Glutamate Uptake by Astrocytic Transporters

Abstract

Astrocytes express glutamate transporters at high density at perisynaptic processes which can tightly control extracellular glutamate levels in proximity of postsynaptic receptors with the potential to modulate functional neuronal activity. Glutamate uptake by these transporters also closely depends on activity-dependent extracellular ion concentrations and may also be regulated by the astrocyte’s intracellular calcium. On the other hand, intracellular $\mathrm{Ca}^{2+}$ dynamics in the astrocyte too can be modulated by glutamate uptake, with potential for functionally relevant interactions with neural activity. Here, we introduce original modeling arguments to study functional implications of glutamate uptake by astrocytes both on their physiology and on that of neurons. In the first case, we consider the contribution of $\mathrm{{Na^+}}$ and $\mathrm{{K^+}}$ homeostasis to astrocytic glutamate uptake, revealing that intracellular anisotropy could account for spatial segregation of transporter- versus receptor-mediated calcium signaling pathways. In the second case, we study how regulation of extracellular glutamate levels by astrocytic transporters could affect tuning responses of primary sensory areas, linking our analysis to experimental observations in the ferret’s primary visual cortex by Schummers et al. (2008, Science 320:1638). We conclude that glutamate uptake by astrocytes can modulate function of neuronal circuits in multiple ways that may look subtle at individual synaptic contacts, but at network level, lead instead to functionally relevant changes in neuronal tuning and stimulus discrimination.

Appendix 1 Network Model

1.1 Neurons

Membrane potential of excitatory (E) and inhibitory (I) neurons changes by the sum of several conductance-based currents following a Hodgkin–Huxley formalism. The currents modulating a neuron’s membrane potential are: (i) an intrinsic voltage-gated current accounting for firing properties of the cell ($i_\mathrm {AP}$); (ii) an ohmic leakage current ($i_l$); (iii) a postsynaptic current mediated by ionotropic glutamate/GABA receptors at lateral (in-network) connections ($i_s$); (iv) a depolarizing current mediated by extrasynaptically located (postsynaptic) NMDA receptors ($i_e$); (v) an external background current accounting for synaptic inputs to the neurons from outside the network ($i_\mathrm {X}$). Equation for the membrane potential of the generic $\alpha$ neuron (i.e., $v_\alpha$, with $\alpha = \mathrm {E,\,I}$) thus reads

$$\begin{aligned} c_m^\alpha \frac{\mathrm {d}v_\alpha }{\mathrm {d}t} = -i_\mathrm {AP}^\alpha - i_l^\alpha - i_s^\alpha - i_e^\alpha - i_\mathrm {X}^\alpha \end{aligned}$$

(13.19)

where $c_m^\alpha$ denoting the neuron membrane capacitance.

1.2 Intrinsic and Leakage Currents

The voltage-gated intrinsic current that is responsible for making the neuron fire ensues from the sum of: (i) a fast $\mathrm{{Na^+}}$-mediated current ($i_{\mathrm{{Na}}}$); (ii) a delayed rectifying $\mathrm{{K^+}}$ current ($i_{\mathrm{{K}},d}$); and (iii) a population-specific, slow noninactivating $\mathrm{{K^+}}$-mediated M-current ($i_{\mathrm{{M}}}^\alpha$), i.e.,

$$\begin{aligned} i_\mathrm {AP}^\alpha = i_{\mathrm{{Na}}} + i_{\mathrm{{K}},d} + i_{\mathrm{{M}}}^\alpha \end{aligned}$$

(13.20)

The generic expression for any of the currents on the right hand side of the previous equation is

$$\begin{aligned} i_y&= g_y^\alpha m_y^{\mu _y} n_y^{\nu _y} (v_\alpha - E_y)&\text {with}\,y=\mathrm{{Na}},\,\mathrm{{K}}d,\,\mathrm{{M}} \end{aligned}$$

