Dynamical analysis of astrocyte-induced neuronal hyper-excitation

This paper introduces a computational model of integrated information in bidirectional neuron–astrocyte communication. Dynamical analysis and information transfer process are studied when stimulation of metabotropic glutamate is loaded into neuron and astrocyte in coupled model, respectively. The results show that the loading glutamate stimulus and coupling strength cause neuronal hyper-excitation, which is in some way linked to the epileptic instabilities. A period of sustained neuronal firing activities could continue after cessation of stimulation because of how long it takes for calcium concentration in astrocyte to decrease to steady state. Additionally, we also found that reducing the feedback of astrocyte could effectively inhibit neuronal seizure-like firing from the aspect of neuronal energy consumption in somatic firing.


Introduction
Neuron plays an essential role in determining how brain learns and processes the neuronal information. The central nervous system (CNS) is distinguished into neuron and glia [1,2]. Astrocyte covers the broadest part of neuroglia in brain and contains molecular mechanism that regulates neuronal signal. Similar to neuron, they form a huge network and communicate with neuronal networks [3]. Interactions between astrocyte and neuron have been extensively found in past experiments to serve a direct and vital function in transmission of information in the neuronal system [4]. There is evidence that synaptic information is integrated and processed by astrocytes, which control the transmission and plasticity of synapses. Astrocytes, serving as positive partners in the function of synapses, are involved concerning the processing, transmission and storage of information by the nervous system [5]. Further, astrocytic effects on short-term synaptic plasticity are controlled by the frequency of calcium oscillations [6]. Astrocyte is found to be responding to external stimuli by increasing intracellular calcium concentration, and calcium signaling is primarily manipulated by the mGluRs (astrocytic glutamate receptors). Response of glutamate to interaction with the metabotropic glutamate receptor in astrocyte can result in production of IP 3 , which in turn induces release of calcium ions from intracellular calcium pool, leading to an increase in calcium concentration [7]. Alternatively, it can also cause calcium ion inward flow via the ionotropic receptors [8]. It is discovered that astrocytes can also transmit calcium signals to neighboring unstimulated astrocytes in the form of intercellular calcium waves [9].
Conventional modeling of CNS has accounted for how neuron is coupled to each other, but the level of involvement of astrocyte signaling has usually been ignored. Contrary to the conventionally acknowledged paradigm where brain function is generated exclusively by neuronal activity, there is a burgeoning view, one that we review herewith, that brain function derives from the coordinated activity of a network of neurons and glial cells [5]. A model that takes into account the interactions between neurons and astrocytes has been proposed by Nadkarni et al. [1,2]. They investigated the influence of astrocyte on two neurons by means of calcium-dependent inward currents through neurons. The calcium-dependent function is based on a model fitted to experimental data [10]. In ref. [11], it further focuses on the relevance of the constructed model to existing physiological data. This bidirectional neuron-astrocyte communication model has been widely adopted [12]. Amiri developed a model to investigate the effect of astrocyte in epilepsy considering the bidirectional interaction between a pyramidal cell and a neuron by coupling Morris-Lecar neuronal model with Li-Rinzel calcium model [13]. The results showed that healthy astrocyte is responsible for compensating the excitatory input by varying the firing frequency, and disruption of signaling function of astrocyte could initiate synchronous pathological firing of neurons. Ma et al. [14,15] considered the information transmission process in a coupled model containing two neurons and one astrocyte, and then, they suggested that lower coupling strength could result in information distortion and bursting-like spikes. Furthermore, they extended the model to a hippocampal tripartite synapse model to study the seizure-like spiking induced by the NMDA glutamate receptors [16,17]. Wang and Zhao [18] studied the regulation effect of astrocyte in two coupled neurons of same excitability, which showed that astrocyte could exert an obvious influence on the dynamical properties of neuron. Han and Wang [19,20] reviewed the dynamics and some effective controlling strategies for epileptic seizures. Their results may provide useful insight into the mechanisms of neurological disease and cognitive dynamics.
