Hindawi Neural Plasticity Volume 2020, Article ID 8864246, 13 pages https://doi.org/10.1155/2020/8864246
Research Article Dynamic Transitions in Neuronal Network Firing Sustained by Abnormal Astrocyte Feedback
Yangyang Yu ,1 Zhixuan Yuan ,1 Yongchen Fan ,1 Jiajia Li ,2 and Ying Wu 1
1State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an 710049, China 2School of Information & Control Engineering, Xi’an University of Architecture & Technology, 710055, China
Correspondence should be addressed to Jiajia Li; [email protected] and Ying Wu; [email protected]
Received 21 August 2020; Revised 24 October 2020; Accepted 28 October 2020; Published 23 November 2020
Academic Editor: Rubin Wang
Copyright © 2020 Yangyang Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Astrocytes play a crucial role in neuronal firing activity. Their abnormal state may lead to the pathological transition of neuronal firing patterns and even induce seizures. However, there is still little evidence explaining how the astrocyte network modulates seizures caused by structural abnormalities, such as gliosis. To explore the role of gliosis of the astrocyte network in epileptic seizures, we first established a direct astrocyte feedback neuronal network model on the basis of the hippocampal CA3 neuron- astrocyte model to simulate the condition of gliosis when astrocyte processes swell and the feedback to neurons increases in an abnormal state. We analyzed the firing pattern transitions of the neuronal network when astrocyte feedback starts to change via increases in both astrocyte feedback intensity and the connection probability of astrocytes to neurons in the network. The results show that as the connection probability and astrocyte feedback intensity increase, neuronal firing transforms from a nonepileptic synchronous firing state to an asynchronous firing state, and when astrocyte feedback starts to become abnormal, seizure-like firing becomes more severe and synchronized; meanwhile, the synchronization area continues to expand and eventually transforms into long-term seizure-like synchronous firing. Therefore, our results prove that astrocyte feedback can regulate the firing of the neuronal network, and when the astrocyte network develops gliosis, there will be an increase in the induction rate of epileptic seizures.
1. Introduction taining the functional stability of the CNS [11, 12]. Function- ally, astrocytes can regulate chemicals in the extracellular Epilepsy is one of the most common neurological diseases, space [13] and maintain the steady state of ion concentra- affecting nearly 70 million people worldwide, and it is charac- tions in the extracellular space [14–17]. And astrocytes can terized by the aberrant synchronous firing of neurons [1–3]. respond to the stimulation of neuronal activity and release It is generally believed that the reason for the aberrant syn- glial transmitters to regulate neuronal firing, and the concept chronous firing of neurons is the imbalance between synaptic of “tripartite synapse” is proposed to describe the bidirec- excitability and inhibition [4–6], which is caused by changes tional communication between astrocytes and neurons [18]. in the structure and function of neurons themselves, includ- At present, many physiological experiments have shown that ing changes in neurotransmitters and mutations in receptors, when astrocytes have abnormal functioning, this will lead to ion channels, and ion transporters, and alterations in net- seizure-like events such as the aberrant synchronous firing of work topology [7]. However, decades of studies have found neurons [10, 17, 19, 20], and through mathematical modeling that the abnormal feedback effect of astrocytes on neurons methods, many researchers have studied the possibility of has a critical effect on the balance of neuronal excitability dysfunctional astrocytes participating in neuronal epilepsy and inhibition and can even cause epilepsy [8–10]. [21–28]. For instance, Amiri et al. studied the effect of Ca2+ Astrocytes are the most important type of glial cells in the oscillations in astrocyte clusters on the firing of neuronal central nervous system (CNS) and play a crucial role in main- populations by constructing a “tripartite synapse” network 2 Neural Plasticity model in the hippocampus, suggesting that astrocytes can neuronal firing. Finally, by analyzing the numerical simula- affect the firing activity of neurons [29, 30]. Fan et al. pro- tion results, the effect of the feedback mechanism of astro- posed a computational mathematical model that can be used cytes on the transition of the firing pattern of the neuronal to study synchronized seizure behavior [21]. Du et al. studied network was summarized, and the possible mechanism of the effect of the K+ concentration on neuronal firing in epi- seizures was discussed. lepsy and the different energy requirements of neurons in normal and epileptic states by constructing a coupling model 2. Model and Methods of astrocytes and neurons in the hippocampus [26]. Li and Rinzel proposed an improved coupling model of astrocytes 2.1. Coupling Network Model of Neurons and Astrocytes. In and neurons, revealing that seizure-like firing occurs when this paper, we constructed a neural network consisting of astrocytes degrade glutamate abnormally [27]. However, pyramidal neurons and interneurons, and each neuron is fi few studies have investigated the effect of astrocyte structural described by the modi ed Morris-Lecar model [43]. The PV IN abnormalities on neuronal epilepsy by modeling methods. membrane potentials of vi and vi for pyramidal neurons In fact, the structure of astrocytes plays an important role and interneurons are as follows [30]: in the CNS and can provide structural support and an energy À ÀÁÀÁ PV PY PY PY supply for neurons [31, 32]. In recent years, physiological C v_ ðÞt = I ðÞt − g m∞ v ðÞt v ðÞt − v m i i CaÀÁi i ÀÁCa experiments have shown that an abnormal astrocyte struc- PYðÞ PY ðÞ− PYðÞ− Þ + gK wi t vi t vK + gL vi t vL , ture can also cause seizure-like events [33, 34]. One typical ÀÁ PV PY example of this is gliosis, which is an important factor caus- w∞ v ðÞt − w ðÞt w_ PVðÞt = ∅ i ÀÁi , ing abnormal astrocyte structure, leading to morphological i τ PVðÞ w vi t and physiological variations in astrocytes after CNS injury À ÀÁÀÁ [35]. Gliosis can cause abnormal feedback from astrocytes _ INðÞ INðÞ− INðÞ INðÞ− Cmvi t = Ii t gCam∞ vi t vi t vCa to neurons and induce epilepsy. Indeed, gliosis has been ÀÁÀÁ INðÞ INðÞ− INðÞ− Þ + gK wi t vi t vK + gL vi t vL found to be a hallmark of epilepsy [7, 33, 34]. Therefore, we ÀÁ introduced a new model based on the actual coupling struc- IN IN w∞ v ðÞt − w ðÞt ture of neuron-astrocytes to investigate the effect of gliosis w_ INðÞt = ∅ i ÀÁi i τ INðÞ on epilepsy. w vi t In the actual brain, a single astrocyte can be connected to ð1Þ numerous neurons and up to 2,000,000 synapses [36, 37], ff and a neuron can be surrounded by di erent astrocytes where vPV and vIN represent the membrane potential of the i – i i [38], thereby forming a complex coupling network [39 41]. th pyramidal neuron and the ith interneuron, respectively; Based on Amiri’s model, we proposed a feedback model from w_ PV and w_ IN are the restoration variables and represent the astrocytes to neurons, which is constructed according to the i i ratio of the number of open potassium channels to the num- actual coupling structure of astrocytes and neurons on the ber of excitatory pyramidal neurons; g , g , and g repre- basis of the hippocampal CA3 [38–42]. Compared with Ca K L sent the channel conductance of the Ca2+,K+, and leak Amiri’s functional model and most of the coupling model current, respectively, which play an important role in form- of neuron-astrocytes, our model more accurately emulates ing the membrane potential; v and v are the Nernst the physiological structure in terms of topology and spatial Ca K potentials of Ca2+ and K+, respectively; v is the reversal distribution and can simulate the structural changes of the L potential of the neuronal leak channels; ∅ is the temperature astrocyte population, which has profound physiological sig- parameter, which is constant; and m∞ðvÞ, w∞ðvÞ, and τ ðvÞ nificance. We used 50 pyramidal neurons and 50 interneu- w PVð Þ INð Þ rons to establish a neuronal network with synaptic (v = vi t or vi t ) describe the role of voltage-dependent connections. An astrocyte network was formed by 50 astro- ion channels in the membrane potential and are given by cytes connected by gap junctions; finally, based on the “tri- − partite synapse” model, neurons and astrocytes were 1 v v1 m∞ðÞv = 1 + tan h , connected to form a coupling network. 2 v 2 In this work, we mainly studied the possible mechanism − 1 v v3 of epilepsy under simulated gliosis conditions by developing w∞ðÞv = 1 + tan h , ð2Þ a mass network model between neurons and astrocytes with 2 v4 an updated feedback model from astrocytes to neurons. The 1 ff τ ðÞv = : e ect of excitatory conductance, the connection probability w cos hvðÞ− v /2v between astrocytes and neurons, and the astrocyte feedback 1 4 intensity on the synchronization of pyramidal neuron popu- PYð Þ INð Þ fi Ii t and Ii t act on pyramidal neurons and interneu- lations were considered separately, and the ring activity of ff the pyramidal neuronal population was analyzed. Then, we rons, respectively, which are a ected by external constant x ð Þ x ð Þ introduced the concept of energy consumption to character- input currents Iconst,i t , system noise currents Inoise,i t , x ð Þ ize the transition of neuronal firing patterns. The study was slowly varying currents Islow,i t , synaptic currents from adja- x ð Þ further expanded by adding electrical synapses to the model cent pyramidal neurons and interneurons Isyn,i t , and feed- ff x ð Þ to verify the robust e ect of neuronal astrocyte feedback on back currents from adjacent astrocytes Ias,i t (x = PY,IN). Neural Plasticity 3
fi PY ð Þ INð Þ The speci c forms of Ii t and Ii t are as follows: follows: hi PYðÞ PY ðÞ PY ðÞ PY ðÞ PY ðÞ PY ðÞ 2+̇ ð Þ Ii t = Iconst,i t + Inoise,i t + Islow,i t + Isyn,i t + Ias,i t , Ca = Jchan + Jleak + Jpump, 7 IINðÞt = IIN ðÞt + IIN ðÞt + IIN ðÞt + IIN ðÞt + IIN ðÞt , i const,i noise,i slow,i syn,i as,i ÂÃðÞIP∗ − ½IP ÀÁ ÀÁ _ 3 3 ½PY ½IN ð Þ _x ∗ x x IP3 = + rip T +T , 8 I ðÞt = ε v − v ðÞt − αI ðÞt , ðÞx = PY, IN : τ 3 slow,i i slow,i ip3 ð3Þ _ α ðÞ− − β ð Þ q = q 1 q qq, 9 where ε and α are the variables that control the bursting where ∗ behavior of neurons, and v is a factor driving the generation ÀÁÂÃÂÃ of bursting. − 3 3 3 2+ − 2+ Jchan = c1V1p∞n∞q Ca ER Ca , In our model, neurons are connected to each other ÀÁÂÃÂÃ − 2+ − 2+ through chemical synapses, wherein pyramidal neurons are Jleak = c1V2 Ca Ca , ER ð10Þ excitatory neurons with unidirectionally connected excit- ÂÃ V Ca2+ 2 atory synapses. Interneurons are inhibitory neurons, and J = − ÂÃ3 , pump 2+ 2 2 they are connected by inhibitory synapses. Pyramidal neu- Ca + k3 rons stimulate the activity of interneurons, and the interneu- rons inhibit the activity of pyramidal neurons, which where combine to form a bidirectional connection. This whole pro- ½ cess is mainly achieved through neurotransmitter transmis- IP3 p∞ = , sion, which is dependent on the membrane potential of ½IP + d each neuron. The concentration of neurotransmitter in the ÂÃ3 1 synaptic cleft that is released from the ith neuron (presynap- Ca2+ n∞ = ÂÃ , tic neuron) is described as follows: 2+ Ca + d5 ½IP + d ð11Þ ½x 1 ðÞð Þ α = a d 3 1 , T i−1 = x , x = PY, IN , 4 q 2 2 ½ 1 + exp ðÞ−ðÞv − θ /σ IP3 + d3 i−1 s s ÂÃ β = a Ca2+ , θ σ q 2 where s is the half-activation voltage and s is the steepness ÂÃ of the sigmoid function. The synaptic currents are modulated ÂÃ c − Ca2+ Ca2+ = 0 , by synaptic variables gx(x = PY, IN), and the equation is as ER i c1 follows [44]: ½ ∗ where IP3 is the concentration of IP3 in astrocytes, IP3 is the _ x x x x g ðÞt = α ½T − ðÞ1 − g ðÞt − β g ðÞt , ðÞx = PY,IN , ð5Þ reversal concentration of IP , τ is the relaxation time con- i s i 1 i s i 3 ip3 stant, r refers to the rate of the increase in IP , ½Ca2+ is ip3 3 _ xð Þ α 2+ where gi t refers to the open level of neuroreceptors and s the concentration of Ca in the cytosol of astrocytes, and q and β are the rate constants that determine the increase and s refers to the proportion of activated IP3 receptors. Jchan, _ xð Þ fi fl decrease in gi t , respectively. Consequently, the speci c Jleak, and Jpump represent the calcium ux from the channel, form of the synaptic currents is as follows [30]: the leakage, and the pump, respectively, and V1, V2, and V3 ÀÁ represent the flux rate of the corresponding calcium flux. PY ðÞ PY ðÞ PY ðÞ− ½ 2+ 2+ Isyn,i t = gsegi−1 t vi t vse Ca ER is the concentration of Ca in the ER of astrocytes. ÀÁÀÁ 2+ + g gIN ðÞt + gINðÞt vPY ðÞt − v , With the action of neurotransmitters, Ca in astrocytes si i−1 i i si oscillates and causes the release of gliotransmitters into syn- ÀÁ ÀÁ IN PY IN IN IN apses to regulate neuronal activity [15, 47]. According to I ðÞt = g g ðÞt v ðÞt − v + g g − ðÞt v ðÞt − v , syn,i se i i se si i 1 i si the work of Volman and colleagues [46], we used a kinetic ð6Þ variable f to describe the astrocyte-neuron interaction, which has the following form: where gse and gsi are the conductance of the excitatory syn- −f ÀÁÂÃ apses and inhibitory synapses, respectively, and vse and vsi _ ðÞ− κΦ 2+ − ½: ð Þ f = τ +1 f Ca Ca th 12 are the excitatory and inhibitory equilibrium potentials, Ca2+ respectively. After neurotransmitters are released into the synaptic In recent years, numerous morphological studies have cleft, some neurotransmitters bind to receptors on adjacent shown that astrocytes form a network via gap junctions [48, 2+ astrocytes, causing Ca oscillations in astrocytes. To 49]. Studies have shown that IP3 is the main messenger that describe the dynamics of this process, we used the improved diffuses throughout the astrocyte network through gap junc- Li-Rinzel model [30, 45, 46]. The mathematical forms are as tions [29, 50], so we used a simplified model to describe gap 4 Neural Plasticity junctions: PY PY PY PY ÀÁ ½ ½− ½ ð Þ JG,i = kg IP3 i+1 +IP3 i−1 2IP3 i , 13
ffi IN IN IN IN where kg is a coupling coe cient representing the coupling coefficient of gap junctions; the coupling model will ulti- mately be added to equation (8). According to the previous description, the astrocyte net- work wraps around neurons to form a complex network. To describe this physiological structure, we proposed a coupling As As As As model of neurons and astrocytes. The specific form is shown in Figure 1. In this model, each neuron receives feedback from the entire astrocyte network; according to numerous Figure 1: Coupling network model of neurons and astrocytes. PY, physiological experimental observations, the feedback effect IN, and As represent the pyramidal neurons, interneurons, and fi of astrocytes inhibits pyramidal neuron excitability [51] but astrocytes, respectively. Neurons are connected by synapses; lled enhances interneuron excitability [52]. The equation for circles represent excitatory synapses, empty circles represent inhibitory synapses, and red rectangles represent the collective describing the feedback effect is as follows: effect of pyramidal neurons and interneurons on astrocytes. Astrocytes are connected by gap junctions; the inhibitory feedback 50 PY ðÞ −γ 〠 from astrocytes to each pyramidal neuron is represented by Ias,i t = 1 P1 f ij, hollow squares, and the excitatory feedback from astrocytes to j=1 ð14Þ each interneuron is represented by solid squares. 50 IN ðÞ γ 〠 Ias,i t = 2 P2 f ij, In this paper, we mainly analyzed the numerical results of j=1 pyramidal neurons. We used the sliding time window method to calculate the average correlation coefficient of all γ γ where 1 and 2 represent the feedback intensity from astro- pairs of pyramidal neurons in each time window and finally cytes to pyramidal neurons and interneurons, respectively. ffi ρ ⋯ averaged the correlation coe cient l, where l =1,2, , L P1 and P2 are the probability of an astrocyte successfully con- (T/δ = L, δ is the interval of the individual time window) of necting with a pyramidal neuron and an interneuron in the all time windows, to obtain the final correlation coefficient network, respectively, representing the degree of tightness ρ. To solve the model equations, the Runge–Kutta method of the connection. f ij represents the interaction of the ith with a fixed time step of 0.01 ms was used. Considering that neuron and the jth astrocyte. To simplify the calculation, all normal firing resumes after a period of time after the connections in the same population have the same probabil- observed neuronal epileptic firing, we set the total time inter- γ fi ity. 1 and P1 are xed variables. val T of the simulation to 25 s. The interval of the individual Without special instructions, the various parameter time window δ was 0.25 s, and 100 time windows were used values used in our simulation are shown in Table 1. in total. 2.2. Methods. Epilepsy is characterized by synchronous 3. Numerical Results and Discussion seizures in neurons, and we used the synchronization of the fi ff abnormal ring of neurons as an indicator to measure 3.1. The E ect of gse on Neuronal Network Synchronization. seizures. To quantify the indicator, we used a cross- In this section, we studied the effect of the conductance of ffi correlational coe cient measurement method based on the the excitatory synapses gse on the synchronization of the method that was used to measure the degree of synchronous pyramidal neuron population with P1 = P2 =0. The result is firing between pairs of neurons [53]. We used MðkÞ and NðkÞ shown in Figure 2. to represent the spike trains of neuronal pairs, where k =1, Figure 2(a) shows that the curve of the correlation coeffi- 2, ⋯,S(T/S = τ; T is the total time interval, and τ is the time cient ρ of the pyramidal neuronal population increases with step). MðkÞ equaled 1 if the neuron fired at the k moment; th the increase in gse. We know that an increase in gse indicates otherwise, MðkÞ equaled 0, which was the same as NðkÞ. an increase in the concentration of neurotransmitters The specific equation is as follows [54]: released by presynaptic neurons, which will affect the firing of postsynaptic neurons, and the connection between neu- ∑S ðÞ ðÞ ρ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik=1MkNk : ð Þ rons can be strengthened. Figures 2(b) and 2(c) provide the ij = 15 time series of the firing of the pyramidal neuronal population ∑S ðÞ∑S ðÞ k=1Mk i=1Nk for gse =0and gse =2, respectively. The results show that the firing state of the pyramidal neuronal population changes The firing threshold is set to -0.1, because it can be seen from an asynchronous to a synchronous state with the fi fi from Figures 2(d) and 2(e) that -0.1 can not only calibrate increase in gse, and we nd that there is a delay in the ring the peak action potential moment of normal firing but also of the neuronal population, which arises from the neuronal avoid the interference of subthreshold oscillation. network being connected by chemical synapses. Finally, to Neural Plasticity 5
