the blockage degree of vascular, massive glutamate release, excessive receptor activation and high extracellular potassium concentration. Our ndings revealed the potential mechanism of generation and propagation of epilepsy after ischemic stroke, and highlighted the neuroprotective effect of astrocytes. It also provides a deeper understanding of the important role of blood oxygen in neuronal ring activities. It’s helpful to formulate new strategies for the treatment of epilepsy after ischemic stroke.


Introduction
Epilepsy is a common neurological disease characterized by aberrant ring of neurons [1,2]. Stroke is an important cause of epilepsy. It is reported that epilepsy patients induced by stroke accounts for about 30-49% of newly diagnosed cases of epilepsy in the aged >65 [3,4], meanwhile epilepsy after stroke further affects the treatment effect of stroke [5,6]. Generally, there are two kinds of strokes: ischemic stroke and hemorrhagic stroke, the former accounts for 71% of all [7], and 9% of epilepsy patients after stroke are induced by ischemic stroke [8].
Ischemic stroke is mainly caused by the blockage of artery, which decreases blood ow and cerebral metabolic rate for oxygen [9]. Oxygen has essential role in supporting neural ring activity and maintaining ion environment in neuronal system of brain [10,11]. Metabolic dysfunction induced by ischemic stroke triggers imbalance of ion concentration inside and outside neurons, which is an important cause of epilepsy [12,13]. Meanwhile, several physiological experiments found that more causes are responsible for the epilepsy after ischemic stroke, including ion channel dysfunction [14], excessive release of neurotransmitters such as glutamate [15], blood-brain barrier destruction [16], and alterations in gene expression [17]. Many physiological studies focused on the probable causes of epilepsy after ischemic stroke but few aimed at the underlying pathogenesis through mathematical modeling methods, especially simulates blood oxygen metabolism in the state of ischemic stroke.
The research of epileptic propagation has been a hotspot. Trevelyan et al. studied the propagation speed of epilepsy by zero magnesium animal model of epilepsy, suggesting that the feedforward inhibition is the prime factor affecting epileptic propagation speed [18]. Khoo et al. proved the generation of epilepsy needs more energy than the one for the propagation process by EEG-fMRI testing method [19]. Reato et al. proposed a network model including of neurons and astrocytes to consider the in uence of astrocyte feedback and pulse simulation on the epileptic propagation and found that the balance between excitatory and inhibitory affects the epileptic propagation speed [20]. Martinet et al. studied the effect of extracellular potassium concentration diffusing on the epileptic propagation by developing mean-eld model and reproduced the temporal and spatial dynamics of epilepsy [21]. Proix et al. explained the spatiotemporal dynamics of epileptic generation, propagation and termination by constructing a neural eld model and veri ed the conclusion by analyzing the data recording from epileptic patients [22]. But for the moment, no computational model focus on the impact of metabolic dysfunction caused by ischemic stroke on the propagation of epilepsy, especially considering the feedback and regulation of astrocytes.
Astrocytes are the most abundant glial cells which play a fundamental role in maintaining the normal function of the cerebral neural system. Astrocytes can regulate synaptic activity by imposing excitability or inhibitory feedback on neurons after stimulated by neurotransmitter released by neurons [23][24][25], and regulate the ion concentration in inside and outside neurons, such as buffering the extracellular K + concentration by a few ion channels [26]. It's also found that astrocytes play a main role in uptaking extracellular neurotransmitter such as glutamate [27][28][29]. Based on these characteristics, astrocytes exert positive physiological effects on neurons after ischemic stroke and have a very important neuroprotective effect [30][31][32]. The above conclusions prove astrocytes can be considered as a new therapeutic target for epilepsy after ischemic stroke. So it's very important to study the effect of astrocytes on epilepsy after ischemic stroke.
In this work, we rst built a minimal network model composed of pyramidal neuron, astrocyte, and introduced an updated blood vessel model that simulates the ischemic stroke state. Based on the minimal network model, we studied the underlying mechanism of inducing epilepsy after ischemic stroke.
In this part, we mainly studied the effects of vascular blockage and the regulation of extracellular K + through astrocyte on neuron ring activity. The numerical results are consistent with the experimental observations [33,34] and veri ed the correctness of our model. Then, a coupled population model was developed by connecting 150 pyramidal neurons, 150 astrocytes and a blood vessel. We studied the impact of metabolic dysfunction caused by ischemic stroke on the propagation of epilepsy. Based on experiment results, the effect of the blockage degree of vascular, the massive release of glutamate, the excessive activation of receptors and the high extracellular concentration of potassium on the propagation of epilepsy after ischemic stroke were considered separately. Finally, the potential mechanism of generation and propagation of epilepsy after ischemic stroke were discussed by analyzing the numerical simulation results.

Model And Methods
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Membrane potential dynamics
The pyramidal neuron is described by the modi ed Hodgkin-Huxley model [35][36][37][38]. The membrane potential V of neuron are as follows [37]: CV = I ext − I Na − I K − I Cl I Na = G Na m 3 h V − E Na + G Nal (V − E Na ) where I ext represents external input current, I Na and I K are sodium and potassium ion currents respectively and containing their respective leakage currents, and I Cl is chloride leakage current. G Na and G K represent the channel conductance of Na + and K + . G Nal , G Kl and G ClL represent the conductance of the three leak currents. m and n describe the sodium and potassium activation degree, h describes the sodium inactivation, and all vary between 0 and 1. α q and β q (q = m, n, h) describe the opening and closing probability of ion channels, these parameters are described as follows [39,40]:

Ion concentration dynamics
In the model, ions concentration are changed follow neuronal activity. K + o is in uenced by K + currents (I K ), Na + -K + pumps (I pump from neuron, I gliapump from glia), diffusion of K + from interstitial volume (I diff ), and uptake by glial cell (I glia ) [35,37], the speci c form of K + o is as follows: where β is the ratio of the intracellular volume to the outer volume. We supposed the quantity of Na + into the cell is equal to K + out of the cell, so K + i can be described as: Na + i is determined by Na + currents (I Na ) and Na + -K + pumps, and Na + o are updated based on Na + i , the speci c form of [Na + ] are as follows: where ρ is Na + -K + pump rate which is updated based on extracellular oxygen concentration ( O 2 o ).
The ability of glial cell handling K + o is modeled by: where G glia is the glial uptake rate of K + o . The diffusion of K + from interstitial volume (I diff ) is modeled by: where ϵ k is K + diffusion constant, and K + ves represents [K + ] in blood vessel.

Oxygen concentration dynamics
In the work, we introduced and modi ed the blood vessel model [41], so that it can simulate blood oxygen metabolism in the state of ischemic stroke. The speci c form of the blood vessel model are as follows: where BFlow represents blood ow in blood vessel, η describes the blockage degree of the blood vessel ( η=1 when the blood vessel in healthy state and η=0 when the blood vessel is completely concluded), and BFlow ves represents maximum blood ow in healthy blood vessel (BFlow ves =1). OEF is oxygen extraction factor, OEF ave is normal oxygen extraction factor, OEF var represents the maximum possible variation of OEF, and τ OEF is an adaptation time that OEF adapt to BFlow changes. O 2 ves represents the concentration of oxygen supplied to the neuron by blood vessel, Figure 1 describes the oxygen concentration O 2 ves varies with the blockage degree of the blood vessel η, and O ini represents the normal oxygen concentration in healthy blood vessel. The extracellular oxygen where α is a conversion factor from changing current to oxygen concentration [37]. ϵ o is diffusion rate of oxygen.

