GABA Regulation of Burst Firing in Hippocampal Astrocyte Neural Circuit: A Biophysical Model

1. Introduction

Spiking neural networks (SNNs) are considered to be the most biologically plausible representation of brain function (Ghosh-dastidar and Adeli, 2009). Additionally, SNNs capture a Hebbian type learning paradigm where the timing between pre- and post-synaptic spikes dictates whether synaptic depression or potentiation occurs (Song et al., 2000). SNNs have also been shown to be effective in time series prediction (Reid et al., 2014), spatiotemporal pattern recognition (Hu et al., 2013), and system control (Liu et al., 2015) in various application domains. In SNNs, the neurons and synapses are fundamental components in the network, where the information is encoded in spikes or action potentials for transmission between neurons (Izhikevich, 2003). In the central nervous system neurons receive input stimuli and respond by firing spike patterns such as bursting, which has been observed in the hippocampus of rodents (Miles and Wong, 1986), electric fish (Gabbiani et al., 1996), and in the primary motor cortex, brainstem and thalamus within the somatomotor system of humans (Arichi et al., 2017). The bursts can, in some cases, represent normal brain function and in other cases abnormal brain function (e.g., epilepsy) (Araque et al., 1999; Halassa et al., 2007).

Research has shown that astrocytes, one type of glial cell, modulate neuronal activity (Halassa et al., 2007; Breslin et al., 2018; Flanagan et al., 2018) where a single astrocyte may enwrap a large number of synapses (~10⁵ synapses), and connect to several neighboring neurons (four-eight). The interplay between an astrocyte and the neighboring neurons is believed to occur at the tripartite synapse (Araque et al., 1999), which is bi-directional and serves, in some cases, to modulate the synaptic transmission probability rate (PR): via the direct/indirect retrograde signaling messenger endocannabinoids (Wade et al., 2012). This gives rise to re-modeling of the SNN connectivity (Wade et al., 2011; Naeem et al., 2015; Johnson et al., 2018; Liu et al., 2018).

It has also been reported that gamma-aminobutyric acidergic (GABA) interneurons participate in astrocyte-mediated control of excitatory synaptic transmission (Perea et al., 2016) and exercises control over the firing frequency of pyramidal cells. Furthermore GABA release synchronizes principal cell population discharge contributing to the generation of rhythmic activity in neuronal networks, such as theta and gamma frequency oscillations (Kullmann, 2011). A recent paper reported that GABA released in proximity to a tripartite synapse can activate GABA-B receptors on the astrocyte leading to gliotransmission, which is known to regulate synaptic transmission probability (Liu et al., 2018). The research reported in Kurosinski and Götz (2002), Kullmann (2011), Liu et al. (2018) provides the underpinning for the work presented here.

In this paper, we investigate the coupling between a GABA interneuron, an astrocyte terminal and the pre and postsynaptic terminals. The main contributions of this paper include (i) a novel biophysical model that describes the signaling pathways at the tripartite synapse and (ii) a novel mechanism that can potentially explain postsynaptic neuron burst firing. The rest of paper is organized as follows. Section 2 presents the biophysical model while section 3 provides simulation results that demonstrate the bursting. Section 4 concludes the paper and discusses future work.

2. Biophysical Model of a Network Bursting

In this section, a detailed discussion of the signaling pathways at the tripartite synapse is presented with a specific focus on GABA signaling between the presynaptic terminal and the nearby astrocyte. It will be shown that this interplay acts as a frequency dependent switch, which modulates the probability of release (PR) at the presynaptic terminal. Our Ca²⁺ dynamics model shows that calcium (Ca²⁺) oscillations only occur over a range of inositol 1, 4, 5-trisphosphate (IP₃) concentrations and furthermore this paper will show that Ca²⁺ oscillations are periodic and this behavior is key to the bursting behavior.

2.1. Signaling Pathways and Activity Regulations

The conventional tripartite synapse has three terminals: the presynaptic axon, postsynaptic dendrite and the astrocyte cell (Wade et al., 2012; Liu et al., 2018). In this paper we consider earlier work where, in a hippocampal astrocyte neural network, GABA interneurons interact with excitatory tripartite synapses to dynamically change the synaptic transmission behavior from inhibitory to excitatory through modulation of PR (Perea et al., 2016). The signaling pathways between the GABA interneuron and tripartite synapse are shown in Figure 1. When an input stimulus of frequency (f_pre) is present at the excitatory presynaptic axon, neurotransmitter (glutamate) is released into the cleft and subsequently binds to receptors at the postsynaptic dendrite causing the depolarization of the postsynaptic neuron. While the authors accept that fast-spiking interneurons can fire at much higher frequencies than glutamatergic neurons, in this work we assume for simplicity that the firing rate of the GABA interneuron (f_GABA) follows f_pre, as the most likely physiological condition would be the activation of GABA interneuron by activation of glutamatergic axons (Serrano et al., 2006; Covelo and Araque, 2018). While GABA initially binds to GABA-A receptors inhibiting synaptic transmission and post-synaptic neuronal activity, recent work (Perea et al., 2016) has shown that with repeated firing of GABA interneurons, GABA also binds to GABA-B receptors on the astrocyte membrane, resulting in a switch from inhibition to excitation at the presynaptic terminal, and an associated excitatory response at the postsynaptic terminal. In this paper we focus on astrocyte-mediated GABA-induced excitation since the postsynaptic inhibition was found negligible, and the transient acute presynaptic inhibition was overpowered by the astrocyte signaling during sustained activity (Perea et al., 2016). Hence, our model dos not incorporate these negligible or transient inhibitory effects, focusing in the sustained mechanisms and effects of inhibitory signaling through astrocyte activation.