It is known that many lesions result from abnormal interactions between neurons and astrocytes, such as epilepsy, Alzheimer's disease and Parkinson's disease [21][22][23][24][25][26]. Epilepsy is much more common in patients mediated predominantly by the changes in excitatory and inhibitory synaptic inputs leading to neuronal hyper-excitable seizures, including spontaneous and evoked spiking [21]. A possible mechanism for the onset of epilepsy was described by Nadkarni et al. [2]. They predicted that such spontaneous seizure-like spiking of neuron would occur in lack of any stimuli, i.e., neuronal hyper-excitation. Furthermore, Nadkarni et al. proposed a couple model of Pinsky-Rinzel (the soma potential of the Pyramidal neuron) and Li-Rinzel (astrocyte) based on experimental data. Glutamate receptors (mGluRs) over-expression on astrocyte has been shown to result in sustained oscillations in somatic potential, which occurs in patients with CNS epilepsy [27]. Many modeling investigations have been concentrated on astrocytic contribution to epilepsy. Stasenko, Postnov et al. [28][29][30] found that larger coupling strength (corresponding to a larger density of mGluRs) could induce neuronal hyperexcitation phenomena. Astrocytes release neurotransmitters to regulate, in a calcium-dependent manner, transmission at adjacent synapses. The effect of astrocytes on neurons is dependent on the amount of neurotransmitters output from astrocytes by firings. In past investigations, more phenomena and details for neuronal hyper-excitation have not been given, but probabilistic mechanisms during astrocyte feedback to neurons are not discussed in the context of neuronal hyper-excitation. In this paper, a mathematical model is introduced and numerically simulated by selecting suitable and controllable parameters based on previous studies. We focus on the rate of IP 3 production and how astrocytes alter the firing pattern of pyramidal cells, on the basis of coupled model suggested by Nadkarni by applying external stimuli to neuron and astrocyte, respectively.

Models
The simplest case we discuss is that of a single neuron coupled to an astrocyte. When the neuron stimulated with a DC current generates an action potential, Ca 2? signal is produced in the synaptic astrocyte, resulting in the liberation of glutamate from the astrocyte which feedback to another synapse of the identical neuron. Considering the wide range of dendrites and their innumerable synapses, this is a realistic, but simple circuit. The Hodgkin-Huxley equation is used to model the behavior of neuron. It describes cell membrane potential dynamics, including sodium, potassium and leakage currents. The system is as follows: where V indicates the neuronal membrane potential, m and h are the odds of activation and inactivation of sodium channels, and n is the odds of activation of potassium channels. I ext corresponds to the input of current injected into the neuron. Here we introduce I as to denote the current received by neuron as feedback from astrocyte. Calcium signal in astrocytes that occurred can be described at two levels. (1) Elevated intracellular calcium refers to a periodic increase in intracellular calcium concentration that manifests itself in the form of oscillations. (2) The propagation of the calcium elevation wave generated by the calcium response is radially propagated and usually originates from individual astrocytes. It is known that astrocytes do not respond to external stimuli in the form of electrical impulses, i.e., it is not electrically exciting because there are no sodium channels in the membrane. Sodium channels are required for generation of action potential. However, there are several neurotransmitter receptors on membrane of astrocyte that responds to neurotransmitters. Releasing of neurotransmitters from presynaptic neuron activates metabotropic receptors on astrocyte, leading to an increase in IP 3 concentrations. The subsequent rise in IP 3 concentration triggers the calcium release from the endoplasmic reticulum, followed by calcium-induced calcium release, as calcium concentration rises to a certain level, which causes the liberation of glial transmitters like glutamate. The released glutamate interacts with the glutamate onto another synapse of the same neuron, affecting the dynamical behaviors of neuron.
