Research ArticleA Computational Model to Investigate GABA-Activated AstrocyteModulation of Neuronal Excitation
Licong Li,1,2 Jin Zhou,3 Hongji Sun,4 Jing Liu,2 Hongrui Wang,2 Xiuling Liu ,2
and Changyong Wang 3
1College of Physics Science and Technology, Hebei University, Baoding 071002, China2Key Laboratory of Digital Medical Engineering of Hebei Province, Hebei University, Baoding 071002, China3Department of Neural Engineering and Biological Interdisciplinary Studies, Institute of Military Cognition and Brain Sciences,Academy of Military Medical Sciences, Beijing 100850, China4Brainnetome Center, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China
Gamma-aminobutyric acid (GABA) is critical for proper neural network function and can activate astrocytes to induce neuronalexcitability; however, the mechanism by which astrocytes transform inhibitory signaling to excitatory enhancement remainsunclear. Computational modeling can be a powerful tool to provide further understanding of how GABA-activated astrocytesmodulate neuronal excitation. In the present study, we implemented a biophysical neuronal network model to investigate theeffects of astrocytes on excitatory pre- and postsynaptic terminals following exposure to increasing concentrations of externalGABA. The model completely describes the effects of GABA on astrocytes and excitatory presynaptic terminals within theframework of glutamatergic gliotransmission according to neurophysiological findings. Utilizing this model, our results showthat astrocytes can rapidly respond to incoming GABA by inducing Ca2+ oscillations and subsequent gliotransmitter glutamaterelease. Elevation in GABA concentrations not only naturally decreases neuronal spikes but also enhances astrocytic glutamaterelease, which leads to an increase in astrocyte-mediated presynaptic release and postsynaptic slow inward currents. Neuronalexcitation induced by GABA-activated astrocytes partly counteracts the inhibitory effect of GABA. Overall, the model helps toincrease knowledge regarding the involvement of astrocytes in neuronal regulation using simulated bath perfusion of GABA,which may be useful for exploring the effects of GABA-type antiepileptic drugs.
1. Introduction
The brain is an adaptive nonlinear dynamic system, in whichexcitatory-inhibitory (EI) balance is vital for normal brainfunction [1]. Regulation of neuronal excitability is one ofthe key factors in achieving EI balance. Among the regulatorytransmitters involved in neuronal excitability, GABA,released by the GABAergic neurons or astrocytes [2, 3], isthe major inhibitory transmitter, which decreases neuronalexcitability and prevents the neurons from overfiring. It iswell established that GABA plays a central role in memoryconsolidation [4], motor coordination [5], and motorlearning [6]. The dysregulation of GABA is closely relatedto certain neurological disorders, such as epilepsy [7, 8], Par-
kinson’s disease [9], and anxiety and depression [10]. Exper-imental studies and the computational models haveincreased our understanding of the significant functions ofGABA. Many reports suggest that elevated GABA concentra-tions exert both inhibitory and excitatory effects on neuronalfiring [11–13]. Such contradictory scenario has also beenreported in bath perfusion of a GABABR agonist [14, 15].One potential mechanism is that elevation in ambient GABAconcentration can reduce the inhibitory effect of GABAergicsynapses on the connected excitatory neurons, resulting in anoverall increase in excitability via the disinhibition of theneurons [12, 14].
Due to the tight morphological arrangement of theastrocytes and neurons, named “tripartite synapses” [16],
HindawiComputational and Mathematical Methods in MedicineVolume 2020, Article ID 8750167, 13 pageshttps://doi.org/10.1155/2020/8750167
astrocytes actively participate in the modulation of neuronalactivity and synaptic behavior (for reviews, see [17–19]).Recent studies have demonstrated that GABA-activatedastrocytes in the hippocampus or cortex can induce depolar-ized current and enhance neuronal excitability via therelease of gliotransmitter glutamate [20, 21]. These findingsinitiate a discussion regarding astrocytes as modulators inthe conversion of inhibitory stimuli to excitatory signals,which provide novel insight into the function of astrocytesin terms of neural network excitation. However, the signal-ing machinery involved in the intracellular cascade ofGABABR-mediated Ca2+ increase and glutamate release inastrocytes remains unclear [20]. In this case, computationmodeling can be used as a viable alternative to understandthe role of GABA-activated astrocytes in the regulation ofneuronal excitation. Most astrocyte models are based onthe Ca2+ dynamics models [22, 23], which are overwhelm-ingly driven by synaptically released glutamate [24–26].According to the existing glutamate-activated astrocytemathematical framework, Li et al. developed a GABA-activated astrocyte model coupled to the seizure-firing neu-rons [27]. In particular, the astrocyte model added a com-plex differential equation and involved dual signaling ofGABA released by interneuron and glutamate released byastrocyte. Nevertheless, an accurate description of the regu-lation of the excitatory neurons by GABA-activated astro-cytes remains elusive.