Table 1: Parameters used in the model. to the stability of the system, normal synchronous firing quickly resumes. Figures 3(d)–3(f) show that the firing of Parameter Value Parameter Value Parameter Value the neuronal population changes from a nonepileptic syn- ∗ -1 Cm 1.0 v -0.22 V3 0.9 μMs chronous firing state with slight depolarization block firing fi g 1.0 α 0.001 k 0.1 μM (Figure 3 (d)) to asynchronous ring with slight local ca 3 fi ρ σ μ seizure-like ring (Figure 3(e)), and then, continues to vca 1.0 s 0.02 d1 0.13 M fi increase until P2 =1.Theseizure-like ring is more severe α μ gk 2.0 s 0.1 d5 0.082 M and synchronized; eventually, the firing of the pyramidal neu- β μ -1 fi vk -0.7 s 0.05 a2 0.2 Ms ronal population changes from asynchronous ring (Figure 3 (e)) to seizure-like synchronous firing (Figure 3(f)). The yel- g 0.5 g 0.1 d 1.05 μM l si 2 low strip area in Figure 3(f) indicates that the neuronal popu- μ vl -0.5 vse -0.85 d3 0.94 M lation is in a state of seizure-like synchronous firing and then fi v1 -0.01 vsi 0 c0 2 μM resumes normal ring later, and experiments have shown that ∗ μ τ epilepsy firing reflects the synchronous firing of the neuronal v2 0.15 IP3 0.16 M Ca2+ 6s τ population [4, 58]. v 0.1 ip 7s ½Ca 0.2 3 3 th Figure 3(g) shows the 25th pyramidal neuron from r μ -1 κ -1 Figure 3(f). We can observe two phenomena from v4 0.145 ip3 7.2 Ms 0.5 s Figure 3(g), which will be analyzed in Figure 4. The first phe- ∅ c k 1.15 1 0.185 g 0.1 nomenon is the spreading depression in the area of box 2, θ -1 γ s 0.2 V1 6s 1 0.05 which corresponds to box 1 in Figure 3(f). Studies have ε -1 shown that spreading depression is closely related to epilepsy 0.0005 V2 0.11 s P1 0.8 [50, 58–60]. The second phenomenon is the depolarization block in the area of box 3, which is one of the typical charac- study the variations in the synchronization of neurons in teristics of epileptic seizures [3]. detail, we show the firing time histories of the 25th and To investigate the structural aberrations of the astrocyte 26th neurons in Figures 2(d) and 2(e). The results show that network during epileptic gliosis, which is different from the the firing of the two neurons changes from asynchronous to regular neighboring connection model used in previous stud- “ ” synchronous with increasing gse. ies [29, 30], we used an all-to-all connection to examine the These results suggest that the synchronization of the gliosis effect on neuronal firing shown in Figure 3, where both pyramidal neuronal population increases with increasing the structural (connection probability, Figures 3(a) and 3(b)) conductance gse. and functional variations (feedback intensity, Figure 3(c)) reflect epileptic gliosis. The simulation results support the 3.2. The Effect of P on Neuronal Network Synchronization. experimental observations that the presence of gliosis in the Epilepsy is characterized by the aberrant synchronous firing astrocyte network accelerates epileptic seizures [30]. of neurons. To explore the reasons for the aberrant synchro- Because of gliosis, reactive astrocytes become hypertro- nous firing of neurons, we established a new model of the phic, and then, the processes overlap until hyperplasia pro- coupling network of neurons and astrocytes that more closely duces astroglial scars [43, 61, 62]. The structural emulates the actual physiology. Because experiments have connections between astrocytes and neurons become closer shown that the increased excitability of interneurons is benefi- and tighter, corresponding to the increase in the connection cial in enhancing the inhibition of the nervous system and probability P in the model. In this process, reactive astrocytes suppressing seizures [55–57], we studied the impact of release substances such as immunomodulators, neurotrophic changes in the connection probability of P1 with P2 =0.8, P2 factors, and growth factors to modulate the excitability of γ with P1 =0.8, and the astrocyte feedback intensity 2 from neurons. For example, cytokines (a kind of immunomodula- astrocytes to interneurons on pyramidal neuronal population tor) released by reactive astrocytes act on neurons, which will fi fi synchronous ring with gse =2. Only the phase of signi cant lead to increases in postsynaptic AMPA receptors and gluta- seizures was selected to analyze the change in synchronization, mate; this causes hyperexcitability of the neuronal network and the phase was located approximately within the time and seizures [7], corresponding to the increase in the feed- interval from 8 s to 18 s. The results are shown in Figure 3. back intensity γ in the model. These results indicate that with First, we studied the law of the firing synchronization of the emergence and development of gliosis, the connection the neuronal population based on the connection probability probability P and feedback intensity γ of astrocytes toward γ – 2 2 P1, P2 and the feedback intensity 2. Figures 3(a) 3(c) show interneurons increase, resulting in an abnormal increase in that the correlation coefficient ρ of the pyramidal neuronal feedback from astrocytes; because astrocytes are connected population decreases first and then increases with the con- to each other through gap junctions, the feedback effect of γ ρ nection probability P1, P2 and feedback intensity 2, and the astrocytes on each neuron tends to be the same, resulting : : γ : fi is minimal when P1 =02, P2 =035,or 2 =008. Second, in the enhancement of the synchronization ring of the neu- fl fi we studied in detail the in uence of changes in P2 on the r- ronal population and the continued expansion of the area of ing transition of pyramidal neurons. Figure 3(d) shows the seizure-like synchronous firing. In other words, astrocytes fi γ abundant ring of the neuronal population. There is slight are in gliosis, which leads to gradual increases in P2 and 2; depolarization block firing in the early stage, and then due this causes astrocyte feedback to become abnormal, which 6 Neural Plasticity
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12 13 14 12 13 14 Time (s) Time (s) 25th pyramidal neuron 25th pyramidal neuron 26th pyramidal neuron 26th pyramidal neuron (d) (e) Figure ffi ρ 2: The synchronization states of the pyramidal neuronal population change with gse. (a) The correlation coe cient of the pyramidal fi neuronal population changes with gse. (b) The time series of the ring of the pyramidal neuronal population at gse =0. (c) The time series of fi fi the ring of the pyramidal neuronal population at gse =2. (d) The time series of the 25th and 26th pyramidal neurons ring at gse =0. (e) The fi – time series of the 25th and 26th pyramidal neurons ring at gse =2. The bottom of (b) (e) shows a partially enlarged view of the corresponding figure.