FIGURE 1

Figure 1. Signaling pathways at the tripartite synapse where GABA released from GABA interneuron binds to GABA-B receptors on the astrocyte membrane and $I{P}_3^{GABA}$ is released into the astrocyte cytosol. 2-AG released from the postsynaptic neuron binds to the receptors on the astrocyte membrane triggering the generation of $I{P}_3^{AG}$. With f_post low and IP₃ reached a level, IP₃ induces Ca²⁺ release from the ER and gliotransmitter is released into the synaptic cleft [Glu (e-SP) pathway] where it binds to mGluR receptors on the presynaptic membrane. This causes an increase in PR and the plasticity window opens.

As f_GABA increases, the GABA concentration level in the extracellular space increases, and a level is reached whereby binding to GABA-B receptors on the astrocyte membrane commences, leading to the production of IP₃: we subsequently refer to IP₃ due to GABA as $I{P}_3^{GABA}$, which contributes to the overall cytosolic IP₃ (Perea et al., 2016). $I{P}_3^{GABA}$ is a secondary messenger which is degraded when released into the cytoplasm: initially cytosolic Ca²⁺ and IP₃ levels are low and therefore degradation of IP₃ will also be low, as the degradation rate correlates with both Ca²⁺ and IP₃ concentrations. This degradation is gradually overcome with increasing levels of GABA and the PLCδ signaling pathway, which is modulated by Ca²⁺ and is accounted for in this work. Finally $I{P}_3^{GABA}$ starts to bind to IP₃ receptors (IP₃R_s) on the Endoplasmic Reticulum (ER). When the total cytosolic IP₃ is sufficiently high, Ca²⁺ is released from the ER (De Pittà et al., 2009). At some point both IP₃ and Ca²⁺ reaches a level at which an oscillating Calcium-Induced Calcium Release (CICR) occurs from the ER (Marchant et al., 1999): hereafter referred to as T_CICR. Several mechanisms are believed to contribute to Ca²⁺ oscillation but there is still much debate around this topic. For example, IP₃R_s have binding sites for both IP₃ and Ca²⁺, and Ca²⁺ release from the ER is believed to rely on coincidence binding of these ions. The time between IP₃ and Ca²⁺ binding depends on the concentration of these ions and therefore this could explain why Ca²⁺ is believed to be a regulator of IP₃R_s activity: at low Ca²⁺ levels IP₃R_s activity is increased, whereas the opposite is true at high Ca²⁺ levels. This is in agreement with other research (Dawson, 1997). However, there is not enough experimental evidence on these receptors to formulate a sufficiently detailed model. Therefore, in this work we revert to a hitherto accepted model (Perea et al., 2016) where Ca²⁺ oscillatory behavior is believed to arise from the feedback interplay between Ca²⁺, IP₃, and IP₃ degradation. As Ca²⁺ and IP₃ rapidly increase there is a complex dependency between the concentrations of both Ca²⁺ and IP₃ and Ca²⁺/IP₃-induced degradation of IP₃, which is the dominant process at elevated Ca²⁺/IP₃ levels. Therefore, a transient elevation of Ca²⁺ and/or IP₃ is followed by a rapid drop in IP₃, which can reduce IP₃ to below T_CICR. At this point degradation of IP₃ is weak because both the Ca²⁺ and IP₃ levels have fallen and therefore IP₃ starts to increase again due to $I{P}_3^{GABA}$. When the T_CICR level is reached again a transient elevation of Ca²⁺ re-occurs. We will demonstrate that our results support this behavior. This oscillatory behavior causes the release of the glutamate from the astrocyte (gliotransmitter) into the synapse [see Glu (e-SP) pathway in Figure 1], which binds to pre-synaptic group I metabotropic Glutamate Receptors (mGluRs) at the presynaptic terminal. This signaling pathway results in an increase in PR at the presynaptic terminal (Navarrete and Araque, 2010).

As PR increases more glutamate is released into the cleft and potentiation/depression of the synaptic weight can commence with the availability of glutamate to bind to N-methyl-D-aspartate (NMDA)-type glutamate receptors (Lüscher and Malenka, 2012). While we acknowledge that the biophysical mechanisms regulating the functional dependency between PR and plasticity are complex and not fully understood, we propose that PR acts as a “switch” which can turn on/off potentiation/depression at synaptic sites. To formulate a tractable mathematical model that captures this relationship we modulate the height of the Spike Timing Dependent Plasticity (STDP) associated plasticity window using PR: with PR≥PR^* (PR^* is defined as the plasticity activation level) the plasticity window fully opens and with PR<PR^* the plasticity window closes. The decision on whether potentiation or depression occurs is governed by the STDP rule (Magee and Johnston, 1997) where potentiation occurs when the presynaptic spike precedes postsynaptic spike, otherwise depression occurs. Additionally, we consider the case where the postsynaptic neuron is sufficiently depolarized such that the retrograde messenger 2-arachidonyl glycerol (2-AG) is released from the postsynaptic neuron. Since the contribution of 2-AG signaling to the observed GABA-mediated regulatory effects of astrocytes on excitatory transmission is negligible (Perea et al., 2016), the authors take the view that 2-AG signaling onto GABAergic terminals would not be a significant factor in network bursting. However, we do consider 2-AG binding to type 1 Cannabinoid Receptors (CB1Rs) on the astrocyte membrane which then initiates the release of the IP₃ into the cytoplasm of the astrocyte: we denote this secondary messenger as $I{P}_3^{AG}$.