In the present study, we addressed this issue by devising anovel neuron-astrocyte interaction computational model andsimulating injection of exogenous GABA (“virtual” GABA,½GABAex� = 0 μM, 1 μM, 5 μM, and 10 μM) into the extracel-lular space. The goal of this paper was to describe neuronalpopulation activity and investigate the modulation of astro-cytes on the neurons in stimulus conditions by lumping themicroscale tripartite synapse model and macroscaleneuronal-astrocytic network. On the cell level, the GABA-activated astrocyte model was incorporated into a modelingframework of glutamatergic gliotransmission [28]. Themodel described the activation of astrocytes by synaptic glu-tamate, in addition to the crosstalk between the GABABRsand mGluRs to induce astrocytic calcium elevation [29]. Byconsidering these two signaling pathways, we may betterunderstand the effect of GABA on astrocytes. Furthermore,synaptic regulation by astrocyte-derived glutamate is not astraightforward process in contrast to the previous synapticmodels [28, 30]. At presynaptic terminals of glutamatergicsynapses, the GABABR can strongly inhibit synaptic activity[31]. Thus, the comodulation of presynaptic behavior byastrocytic glutamate and increasing concentrations of exoge-nous GABA was considered. Using the model, we can collectneuronal spikes and network data simultaneously and under-stand the role of astrocytic modulation in response to exoge-nous GABA by recording the changes in a set of synapticparameters over time. Numerical simulation shows that thefrequency and amplitude of GABA-evoked astrocytic Ca2+
oscillations increase with increasing GABA concentrations.Moreover, the enhanced astrocytic glutamate release pro-motes astrocyte-regulated synaptic release and increasesastrocyte-mediated postsynaptic slow inward currents (SICs,i.e., depolarizing currents); both of which can finely tune
neuronal excitation. These are efficient and adjustable regula-tory mechanisms for balancing the excitation and inhibitionof neuronal networks.
2. Model and Methods
Based on neurophysiological features and experimentalobservations, the complete model we developed includesthe inhibitory and excitatory neurons, synapses, and astro-cytes, as well as the glutamate and exogenous GABA signal-ing pathways. The response of astrocytes to GABA occursthrough activation of GABABRs. Similar to the activation ofastrocytes by glutamate released from presynaptic terminals(red arrows in Figure 1), production of the second messengerinositol 1,4,5-trisphosphate (IP3) is triggered, thereby evok-ing Ca2+ oscillations [20]. Ca2+ elevation in astrocytes causesglutamate releasing into the extracellular space and ulti-mately modulating presynaptic vesicle release probabilityand postsynaptic neuronal excitability (orange arrows).Another important signaling pathway involves the opposingeffects of astrocytic glutamate and exogenous GABA at pre-synaptic terminals of excitatory synapses (blue arrows),where they comodulate synaptic release. Moreover, exoge-nous GABA can directly regulate the probability of presynap-tic GABA release and reshape the activity of postsynapticGABAARs [13]. Given that GABA is mainly taken up bythe GABAergic neurons [32], the clearance of extracellular
GABABR AMPAR NMDAR
NR2B-containingNMDAR
mGluR1/5
IP3
[Ca2+]
GABAex
Presynapticneuron
Postsynapticneuron
Synapticcleft
[Ca2+]
Glusyn
Gluastro
Figure 1: Themain pathways of exogenous GABA (GABAex) actionin the tripartite glutamatergic synapse model. Red arrows indicatethe glutamatergic gliotransmission pathways, including astrocyticmetabotropic glutamate receptor (mGluR) activation, IP3-gatedCa2+ mobilization, astrocytic glutamate release, presynapticmGluR activation, and postsynaptic AMPAR and NMDAR-mediated depolarization currents. The orange and blue arrowsrepresent the regulation of the astrocyte and presynaptic neuronby exogenous GABA, respectively. GABAex represents simulatedGABA injection to the neuronal-astrocytic model. ER refers to theendoplasmic reticulum. It is necessary to note that locations ofastrocytic and neuronal receptors in this figure do not representtheir exact physiological distribution.
2 Computational and Mathematical Methods in Medicine
GABA by astrocytic uptake is ignored in our model. Themodel architecture is presented in Figure 1, and subsequentsections describe the mathematical formulas in detail.
2.1. Neuron Model. In the present study, the dynamics of asingle neuron receiving different stimuli were simulatedusing a conductance-based leaky integrate-and-fire (LIF)model. This simple spiking model greatly reduces the com-plexity of neuronal dynamic equation and computationalexpense, in particular when large groups of the neurons arecoupled together into neural astrocytic networks. Both theexcitatory and inhibitory neurons are given by different ini-tial values of the model variables. The neuronal model isdescribed by the following equation:
τmdvdt
= vrest − vð Þ + ge ve − vð Þ + gi vi − vð Þ + Iexð Þ × 1gleak
,
ð1Þ
where ge, gi, and gleak refer to the total excitatory, inhibitory,and leak conductance, with a reversal potential ve = 0mVand vi = −80mV, respectively. Iex is a constant input current,denoting the background current to each neuron. The neu-ron’s membrane potential v resets to the default value vrest= −60mV when it reaches the set threshold vth = −50mV.Each spike originating from presynaptic neuron i causes aconductance change △g in postsynaptic neuron j, irrespec-tive of whether the inputs are excitatory or inhibitory [33],i.e., gj → gj +△gij. The postsynaptic conductance can becalculated as follows:
dgedt
= −geτe
,
dgidt
= −giτi,
ð2Þ
△g = �gj∙r tð Þ, ð3Þwhere τe = 5ms and τi = 10ms are synaptic time constants,and �gj is the maximal postsynaptic conductance. Excitatorysynaptic action was simulated by modeling AMPA andNMDA conductance, while GABAA conductance wasmodeled to simulate inhibitory synaptic inputs. The itemrðtÞ is postsynaptic channel open probability, dependingon the neurotransmitter concentration in the synaptic cleft(see Equation (22)). The values for neuronal parametersare listed in Table 1.