In this paper, the Li-Rinzel model is chosen for the description of oscillatory dynamics of astrocytes. The system is as follows: where J Channel (q) is the flow of calcium ions from the endoplasmic reticulum to the cytosol through the IP 3 channels. J pump is an ATP-dependent pumping stream on the endoplasmic reticulum that pumps cytoplasmic calcium ions back to the endoplasmic reticulum calcium pool. J leak denotes the leakage of calcium ions from the endoplasmic reticulum to the cytosol. q is the probability that the calcium channel is in the open state. The specific expressions for the various calcium ion flows in the equation are shown below: where [Ca 2? ] corresponds to calcium concentration in the cytosol. q is the proportion of activated IP 3 R. The concentration of IP 3 is indicated by [IP 3 ]. Calcium oscillations cause the glutamate to be released in the intercellular space. The released glutamate binds to the NMDA glutamate receptors, causing changes in membrane potential. The relationship between astrocytic calcium concentration and the inward current loaded to the adjacent neuron is expressed by I astro , whose fitted functional form can be expressed as [27]: y ¼ ½Ca 2þ À 196:69; Experiments have been carried out to demonstrate that astrocyte can influence the excitability of neuron. Failure of synaptic transmission can occur within the synaptic transmission process due to the probability of synapses with a very low poor ability to release neurotransmitters [31]. What's more, the impaired glutamate uptake by neuron also might alter the excitability of neuron-astrocyte network [32]. For consideration of these probabilistic mechanisms, a constant coefficient a is multiplied by I as that has the following form: A new parameter a 2[0,1] is introduced in the coupled model, which accounts for the strength of astrocytic feedback to neuron. Given that a = 0, it represents no effect on neuron, whereas a = 1, the effect has full consideration of astrocyte. The model for intracellular IP 3 production is [1]: where r mGluR is the IP 3 production rate and H(y) is Heaviside function, when y [ 0, H(y) = 1, otherwise H(y) = 0. In this paper, the parameters r mGluR and a are chosen to analyze firing activities of neuron. In the experiment, astrocyte responded to various concentrations of glutamate applied in the culture with an oscillatory elevation in intracellular calcium. It is possible to map glutamate concentrations to [IP 3 ] as each specific [IP 3 ] corresponds to each oscillation frequency. In this way, the parameter r mGluR can be considered as the rate of IP 3 production per unit volume of the transporter (mGluRs). As described above, r mGluR is positively proportional to the glutamate receptor expression levels on the soma from [1]. Details of all parameters are given in Table 1.

Results
In what follows, the input part of coupled system is divided into pyramidal neuron and astrocyte. The

Stimuli are received by neuron
To study the firing activities of neuron with the external current stimulus, the injected current input to the coupled system is selected to be 30 lA/cm 2 lasting 40 s, i.e., I ext = 30 lA/cm 2 . The bifurcation diagram for the Hodgkin-Huxley model is plotted in Fig. 2 using external stimulus current as control parameter. The maximum and minimum values of oscillations are depicted in curves a and b. It is seen that when the  Fig. 3. It is apparent from these figures that an increase in r mGluR triggers neuronal firing. In Fig. 3a, neuronal firing activities stopped when stimulus is stopped after t = 40 s at the IP 3 production rate r mGluR = 0.0002 lmol/ms. In Fig. 3b, after the cessation of current input, neuron undergoes a period of continuous oscillation before returning to a resting state at IP 3 production rate r mGluR = 0.0006 lmol/ms. In Fig. 3c, phenomenon of continuous oscillation of neuron, i.e., hyper-excitation phenomenon, occurs at IP 3 production rate r mGluR-= 0.0008 lmol/ms. Given that the rate of IP 3 production is proportional to the level of glutamate receptor, over-expression of glutamate receptors causes the hyper-excitation of neuron, a characteristic of epilepsy. This finding is in agreement with the overexpression of mGluRs. In pathological conditions, like mesial temporal lobe epilepsy (TLE), up to 20-fold mGluRs were observed in astrocytes.