During the synapse learning phase, the frequency of the postsynaptic neuron, f_post, is increasing, as is the 2-AG signal and consequently $I{P}_3^{AG}$. As $I{P}_3^{AG}$ contributes to the total IP₃, IP₃ will eventually reach a level where degradation of IP₃ no longer reduces IP₃ (and therefore Ca²⁺) to below T_CICR. In this instance, both the oscillatory Ca²⁺ transient and Glu (e-SP) pathways cease (Liu et al., 2018) (see Figure 2). In addition, the released 2-AG also binds to CB1Rs on the presynaptic terminal triggering the Suppression of Excitation (DSE) pathway, and results in a decrease in PR (Alger, 2002). Due to the reduction in the Glu (e-SP) pathway and the increase in the DSE pathway, PR decreases at the presynaptic terminal, the level of neurotransmitter in the cleft then falls to baseline and the frequency of the postsynaptic f_post diminishes which in turn causes $I{P}_3^{AG}$ to reduce. Furthermore, the total IP₃ degrades due to cytosolic degradation pathways including IP₃ 3-kinase $I{P}_3^{3K}$, and dephosphorylation by inositol polyphosphate 5-phosphatase ($I{P}_3^{5P}$) (see Equation 11). Together, these processes reduce IP₃ levels below T_CICR, and the rate of degradation diminishes sufficiently to allow IP₃ to increase again due to $I{P}_3^{GABA}$. When the T_CICR level is again exceeded, Ca²⁺ oscillations re-commence, the Glu (e-SP) pathway is re-established, PR increases and the level of neurotransmitter in the cleft is raised. In this post-learning phase PR cannot be elevated to a level where the plasticity window opens (PR<PR^*), as the postsynaptic neuron is active and therefore the 2-AG pathway leads to a reduction in PR due to the DSE pathway. Consequently, the postsynaptic neuron firing rate reaches a maximum when the T_CICR level is reached but it subsequently falls afterwards: a postsynaptic burst has occurred. This is followed by repeated bursts at each Ca²⁺ oscillatory period. We therefore propose that neuronal burst firing directly correlates with astrocytic Ca²⁺ oscillation.

FIGURE 2

Figure 2. Signaling pathways of the tripartite synapse where for high f_post T_CICR is reached and hence the Ca²⁺ oscillation and the Glu (e-SP) pathway ceases causing PR to fall and the plasticity window shuts: PR also falls due to the increase in the 2-AG (DSE) pathway. Under this condition the level of neurotransmitter in the cleft falls to baseline and f_post diminishes.

Moreover, it should be noted that the duration of the burst correlated with the frequency of presynaptic terminal f_pre at the excitatory presynaptic axon. As f_pre increases so does the rate of increase of IP₃ and the burst period is reduced. Therefore, network bursting is f_pre dependant and will only occur over a range of f_pre.

2.2. Postsynaptic Neuron Model

In this paper, the Leaky Integrate and Fire (LIF) model (Gerstner and Kistler, 2002) is used due to the relatively low computing requirement and minimal parameters tuning. The LIF model is given by

$\begin{array}{ll}{\tau }_m\frac{dv}{dt}=-v(t)+R_m{\displaystyle \sum _{i=1}^n}{I}_{syn}^i(t),& (1)\end{array}$

where τ_m is the neuron membrane time constant, v is the neuron membrane potential, R_m is the membrane resistance, $I_{syn}^i$ is the current injected to the neuron membrane by ith synapse, and n is the total number of synapses associated with the neuron. When the neuron membrane potential v is greater than the firing threshold value, v_th, the neuron fires and outputs a spike followed by a reset state or a refractory period (~2ms). The release of 2-AG correlates with the postsynaptic neuron activity (Naeem et al., 2015) and this is expressed as

$\begin{array}{ll}\frac{d(AG)}{dt}=\frac{-AG}{{\tau }_{AG}}+r_{AG}\delta (t-t_{sp}),& (2)\end{array}$

where AG denotes the released amount of 2-AG, τ_AG, and r_AG are the 2-AG decay and production rates, and t_sp is the postsynaptic spike time. The released 2-AG binds to the CB1Rs at the presynaptic terminal and at the astrocyte terminal, and this will be discussed in section 2.4.

2.3. GABA Interneuron

The spike train at the presynaptic axon also presents at the GABA interneuron causing the release of GABA neurotransmitter (Perea et al., 2016), which can be described by

$\begin{array}{ll}\frac{d(GABA)}{dt}=\frac{-GABA}{{\tau }_{GABA}}+r_{GABA}\delta (t-t_{sp}),& (3)\end{array}$

where GABA denotes the released amount of the neurotransmitter GABA, τ_GABA, and r_GABA are the GABA decay and production rates, and t_sp is the presynaptic spike arrival time. GABA binds to the GABA-B receptors at the astrocyte cell and this is modeled in the next subsection.