2.2. Astrocyte Model. Unlike the generation of action poten-tials in the neurons, the computational models describingastrocytic functions always employ astrocytic Ca2+ signaling[35]. Following GABA application, GABABR-mediatedCa2+ waves depend on the intracellular IP3 cascade and areabsent in IP3R2
-/- mice [20]. However, a paucity of exactexperimental measurement for GABA-induced IP3 activa-tion limits the precise parameters of the IP3 dynamic model.Thus, according to experimental results of astrocyte responseto GABA [20, 29] and the existing glutamate-activated astro-
cyte models [24–26], we simplified the model to concentrateonly on the IP3-evoked Ca2+ signaling pathway, in which therate of IP3 production depended on the amount of synapticglutamate and exogenous GABA in the synaptic cleft. Tomodel the dynamics of Ca2+ oscillations, we employed theclassical Li–Rinzel model [23]. The modified model of astro-cytic IP3 production can be explained with the following setof equations:
d IP3½ �dt
=IP3½ �∗ − IP3½ �ð Þ
τIP3+ JGABAex
+ Jglutamate + JG, ð4Þ
JGABAex=
vIP3gaba∙ GABAex½ �n1kn1gaba + GABAex½ �n1 ,
ð5Þ
Jglutamate =vIP3glu ∙ Glu½ �n2kn2glu + Glu½ �n2 ,
ð6Þ
where the first equation describes the process of IP3 degrada-tion to a steady state at a degradation rate τIP3 = 7 s and anequilibrium concentration ½IP3�∗ = 0:16μM. The secondand third equations describe the production rate of cytoplas-mic IP3 induced by exogenous GABA and presynaptic gluta-mate, respectively. ½GABAex� and ½Glu� are the exogenousGABA and synaptic glutamate concentration, respectively.vIP3gaba and vIP3glu represent the rate of IP3 production via GABA
and glutamate, respectively, and kn1gaba and kn2glu are theirrespective dissociation constants with Hill coefficients n1and n2. Given the amplification interaction between theGABA and glutamate signaling pathways [29], we added aGABA concentration-dependent proportional coefficient kto glutamate-induced IP3 production:
Jglutamate = Jglutamate∙ 1 + k∙ GABAex½ �ð Þ: ð7Þ
In our model, the available magnitude of extracellularglutamate and GABA abruptly increases following presynap-tic release and exogenous input, respectively, and then expo-nentially decays. The estimated glutamate (Equation (20))
Table 1: Parameters used in the neuronal model [34].
Parameter Description Value
τm Membrane time constant 20ms
vth Firing threshold -50mV
vrest Resting potential -60mV
ve Excitatory reversal potential 0mV
vi Inhibitory reversal potential -80mV
gleak Leak conductance 10 nS
Iex Background current 105 pA
τref Absolute refractory period 5ms
τe Excitatory conductance decay time constant 5ms
τi Inhibitory conductance decay time constant 10ms
3Computational and Mathematical Methods in Medicine
and GABA concentration in the synaptic cleft can be repre-sented mathematically as follows:
d GABAex½ �dt
= −gcGABA∙ GABAex½ �∙Θ t − T stimð Þ, ð8Þ
where gcGABA is the clearance rate of exogenous GABA with
value ln ð2Þ/3 s−1 [36].Θ is the Heaviside function. Tstim rep-resents the constant stimulus of GABA concentration with aduration 0.5 s [27].
The above dynamic IP3 model is similar to the for-mulation of IP3 production in [27]; however, there aresome important differences. Firstly, compared with theautoreceptor-mediated astrocyte activation model, theastrocyte model we developed was coactivated by exoge-nous GABA and synaptically released glutamate. Also,we considered the generation of IP3 via gap junction dif-fusion from neighboring astrocytes in a nonlinear cou-pling manner [37]. JG can be described by the followingtwo equations:
JG = −Fex2
1 + tanhΔijIP3 − IPthr
3ω
! !sgn ΔijIP3
� �, ð9Þ
ΔijIP3 = IP3½ �i − IP3½ �j, ð10Þ
where IPthr3 is the threshold of IP3 diffusion at 0.3μM, ω is
the scaling factor of diffusion, and Fex is the IP3permeability.
Finally, the dynamic equations of astrocytic IP3ðtÞ areintegrated into the Li–Rinzel model to describe the kineticsand properties of Ca2+ oscillations in astrocytes:
d Ca2+� �dt
= Jchannel − Jpump + J leak, ð11Þ
where the three terms denote the calcium flux from the endo-plasmic reticulum (ER) to the cytosol gated by IP3 (Jchannel),the leakage flux from the ER (J leak), and Ca2+ reuptake fromthe cytosol to the ER via ATP-dependent pumps (Jpump),respectively. For the sake of simplicity, the expansion ofterms is omitted and the three fluxes and parameters referto those in [38].