Coupled population model
To study the propagation of epilepsy, we built a coupled population model based on the simple network model and consider the astrocyte feedback, including 150 pyramidal neurons, 150 astrocytes and a blood vessel. In the model, pyramidal neurons are connected by synapses, and astrocytes are connected by gap junctions, pyramidal neurons and astrocyte cells are connected with the bidirectional feedback, and equal concentration of oxygen are transported to every neurons and astrocytes by blood vessel. The schematic of model is shown in Figure 2. In this model, the membrane potential V of neuron is changed as follows: where I syn represents synaptic current, g se describes the activation level of glutamate receptors on the postsynaptic neuron, and V e is reversal potential of synapse. The synaptic input s from previous pyramidal neuron is modeled by: In coupled population model, K + o also is in uenced by the K + diffusion from neighboring neurons, the speci c form of the lateral diffusion term is as follows: where D k is the potassium diffusion coe cient, and Δx is the distance between two neurons. So the nal manifestation of K + o is as follows: The binding neurotransmitters to receptors on adjacent astrocytes will cause the astrocytes to produce IP 3 , and eventually cause the concentration of Ca 2 + increasing in astrocytes. In order to describe this process, we introduced the improved Li-Rinzel model [42][43][44] as follows: The bidirectional feedback between astrocytes and neighbor neurons is achieved by "tripartite synapse" which consist of pre-and postsynaptic neurons and astrocyte [45][46][47]. It's reported that gliotransmitters inhibit the release of neurotransmitters when gliotransmitters act on presynaptic neurons and inhibit synaptic activity [48,49]. So in the model, the astrocyte feedback is inhibitory.

Methods
We integrated the model numerically using Euler method with a xed time step of 0.08 ms, the total calculation time was 1500s. Considering the simulation accuracy, the rst 160s was abandoned. And the values of various parameters used in the model are shown in Table 1. It's convinced that abnormal ion concentration of intra-and extra-cellular space caused by ischemic stroke, especially potassium ion, affecting the excitation of neurons and even induces epilepsy [50,51]. Therefore, we rst study the effect of blood vessel blockage on {\left[{K}^{+}\right]}_{o} to investigate the mechanism of epilepsy after ischemic stroke in the minimal network model. The result is shown in Figure  3.  (Figure 3 (b)). In \eta = 0.36, insu cient blood oxygen leads to inhibition of the activity of the sodium-potassium pump [13,37], making it unable to fast enough to adjust the concentration gradient of intracellular and extracellular ions, causing {\left[{K}^{+}\right]}_{o} uctuates in large region (shown in Figure 3 (a) and (c)), and further induces the neuron epileptic-like ring (shown in Figure 3 (c)), the epileptic-like ring form is similar to the epileptic ring recorded in the experiment [33], the region of {\left[{K}^{+}\right]}_{o} oscillations also is similar to experimental results [52]. At \eta = 0.2, the sodium-potassium pump strength is too weak to support neuron ring, leading to the neuron to fall into rest state and so does {\left[{K}^{+}\right]}_{o}, the details are shown in Figure 3  These results suggest that blood oxygen is crucial to neuron activity, and insu cient blood oxygen induced by ischemic stroke can cause epilepsy. The results verify the experimental observation that hypoxia in a certain concentration range can induce epilepsy [33].

The effect of {G}_{glia} on epilepsy dynamics
Besides of the sodium-potassium pump adjusting, astrocyte potassium uptake is also very important for extracellular potassium homeostasis [26,36,53,54]. In this section, we investigate the effect of the astrocyte potassium uptake on neuron ring activity used the minimal network model.  Figure 4 (b) shows that the neuron ring frequency in single epileptic-like ring event decreases with the increase of {G}_{glia}, which indicates neuronal excitability decreasing as potassium uptake of astrocyte enhanced.
The results demonstrate that astrocyte could adjusts neuronal excitability and in uences epilepsy dynamics in neuron by taking extracellular potassium which is consistent with the experiment results that astrocyte affects neuronal epilepsy by regulating extracellular potassium [34].
The comparison between the above numerical simulation results and related experimental results proves the correctness of the model and simulation results.