The corresponding oscillations of [Ca 2? ] (Fig. 4a) and [IP 3 ] (Fig. 4b) in astrocyte can be observed in Fig. 4 with the same values of r mGluR in Fig. 3. Neuronal firings lead the glutamate to be released, which interacts with glutamate receptors on astrocyte, resulting in the altering of [IP 3 ] in the neighboring astrocyte and causing an increase in calcium concentrations, which triggers calcium-induced calciumreleasing mechanism. As a result, the concentration of calcium and IP 3 in astrocyte is increased. Corresponding [Ca 2? ] and [IP 3 ] are always maintained in a high level of steady state when r mGluR reaches the conditions for occurrence of hyper-excitation. Comparison of Figs. 3 and 4 reveals that even when stimulation is terminated, neurons continue to fire in the absence of any external stimulation due to elevated calcium levels within the astrocyte. Consequently, we see fully self-sustained oscillations. Figure 5 depicts the output of astrocyte to neuron in the coupled system. As the external current is constant, only the feedback to neuron from astrocyte modulation is discussed. In Fig. 5a, it is seen that the input to neuron remains below 6.3 lA/cm 2 after t = 40 s of Fig. 2 Bifurcation of neuronal membrane potential V about input current I ext in the coupled system. a and b represent the maxima and minima of neuronal membrane potential, respectively, during the oscillations stimulation. Similar to Fig. 3a, the systematic oscillations disappear after the cessation of stimulation. In Fig. 5b, after 40 s of stimulation, the input to neuron falls below 6.3 lA/cm 2 , where the oscillations take a number of periods before ceasing to the steady state corresponding to Fig. 3b. In Fig. 5c, the current input to neuron remains above 6.3lA/cm 2 ; it appears the neuronal hyper-excitation corresponding to Fig. 3c. Figure 6 shows the bifurcation of the variation of V with parameter r mGluR after stimulation, and it is seen that neuron undergoes the hyper-excitation when r mGluR = 0.00064 lmol/ms.

Stimuli are received by astrocyte
It is well established that astrocyte receives external stimuli and neuronal information [33,34], which activate the metabotropic glutamate receptors and cause electrical activity of pyramidal cell. In past studies, the production term of IP 3 induced by external stimulation of glutamate has been reported [35][36][37]. In this section, we stopped the external stimulation of the neurons and placed the astrocyte in a culture medium of a uniform glutamate bath, which activates astrocyte by binding the glutamate to the glutamate receptors [35]. The external glutamate stimulation of astrocyte is used to model the intracellular production of IP 3 : where G glutamate is modeled by [37] G glutamate ¼ v g S n k n g þ S n ; where v g = 0.00062 lmol/ms is the maximum production rate of IP 3 , S denotes the amount of glutamate applied to the culture (S = 100 lmol), n is the Hill coefficient (n = 0.3), and k g is the dissociation constant for glutamate stimulation of IP 3 production. Figure 7 shows a temporal sequence of V with  Fig. 3a, it reveals that a period of bursting-like spikes occurs before stimulus stops at t = 40 s during a time of low expression of glutamate receptors in Fig. 7a. In Fig. 7b, it shows that the duration of neuron that returns to rest state is lengthened after stimulation is stopped. Figure 7c, similar to Fig. 3c, shows that overexpression of glutamate leads to generation of hyperexcitation in neuron. It can be observed that calcium ion (Fig. 8a) and IP 3 concentration (Fig. 8b) during hyper-excitation maintain a higher level of steady state. Comparing with Fig. 4, Fig. 8 depicts the development of intracellular calcium (Fig. 8a) and IP 3 (Fig. 8b) dynamics when the parameter r mGluR increases. As external stimulation is a constant, the feedback on astrocyte modulation is studied. Due to the trains of outer stimuli for t = 40 s, [Ca 2? ] and [IP 3 ] build up and decay and then maintain in a lower level of steady state when the parameter r mGluR reaches the condition for occurrence of hyperexcitation.