When IP3-induced elevation in Ca2+ concentrationexceeds the threshold Cθ at time t j, a fraction rAðt jÞ of astro-cytic glutamate resources can be released into the extracellu-lar space by vesicle exocytosis [39]. The fraction of readilyreleasable is given as follows:
rA tð Þ =UA∙xA tð Þ, ð12Þ
where UA is the resting glutamate release probability and xAis the fraction of available glutamate resources for release,according to the following:
dxAdt
=1 − xAτG
− rA tð Þ∙δ t − t j� �
, ð13Þ
where τG is the glutamate resource reintegration time con-stant. The astrocyte-derived glutamate concentration GA inthe extrasynaptic cleft is described as follows:
dGAdt
= −gcA∙GA + rA∙ϱe∙GT∙δ t − t j� �
∙ 1 + k∙ GABAex½ �ð Þ,ð14Þ
where gcA is the clearance rate of glutamate, ϱe denotes thevolume ratio of vesicles to periastrocytic space, and GT isthe total vesicular glutamate concentration in astrocytes.The parameters are given in Table 2.
2.3. Synapse Model. The effects of astrocytic glutamate andexogenous GABA on presynaptic terminals are mediated byactivation of mGluRs and GABABRs, respectively. Both ofthese receptor types regulate the amount of Ca2+ influx intopresynaptic terminals; the former promotes [40] and thelatter retards [41]. Release of vesicles is dependent on ele-vation in intracellular Ca2+ concentration [42]. Therefore,open proportion U of Ca2+ channels modulated by thetwo transmitters is a critical parameter that decides theprobability of presynaptic neurotransmitter release. Tomimic the effects of astrocytic glutamate and exogenousGABA comodulation on synaptic release probability, weused a modified kinetic model of synaptic release regulatedby astrocyte [28], which is based on the Tsodyks–Mark-ram (TM) phenomenological model of synaptic activity[43], and added GABA activation kinetic scheme. The cor-responding kinetic equation reads as follows:
U =U0 + ξ −U0ð Þ∙Γ −U0∙rGABAB, ð15Þ
where U0 is defined as a constant in the TM model [44],denoting the open ratio of Ca2+ channels induced byaction potentials, i.e., resting synaptic release probability.The second term is the effect of astrocyte-released glutamateon presynaptic glutamate receptors, in which Γ is the propor-tion of activated receptors and parameter ξ ð0 ≤ ξ ≤ 1Þ deter-mines the type and strength of the action of astrocytes onpresynaptic terminals. The third term denotes the probabilityof modulation of synaptic basal release by GABA, and rGABAB
is the fraction of GABAB-mediated Ca2+ channels that close.The probability that channels are open or closed is deter-mined by the following:
dΓdt
=OG∙GA∙ 1 − Γð Þ −ΩG∙Γ, ð16Þ
drGABAB
dt= αGABAB
∙ GABAex½ �∙ 1 − rGABAB
� �− βGABAB
∙rGABAB,
ð17Þwhere ½GABAex� and GA represent the concentrations ofexogenous GABA and astrocyte-released glutamate, respec-tively, given by Equation (8) and Equation (14). The param-eters OG and ΩG are the forward and backward rateconstants for astrocytic glutamate binding to excitatory pre-synaptic receptors, respectively, and αGABAB
and βGABABare
4 Computational and Mathematical Methods in Medicine
the rate constants of GABABR-mediated binding or unbind-ing, respectively.
Accordingly, the modified resting synaptic release proba-bility U will determine the fraction uS of available neuro-transmitter to be utilized occurring at each presynapticspike time tk:
duSdt
=−uSτfac
+U∙ 1 − uSð Þ∙δ t − tkð Þ, ð18Þ
where τfac is the relaxation time constant of facilitation anddetermines the rate of uS decaying to zero in the absence ofspikes. The fraction of total neurotransmitter available forrelease xS is described by the following:
dxSdt
=1 − xSτrec
− rS∙δ t − tkð Þ, ð19Þ
where τrec is the recovery time constant and determines thereintegration rate of resources to the available pool. rSdenotes the fraction of neurotransmitter effectively releasedinto the extracellular space, i.e., probability of neurotransmit-ter release (Pr), defined as rS = uS∙xS. The estimated gluta-mate concentration GS from presynaptic release can berepresented mathematically as follows:
dGSdt
= −gcS∙GS + rS∙ϱc∙YT∙δ t − tkð Þ, ð20Þ
where gcS is the clearance rate of glutamate via neuronal andastrocytic uptake or spillover (flowing out of the synapticcleft). ϱc is the vesicular to mixing volume ratio, and YT rep-resents the total vesicular glutamate concentration in the pre-synaptic bouton. Akin to those described in glutamate releaseprocess in Equations (18)–(20), when a single action poten-tial arrived at GABAergic synapse, GABA in synaptic vesiclesare released from presynaptic terminal and then diffuse in thesynaptic cleft, where they are retrieved by endocytosis [45].Because this study focused on the regulation of astrocyteson the excitatory neurons, the open proportion U in presyn-aptic terminal of GABAergic synapse was set a constant forsimplification.
Once vesicles are released from presynaptic terminals orastrocytes, glutamate diffuses across the synaptic cleft andbinds to postsynaptic AMPA and NMDA receptors, resultingin an increase in conductance. The dynamics of the synapticconductance gating variable rðtÞ in Equation (3) can be rep-resented using a two-state model [46]. Transition betweenclosed and open states of the channel is considered a Markovprocess based on the following diagram:
C + T
α
⇌β
O, ð21Þ
where C and O indicate the unbound and bound states of thechannel to neurotransmitter, respectively, T is the glutamate
Table 2: Parameters used in the astrocyte model [24, 28, 37].