The effect of \eta on the generation and propagation of epilepsy
In the 3.1 section, we used the minimal neuron-astrocyte-blood vessel network model to study the impact of \eta and {G}_{glia} on epilepsy dynamics in single neuron. In the following research, we will study the generation and development of the epilepsy after ischemic stroke at network level. Firstly, we investigate the effect of \eta on the generation of epilepsy in neuronal population with synaptic conductance {g}_{se}= 0.145 and the strength of the astrocyte feedback{\lambda }= 0 (without the astrocyte feedback), and we select the time period from 480s to 1280s for more intuitive research. Figure 5 (a), (c) and (e) show the time series of neuronal network ring for \eta = 0.5, 0.36, and 0.2, respectively, and the 50th neuron is selected to show the ring state as Figure 5 (b), (d) and (f). As seen in Figure 5 (a), when \eta = 0.5, neuronal population re normally because of su cient blood oxygen supply, and the ring state of the 50th neuron is given in Figure 5 (b). But with the decrease of \eta, the ring state changes to epileptic-like ring ( Figure 5 (c), \eta = 0.36) and the epilepsy spread to the tail of the neuronal population because the limited blood oxygen supply weakens the ability of synaptic transmission, and the detail of epileptic-like ring is shown in Figure 5 (d). When \eta is further reduced, neuronal population go to rest state ( Figure 5 (e), \eta = 0.2) because the blood oxygen too limited to maintain neuronal ring, and the detail of rest state is shown in Figure 5 (f).
The results con rm that ischemic stroke can trigger the generation and propagation of epilepsy. In the following research the value of \eta is set to 0.36 to simulate the condition of ischemic stroke.

The effect of abnormal neurotransmitter release on epileptic propagation
Studies have shown that ischemic stroke leads to the release of a large amount of excitatory neurotransmitter glutamates, which signi cantly in uences the excitability of neurons [55,56]. For simulating the blood blockage caused by ischemic stroke, the value of \eta is set to 0.36 and the increase of the synaptic input intensity to postsynaptic neurons {p}_{neu} and the synaptic input intensity to astrocytes {p}_{as} are used to simulate the effect of the massive release of glutamates on neurons and astrocytes, respectively.
The effect of {p}_{neu} on epileptic propagation are shown in Figure 6 with {p}_{as} = 1. From Figure 6 (a), we observe the synaptic current {I}_{syn} (the averaged synaptic current at epileptic ring times) received by each neuron in neuronal population gradually increases as {p}_{neu} increases from 1.00 to 1.24 (Figure 6 (a)), the reason is the increase of excitatory neurotransmitter glutamates in synapse enhances the excitability of neurons. The increase of synaptic current leads to a gradual raise in the network activation of the neuronal population (network activation refers to the proportion of the number of ring neurons to all neurons in neuronal population, which is averaged value of the network activation at epileptic ring times) (Figure 6 (b)), and eventually cause more neurons are recruited to epileptic-like ring (Figure 6 (c)) and the propagation speed of epilepsy also is enhanced (Figure 6 (d)), the region of the propagation speed is compatible with experimental evidence [57].  These results con rm that the massive release of glutamate caused by ischemic stroke strengthens the distance and speed of epileptic propagation. Figure 7 shows Astrocytes suppress synaptic activity by imposing inhibitory feedback on presynaptic neurons after stimulated by neurotransmitter released by neurons [48,49]. Utilizing this feedback, astrocytes can more strongly participate in the regulation of synaptic transmission under condition of an increase of glutamate caused by ischemic stroke. As seen in Figure 7 (a), the astrocyte feedback current {I}_{as} received by each neuron in neuronal population gradually increases as {p}_{as} increases from 1.00 to 1.20. And the increase of astrocyte feedback current suppresses the activity of presynaptic neuron, resulting in a gradual reduction in the release of glutamate in the synapse (Figure 7 (b)). The reduction in releasing glutamate means the synaptic activity is suppressed, which ultimately leads to the number of neurons with epileptic-like ring is reduced (Figure 7 (c)), and with slower the propagation speed of epilepsy (Figure 7 (d)).
The results demonstrate astrocytes restrain epileptic propagation by inhibiting synaptic transmission during ischemic stroke, and play an important neuroprotective role.
In the above two section, we separately studied the effect of increased glutamate acting on neurons and astrocytes on epilepsy propagation. In this section, we simultaneously consider the effect of increased glutamate acting on neurons and astrocytes on epilepsy propagation in neuronal population under the competition between the two in uences. The results are shown in Figure 8. Figure 8  The results in the section indicate that the massive release of glutamate caused by ischemic stroke promotes the propagation of epilepsy by enhancing the synaptic current, while the feedback effect of astrocytes inhibit synaptic transmission and thus suppress the propagation of epilepsy, which proves the neuroprotective effect of astrocytes, but this effect is limited and cannot prevent the generation and propagation of epilepsy.  It's reported that astrocyte potassium uptake plays a very important role in removing extracellular potassium [26,53,54], and this effect becomes more pronounced during ischemic stroke process. The increase in {\left[{K}^{+}\right]}_{o} caused by ischemic stroke stimulates the increase in potassium uptake capacity of astrocytes, thereby inhibiting neuronal excitability and reducing neuronal damage [63]. For simulating the blood blockage caused by ischemic stroke, the value of the blockage degree of the blood vessel \eta is set to 0.36, and {G}_{glia} is used to depict the potassium uptake capacity of astrocytes.  Figure 11 (b). In Figure 11 (Figure 11 (e)) and the speed of epileptic propagation is reduced (Figure 11 (f)).
The results con rm that extracellular potassium concentration signi cantly in uences the excitability of neurons and astrocytes play a very important role in removing extracellular potassium. Due to the scavenging effect of extracellular potassium by astrocytes, the epileptic propagation is effectively weakened.

