Abstract
A coupled model containing two neurons and one astrocyte is constructed by integrating Hodgkin-Huxley neuronal model and Li-Rinzel calcium model. Based on this hybrid model, information transmission between neurons is studied numerically. Our results show that when the successive spikes are produced in neuron 1 (N1), the bursting-like spikes (BLSs) occur in two neurons simultaneously during the spikes being transferred to neuron 2 (N2). The existence of the astrocyte and a higher expression level of mGluRs facilitate the occurrence of BLSs, but the rate of occurrence is not sensitive to the parameters. Furthermore, time delay τ occurs during the information transmission, and τ is almost independent of the effect of the astrocyte. Additionally, we found that low coupling strength may result in the distortion of the information, and this distortion is also proven to be almost independent of the astrocyte.
Introduction
Although the number of the glial cells is several times larger than that of the neurons in most parts of the brain, few studies have focused on the effect of glial cells on neuronal behavior. Over the past decades, an increasing number of works have demonstrated that the interaction between glial cell and neuron serves an important function in information transmission in the neuron system [1]–[3]. Astrocytes are the most numerous type and the best studied glial cells. Astrocytes modulate synaptic transmission through many different pathways [4], [5]. The most discussed one is that the presynaptic neuron release a kind of neurotransmitter, glutamate, which activates glutamate ionotropic receptors (i-GluRs) on the postsynaptic membrane. Astrocytes participate in this synaptic transmission by responding to the glutamate in the synaptic cleft through calcium elevation; this elevation of Ca2+ above a certain threshold triggers the release of glutamate to the synaptic cleft [6]–[9]. Calcium elevation also results in the release of other transmitters such as Adenosine Triphosphate (ATP). ATP activates the purinergic ionotropic receptors, which facilitates the enhancement of neuronal excitability [10]–[15]. Moreover, astrocytes absorb excess potassium released by neurons in the synaptic space and thus regulate excitation [16], [17]. This bidirectional coupling between neurons and astrocytes indicates the concept of the “tripartite synapse” [18]–[21].
Traditional modeling studies of neuron system consider the coupling between the neurons, but ignore the participation of glial cells [22]–[25]. Nadkarni and Jung introduced a model accounting for the interaction between the neurons and the astrocytes [26]. They model the effect of astrocyte on neuron through a calcium-dependent inward current in the neuron. The calcium-dependent function is fitted from experimental data [27]. This kind of modeling scheme is extensively employed by many researchers [14], [15], [20], [28], [29]. The model proposed by Nadkarni and Jung predicted the seizure-like spontaneous oscillations in the absence of stimuli. Following Nadkarni and Jung, many modeling studies focus on the contribution of astrocytes to epilepsy [30]–[34] motivated by experimental findings [35]. For example, Amiri et al. concluded that disruption of the homeostatic function of astrocytes may initiate the hypersynchronous firing of neurons through successive research works [32]–[34]. This finding suggests that the neuron-astrocyte interaction may represent a novel target to develop effective therapeutic strategies for epilepsy.
Neurons are widely accepted to be organized into networks, and neuronal networks exchange information through electrical and chemical synapses. Increasing evidences indicate that astrocytes are also organized into networks [4], and astrocyte networks are interconnected through gap junction channels. The channels are regulated by extra- and intracellular signals that enable the exchange of information. Based on these two networks, a recent review paper suggests the concept of “astroglial networks” [2]. Many recent modeling works focus on the neuronal synchronization in the astroglial network [28], [36]–[40]. As an example, by integrating Norris-Lecar neuron model and Li-Rinzel calcium model, Amiri et al. constructed a model to study how astrocytes participate in the interplay between the pyramidal cells and interneurons [40]. Furthermore, they extended their three-unit model to a neuronal population model to study the effect of astrocyte on neuronal synchronization. Astrocytes are concluded to be capable of changing the threshold value of transition from synchronous to asynchronous behavior among neurons [39].
Postnov et al. proposed a model containing three units (the presynaptic, and postsynaptic neurons and the glial cell) [5]. Their model can predict the long-term potentiation of the postsynaptic neuron. In the modeling study of astrocyte-neuron interaction, pyramidal cells and interneurons are often the focus [5], [15], [39], [40]. Our study is likewise based the same coupled neurons. We will focus on the effect of astrocyte when information is transferred from pyramidal cell to interneurons, i.e., how the existence of astrocyte changes the response of interneuron to the firing pattern of pyramidal cell.