2.4. Astrocyte Cell

When GABA binds to GABA-B receptors on the astrocyte membrane, the amount of IP₃ released is given by

$\begin{array}{ll}\frac{d(I{P}_3^{GABA})}{dt}=\frac{I{P}_3^{GABA*}-I{P}_3^{GABA}}{{\tau }_{ip3}^{GABA}}+r_{ip3}^{GABA}GABA,& (4)\end{array}$

where $I{P}_3^{GABA}$ is the quantity of IP₃ generated by GABA within the cytoplasm, $I{P}_3^{GABA*}$ is the baseline GABA level, ${\tau }_{ip3}^{GABA}$ is the decay rate of $I{P}_3^{GABA}$ and $r_{ip3}^{GABA}$ is the production rate of $I{P}_3^{GABA}$. When the postsynaptic neuron fires, the released 2-AG can also trigger IP₃ generation (Wade et al., 2012) and this is modeled by

$\begin{array}{ll}\frac{d(I{P}_3^{AG})}{dt}=\frac{I{P}_3^{AG*}-I{P}_3^{AG}}{{\tau }_{ip3}^{AG}}+r_{ip3}^{AG}AG,& (5)\end{array}$

where $I{P}_3^{AG}$ is the quantity of IP₃ generated by 2-AG within the cytoplasm, $I{P}_3^{AG*}$ is the baseline level, ${\tau }_{ip3}^{AG}$ is decay rate of $I{P}_3^{AG}$, and $r_{ip3}^{AG}$ is production rate of $I{P}_3^{AG}$.

In addition, the IP₃ production is also increased by the hydrolysis of the highly phosphorylated membrane lipid phosphatidylinositol 4, 5–bisphosphate (PIP₂), such as the phosphoinositide-specific phospholipase C (PLC) isoenzyme of PLCδ (De Pittà et al., 2009). The PLCδ signaling is agonist independent and modulated by Ca²⁺ (De Pittà et al., 2009), and its activation rate can be modeled by

$\begin{array}{ll}PLC\delta =PLC{\delta }^{\prime }Hill(C{a}^{2+},K_{PLC\delta },2),& (6)\end{array}$

where the maximum PLCδ-dependant IP₃ production rate (De Pittà et al., 2009) can be modeled by

$\begin{array}{ll}PLC{\delta }^{\prime }=\overline{PLC{\delta }^{\prime }}/(1+I{P}_3/K_{\delta }),& (7)\end{array}$

and K_δ is the inhibition constant of PLCδ activity. The Hill function (De Pittà et al., 2009) is described by

$\begin{array}{ll}Hill(x,K,n)\equiv \frac{x^n}{x^n+K^n},& (8)\end{array}$

where n is the Hill coefficient and K is the midpoint of the Hill function, namely the value of x at which Hill(x, K, n)|_{x = K} = 1/2.

The degradation of IP₃ mainly occurs through phosphorylation into inositol 1, 3, 4, 5-tetrakisphosphate (IP₄), catalyzed by IP₃ 3-kinase (3K), and dephosphorylation by inositol polyphosphate 5-phosphatase (5P). The rate of IP₃ degradation by $I{P}_3^{5P}$ (De Pittà et al., 2009) can be modeled by

$\begin{array}{ll}I{P}_3^{5P}\approx {\overline{r}}_{5P}I{P}_3,& (9)\end{array}$

where ${\overline{r}}_{5P}$ is the IP₃ degradation rate by IP-5P. The activity of $I{P}_3^{3K}$ is regulated by Ca²⁺ in a complex fashion (De Pittà et al., 2009). The rate of IP₃ degradation by $I{P}_3^{3K}$ can be modeled by

$\begin{array}{ll}I{P}_3^{3K}={\overline{v}}_{3K}Hill(C{a}^{2+},K_D,4)Hill(I{P}_3,K_3,1),& (10)\end{array}$

where ${\overline{v}}_{3K}$ is the maximum degradation rate by $I{P}_3^{3K}$, K_D is the Ca²⁺ affinity of $I{P}_3^{3K}$, and K₃ is the IP₃ affinity of $I{P}_3^{3K}$. Based on the previous contributions of IP₃, the total IP₃ is given by

$\begin{array}{ll}I{P}_3=I{P}_3^{GABA}+I{P}_3^{AG}+PLC\delta -I{P}_3^{5P}-I{P}_3^{3K}.& (11)\end{array}$

The Li-Rinzel model (Li and Rinzel, 1994) is used to model the Ca²⁺ dynamics within the astrocyte cell. The model consists of three channels, J_chan, J_leak, and J_pump, where J_chan models the Ca²⁺ channel opening based on the mutual gating of the Ca²⁺ and IP₃, J_leak models the Ca²⁺ leakage from the ER into the cytoplasm and J_pump models how Ca²⁺ is pumped out from the cytoplasm into the ER via Sarco-Endoplasmic-Reticulum Ca²⁺-ATPase (SERCA) pumps. The Ca²⁺ model in the approach of De Pittà et al. (2009) is used in this work, and it is described by

$\begin{array}{llll}\frac{d(C{a}^{2+})}{dt}& =& J_{chan}(C{a}^{2+},h,I{P}_3)+J_{leak}(C{a}^{2+})-J_{pump}(C{a}^{2+}),& (12)\end{array}$

$\begin{array}{llll}\frac{dh}{dt}& =& \frac{h_{\infty }-h}{{\tau }_h},& (13)\end{array}$

where J_chan is Ca²⁺ release depending on the Ca²⁺ and IP₃ concentrations, J_pump is the amount of stored Ca²⁺ within the ER via the SERCA pumps, J_leak is the Ca²⁺ leaking out of the ER and h is the fraction of activated IP₃R_s. The parameters h_∞ and τ_h are given by

$\begin{array}{llll}{h}_{\infty }& =& \frac{Q_2}{Q_2+C{a}^{2+}},& (14)\end{array}$