Parameter Description Value
IP3∗ Steady-state concentration of IP3 0.16μM
τIP3 IP3 degradation constant 7 s
vIP3gaba Rate of IP3 production via GABA 0.1625μMs-1
vIP3glu Rate of IP3 production via glutamate 0.062μMs-1
n1 GABA Hill coefficient 0.3
n2 Glutamate Hill coefficient 0.2
knGABA Dissociation constant for GABA-stimulated IP3 production 0.6 μM
knglu Dissociation constant for glutamate-stimulated IP3 production 0.78μM
IPthr3 Threshold for IP3 diffusion 0.3 μM
Fex IP3 permeability 0.09μMs-1
ω Scaling factor of IP3 diffusion 0.05μM
Cθ Ca2+ threshold for exocytosis 0.2 μM
τG Glutamate reintegration time constant 1.66 s
UA Resting glutamate release probability 0.6
ϱe Volume ratio of vesicles to periastrocytic space 6:5 × 10−4
gcA Clearance rate of astrocytic glutamate 60 s-1
GT Total vesicular glutamate concentration 200mM
k GABA concentration-dependent proportionality coefficient 0.3
5Computational and Mathematical Methods in Medicine
or GABA concentration, and α and β are the forward andbackward rate constants, respectively. The two-state kineticmodel describes the process of direct receptor binding toneurotransmitter, changing the ion channel. The fraction ofthe receptors in the open state r is described by the first-order dynamic equation:
drdt
= α∙ T½ �∙ 1 − rð Þ − β∙r: ð22Þ
Finally, NMDA-, AMPA-, and GABAA-mediated post-synaptic currents are expressed as functions of rðtÞ and themembrane potential v by the following equations [46]:
IAMPA = �gAMPA∙r tð Þ∙ v − EAMPAð Þ, ð23Þ
INMDA = �gNMDA∙Mg vð Þ∙r tð Þ∙ v − ENMDAð Þ, ð24Þ
IGABAa= �gGABAA
∙r tð Þ∙ v − EGABAA
� �, ð25Þ
where �g is the maximum conductance, �gAMPA = 0:35nS,�gNMDA = 0:1nS, �gGABAA
= 0:62nS, v is the postsynapticmembrane potential, E is the synaptic reversal potential,EAMPA = ENMDA = 0mV, and EGABAA
= −80mV. Notably, aunique property of NMDA currents is that NMDA receptorscontain a voltage-dependent term representing magnesium(Mg2+) block at ½Mg2+� = 1mM [47]:
Mg vð Þ = 11 + exp −0:062 ∗ vð Þ Mg2+
� �/3:57
: ð26Þ
The slow inward current Isic elicited by astrocytic gluta-mate can be defined as follows [48, 49]:
Isic tð Þ = INMDA tð Þ + IAMPA tð Þ: ð27Þ
In this fashion, postsynaptic mechanisms of astrocyticglutamate regulation mediated by NMDA and AMPAreceptors complicate the characterization of regular post-synaptic currents by glutamate released from presynapticterminals. Thus, the two types of excitatory events (SICsand EPSCs) can be distinguished according to their origin,astrocyte-derived glutamate GA (Equation (14)) and syn-aptically released glutamate GS (Equation (20)), respec-tively. Noted worthily, although the NMDA receptorsthat mediate SICs and EPSCs are different [50], NMDAreceptor subtypes are often not distinguished in the com-putational models for simplicity. The parameters for thesynapse model are listed in Table 3.
2.4. Neuronal–Astrocyte Network. Based on the networkcomponents mentioned above, we simulated a two-dimensional network model (Figure 2). Each neuron andastrocyte were assigned a spatial location in a square gridwith dimensions of 10 × 10mm planar units, providing asuitable physical arrangement for signaling transmissionamong them [51]. The neuronal network model comprises400 excitatory and 100 inhibitory neurons with 0.2 connec-tion probability as well as 400 astrocytes. Experimentalresults show that individual cortical astrocyte can enwrapthe nearby neurons within its territory and modulate the syn-apses associated with hundreds of dendrites [52]. According
Table 3: Parameters used in the synapse model [28, 46].
Parameter Description Value
U0 Resting synaptic release probability 0.3
ξ Type and strength of astrocytes acting on presynapses 0.8
OG Activation rate of astrocytic glutamate 1.5M-1 s-1
ΩG Inactivation rate of astrocytic glutamate 0.2 s-1
αGABAB GABAB forward rate constant 16 μM-1 s-1
βGABAB GABAB backward rate constant 6 s-1
τfac Facilitation time constant 0.3 s
τrec Recovery time constant 0.5 s
gcS Presynaptic glutamate clearance rate 40 s-1
ϱc Vesicular to mixed volume ratio 0.005
YT Total vesicular glutamate concentration 500mM
αAMPA AMPA forward rate constant 1.1 μM-1 s-1
βAMPA AMPA backward rate constant 190 s-1
αNMDA NMDA forward rate constant 0.072μM-1 s-1
βNMDA NMDA backward rate constant 6.6 s-1
αGABAA GABAA forward rate constant 0.53 μM-1 s-1
βGABAA GABAA backward rate constant 180 s-1
guptakeGABA GABA reuptake rate constant 6 s-1
6 Computational and Mathematical Methods in Medicine
to their spatial coordinates, therefore, we calculated the min-imum distance between astrocytes and the pre- and postsyn-aptic neurons and then determined which astrocyte shouldbe responsible for this synapse. Connection between astro-cytes was also based on their coordinates. Considering amean numerical density of astrocytes of 17,575 cells permm3 in the rat visual cortex [53], average intersomal distancebetween astrocytes was set 675μm in our model by enlargingin proportion. In this way, an astrocyte can be connected toabout 100 nearby excitatory synapses and 4 astrocytes onaverage. Multiple variables were calculated and recorded dur-ing each 10 s simulation, including neuronal firing, astrocyticcalcium oscillations, and excitatory presynaptic release. Themodel was implemented in the Brian 2.0 simulator [54, 55].