Models
Our model, which is schematized in Fig. 1, contains two conductance-based neurons and one astrocyte. Pyramidal cells are known to excite interneurons. By contrast, the interneurons inhibit the pyramidal cells. Thus, the two neurons in the model are coupled by excitatory and inhibitory synapses. The Hodgkin-Huxley equations have served a vital function in the theoretical understanding of neuronal behavior [41]. Following Ref. [26], we use the Hodgkin-Huxley equations to model the two neurons. The model equation describing the transmembrane potential contains sodium, potassium, and leak currents. The equations are given by
 |
 |
 |
 |
(1) |
where 
denotes the transmembrane potential of xth neuron (x = 1,2), and 
represents the fraction of open Na+ channels, and 
represents the fraction of open potassium channels. The values of parameters are listed in Table 1. The closing and opening rates of the gates are given by
 |
 |
 |
(2) |
where 
denotes the injected current input in xth neuron, 
is the feedback current received from the astrocyte by xth neuron, and 
is the synaptic current received by xth neuron. Terman et al.
[42] suggest that the neuron releases a neurotransmitter to the synaptic cleft depending on the membrane potential, and the concentration of neurotransmitter released by xth neuron is given by
Figure 1. Schematic of the three-unit model.
N1: pyramidal cell; N2: interneuron; A: astrocyte.
Table 1. Parameter values.
| parameter | value |
| Cm | 1 µF/cm2 |
| gK | 36.0 mS/cm2 |
| gNa | 120.0 mS/cm2 |
| gL | 0.3 mS/cm2 |
| vK | −12.0 mV |
| vNa | 115 mV |
| vL | 10.6 mV |
| θs | 85.0 |
| σs | 2.0 |
| αs | 0.1 |
| βs | 0.05 |
| gsi | 0.1 |
| vsi | 0.0 mV |
| vse | −85.0 mV |
| c 0 | 2.0 µM |
| c 1 | 0.185 |
| va | 6 s−1 |
| vb | 0.11 s−1 |
| vc | 0.0 µM/s |
| d 1 | 0.13 µM |
| d 2 | 1.049 µM |
| d 3 | 0.9434 µM |
| d 5 | 0.08234 µM |
| a 2 | 0.2 µM−1s−1 |
| k 3 | 0.1 µM |
| P 0 | 160.0 nM |
| τP | 0.00014 ms−1 |
| gse | variable |
| rp | variable |
The parameter values are obtained from references with slight modification.
 |
(3) |
Following Terman et al.
[42], the synaptic variable 
is introduced to explain the effect of the neurotransmitter release 
by yth neuron on the xth neuron, and the dynamic equation is given by
 |
(4) |
Then, the synaptic current 
received from each other by the two neurons in our model is
 |
 |
(5) |
where 
and 
are the maximal conductance of the excitatory and inhibitory synapses. 
and 
are the corresponding reversal potential.
Astrocytes do not generate action potentials, i.e., the astrocytes are non-excitable electrically. The astrocytes respond to the neurotransmitter release in the synaptic cleft through IP3 production[see Fig. 1]. Subsequently, elevation of IP3 concentration induces the release of Ca2+ from endoplasmic reticulum (ER), and then more Ca2+ are released depending on the IP3-induced Ca2+ elevation. The elevation of Ca2+ above a certain threshold triggers the release of glial transmitters, which, in turn, will influence the dynamics of the neurons. We use Li–Rinzel model to describe the Ca2+ exchange in the astrocyte[43]. This process contains three fluxes across the ER membrane: flux release through the ion channels (IP3Rs), removal of Ca2+ by an ATP-dependent pump, and a leak.
 |
(6) |
with
 |
(7) |
where C denotes the Ca2+ concentration in the intracellular space, q is the fraction of activated IP3R, and P is the IP3 concentration in the intracellular space. The values of parameters are listed in Table 1. The production of intracellular IP3 is modeled by
 |
(8) |
Nadkarni and Jung fit the experimental data[27] using the function of the current versus astrocytic Ca2+ concentration
 |
(9) |
Numerous physiological studies show that astrocytes release ATP, which has direct excitatory effects on hippocampal interneurons [44], [45]. By contrast, astrocytes decrease pyramidal neuron excitability (Fig. 1) [36], [46]. These findings suggest the following current 
in Equation (1):
 |
(10) |
where we introduce parameter 
to account for the effect of astrocytes. Given that 
= 0, the effect of astrocyte is ignored, whereas when 
= 1, the effect of astrocyte is considered fully.