$\begin{array}{llll}{\tau }_h& =& \frac{1}{a_2(Q_2+C{a}^{2+})},& (15)\end{array}$

where

$\begin{array}{ll}{Q}_2=d_2\frac{I{P}_3+d_1}{I{P}_3+d_3}.& (16)\end{array}$

J_chan is given by

$\begin{array}{ll}{J}_{chan}=r_C{m}_{\infty }^3{n}_{\infty }^3{h}_{\infty }^3(C_0-(1+C_1)C{a}^{2+}),& (17)\end{array}$

where r_C is the maximal Calcium-Induced Calcium Release (CICR) rate, C₀ is the total free Ca²⁺ cytosolic concentration, C₁ is the ER/cytoplasm volume ratio, and m_∞ and n_∞ are the IP₃ Induced Calcium Release (IICR) and CICR channels respectively, which are given by

$\begin{array}{ll}{m}_{\infty }=\frac{I{P}_3}{I{P}_3+d_1},& (18)\end{array}$

and

$\begin{array}{ll}{n}_{\infty }=\frac{C{a}^{2+}}{C{a}^{2+}+d_5}.& (19)\end{array}$

J_leak and J_pump are described by

$\begin{array}{ll}{J}_{leak}=r_L(C_0-(1+C_1)C{a}^{2+}),& (20)\end{array}$

and

$\begin{array}{ll}{J}_{pump}=v_{ER}\frac{{(C{a}^{2+})}^2}{k_{ER}^2+{(C{a}^{2+})}^2},& (21)\end{array}$

where r_L is the Ca²⁺ leakage rate, v_ER is the maximum SERCA pump uptake rate and k_ER is the SERCA pump activation constant.

The intracellular astrocytic calcium dynamics are used to regulate the release of glutamate from the astrocyte: the Glu pathway. To model this release, it is assumed that when Ca²⁺ crosses the CICR threshold, a quantity of glutamate is released (Wade et al., 2012). It is described by

$\begin{array}{ll}\frac{d(Glu)}{dt}=\frac{-Glu}{{\tau }_{Glu}}+r_{Glu}\delta (t-t_{Ca}),& (22)\end{array}$

where Glu is the quantity of released glutamate, τ_Glu is decay rate of glutamate, r_Glu is production rate of glutamate, and t_Ca is the time at which Ca²⁺ crosses the threshold. The released glutamate drives the generation of e-SP (Wade et al., 2012). The level of e-SP is modeled by

$\begin{array}{ll}{\tau }_{eSP}\frac{d(eSP)}{dt}=-Glu+m_{eSP}Glu(t),& (23)\end{array}$

where τ_eSP is the decay rate of Glu, and m_eSP is a constant weight used to control the height of e-SP. It shows that the e-SP level depends on the glutamate released from the astrocyte cell.

The model of DSE in the approach of Wade et al. (2012) is used to describe the relationship between the DSE and the released 2-AG from postsynaptic neuron. The DSE is assumed to change linearly with the cytosolic concentration of 2-AG, which is described by

$\begin{array}{ll}DSE=AG\times K_{AG},& (24)\end{array}$

where AG is the concentration of 2-AG and K_AG is the scaling factor for the DSE.

2.5. Synapse Model

For the synapse, a probabilistic model is employed which is based on the failure and success mechanisms of synaptic neurotransmitter release (Navarrete and Araque, 2010; Wade et al., 2012). A uniformly distributed pseudo-random number generator is used. If the generated random number rand is less than or equal to the PR, a current I_inj is injected into the neuron which is shown by

$\begin{array}{ll}{I}_{syn}^i(t)=\{\begin{array}{ll}{r}_I*w_{syn}^i(t),\hfill & rand\le PR\hfill \\ 0,\hfill & rand>PR\hfill \end{array}& (25)\end{array}$

where r_I is the current production rate, and $w_{syn}^i$ is the weight of the ith synapse. The associated PR of each synapse is determined by the DSE and e-SP together, which is given by

$\begin{array}{ll}PR(t)=PR(t_0)+DSE(t)/100+eSP(t)/100,& (26)\end{array}$

where PR(t₀) is the initial PR for each synapse. As discussed in section 2.1, the PR can switch on/off learning at the synaptic terminal by modulating the height of the plasticity learning window. The authors are not aware of any biophysical model that relates PR to the plasticity window weighting parameter A₀ and therefore in this work it is assumed that A₀ is modulated according to

$\begin{array}{ll}{A}_0=\{\begin{array}{ll}0,\hfill & PR\le P{R}^{\ast }\hfill \\ (PR-P{R}^{\ast })*r,\hfill & PR>P{R}^{\ast }\hfill \end{array}& (27)\end{array}$

where PR^* is the learning activation level and r is a constant value which controls the maximum height of the learning window. The STDP rule used in this approach to update the synaptic weights according to the timing difference between the post and presynaptic spikes is described by

$\begin{array}{ll}\delta w(\Delta t)=\{\begin{array}{ll}-A_{0}exp(\frac{\Delta t}{{\tau }_{+}}),\hfill & \Delta t\le 0\hfill \\ A_{0}exp(\frac{-\Delta t}{{\tau }_{-}}),\hfill & \Delta t>0\hfill \end{array}& (28)\end{array}$

where δw(Δt) is the weight update, Δt is the time difference between the post and presynaptic spikes, A₀ is the height of the plasticity window which limits the maximum levels of weight potentiation and depression, and τ₊ and τ₋ control the width of the plasticity window. A symmetrical plasticity window is assumed in this approach, and τ₊ = τ₋ = 40ms.