PY leads to the development of seizure-like activity. The results Figure 4(b), at the initial stage, the growth rate of Ias is are consistent with the observations in clinical trials that PY greater than that of Islow, which then reverses. Since astrocyte when the astrocyte network develops gliosis after severe epi- feedback has an inhibitory effect on pyramidal neurons, and leptic seizures in a population of epileptic patients, seizures the slow-varying current has an excitatory effect, the compe- will be further induced and aggravated [33, 63]. tition between the two effects causes the total stimulation 2+ Moreover, Ca plays a vital role in the feedback from current IPY to first become negative and then to recover astrocytes to neurons, and studies in recent years have shown 2+ (shown in the bottom of Figure 4(b)), so that the neuron that Ca signals are closely related to epilepsy activity [59, appears to repolarize first and then depolarize. 64, 65]. In this work, we studied the relationship between var- Then, the self-feedback process of the slow-varying cur- 2+ iations in the calcium concentration [Ca ] and neuronal rent IPY leads to epileptic firing in neurons, which is process fi slow epileptic ring by examining the 25th pyramidal neuron 2. In process 3, the neuron is affected by a depolarizing block. when the connection probability P2 =1 (Figure 3(g)) as an Figure 4(a) shows that the calcium concentration is lower than fi example. In the models of neuronal ring, the collective cur- 0.2 after 50.6 seconds, causing astrocytes to stop releasing glial PY PY rents of Islow and Ias and the total neuronal external current transmitters, so that IPY and IPY decrease rapidly. The com- PY fi as slow I were introduced to study the current-sensitive ring in bined effect of the currents causes IPY to be abnormal, leading fi view of neuronal ring bifurcation versus current in previous to a depolarization block. These phenomena indicate that the PY PY 2+ dynamical studies [54, 66]. I is mainly regulated by Islow transition in neuronal firing activity is regulated by Ca , PY 2+ and Ias . To observe the regulation of Ca completely, we which proves that abnormal astrocytes can cause neurons to extended the total time to 70 s. The result is shown in Figure 4. exhibit epileptic firing and other abnormal behaviors. Figure 4(a) shows the abundance of firing behaviors. We The above results indicate that the connection probability divided these behaviors into three main phenomena. The P and the feedback intensity γ play important roles in the first phenomenon is spreading depression after high- regulation of neuronal population synchronous firing activity frequency firing. The main cause of this is that after 2.7 sec- and reveal the role of gliosis in the astrocyte network in the onds, the calcium concentration is higher than 0.2 (shown occurrence and development of epileptic seizures. in Figure 4(a)), which causes astrocytes to release glial trans- mitters into nearby synapses [46] and the astrocyte feedback 3.3. The Effect of P on Neuronal Network Energy PY current Ias to rise rapidly and to stimulate the slow-variation Consumption. Neuronal epilepsy firing consumes much PY current Islow to also rise; however, as shown at the top of energy [67]. During the past few years, many researchers 8 Neural Plasticity
0.50 0.6 0.5 0.45 0.5 0.4 0.40 0.4 0.35 0.3 � �
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–0.6 V –0.5
30 25th 3 –0.8 40 –1
Pyramidal neurons –1 50 –1.2 –1.5 5101520 25 015 01520 25 Time (s) Time (s) (f) (g) Figure γ fi 3: The impact of changes in P and 2 on the synchronous ring of the pyramidal neuronal population. (a) The correlation ffi ρ ffi ρ coe cient of the pyramidal neuronal population changes with P1. (b) The correlation coe cient of the pyramidal neuronal ffi ρ γ population changes with P2. (c) The correlation coe cient of the pyramidal neuronal population changes with 2. (d) The time fi fi : series of pyramidal neuronal population ring at P2 =0. (e) The time series of pyramidal neuronal population ring at P2 =035. (f) The fi : fi time series of pyramidal neuronal population ring at P2 =1 (g) The time series of the ring of the 25th pyramidal neuron is shown in (f). have studied this phenomenon [68–70] and proposed many neous power value of neuronal energy consumption. We methods to calculate the energy consumption of neurons mainly studied pyramidal neurons, so only the energy con- [71–74]. To describe this feature, we used the energy con- sumption formulas of the pyramidal neuron population are sumption formula based on M-L neurons [27, 75]: listed. In this section, we studied the pyramidal neuronal popu-
Ð lation energy consumption variations with the change in the T ′ðÞ 0 H t dt connection probability P . The results are shown in Figure 5.
2
I 1 (50.6,1.6)
] 0.6 –0.3
2+ (2.7,0.25) 25th
3 PY 0 V [Ca 2 0.5 PY 0.2 (2.7,0.2) (50.6,0.2) –1.1 I 0 1 –0.5 10 25 40 55 70 0 10203040506070 Time (s) Time (s)
[Ca2+] PY I slow VPY PY 25th I as (a) (b)
Figure 4: Calcium modulates neuronal epileptic firing. (a) The comparison diagram of the Ca2+ concentration and neuronal firing. (b) The PY PY PY top of (a) shows the time series of Islow and Ias , and the bottom of (a) shows the time series of I .
VPY 0 10 –0.2
20 –0.4 –0.6 0.45 30 –0.8 0.40
Pyramidal neurons 40 –1 0.35 50 –1.2 0.30
5101520 25 H >
Time (s) < 0.25 (a) H′ 0.20 0 0.15 10 0.35 0.10 20 0.0 0.2 0.4 0.6 0.8 1.0 –0.5 P H′ 2 30 (c) 40 –1
50 5101520 25 Time (s) (b) Figure fi 5: Variation in the energy consumption of the pyramidal neuronal population with the connection probability P2. (a) The ring : pattern of the pyramidal neuronal population at P2 =1 (b) The instantaneous power pattern of the energy consumption of the pyramidal : neuronal population at P2 =1 (c) The average energy consumption
0.30 0.4 0.25 0.3 0.20 �
� 0.2 0.15
0.10 0.1
0.05 0.0 0.000 0.005 0.010 0.015 0.020 0.0 0.2 0.4 0.6 0.8 1.0 P D 2 (a) (b)
VPY
0.1 10 0 20
30 –0.1
Pyramidal neurons 40 –0.2
50 5101520 25 Time (s) (c)
VPY 0 –0.2 10 –0.4 20 –0.6 30 –0.8 –1
Pyramidal neurons 40 –1.2 50 5101520 25 Time (s) (d)
VPY 0 –0.2 10 –0.4 20 –0.6 30 –0.8