3. Results
3.1. Astrocytic Glutamate Increases the Average NetworkFiring Rate. We began our analysis by describing the firingbehavior of the neurons. The neural-astrocytic network wastreated with increasing GABA concentrations (0, 1, 5, and10μM). A quarter of the neurons were chosen for clarifica-tion using raster plot (Figure 3). The network firing rate,measured as an average across the spikes of all neurons (timebin width Δt = 1ms), is shown in the bottom panel of the ras-ter plot of neuronal firing. Furthermore, we fixed inhibitorysynaptic release probability at a constant of 0.5 [56] to ignorethe effect of GABABR-mediated disinhibition at inhibitorypresynaptic terminals. In order to produce weak and stronginhibitory effects on excitatory inputs, maximal conductanceof GABAA increases with the change of exogenous concen-tration depending on the experimental results [13].
Figures 3(a)–3(d) (top panel) show that the firing activity ofthe excitatory (red) and inhibitory (blue) neurons duringtreatment with different concentrations of GABA decreased,in line with the previous experimental observation that weperformed acute GABA treatment on the cortical cultures[57]. The release time of gliotransmitter glutamate (green)from astrocytes was significantly affected. Astrocytic gluta-mate triggered synchronized responses in the neurons, whichincreased the average network firing rate (bottom panel). Thesynchronized neuronal activities are consistent with theresults of electrophysiological experiments [50, 58] and com-putational simulation [59], which imply that astrocytes havea vital role in modulation the firing of the neurons.
3.2. GABA-Activated Astrocytes Augment CalciumOscillations and Glutamate Release. The key modelparameters for GABA-induced astrocytic activity areshown in Figure 4, including the intracellular cascade ofIP3-mediated Ca2+ response and Ca2+-dependent gluta-mate release. The dynamic changes in intracellular Ca2+
in astrocyte were regulated by the integration of theexogenous GABA and presynaptic glutamate signalingpathways (Figure 1), akin to spatial and temporal integra-tors [60]. With the exogenous GABA concentrationincreased (½GABAex� = 0, 1, 5, and 10 μM), the concentra-tion of glutamate resulting from presynaptic releasedecreased significantly (Figure 4(a)), while the activation ofglutamate receptors on astrocyte increased (Figure 4(b)).This may be attributed to the amplification of mGluRs viacrosstalk with the GABABRs [29]. In these cases, comparedto the condition without exogenous stimuli (Figure 4(c),black line), the time course of IP3 production differed acrossthe three paradigms due to the increase of GABA concentra-tions (½GABAex� = 1, 5, and 10μM) in the synaptic cleft(Figure 4(c), colored lines), which resulted in greater Ca2+
oscillatory behavior (Figure 4(d)). In the Ca2+ variationcurve, the single pulse elevation may arise from direct stimu-lation with exogenous GABA, and the subsequent long-lasting Ca2+ oscillations could be evoked by glutamate [27].Indeed, the astrocytic Ca2+ response closely depends on theamount of glutamate released from presynaptic terminals inthe synaptic cleft [51]. Although the neurons were sup-pressed, an amplification interaction between the GABAand glutamate signaling pathways augmented the amplitudeand frequency of Ca2+ oscillations in astrocyte. Simulatedresults indicate that a higher concentration of GABA leadsto a stronger and prolonged Ca2+ wave, which changes therate of glutamate release (Figure 4(e)). Nevertheless, thereare no experimental data available to precisely quantitatethe effect of these two receptors. Further experimentalstudies are required to elucidate the exact mechanismand quantitative properties of Ca2+ oscillations evoked bymultiple neurotransmitters.
3.3. Enhanced Astrocytic Glutamate Promotes PresynapticRelease. The variations in presynaptic release probability inour model were regulated by astrocytic glutamate and exoge-nous GABA (Figure 1). As a contrast, the direct response ofpresynaptic terminal to exogenous GABA in the absence of
0
0
2
4
6
8
2 4 6 8
Figure 2: The 2D neural astrocytic network model. Astrocytes, theinhibitory neurons, and the excitatory neurons are represented bygreen stars and blue and red dots, respectively. Green linesrepresent astrocyte network and the neuronal connections areomitted for clarity.