From the proposed models, it can be seen that if the f_pre is large enough, IP₃ is generated sufficiently to cause Ca²⁺ oscillations. Then the Ca²⁺-induced glutamate binds to the mGluRs receptors at the presynaptic terminal resulting in an increase of the synaptic transmission probability PR. Note that the authors wish to point out that astrocytes are believed to gate LTP and LTD by regulating glutamate levels in the synaptic cleft (Foncelle et al., 2018). Since there are many complex biophysical mechanisms involved in the regulation of glutamate, which are still under debate, the authors take the view that modulating the STDP plasticity window using PR is an effective way to capture this gating function. Elevating PR opens the synaptic plasticity learning window and over time f_post gradually increases which, via the 2-AG pathway, contributes to astrocytic IP₃ level until the Ca²⁺ oscillation stops. This is accompanied by a reduction in the Glu (e-SP) pathway and PR falls causing a reduction in f_post. Therefore, the bursting activity of the postsynaptic neuron is regulated by the GABA interneuron and the astrocyte cell. The results in the next section show the signaling pathways leading to a bursting postsynaptic neuron.

3. Results

This section provides simulation results which highlight the dynamic behavior at the synapse terminals and how the interactions between an astrocyte and GABA interneuron can give rise to bursting behavior. The MATLAB simulation platform is used in this work together with the Euler method with the time step of 1 ms. Tables A1, A2 give all the model parameters.

3.1. Bursting Output Spike Pattern

In this simulation both the presynaptic excitatory neuron and the GABA interneuron are stimulated by the same spike train at frequency f_pre = f_GABA which causes the release of GABA and glutamate (Perea et al., 2016). The presynaptic excitatory neuron/GABA interneuron stimulus is 40 Hz in the following simulations as this is sufficient to produce a cytosolic [IP₃]>0.5μM. With PR>PR^* (see Figure 3), a significant increase occurs in the level of neurotransmitter in the cleft, the learning window opens (Figure 4) and weight potentiation starts (Figure 5) resulting in postsynaptic firing. Note that in Figure 4, the plasticity window height parameter A₀ increases periodically with a corresponding potentiation of the synaptic weight (Figure 5). In our model the resting level for PR is 0.1 and based on the model in Equation (27), if PR>PR^* (PR^* = 0.45 in this work), the STDP learning window opens (A₀>0) at ~80 s, as shown in Figure 4. After the synaptic weight is potentiated, the synapse generates a depolarising current only when the input stimulus is presented at the presynaptic terminal and the PR value is greater than the value of a random number (see probabilistic-based synapse model in Equation 25): this current is injected into the postsynaptic LIF neuron. This injected current increases the postsynaptic potential and the neuron fires a spike if the membrane potential is greater than the firing threshold, v_th.

FIGURE 3

Figure 3. PR as a function of Glu (e-SP) and DSE signals over time. Note that the plasticity window will only be open when PR>PR^* and this can be observed in Figure 4.

FIGURE 4

Figure 4. Plasticity window height A₀ as a function of time. Note that PR acts as a switch to open/close the plasticity window controlling the learning period.

FIGURE 5

Figure 5. Time dependant synaptic weight update governed by the STDP learning rule. When the plasticity window is open, learning commences, the synaptic weight begins to potentiate and after a period of learning the window shuts off and the synaptic weight stabilizes.

After a period of learning the postsynaptic neuron activity has stabilized and PR drops sufficiently, toward the end of the first set of PR “spikes” (see Figure 3), closing the plasticity window (A₀ = 0) and the weight stabilizes to ~610 at 110 s, as shown in Figure 5: note that because the postsynaptic neuron is now active, PR<PR^* for all subsequent Ca²⁺ oscillations as the DSE pathway is also active. Figure 6 shows the amount of GABA released by the GABA interneuron as a function of time where, as expected, GABA increases gradually and then stabilizes at 0.027 μM under the input spike stimulus. $I{P}_3^{GABA}$ is shown in Figure 7 (blue) as a function of time and stabilizes at ~0.58μM which is consistent with the input stimuli profile. Figure 7 shows the other IP₃ sources that contribute to the total IP₃ in the cytosol.

FIGURE 6

Figure 6. GABA released from the GABA interneuron as a function of time. Under the input spike stimulus of f_GABA, GABA increases and then stabilized.

FIGURE 7

Figure 7. IP₃ dynamics within the astrocyte cell over time. The overall IP₃ includes contributions from $I{P}_3^{GABA}$, $I{P}_3^{AG}$, PLCδ, $I{P}_3^{5P}$, and $I{P}_3^{3K}$, where the degradations of IP₃, $I{P}_3^{5P}$, and $I{P}_3^{3K}$, are shown as negative values.

Initially the total IP₃ increases with $I{P}_3^{GABA}$ until the T_CICR level is reached triggering the release of Ca²⁺, as shown in Figure 8. We have observed from our model that T_CICR is consistent with an IP₃ level of approximately 0.5 μM and whenever IP₃ exceeds this threshold a transient elevation in Ca²⁺ occurs, as can be seen in Figure 8. Note however that as the IP₃ level increases with $I{P}_3^{AG}$ the degradation in IP₃ due to elevated Ca²⁺/IP₃ levels is insufficient to reduce IP₃ to below 0.5 μM and consequently the transient elevations of Ca²⁺ stops just after 100 s followed by a relatively slow degradation of IP₃ and Ca²⁺: these periodic bursts in Ca²⁺ gives rise to a Ca²⁺ oscillatory wave where the initial Ca²⁺ burst is longer due to synaptic potentiation. At the onset of each subsequent Ca²⁺ burst the IP₃ level drops sharply and we attribute this to strong dependence of $I{P}_3^{3K}$ on Ca²⁺ (Equation 10). As the Ca²⁺ level drops J_chan (Equation 17) reverses direction perturbing the rate of change in Ca²⁺ (Equation 12) and this causes a rapid increase in $I{P}_3^{3K}$ and a corresponding decrease in IP₃ (Equation 11).