Pyramidal neurons –1 40 –1.2 50 5101520 25 Time (s) (e) Figure fi 6: The impact of changes in D and P2 on the synchronous ring of the pyramidal neuronal population. (a) The change in the correlation coefficient ρ of the pyramidal neuronal population with the change in coupling strength D. (b) The change in the correlation ffi ρ coe cient of the pyramidal neuronal population with the change in connection probability P2. (c) The time series of pyramidal fi fi : neuronal population ring at P2 =0. (d) The time series of pyramidal neuronal population ring at P2 =035. (e) The time series of fi : pyramidal neuronal population ring at P2 =1 Neural Plasticity 11 postsynaptic neurons more sensitive to changes in the firing by electrical synapses according to the connection probability fi fi fl of presynaptic neurons, resulting in a change in the ring P2, and the results further con rmed the in uence of astro- of the pyramidal neuronal population from asynchronous cyte feedback on neuronal firing activity and the universality to synchronous; relative to that of chemical synapses, the and stability of the feedback model. Therefore, the results of synchronous firing shown in Figure 6(c) has no delay, which this study demonstrate the ability of the astrocyte population is consistent with the true characteristics of the electrical syn- to regulate the firing of neurons and the key role of astrocyte apse and is more conducive to the firing synchronization of network gliosis in neuronal seizures. In summary, our results the neuronal network. reveal a potential mechanism of seizure firing and provide a Figure 6(b) shows that the correlation coefficient ρ of the new direction for the treatment of brain diseases such as pyramidal neuronal population decreases first, correspond- epilepsy. ing to the process shown in Figure 6(c) and Figure 6(d), fi and then increases until P2 =1; the ring of the neuronal pop- Data Availability ulation changes from asynchronous firing (Figure 6(d)) to seizure-like synchronous firing (Figure 6(e)). The data used to support the findings of this study are avail- The above results indicate that the coupling intensity D able from the corresponding author upon request. and the connection probability P2 play vital roles in the reg- ulation of neuronal population synchronous firing activity, Conflicts of Interest similar to those of the above chemical synapse study. The fl model is suitable for all synapse types (two types), and the The author declare that they have no con icts of interest. results do not change with the connection method, proving the universality and stability of the feedback model. Acknowledgments This work was supported by the National Natural Science 4. Conclusion Foundation of China (Grant Nos.11772242, 11972275, and 12002251) and China Postdoctoral Science Foundation An abnormal astrocyte structure may cause epilepsy, but few (Grant No. 2018M631140). works have investigated the effects of astrocyte structural abnormalities such as gliosis on epilepsy through modeling methods. In this work, we used a new model to study the References ff fi e ects of astrocyte feedback on neuronal population ring [1] H. Yu, L. Zhu, L. Cai et al., “Variation of functional brain con- and the generation and development of epilepsy in gliosis. nectivity in epileptic seizures: an EEG analysis with cross- In the current research on the coupling model of neuron- frequency phase synchronization,” Cognitive Neurodynamics, astrocytes, most astrocytes are functional, and there is no vol. 14, no. 1, pp. 35–49, 2020. comprehensive consideration of the actual spatial structure [2] R. D. Thijs, R. Surges, T. J. O'Brien, and J. W. Sander, “Epilepsy and relative distribution of astrocytes and neurons. In this in adults,” The Lancet, vol. 393, no. 10172, pp. 689–701, 2019. work, we constructed a feedback model, which is very similar [3] M. Bikson, P. J. Hahn, J. E. Fox, and J. G. R. Jefferys, “Depolar- to the physiological structure, to describe the feedback cou- ization block of neurons during maintenance of electrographic pling between astrocytes and neurons based on physiological seizures,” Journal of Neurophysiology, vol. 90, no. 4, pp. 2402– and anatomical features. 2408, 2003. “ ‐ We showed that the increase in the conductance gse of [4] X. Lu, M. Yang, Y. Yang, and X. F. Wang, Atlastin 1 modu- the excitatory synapses strengthens the connection between lates seizure activity and neuronal excitability,” CNS Neurosci- pyramidal neurons, causing each neuron to be activated by ence & Therapeutics, vol. 26, no. 3, pp. 385–393, 2020. the previous neuron and the firing state to become synchro- [5] J. Ziburkus, J. R. Cressman, E. Barreto, and S. J. Schiff, “Inter- nized. More importantly, to study the transition of the firing neuron and pyramidal cell interplay during in vitro seizure- like events,” Journal of Neurophysiology, vol. 95, no. 6, state of the pyramidal neuronal population connected by – chemical synapses, we changed the connection probability pp. 3948 3954, 2006. γ [6] B. Christian and F. Marco, “Plasticity of hippocampal P and astrocyte feedback intensity 2, and the result showed fi excitatory-inhibitory balance: missing the synaptic control in that the participation of astrocytes in neuronal ring activi- ” fi the epileptic brain, Neural Plasticity, vol. 2016, Article ID ties will lead to the transition of the neuron ring state from 8607038, 13 pages, 2016. a synchronous state with slight depolarization block firing to fi fi [7] D. C. Patel, B. P. Tewari, L. Chaunsali, and H. Sontheimer, asynchronous ring with slight local seizure-like ring, when “Neuron–glia interactions in the pathophysiology of epilepsy,” astrocytes are in gliosis and gradually become more severe Nature Reviews Neuroscience, vol. 20, no. 5, pp. 282–297, 2019. with an increase in P2, causing astrocyte feedback to become [8] J. De Keyser, J. P. Mostert, and M. W. Koch, “Dysfunctional abnormal and leading to the transition of neurons from asyn- astrocytes as key players in the pathogenesis of central nervous chronous firing to seizure-like synchronous firing. Addition- system disorders,” Journal of the Neurological Sciences, ally, the analysis of the firing activity of the neuron vol. 267, no. 1–2, pp. 3–16, 2008. population showed that the transition of neuronal firing [9] J. Jefferys, “Basic mechanisms of focal epilepsies,” Experimen- activity is regulated by Ca2+. Then, we analyzed the energy tal Physiology, vol. 75, no. 2, pp. 127–162, 1990. consumption of pyramidal neurons connected by chemical [10] G.-F. Tian, H. Azmi, T. Takano et al., “An astrocytic basis of synapses and the firing state of pyramidal neurons connected epilepsy,” Nature Medicine, vol. 11, no. 9, pp. 973–981, 2005. 12 Neural Plasticity