7Computational and Mathematical Methods in Medicine
astrocytic modulation is shown in Figure 5(a). The releaseprobability Pr significantly decreased with GABA inhibition,while Pr in central excitatory synapses was between 0.1 and0.9 [61]. This indicates that increased GABA concentrationsrobustly inhibit excitatory presynaptic release [31]. In thepresence of astrocyte-derived glutamate, the scenario couldbe reversed (Figure 5(b)). In particular, the regulation of pre-synaptic release by astrocyte-derived glutamate was signifi-cantly enhanced as the concentration of exogenous GABAincreased. The rate of astrocytic glutamate potentiation waslarger than the rate of GABA depression, which gave rise toan increase in Pr, illustrating that glutamate exocytosis fromastrocytes promoted presynaptic transmitter release [40].Furthermore, the variations in excitatory postsynaptic cur-rents (EPSCs) elicited by presynaptic glutamate release areconsistent with the quantal size variability in presynaptic ves-icle release (Figure 6). The results suggest that exogenousGABA causes a large decrease in the EPSC amplitude [14],whereas the release of astrocytic glutamate enhances excit-atory postsynaptic currents.
3.4. Astrocyte-Mediated Slow Inward Currents IncreasePostsynaptic Neuronal Excitability. Both excitatory andinhibitory inputs to the postsynaptic neurons are accompa-nied by a decrease in impedance and an increase in mem-brane conductance [33]. The inhibitory and excitatorysynaptic conductance values gi and ge were consistently bal-anced at a certain ratio [62]; the former was several fold
higher than the latter [63]. In our model, postsynapticevents were characterized by changes in current insteadof changes in conductance. To investigate the impact ofNMDAR- and AMPAR-mediated SICs on the postsynapticneurons, we removed the excitatory effect of regular syn-aptic currents driven by synaptically released glutamate.These results could be the direct response of the postsyn-aptic neuron to astrocyte.
GABA-evoked astrocytic Ca2+ oscillations triggered therelease of glutamate (Figure 7(a)), which diffused into theextracellular space and bound to post-extrasynaptic gluta-mate receptors, eliciting depolarizing SICs (Figures 7(b)–7(e)). Astrocyte-mediated SICs caused membrane depolari-zation in the postsynaptic neurons, which was sufficient toreach the firing threshold and ultimately influenced thetime course of postsynaptic neuronal firing (Figure 7).Pathological conditions, such as neuronal depolarization,termed paroxysmal depolarization shifts, may contributeto seizure generation [64]. Indeed, intracellular Ca2+ eleva-tion in astrocytes is a crucial factor in the regulation ofSICs, which has been previously reported in in vitro [49]and numerical studies [48, 65]. Moreover, the SIC ampli-tude is determined by the extracellular glutamate concen-tration. Both an enhanced astrocytic glutamate release anda downregulated glutamate clearance rate can dramaticallymodulate neuronal excitability, therefore increasing seizuresusceptibility [48, 66, 67]. The results of this simulationshow that the higher concentration of exogenous GABA is
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Figure 3: The raster plot and mean firing rate of neural activity during treatment with different concentrations of GABA: (a) 0μM, (b) 1 μM,(c) 5 μM, and (d) 10 μM. The top panel shows astrocytic glutamate release (green) and excitatory (red) and inhibitory (blue) neuronal firing.The average network firing rate is shown at the bottom.
8 Computational and Mathematical Methods in Medicine
injected to neuronal-astrocytic network, the larger ampli-tude of astrocyte-mediated SICs is evoked due to theenhanced release of astrocytic glutamate. The increase inexcitability could partly counteract the resulting decreasein EPSCs induced by presynaptic glutamate, thereby fine-tuning neuronal network excitation.
4. Discussion and Conclusions
Numerous in vitro and in vivo studies indicate that astrocytesplay a vital role in neuronal excitability and synaptic trans-mission. Based on neurophysiological findings [20], wedeveloped a biophysical neuronal-astrocytic network modelto quantitatively analyze the impact of astrocytes on themodulation of neuronal excitability at different concentra-tions of exogenous GABA. This is an important issue in viewof recent studies regarding the mechanism of the GABAergicneuron to astrocyte signaling, and the simulation results giveimportant clues to the involved GABA signaling mechanism.
Our research yielded several predictions. Firstly, theperiod and amplitude of Ca2+ signaling were dramatically
elevated by increasing GABA dose strength. Without exoge-nous stimulus, i.e., ½GABAex� = 0 μM, neuronal activity canalso trigger relatively slower astrocytic Ca2+ oscillations. Infact, astrocyte Ca2+ events occur in a slow and prolongedmanner under physiological condition [68]. This is probablydue to the full intracellular cascade of glutamate-activatedastrocyte requiring successive events including astrocyticmGluR activation, second messenger IP3 production, astro-cyte Ca2+ elevation, and astrocytic gliotransmitter release.Consistent with [20], our results show that exogenous GABA
Syn.
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Figure 4: The dynamic response of astrocyte to presynapticglutamate and exogenous GABA. (a) The amount ofpresynaptically released glutamate (Glusyn) decreases as theconcentration of exogenous GABA increases, but (b) there isgreater activation of mGluRs on astrocyte due to the crosstalkwith GABAB receptors, which results in (c) IP3 production and (d)IP3-gated [Ca2+] elevation. (e) Once the [Ca2+] threshold has beenreached (0.2 μM, red dashed line), astrocyte releases glutamate(Gluastro) into the extracellular space. To avoid triggeringrepeatedly the release in all the time steps, gliotransmitter releaseevent is triggered only once in threshold conditions, a mechanismsimilar to the refractory period of the neuron.