Like previous cases, time course of current received by neuron as feedback from astrocyte I as with different values of r mGluR is shown in Fig. 9. At the end of external stimulation, calcium concentration is in a low steady state, which is insufficient to maintain the output oscillation of astrocyte I as (Fig. 9a, b). As r mGluR increases further (Fig. 9c), the current input to astrocyte remains above 6.3 lA/cm 2 . This is in agreement with the case in Fig. 5. Figure 10 denotes the bifurcation of r mGluR and V. When r mGluR-[ 0.0007 lmol/ms, the neuronal hyper-excitation is obtained after 40 s of stimulation of astrocyte. It is worth noting that the parameter value required for the occurrence of oscillation is larger than that in neuronstimulation properties. It is suggested that more functionally active glutamate receptors are needed to induce hyper-excitation when stimulating astrocyte.

The feedback strength of astrocyte in antiepileptic processes
Astrocyte transmits signals through the release of glutamate to neuron. It also receives glutamate released by neuron to trigger calcium oscillations. Accordingly, failure of synaptic transmission and the alteration the glia-derived excitatory pathway in concert with impaired glutamate uptake might result in lower feedback strength. In this section, we will investigate the effect of hyper-excited by astrocyte. It   Fig. 7 Time courses of soma membrane potential V with different values of r mGluR . Stimulation is loaded in astrocyte, and the arrow means that the system is unstimulated. a r mGluR = 0.0002 lmol/ms; b r mGluR = 0.0006 lmol/ms; c r mGluR = 0.0008 lmol/ms. In the top panel a, the neuron falling back into its resting state has a shorter time after termination of the stimulus. The second panel b has a longer time to return to resting state. The third panel c occurs hyper-excitation. The arrow means that the coupled system is unstimulated by the external current at 40 s is mentioned that the feedback of astrocyte to pyramidal cell is reduced to a = 0.5 in Fig. 11 and a = 0.6 in Fig. 12. It represents the time sequence of membrane voltage (Figs. 11a and 12a), calcium concentration (Figs. 11b and 12b) and feedback current (Figs. 11c and 12c) after reducing involvement of astrocyte in original hyper-excitation state, respectively. It shows that the original hyper-excitation disappears and corresponding calcium concentration and current values are above the threshold. Comparing Fig. 11a, after the cessation of current input, neuron undergoes a longer period of continuous oscillation   (c) Fig. 9 Time courses of the output of astrocyte I as with different values of r mGluR in coupled system. The arrow means that the system is unstimulated at 40 s. The red dotted line represents I as = 6.3 lA/cm 2 , i.e., the minima of leading to neuronal firing. In the first (a) and second (b) panels, the neuron falls back into its resting state after I as is less than 6.3 lA/cm 2 . a r mGluR = 0.0002 lmol/ ms; b r mGluR = 0.0006 lmol/ ms; c r mGluR = 0.0008 lmol/ms before returning to a resting state at a = 0.6 in Fig. 12a. It suggests that astrocyte is actively influenced by the dynamical behavior of the neuron. Figure 13 depicts in detail the ISI of the soma with r mGluR = 0.0008 lmol/ms. Firing patterns manifest themselves originally with long irregular intervals a \ 0.645, which, whereafter, shifts to fast-spiking with a shorter period. Figures 11a and 12a can be used as examples to observe that the potential evolution possesses long irregular intervals caused by sustained oscillations of a certain duration that would occur with the non-hyper-excitation. Short-lasting oscillations are accompanied by long intervals in which a large number of ISI are available. Therefore, this can be used to determine whether a neuron has been hyperexcited or not. When a [ 0.645, the potential evolution has shorter ISI with a fewer number corresponding to hyper-excited, as also can be observed in Fig. 3c.