[11] C. Agulhon, J. Petravicz, A. B. McMullen et al., “What is the work,” Science China Technological Sciences, vol. 60, no. 7, role of astrocyte calcium in neurophysiology?,” Neuron, pp. 1011–1018, 2017. vol. 59, no. 6, pp. 932–946, 2008. [29] M. Amiri, F. Bahrami, and M. Janahmadi, “Functional contri- [12] B. G. Eric and P. Yuriy, “Modulation of central synapses by butions of astrocytes in synchronization of a neuronal network astrocyte-released ATP and postsynaptic P2X receptors,” Neu- model,” Journal of Theoretical Biology, vol. 292, pp. 60–70, ral Plasticity, vol. 2017, Article ID 9454275, 11 pages, 2017. 2012. [13] M. F. Bear, B. W. Connors, and M. A. Paradiso, Neuroscience: [30] M. Amiri, N. Hosseinmardi, F. Bahrami, and M. Janahmadi, Exploring the Brain, Lippincott Williams & Wilkins, Balti- “Astrocyte-neuron interaction as a mechanism responsible more, Md, 2001. for generation of neural synchrony: a study based on modeling ” [14] P. Bezzi, G. Carmignoto, L. Pasti et al., “Prostaglandins stimu- and experiments, Journal of Computational Neuroscience, – late calcium-dependent glutamate release in astrocytes,” vol. 34, no. 3, pp. 489 504, 2013. Nature, vol. 391, no. 6664, pp. 281–285, 1998. [31] S. Liddelow and B. Barres, “SnapShot: astrocytes in health and ” – [15] E. A. Newman, “New roles for astrocytes: regulation of synap- disease, Cell, vol. 162, no. 5, pp. 1170 1170.e1, 2015. tic transmission,” Trends in Neurosciences, vol. 26, no. 10, [32] A. Volterra, N. Liaudet, and I. Savtchouk, “Astrocyte Ca2+ sig- pp. 536–542, 2003. nalling: an unexpected complexity,” Nature Reviews Neurosci- – [16] C. Henneberger, “Does rapid and physiological astrocyte- ence, vol. 15, no. 5, pp. 327 335, 2014. neuron signalling amplify epileptic activity?,” The Journal of [33] S. Robel, et al.S. C. Buckingham, J. L. Boni et al., “Reactive Physiology, vol. 595, no. 6, pp. 1917–1927, 2017. astrogliosis causes the development of spontaneous seizures,” – [17] H. R. Parri, T. M. Gould, and V. Crunelli, “Spontaneous astro- Journal of Neuroscience, vol. 35, no. 8, pp. 3330 3345, 2015. cytic Ca2+ oscillations in situ drive NMDAR-mediated neuro- [34] N. A. Oberheim, G. F. Tian, X. Han et al., “Loss of astrocytic nal excitation,” Nature Neuroscience, vol. 4, no. 8, pp. 803–812, domain organization in the epileptic brain,” Journal of Neuro- 2001. science, vol. 28, no. 13, pp. 3264–3276, 2008. [18] A. Araque, V. Parpura, R. P. Sanzgiri, and P. G. Haydon, “Tri- [35] S. Robel, “Astroglial scarring and seizures,” The Neuroscientist, partite synapses: glia, the unacknowledged partner,” Trends in vol. 23, no. 2, pp. 152–168, 2017. Neurosciences, vol. 22, no. 5, pp. 208–215, 1999. [36] N. A. Oberheim, X. Wang, S. Goldman, and M. Nedergaard, [19] M. Gómez-Gonzalo, G. Losi, A. Chiavegato et al., “An excit- “Astrocytic complexity distinguishes the human brain,” atory loop with astrocytes contributes to drive neurons to sei- Trends in Neurosciences, vol. 29, no. 10, pp. 547–553, 2006. zure threshold,” PLoS Biology, vol. 8, no. 4, 2010. [37] M. R. Freeman, “Specification and morphogenesis of astro- ” – [20] N. Kang, J. Xu, Q. Xu, M. Nedergaard, and J. Kang, “Astrocytic cytes, Science, vol. 330, no. 6005, pp. 774 778, 2010. glutamate release-induced transient depolarization and epilep- [38] M. M. Halassa, T. Fellin, H. Takano, J.-H. Dong, and P. G. tiform discharges in hippocampal CA1 pyramidal neurons,” Haydon, “Synaptic islands defined by the territory of a single Journal of Neurophysiology, vol. 94, no. 6, pp. 4121–4130, 2005. astrocyte,” Journal of Neuroscience., vol. 27, no. 24, pp. 6473– [21] D. Fan, Q. Wang, and M. Perc, “Disinhibition-induced transi- 6477, 2007. tions between absence and tonic-clonic epileptic seizures,” Sci- [39] C. Giaume, A. Koulakoff, L. Roux, D. Holcman, and entific Reports, vol. 5, no. 1, 2015. N. Rouach, “Astroglial networks: a step further in neuroglial ” [22] J. Li, R. Wang, M. du, J. Tang, and Y. Wu, “ Dynamic transition and gliovascular interactions, Nature Reviews Neuroscience, – on the seizure-like neuronal activity by astrocytic calcium vol. 11, no. 2, pp. 87 99, 2010. channel block,” Chaos, Solitons & Fractals, vol. 91, pp. 702– [40] A. Covelo and A. Araque, “Lateral regulation of synaptic trans- 708, 2016. mission by astrocytes,” Neuroscience, vol. 323, pp. 62–66, 2016. [23] M.-M. Du, J.-J. Li, Z.-X. Yuan, Y.-C. Fan, and Y. Wu, “Astro- [41] A. Verkhratsky and M. Nedergaard, “Physiology of astroglia,” cyte and ions metabolism during epileptogenesis: a review Physiological Reviews, vol. 98, no. 1, pp. 239–389, 2018. for modeling studies,” Chinese Physics B, vol. 29, no. 3, article [42] D. Rossi, “Astrocyte physiopathology: at the crossroads of 038701, 2020. intercellular networking, inflammation and cell death,” Prog- [24] B. Hu, D. Wang, Z. Xia et al., “Regulation and control roles of ress in Neurobiology, vol. 130, pp. 86–120, 2015. the basal ganglia in the development of absence epileptiform [43] C. Morris and H. Lecar, “Voltage oscillations in the barnacle activities,” Cognitive Neurodynamics, vol. 14, no. 1, pp. 137– giant muscle fiber,” Biophysical journal, vol. 35, no. 1, 154, 2020. pp. 193–213, 1981. [25] Z.-X. Yuan, P.-H. Feng, M.-M. Du, and Y. Wu, “Dynamical [44] D. Terman, J. E. Rubin, A. C. Yew, and C. J. Wilson, “Activity response of a neuron–astrocyte coupling system under electro- patterns in a model for the subthalamopallidal network of the magnetic induction and external stimulation,” Chinese Physics basal ganglia,” Journal of Neuroscience, vol. 22, no. 7, pp. 2963– B, vol. 29, no. 3, 2020. 2976, 2002. [26] M. Du, J. Li, R. Wang, and Y. Wu, “The influence of potassium [45] Y. X. Li and J. Rinzel, “Equations for InsP3 receptor-mediated concentration on epileptic seizures in a coupled neuronal [Ca2+]i oscillations derived from a detailed kinetic model: a model in the hippocampus,” Cognitive Neurodynamics, Hodgkin-Huxley like formalism,” Journal of Theoretical Biol- vol. 10, no. 5, pp. 405–414, 2016. ogy, vol. 166, no. 4, pp. 461–473, 1994. [27] J. Li, J. Tang, J. Ma, M. Du, R. Wang, and Y. Wu, “Dynamic [46] V. Volman, E. Ben-Jacob, and H. Levine, “The astrocyte as a transition of neuronal firing induced by abnormal astrocytic gatekeeper of synaptic information transfer,” Neural Compu- glutamate oscillation,” Scientific Reports, vol. 6, no. 1, 2016. tation, vol. 19, no. 2, pp. 303–326, 2007. [28] J. Tang, J. Zhang, J. Ma, G. Y. Zhang, and X. Q. Yang, “Astro- [47] M. M. Halassa, T. Fellin, and P. G. Haydon, “Tripartite synap- cyte calcium wave induces seizure-like behavior in neuron net- ses: roles for astrocytic purines in the control of synaptic Neural Plasticity 13