Conclusion
Epilepsy after ischemic stroke seriously affects people's health and quality of life, so that it's very important to study its pathogenesis and propagation in the neuronal network. Few work investigated the pathogenesis and propagation of epilepsy after ischemic stroke by utilizing modeling methods. In this work, we constructed a minimal network model consisting of neuron, astrocyte, and a vessel model. By utilizing the minimal model, we investigated the underlying mechanism of epilepsy after ischemic stroke. And based on the minimal model, we constructed a coupled network model including neuronal population, astrocytes population and a blood vessel to study the propagation characteristics of epilepsy after ischemic stroke, and considered the feedback and regulation of astrocytes.
First, we studied the effects of the blockage degree of the blood vessel and the potassium uptake ability of astrocyte on the generation of epilepsy after ischemic stroke in minimal network model. The results showed that ischemic stroke can trigger the generation of epilepsy, and the disruption of ion concentration gradient is an important reason. The speci c reason is that the insu cient blood oxygen induced by ischemic stroke cannot meet the demands of sodium-potassium pump to restore the ion concentration gradient, resulting in the imbalance of ion concentration inside and outside the neuron.
Secondly, we studied the effect of the massive release of glutamate caused by ischemic stroke on epileptic propagation. The results showed that the massive release of glutamate acting on different cell leads to opposite effect, and highlighted the neuroprotective effect of astrocytes. Excessive glutamates acting on post-synaptic neurons could promote the propagation of epilepsy in neuronal population by increasing the synaptic transmission. But on the contrary, excessive glutamates act on astrocytes will inhibit epileptic propagation by inhibiting neurotransmitter transmission. And under the competition between the two effects, the massive release of glutamate still strength epileptic propagation. The result veri ed that ischemic stroke can induce epilepsy in real state.
Thirdly, we studied the in uence of excessive activation of glutamate receptors in postsynaptic neurons and metabotropic glutamate receptors in presynaptic neurons caused by ischemic stroke on the propagation of epilepsy after ischemic stroke. The results showed that the former enhances the propagation of epilepsy after ischemic stroke by strengthening synaptic transmission, and the latter showed the opposite effect by inhibiting the excitability of presynaptic neurons through astrocyte feedback, which illustrates the neuroprotective effect of astrocytes.
Finally, by analyzing the effect of the potassium uptake of astrocytes on epileptic propagation, we revealed the abnormal increase of extracellular potassium concentration not only induces epilepsy but also promotes its propagation, and the high capacity of potassium uptake of astrocytes can weaken epileptic propagation.
In this study, by constructing neuron-astrocyte-blood vessel coupled network model, we mainly investigated four factors which in uence the generation and propagation of epilepsy after ischemic stroke: the blockage degree of vascular, massive glutamate release, excessive receptor activation and high extracellular potassium concentration. Our ndings revealed the potential mechanism of generation and propagation of epilepsy after ischemic stroke, and highlighted the neuroprotective effect of astrocytes. It also provides a deeper understanding of the important role of blood oxygen in neuronal ring activities. It's helpful to formulate new strategies for the treatment of epilepsy after ischemic stroke. Tables   Table 1 is              Please see the Manuscript le for the complete gure caption.

Figure 10
Please see the Manuscript le for the complete gure caption.

Figure 11
Please see the Manuscript le for the complete gure caption.