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Figure 5: Variations in presynaptic glutamate release probability(Pr) during (a) astrocyte-independent and (b) astrocyte-dependentpathways. The dots represent each presynaptic release event.
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Figure 6: Variations in excitatory postsynaptic currents (EPSCs)induced by glutamate released from presynaptic terminal duringtreatment with different concentrations of GABA: (a) 0 μM, (b)1 μM, (c) 5 μM, and (d) 10μM.
9Computational and Mathematical Methods in Medicine
produces a concentration-dependent increase in the eleva-tion rate of astrocyte Ca2+.
It is known that astrocytes release glutamate through mul-tiple pathways [69], in which Ca2+-dependent exocytosis maybe the most widely examined. Considering the spontaneousastrocytic Ca2+ oscillations [70], neuronal activity, or externalstimulus-induced Ca2+ increase [71], there seems to be a con-sensus that increased Ca2+ concentration results in glutamaterelease and subsequent neuronal regulation. Yet, Ca2+-depen-dent mechanism remains a controversial topic. Recent studysuggests that Ca2+-dependent gliotransmission is a pharmaco-logical phenomenon rather than a physiological process [72].Given the complexity of astrocyte Ca2+ activity [73], it is notsurprising that the literatures report some discrepant resultsregarding the mechanisms of glutamate release and gliotrans-mission. Although the mechanism is not fully settled, a num-ber of published astrocyte computational models have beenbased on astrocytic Ca2+ signaling [35]. Our model also con-siders Ca2+-dependentmechanism as themethod of glutamaterelease from astrocytes. Numerical results in this paper verifyastrocyte function as the conversion of GABA inhibition intoglutamatergic excitation, in which astrocytic Ca2+ activity is
a crucial factor contributing to network excitability. Althoughthe release of synaptic glutamate was inhibited by GABA,astrocyte Ca2+ response in this model was enhanced, whichmay be the amplifying interaction between the GABABR andmGluR-induced Ca2+ signaling [29]. If these are further veri-fied by experimentation, irrespective of which neurotransmit-ter excites astrocytes, an intervention in Ca2+ signaling may bea potential mechanism for some drug targets. For instance,anticonvulsants reduce neuronal excitability by blockingastrocytic Ca2+ signaling [64].
Another significant finding of our model is that GABAand astrocytic glutamate play opposing roles at excitatorypresynaptic terminals. We used neurotransmitter depletionand reintegration to describe the modulation of basal releaseprobability by these two factors. The net effect of GABA andastrocytic glutamate on excitatory presynaptic terminals isdue to the coexistence of mixed scenarios. The results suggestthat the enhanced release of astrocytic glutamate results instronger vesicle recruitment and release as compared withthe inhibitory effect of GABA. From the perspective of infor-mation transmission, the probability of presynaptic gluta-mate release depends on the pattern of presynaptic activitythat integrates the contradictory effects of GABA and astro-cytic glutamate. Thus, the synapse can be thought of as a filter[74], determining whether synaptic information is transmit-ted to the postsynaptic neurons.
Our model illustrates that astrocyte-elicited SIC by acti-vation of extrasynaptically located postsynaptic NMDAreceptors is the other mechanism that affects neuronal activ-ity. Unlike the results of postsynaptic high frequency activityproduced by SICs in pathological conditions [48, 66, 75],SIC-mediated increase of neuronal excitability in our studyseems to be slightly in modulation of amplitude and period.This scenario may result from integrated inhibitory inputsto the postsynaptic neurons, which are connected to about20 inhibitory neurons in our model. In fact, SICs can beamplified by upregulating mGluRs on astrocytes, but thiswas commonly observed in pathological tissues such as epi-lepsy [76]. Although it is hard to fight against the extraamount of GABA, these results evidence the involvement ofastrocytes in modulation of neuronal activity, suggesting thatastrocytic glutamate mediates the increase of excitation in theneural-astrocyte network.
Overall, the model described in the present study attemptsto mimic the bidirectional neuron-astrocyte interaction underexogenous stimuli. These results demonstrate the involvementof astrocytes in synaptic transmission and suggest the excit-atory drive of GABA-activated astrocytes in the neuronal net-work. Indeed, there exists a fine balance between excitationand inhibition in the central nervous system. Astrocytes havethe potential ability to modulate synaptic activity and main-tain this excitatory-inhibitory balance in particular brainstates, which is far-reaching within brain function.
Data Availability
The data in this study are available on request from the cor-responding author. The data are not publicly available due toprivacy or ethical restrictions.
–5
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Figure 7: Astrocyte-mediated SICs and the change in postsynapticmembrane potential. (a) Enhanced glutamate release by astrocytesdue to the elevated concentrations of GABA ((b–e) ½GABAex� =0μM, 1μM, 5μM, and 10μM) results in a stronger Isic andsubsequently makes the neurons more excitable to resist theincreased GABA inhibition.
10 Computational and Mathematical Methods in Medicine
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper.
Acknowledgments
This study was supported by the National Natural ScienceFoundation of China (Grant nos. 61703133 and 61673158)and the Hebei Province Nature Science Fund for YoungScholars (Grant no. F2018201070).
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13Computational and Mathematical Methods in Medicine