We now investigate the feedback of astrocyte on neuronal seizure-like firing from aspect of energy consumption in somatic firing in two-dimensional parameter space r mGluR and a. Many past experiments have been used to show that epilepsy consumes much more energy than normal neuronal firings. Evolution of seizure-like firing is described quantitatively in terms of Hamilton energy according to a measure \ H [ developed in the literature [15,38]: We can see that the feedback of astrocyte is not enough to sustain the constant firing of neuron after stopping the stimulation hHi ¼ H 0 ¼ g k n 4 ðV k À VÞ 2 þ g Na m 3 hðV Na À VÞ 2 þ g L ðV 1 À VÞ 2 À VI as where \ H [ describes the mean energy consumption for neuronal firing on a temporal scale of T. Energy consumption of K ? , Na ? and leakage channels is described in the first three items of the equations. The last term in the equation describes the total energy expenditure of the external input to neuron. The input current is derived from pyramidal cell and astrocyte. In Fig. 14, time scale of part of Fig. 3 (88,280-88450 ms) and the corresponding energy consumed by neuron are measured separately for a while at r mGluR = 0.0002 lmol/ms (Fig. 14 a 1 a 2 ), r mGluR = 0.0006 lmol/ms (Fig. 14b 1 -b 2 ) and r mGluR = 0.0008 lmol/ms (Fig. 14c 1 -c 2 ). It should be mentioned that both neuron and energy consumption undergo oscillations in Fig. 14c 1 -c 2 . Figures 15 and 16 (Projection of Fig. 15 in xoy plane) represent the average energy consumption under the variations of parameters r mGluR and a by simulating neuron. As the average energy consumption \ H [ is under approximately 3000 nJ/ (ms 9 cm 2 ), the neuronal firing is maintained normal. However, when two parameters r mGluR and a increase to the threshold, a high level of average energy consumption is achieved at approximately 11,000 nJ/ (ms 9 cm 2 ), corresponding to epileptic seizures. This confirms it remains relatively high due to the energyconsuming characteristic of epilepsy. Furthermore, it is shown that seizure-like firing is enhanced by increasing the values of r mGluR as a reaches a threshold   Fig. 17 in xoy plane), the average energy consumed is significantly higher (approximately 10,000 nJ/ (ms 9 cm 2 )) with epilepsy. It is demonstrated that the astrocyte can actively participate in the synaptic plasticity and has the process of bidirectional information exchange with the adjacent neuron. Experimental evidence has shown that it has an important role in increasing integrated neuronal information in CNS. In this paper, we  Fig. 17 Neuronal energy consumption in somatic firing with variations of twodimensional parameters a and r mGluR with stimulating astrocyte after 40 s. A high level of average energy consumption, approximately 10,000 nJ/ (ms 9 cm 2 ), corresponds to seizures introduced a mathematical model of bidirectional information exchange in a simple neuron-astrocyte circuitry that can be rendered neuronal hyper-excitation to explore the dynamical behaviors underlying the IP 3 production rate and the strength of astrocytic feedback to neuron mechanism. This modified model allows us to better understand the dynamical mechanisms of seizures arising from a brain system. To test the hypothesis that hyper-excitation is trigged by the glutamate pathway, we stimulated neuron and astrocyte as varying the relationship between astrocytic calcium ion concentration and the inward current loaded to the adjacent neuron. Firstly, seizure-like firing is identified by elevating the rate of IP 3 production in neuron, which is in agreement with previously well-conducted studies; meanwhile, hyperexcitation has been found, given that the rate of IP 3 production is proportional to the level of glutamate receptor.
In addition, when astrocyte was loaded with glutamate stimulation, we studied the effects of astrocyte-enhanced excitation of adjacent neuron. Detailed discussions reveal that more functionally active glutamate receptors are needed to induce hyperexcitation when stimulating astrocyte. The role of feedback from astrocyte to neuron was also discussed. When the feedback strength is reduced, calcium oscillation in astrocyte and neuronal hyper-excitation is inhibited. Finally, neuronal average energy consumption was explored from the aspect of energy consumption. We predicted that astrocyte is actively involved in neuronal firing, and it may be a major factor in containing neuronal hyper-excitability. This result may have some interesting implications for a rational antiepileptic pathway, providing clinically valuable information. Hence, we hope that this study will provide an insight into a new pathway that could induce neuronal hyper-excitation. A challenge of future research is to account for the extent to which astrocyte-induced neuronal hyper-excitability can spread across a simple spatial grid due to the overexpression of mGluRs.
Funding The authors have not disclosed any funding.