physiology and behavior,” Neuropharmacology, vol. 57, no. 4, [63] R. S. Briellmann, R. M. Kalnins, S. F. Berkovic, and G. D. Jack- pp. 343–346, 2009. son, “Hippocampal pathology in refractory temporal lobe epi- [48] A. Wallraff, R. Köhling, U. Heinemann, M. Theis, K. Willecke, lepsy,” Neurology, vol. 58, no. 2, pp. 265–271, 2002. and C. Steinhäuser, “The impact of astrocytic gap junctional [64] O. K. Steinlein, “Calcium signaling and epilepsy,” Cell and tis- coupling on potassium buffering in the hippocampus,” Journal sue research, vol. 357, no. 2, pp. 385–393, 2014. – of Neuroscience, vol. 26, no. 20, pp. 5438 5447, 2006. [65] K. Heuser, C. G. Nome, K. H. Pettersen et al., “Ca2+ signals in [49] H. Chai, B. Diaz-Castro, E. Shigetomi et al., “Neural circuit- astrocytes facilitate spread of epileptiform activity,” Cerebral specialized astrocytes: transcriptomic, proteomic, morpholog- Cortex, vol. 28, no. 11, pp. 4036–4048, 2018. ” – ical, and functional evidence, Neuron, vol. 95, no. 3, pp. 531 [66] S. Nadkarni and P. Jung, “Spontaneous oscillations of dressed 549.e9, 2017. neurons: a new mechanism for epilepsy?,” Physical Review Let- [50] G. ULLAH, P. JUNG, and A. CORNELLBELL, “Anti-phase ters, vol. 91, no. 26, 2003. calcium oscillations in astrocytes via inositol (1, 4, 5)-trisphos- “ ” ” – [67] S. Rahman, Mitochondrial disease and epilepsy, Develop- phate regeneration, Cell Calcium, vol. 39, no. 3, pp. 197 208, mental Medicine & Child Neurology, vol. 54, no. 5, pp. 397– 2006. 406, 2012. [51] S. Koizumi, K. Fujishita, M. Tsuda, Y. Shigemoto-Mogami, “ – “ [68] D. Janigro, Blood brain barrier, ion homeostasis and epi- and K. Inoue, Dynamic inhibition of excitatory synaptic lepsy: possible implications towards the understanding of transmission by astrocyte-derived ATP in hippocampal cul- ketogenic diet mechanisms,” Epilepsy Research, vol. 37, no. 3, tures,” Proceedings of the National Academy of Sciences of the – – pp. 223 232, 1999. United States of America, vol. 100, no. 19, pp. 11023 11028, “ 2011. [69] C. R. P. Dechandt, G. D. Ferrari, J. R. dos Santos et al., Energy metabolism and redox state in brains of Wistar audiogenic [52] T. Fellin, O. Pascual, and P. G. Haydon, “Astrocytes coordinate rats, a genetic model of epilepsy,” Frontiers in Neurology, synaptic networks: balanced excitation and inhibition,” Physi- vol. 10, 2019. ology, vol. 21, no. 3, pp. 208–215, 2006. [70] Y. Wang, X. Xu, Y. Zhu, and R. Wang, “Neural energy mech- [53] J. P. Welsh, E. J. Lang, I. Suglhara, and R. Llinás, “Dynamic anism and neurodynamics of memory transformation,” Non- organization of motor control within the olivocerebellar sys- linear Dynamsics, vol. 97, no. 1, pp. 697–714, 2019. tem,” Nature, vol. 374, no. 6521, pp. 453–457, 1995. [71] F. Zhu, R. Wang, X. Pan, and Z. Zhu, “Energy expenditure [54] X.-J. Wang and G. Buzsáki, “Gamma oscillation by synaptic computation of a single bursting neuron,” Cognitive Neurody- inhibition in a hippocampal interneuronal network model,” namics, vol. 13, no. 1, pp. 75–87, 2019. The Journal of neuroscience : the official journal of the Society “ ffi fi for Neuroscience, vol. 16, no. 20, pp. 6402–6413, 1996. [72] F. Zhu, R. Wang, K. Aihara, and X. Pan, Energy-e cient r- ing patterns with sparse bursts in the Chay neuron model,” [55] D. S. Shin, W. Yu, A. Sutton, M. Calos, E. Puil, and P. L. Car- Nonlinear Dynamics, vol. 100, no. 3, pp. 2657–2672, 2020. len, “Isovaline, a rare amino acid, has anticonvulsant proper- ties in two in vitro hippocampal seizure models by increasing [73] R. Wang, Z. Wang, and Z. Zhu, “The essence of neuronal activ- interneuronal activity,” Epilepsia, vol. 52, no. 11, pp. 2084– ity from the consistency of two different neuron models,” Non- 2093, 2011. linear Dynamics, vol. 92, no. 3, pp. 973–982, 2018. [56] T. P. Ladas, C. C. Chiang, L. E. Gonzalez-Reyes, T. Nowak, and [74] H. Zheng, R. Wang, and J. Qu, “Effect of different glucose sup- D. M. Durand, “Seizure reduction through interneuron- ply conditions on neuronal energy metabolism,” Cognitive mediated entrainment using low frequency optical stimula- Neurodynamics, vol. 10, no. 6, pp. 563–571, 2016. tion,” Experimental Neurology, vol. 269, pp. 120–132, 2015. [75] A. Moujahid, A. d’Anjou, F. J. Torrealdea, and F. Torrealdea, [57] P. Winkler, H. J. Luhmann, and W. Kilb, “Taurine potentiates “Energy and information in Hodgkin-Huxley neurons,” Phys- the anticonvulsive effect of the GABAAagonist muscimol and ical Review E Statistical Nonlinear & Soft Matter Physics, pentobarbital in the immature mouse hippocampus,” Epilep- vol. 83, no. 3, 2011. sia, vol. 60, no. 3, pp. 464–474, 2019. [76] S. M. Molchanova, J. Huupponen, S. E. Lauri, and T. Taira, [58] Y. Wei, G. Ullah, and S. J. Schiff, “Unification of neuronal “Gap junctions between CA3 pyramidal cells contribute to spikes, seizures, and spreading depression,” Journal of Neuro- network synchronization in neonatal hippocampus,” Neuro- science, vol. 34, no. 35, pp. 11733–11743, 2014. pharmacology, vol. 107, pp. 9–17, 2016. [59] G. G. Somjen, H. Kager, and W. J. Wadman, “Calcium sensi- tive non-selective cation current promotes seizure-like dis- charges and spreading depression in a model neuron,” Journal of Computational Neuroscience, vol. 26, no. 1, pp. 139–147, 2009. [60] T. Rathmann, M. K. Ghadiri, W. Stummer, and A. Gorji, “Spreading depolarization facilitates the transition of neuronal burst firing from interictal to ictal state,” Neuroscience, vol. 441, pp. 176–183, 2020. [61] Y. Bernardinelli, J. Randall, E. Janett et al., “Activity-dependent structural plasticity of perisynaptic astrocytic domains pro- motes excitatory synapse stability,” Current Biology, vol. 24, no. 15, pp. 1679–1688, 2014. [62] M. V. Sofroniew, “Astrogliosis,” Cold Spring Harbor Perspec- tives in Biology, vol. 7, no. 2, 2015.