FIGURE 8

Figure 8. Ca²⁺ oscillations in the astrocyte cell as a function of time. Note a longer oscillatory period at the start due to learning but thereafter the oscillator period stabilizes with constant on/off ratio.

The Ca²⁺ oscillation is initiated at ~20 s (T_CICR is exceeded; Figure 8) and this triggers the release of glutamate targeting group I mGluRs on the presynaptic terminal, i.e., Glu (e-SP) pathway is activated (Figure 9). Figure 9 shows that the Glu (e-SP) signal accumulates at each CICR and rapidly decays after the Ca²⁺ transients have ceased at ~120 s. Also the DSE pathway increases as the activity of the postsynaptic neuron is increasing, and competes with the Glu (e-SP) pathway to restrict PR to a relatively stable low value for all subsequent Ca²⁺ oscillations that occur post-learning, as shown in Figure 10. Again note a longer period of elevation of the DSE signal at the start due to synaptic potentiation but thereafter the DSE profile repeats in time.

FIGURE 9

Figure 9. Glu (e-SP) signaling pathway as a function of time. Note a longer period of elevation of Glu (e-SP) at the start due to synaptic potentiation but thereafter the Glu (e-SP) profile repeats in time.

FIGURE 10

Figure 10. DSE signaling pathway from the postsynaptic neuron, which correlates with the activity of the postsynaptic neuron.

Figure 11 shows the firing rate of the postsynaptic neuron, which is calculated based on a sliding time window of 10 s (blue) and 40 s (red), respectively. Note that the first burst reaches a higher level of postsynaptic neuron activity when compared to subsequent bursts. Also, between bursts the activity never falls back to zero. This is because the first burst occurs during the weight potentiation phase when the synaptic weight is continually updated and eventually stabilized, whereas in all subsequent neuronal bursts no weight potentiation occurs. Clearly from Figure 11 a continual postsynaptic bursting behavior is evident.

FIGURE 11

Figure 11. Firing rate of the postsynaptic neuron where a continual bursting behavior is evident. The firing activity was calculated using a sliding time window of 10 s (blue) and again for a sliding window of 40 s (red) where the latter gives a better average.

Referring to Figure 12, we show simulations for f_pre of 20, 40, and 80 Hz where clearly only f_pre = 40Hz results in repeated Ca²⁺ oscillations. This is because at 20Hz the T_CICR level cannot be reached whereas at 80Hz the astrocyte cytosol is quickly swamped with both IP₃ and Ca²⁺ and subsequent degradation in IP₃ is insufficient to allow further CICR. Consequently our model shows presynaptic frequency selectivity which is consistent with work reported elsewhere (Bienenstock et al., 1982; Dong et al., 2015).

FIGURE 12

Figure 12. Astrocytic Ca²⁺ as a function of time with f_pre of 20, 40, and 80 Hz as a parameter. Note that for the extreme cases of (20 or 80 Hz) no Ca²⁺ oscillations occur: for f_pre = 20 Hz T_CICR can never be achieved and at 80Hz degradation of IP₃ is insufficient to allow CICR to repeatedly occur. Consequently, there is a frequency window over which oscillations can occur.

In addition, as the morphology of GABA interneurons and receptor density at the astrocyte cell differ, the $I{P}_3^{GABA}$ levels vary under the same input f_pre. $I{P}_3^{GABA}$ is a main contributor to the total IP₃, thus the greater $I{P}_3^{GABA}$, the longer the process of IP₃ degradation.

Figure 7 shows that when IP₃ degrades sufficiently to once again enable IP₃ to cross T_CICR from below, a transient elevation in Ca²⁺ results and PR increases with a corresponding increase in the postsynaptic neuron burst frequency. Therefore, different $I{P}_3^{GABA}$ levels lead to different burst frequencies of the postsynaptic neuron. To determine the dependency of neuronal burst frequency on the production rate of $I{P}_3^{GABA}$, $r_{ip3}^{GABA}$, a simulation was carried out (Figure 13) which shows the firing rates of the postsynaptic neuron under different production rates with f_pre fixed at 40 Hz. It can be seen that when the $r_{ip3}^{GABA}$ increases, the frequency of the bursting decreases. For example, for the first 1,000 s, there are 6, 5, 4 bursts under the $I{P}_3^{GABA}$ production rates ($r_{ip3}^{GABA}$) of 1.8, 2, and 2.2, respectively. This is because a high $I{P}_3^{GABA}$ level requires a significant time period to degrade the total IP₃, and to restart the Ca²⁺ oscillation and bursting behavior, thus the bursting frequency is low. Note that a fixed frequency of the input stimulus (i.e., f_pre = 40 Hz) is used in this experiment, however the same results are observed for other f_pre values such as 50 Hz, and the burst frequency variation is not constrained for specific f_pre values. The results in Figures 12, 13 demonstrate the functionalities of the GABA interneuron including the presynaptic frequency selectivity and postsynaptic bursting frequency regulation.