All parameter values are listed in Table 1. We solve the model Equations (1) to (10) by using a fourth-order Runge-Kutta integration scheme with a time step 0.05, and simulations verify that further time step reduction does not significantly improve accuracy.
Results and Discussion
Ignoring the effects of astrocyte and synaptic current, i.e., 
= 0 and 
= 0, 
6.24 µA/cm2 is needed to generate persistent action potentials in the isolated H–H neuron. To study the information transmission from N1 to N2, we let 
= 10.0 µA/cm2, and 
= 0.0 µA/cm2, for which the persistent action potentials are generated in N1, but cannot be generated in N2 on its own. While the effect of the astrocyte is ignored (
= 0), the persistent action potentials are found in N2 for large coupling strength 
, that means the information implied in the action potentials of N1 are transferred to N2. Comparing Figs. 2 (a) and (b), 
= 0.9 is sufficient for the information transmission, but 
= 0.5 is not. The results of calculation show that the critical value of 
for the information transmission is 0.56. When the effect of the astrocyte is considered, i.e., 
, the results are not significantly changed by the astrocyte[Figs. 2(c) and (d)]. As an example, in Fig. 3, the critical values are calculated for different values of 
. Both for 
= 10.0 and 20 µA/cm2, the critical value is independent of 
but varies with different values of 
.
Figure 2. Time series of membrane potential in N1 and N2 for different parameter values.
The successive spikes in N1 are induced by the injected current 
= 10.0 µA/cm2, and 
= 0.0 µA/cm2 by which the spikes can not be induced in N2; 
= 0.8 µM/s. (a)
= 0, 
= 0.5; (b)
= 0, 
= 0.9; (c)
= 0.5, 
= 0.5; (d)
= 0.5, 
= 0.9.
Figure 3. The critical value of 
for which the information is transferred from N1 to N2.
= 0.0 µA/cm2 by which the spikes can not be induced in N2; 
= 0.8 µM/s.
Bursting-like Spikes
We now focus on a longer time scale. In Fig. 4, the value of 
is 0.9, for which the persistent action potentials in N1 are successfully transferred to N2. When 
, the successive action potentials are transferred. Notably, bursting-like action potentials are found both in the N1 and N2 for 
. Bursting-like spikes (BLSs) are extensively found in experimental and modeling studies. Cressman Jr. et al. have studied the influence of sodium and potassium dynamics on neuronal behaviors using a single neuron model containing the effect of the glial cell [16], [17]. They found the BLSs in some parameter regions, and the glial cell serves an important function in the appearance of BLSs. However, compared with our model, the glial cell in Ref. [16], [17] modulates neuronal behaviour behavior by removing excess potassium from the extracellular space. Postnov et
al. have found the postsynaptic neuron response to the presynaptic neuron by bursting-like firing in their modeling work regarding the effect of glial cell, but the presynaptic neuron fires successively [37]. This finding differs from our results because the BLSs always appear in N1 and N2 simultaneously. Theoretically, the resting membrane potential during the bursting spikes is attributed to the inhibitory effect of the astrocyte to N1. In Fig. 5, the time series of the calcium concentration in the astrocyte and total current in N1(
) are depicted to correspond to Fig. 4. The calcium concentration is oscillating. When 
is larger than 196.69 nM, the increase of the inhibitory current 
(negative) will cause 
to decrease to a low level. Otherwise, when 
is larger than 196.69 nM, the inhibitory current vanishes, and 
approaches 10 µA/cm2. The red dashed lines in Fig. 5(b) and (d) represent 6.24 µA/cm2. Obviously, while 
decreases to a value less than 6.24 µA/cm2, the N1 will possess the resting membrane potential. In Fig. 5(b), although 
decreases owing to the larger 
, 
is always larger than 6.24 µA/cm2. As a result, the successive firing of N1 will not be stopped. In Fig. 5 (d), 
decreases to values less than 6.24 µA/cm2 periodically. When 
is less than 6.24 µA/cm2, the N1 possesses the resting membrane potential, and BLSs are produced. Then, the BLSs are transferred to N2 through the excitatory synapse.
Figure 4. Time series of membrane potential in N1 and N2 for different parameter values.
The values of parameters 
, 
and 
are same as in Fig. 2. (a)(b)
= 0.3, 
= 0.9; (c)(d)
= 0.5, 
= 0.9. Note that the time scales are much longer than that in Fig. 2.
Figure 5. Time series of calcium concentration and total current in N1 corresponding to that of membrane potential in Fig. 4.