FIGURE 13

Figure 13. Postsynaptic neuron burst firing as a function of time with $I{P}_3^{GABA}$ production rates ($r_{ip3}^{GABA}$) as a parameter. When $r_{ip3}^{GABA}$ increases, $I{P}_3^{GABA}$ level increases and the frequency of the bursting decreases. Due to a high $I{P}_3^{GABA}$ level, a long time period is required to degrade the overall IP₃ and to restart the Ca²⁺ oscillation. Thus, the bursting frequency is low.

4. Conclusions

In this paper, a biophysical model is proposed where it is shown that GABA interneuron regulates the astrocytic IP₃ secondary messenger and thus the probability of release (PR) at the presynaptic terminal. In our model we propose that PR modulates the height of the plasticity window and therefore controls when synaptic potentiation/depression occurs. Specifically, the simulations show that during the weight potentiation phase, increasing IP₃ leads to a cycle of CICR events where each is followed by rapid degradation in IP₃. Over time the firing frequency of the postsynaptic neuron continually increases and eventually the synaptic weights stabilize. Postsynaptic firing results in the release of 2-AG into the extracellular space and this messenger binds to CB1R receptors on the astrocyte membrane. The associated $I{P}_3^{AG}$ contributes to the total cytosolic IP₃ and eventually Ca²⁺ oscillations, and therefore the Glu (e-SP) pathway ceases: 2-AG also binds to CB1Rs on the presynaptic terminal causing a decrease of the synaptic transmission PR via the DSE pathway. PR therefore decreases at the presynaptic terminal which reduces the level of neurotransmitter in the cleft, and consequently the firing frequency of the postsynaptic neuron diminishes, as does $I{P}_3^{AG}$. Thereafter, the total IP₃/Ca²⁺ degrades significantly over time but is replenished by $I{P}_3^{GABA}$ and a subsequent cycle of CICR events commences—the Glu (e-SP) pathway is re-established with an associated increase in PR and the level of neurotransmitter in the cleft is raised. However, in this instance weight potentiation does not occur as PR<PR^*. The postsynaptic neuron firing rate increases again until the Ca²⁺ transients stop and thereafter the activity of the postsynaptic neuron falls off again. A network burst has occurred and this is followed by repeated bursts where each coincides with Ca²⁺ transients: the network burst frequency correlates with the Ca²⁺ oscillatory wave. In addition, the GABA released by the GABA interneuron controls the frequency range within which the network bursts can occur. Future work will further explore other neurotransmitters released by astrocytes such as D-serine and ATP, and also slow inward currents at the postsynaptic terminal as a result of glutamate release by astrocytes.

The authors recognize that this study is based on biological findings of the simplest signaling mechanisms involving astrocytic GABA responses and astrocytic glutamate signaling in presynaptic terminals that regulate network function. Other factors, such as astrocytic ATP/adenosine release from astrocytes (Covelo and Araque, 2018), are not considered in the present model, but may also contribute to further shape of network activity, adding further complexity of the network effects of astrocyte signaling. Further studies incorporating these additional elements are therefore required to get a complete view of the astrocyte roles in network function. Despite this the present findings have potential implications for the generation of normal and pathologic circuit behavior in the brain, relevant to brain diseases that feature altered synaptic properties or where there is a propensity for the episodic synchronized bursting behavior of neurons. The electroencephalogram (EEG) is a composite product of population-level neuronal firing patterns of differing frequencies. Our findings suggest that GABA-B signaling via astrocytes may be relevant to the generation of certain frequencies and behaviors in the EEG. Seizures are the hallmark of the common brain disease epilepsy and are generated by hyper-synchronous discharges of populations of neurons. Notably, gene expression levels of key components modeled here, including the IP₃ receptor and GABA-B receptor, are dysregulated in human epileptic brain tissue or animal models (Matsumoto et al., 1996; Nishimura et al., 2005; Sheilabi et al., 2018) of epilepsy. Mutations in these genes have also been identified in individuals with epilepsy (Møller et al., 2017; Yoo et al., 2017). Indeed, the GABA-B receptor is a long-standing therapeutic target for the treatment of epilepsy (Bowery, 2006), and more recently the IP₃ receptor was reported to be a target of levetiracetam, one of the most effective anti-epileptic drugs (Nagarkatti et al., 2008). The present model offers a novel mechanism to explain how astrocyte-neuron interactions regulate seizure-like activity (Gómez-Gonzalo et al., 2010), and how alterations in the described pathways may contribute to hyper-synchronous firing. It may also offer therapeutic insights through targeted manipulation of the astrocytic GABA-B or IP₃ systems followed by evaluation of the resting electroencephalogram (EEG) and investigating whether this alters the frequency or occurrence of pathophysiological neuronal firing and seizures.

Data Availability

The raw data supporting the conclusions of this manuscript will be made available by the authors, without undue reservation, to any qualified researcher.

Author Contributions

JL, LM, AA, JW, JHa, SK, DCH, NC, AT, JT, and DMH investigated and proposed the biophysical model. JL, LM, AA, JW, JHa, DCH, NC, and DMH wrote and revised the manuscript. JL, LM, AA, JW, JHa, DCH, NC, AJ, AT, JT, AM, JHi, and DMH analyzed the results and reviewed the manuscript.

Funding

This work acknowledges funding supports from EPSRC (EP/N007141X/1) (EP/N007050/1) and HFSP (RGP0036/2014).

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Appendix

TABLE A1

Table A1. GABA interneuron, neuron, and synapse parameters.

TABLE A2

Table A2. Astrocyte cell parameters.