The values of parameters Ie 1, Ie 2and rP are same as in Fig. 2. (a)(b) λ = 0.3, gse = 0.9; (c)(d) λ = 0.5, gse = 0.9. The red dashed lines indicate the value 6.24 µA/cm2.
The effect of astrocyte serves an important function in the production of the BLSs. As previously mentioned, the excitatory coupling strength determines the the information transmission from N1 to N2 significantly. Thus, we will identify the region of parameter 
and 
, in which the BLSs are produced. Additionally, the IP3 production rate 
has been proven to be associated with the expression level of mGluRs in astrocytes. The enhanced production of IP3 corresponds to over-expressed mGluRs. Over-expression of mGluRs has been reported to facilitate the seizure-like oscillations in the neurons [26]. In our study, three typical values of 
, 0.4, 0.5 and 0.8 µM/s, are selected to represent the normal, intermediate, and enhanced expression level of mGluRs, respectively. The shadow regions in Fig. 6 are the parameter regions in which the BLSs can be found. First, the BLSs appear for an intermediate value of 
. Extremely large or small 
both make the calcium concentration approach a steady value less than 196.69 nM. In our model, the astrocyte fails to feedback to the neurons by 
when 
is less than 196.69 nM. Thus, the BLSs are not produced. Second, only if 
is larger than a critical value do BLSs appear. Thus, we can conclude that the existence of astrocyte is an important condition for the production of the BLSs. Finally, the area of the shadow regions decreases sharply with decreasing 
. Enhanced expression level of mGluRs favors the BLSs. Although the models are different in previous literatures, the similar results have been obtained that the calcium dynamics in the astrocyte strongly affect the neural activity [5], [15].
Figure 6. Parameter region in which BLSs are produced.
The values of parameters 
and 
are same as in Fig. 2. The value of 
equals to (a) 0.4 µM/s; (b) 0.5 µM/s; (c) 0.8 µM/s.
The rate of occurrence of the BLSs is then calculated. In Fig. 7, the rate 
is approximately 0.12 
and is not very sensitive to the parameters, once the values of the parameters are within the shadow regions in Fig. 6. More accurately, 
is maximum, and remains constant in the center of the shadow regions. 
decreases when the parameter values change from the center to the edge of the regions. Furthermore, 
increases with the enhancement of the expression level of mGluRs. Although Cressman Jr. et
al. have not investigated the effect of astrocyte on the rate of the BLSs clearly, Ref. [16] shows that the rate increases with the enhancement of glial strength, and the rate is at the scale from 0.01 
to 0.1 
. This finding is in accordance with our results qualitatively.
Figure 7. Rate of occurrence of the BLSs vs. the values of parameters 
and 
.
The values of parameters 
, 
, and 
are same as in Fig. 6.
Time Delay and Information Distortion
Synaptic transmission is widely accepted to involve time delay attributed to the signal propagation time [47]. Theoretically, neuronal models with time delay have received considerable attention. Delay-induced coherent oscillation [48] is found in neuronal network as well as in other coupled systems. Delay-enhanced synchronization [23], [49] may be relevant for neuronal networks to establish a concept of collective information processing in the presence of delayed information transmission. Our recent works find that delay cooperating with diversity can induce fruitful synchronization transitions [22]. Herein, the delay in the information transmission between two neurons will be verified in the presence of of astrocytes. As an example, in Fig. 8, the time series of 
and 
are recorded to show delay in the information transmission from N1 to N2. The spiking times in N2 always lag behind that in N1. The time delay 
is the time interval between two closest spikes in the two neurons. No matter whether the BLSs are produced or not, the time delay does exist in the information transmission. This time delay are also found in the previous modelling work studying the effect of astrocytes in neuron system [40]. Fig. 8 (b) and (d) show the time delay 
corresponding to (a) and (c), respectively. 
is not constant but oscillates irregularly. However, the oscillatory amplitude is not large; 
possesses low-amplitude changing around a average value.
Figure 8. Time series of membrane potential in N1, N2 and corresponding time delay τ for different parameter values.
(a)(b) 
= 0.3, 
= 0.9; (c)(d) 
= 1.0, 
= 0.9. Note that in the broken regions of (c)and (d), the membrane potential remain on the resting states.
The average value of 
is calculated for different parameter values. Fig. 9 shows that with increasing 
, 
decreases to a minimum first and then increases to a saturated value. The decrease of 
for small 
corresponds to the increase of synchronization in Ref. [40]. The intermediate value of 
corresponding to the minimum 
is about 2.96, and this value is independent of the expression level of mGluRs. Furthermore, for low expression level of mGluRs (
= 0.2), 
is totally independent on the value of 
. With increasing 
, the minimal 
will be influenced by 
. Fig. 9(b) and (c) show that the minimal 
reaches a maximum for an intermediate value of 
, which is similar to the phenomenon of resonance found in random systems.
Figure 9. Average τ vs. the values of parameters 
and 
.
The value of 
equals to (a) 0.2 µM/s; (b) 0.5 µM/s; (c) 0.8 µM/s.
Then, we will turn to another interesting phenomenon implied in Fig. 2 and 8. To exhibit this phenomenon clearly, the time series of 
and 
for different parameter values are depicted in Fig. 10. Notably, N2 does not respond to every spike in N1 through a corresponding spike accurately, i.e., large amounts of spikes are “missed” during the transmission from N1 to N2. Generally, the neuronal information is deemed to be coded in the spike timing or rate. Thus, the missing of spikes may relate to the distortion of the information transmission. Herein, we define the distortion ratio 
by the ratio between the spike number in N1 and N2. Obviously, all the spikes in N1 respond by spiking in N2 for sufficient coupling strength. If 
is reduced, the number of missing spikes increases, i.e., 
increases. In Fig. 10, the values of the parameter 
and 
are set as 0.5 and 0.8, respectively. For these parameter values, the BLSs are produced. In fact, the calculation shows that this kind of missing spike may occur whether the BLSs are produced or not.
Figure 10. Time series of membrane potential in N1 and N2 illustrating the missing spikes.
The parameter value 
= 0.5, 
= 0.8 µM/s. The value of 
equals to (a) 0.57; (b)0.6; (c) 0.7; (d) 0.9; (e) 3.0.
The distortion rate 
is calculated for different parameter values. Fig. 11 shows that the missing spike occurs mainly for small coupling strength 
. 
decreases to zero sharply if 
is increased from 0.56. The critical value of 
, for which 
decreases to zero, is about 1.06. Comparing the three figures in Fig. 11, 
is almost independent of 
and 
. Even the critical value of 
1.06 does not change with the changing of 
and 
. Thus, we can conclude that the effect of astrocyte does not serve an important function in the occurrence of missing spike. To exhibit the slight effect of astrocyte, the values of 
are amplified in Fig. 12. For intermediate or enhanced expression level of mGluRs, 
is non-zero with an intermediate value of 
, whereas 
is larger than the critical value 1.06. We conclude that the effect of astrocyte induces the occurrence of miss of very few spike. Obviously, the accidental miss of spike does not result in the distortion of the information.
Figure 11. Distortion ratio 
vs. the values of parameters 
and 
.
The value of 
equals to (a) 0.2 µM/s; (b) 0.5 µM/s; (c) 0.8 µM/s.
Figure 12. Distortion ratio 
vs. the values of parameters 
and 
.
The value of 
equals to (a)0.2 µM/s; (b) 0.5 µM/s; (c) 0.8 µM/s. Note that the scale of 
axis differs from that in Fig. 11.
Conclusions
In this paper, the information transmission between neurons is studied by using a model that contains two neurons and one astrocyte. First, we identify the parameter region in which the information can be transferred from N1 to N2. The effect of astrocyte does not influence this parameter region. Secondly, in the parameter region for information transmission, we find BLSs in two neurons simultaneously. The parameter values for the occurrence of BLSs are also identified, and the results show that the higher expression level of mGluRs and the existence of astrocyte facilitate the occurrence of BLSs. Meanwhile, the rate for the occurrence of BLSs is calculated, and the rate is not very sensitive to the parameters. Third, time delay in information transmission is studied. The results show that 
is not constant but oscillate with small amplitude. The average value of 
is dependent on 
sensitively, but almost independent of 
and 
. Finally, we found amounts of spikes are “missed” during the transmission from N1 to N2. This distortion occurs mainly for small coupling strength 
. Although the astrocyte also induces very few missing spikes, it does not result in the distortion of the information.
Although glial cells have been widely accepted to serve an important function in synaptic transmission in neuron system, theoretical knowledge on the mechanism of interaction between glial cell and neurons is lacking. The modelling studies in this paper can help us to understand the mechanism by which the astrocytes participate in neuronal information transmission.
Funding Statement
This work is supported by the National Nature Science of Foundation of China under the Grant No. 11105219 (JT) (http://www.nsfc.gov.cn/Portal0/default152.htm). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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