(13.21)

where activating (inactivating) gating variables $m_y$ ($n_y$) are in the form

$$\begin{aligned} \frac{\mathrm {d}\chi _y}{\mathrm {d}t}&= \beta _{y,o}^\chi (1-\chi _y)-\beta _{y,c}^\chi \chi _y&\text {with}\, \chi =m,\,n \end{aligned}$$

(13.22)

$$\begin{aligned} \beta _x&= \frac{b_1 b_2(v_\alpha )}{\exp (b_3(v_\alpha ))+b_4}&\text {with}\, x=o,\,c \end{aligned}$$

(13.23)

and current-specific voltage dependence of gating variables is detailed in Table 13.8.

Table 13.8 Voltage dependence of intrinsic gating variables (Destexhe and Paré 1999)

The leakage current is modeled by a standard ohmic current, i.e.,

$$\begin{aligned} i_l^\alpha = g_l^\alpha (v_\alpha - E_\alpha ) \end{aligned}$$

(13.24)

1.3 Synaptic and Extrasynaptic Currents

Each neuron receives glutamatergic and GABAergic lateral synaptic inputs and is stimulated by glutamatergic afferents. Lateral glutamatergic connections are mediated by both AMPARs and NMDARs, whereas for afferent synapses we only consider AMPAR-mediated currents. Accordingly, the total synaptic drive to a generic $\alpha$ neuron ($\alpha = E, I$) is given by

$$\begin{aligned} i_s^\alpha = \frac{1}{N_{\alpha \mathrm {X}}}\sum _{j \in \mathrm {X}} i_\mathrm {AMPA}^{\alpha j} + \frac{1}{N_{\alpha \mathrm {E}}} \sum _{j \in {E}} \left( i_\mathrm {AMPA}^{\alpha j} + i_\mathrm {NMDA}^{\alpha j}\right) + \frac{1}{N_{\alpha I}}\sum _{j \in I} i_\mathrm {GABA}^{\alpha j} \end{aligned}$$

(13.25)

where $N_{\alpha \phi }$ ($\phi =E, I, X)$, and receptors-specific currents are given by (Destexhe et al. 1998; Saftenku 2005):

$$\begin{aligned} i_\mathrm {AMPA}^{\alpha j}&= g_\mathrm {AMPA}^{\alpha j} O_\mathrm {AMPA}^{\alpha j} (v_\alpha -E_\mathrm {AMPA}) \end{aligned}$$

(13.26)

$$\begin{aligned} i_\mathrm {NMDA}^{\alpha j}&= g_\mathrm {NMDA}^{\alpha j} B_\alpha (\mathrm{{[Mg^{2+}]_e}}) O_\mathrm {NMDA}^{\alpha j} (v_\alpha -E_\mathrm {NMDA}) \end{aligned}$$

(13.27)

$$\begin{aligned} i_\mathrm {GABA}^{\alpha j}&= g_\mathrm {GABA} O_\mathrm {GABA}^{\alpha j} (v_\alpha -E_\mathrm {GABA}) \end{aligned}$$

(13.28)

with $B_\alpha (\mathrm{{[Mg^{2+}]_\textit{e}}}) \,{=}\, 1/\left( 1+\exp (-0.062v_\alpha )\mathrm{{[Mg^{2+}]_\textit{e}}}/3.57\right)$ (Jahr and Stevens 1990), where $g_x,\,E_x$ ($x= \mathrm {AMPA,\,NMDA,\,GABA}$), respectively, stand for receptor-specific conductances and reverse potentials, and $\mathrm{{[Mg^{2+}]_e}}$ is the extracellular magnesium concentration.

Currents mediated by extrasynaptic NMDARs in neuron $\alpha$ are modeled akin to synaptic currents, yet considering an expression for the fraction of open receptors that is a power law function of the extracellular ambient glutamate concentration in the neuron’s surroundings ($G_0^\alpha$) (Bentzen et al. 2009), i.e.,

$$\begin{aligned} i_e^\alpha&= g_e^\alpha B_\alpha (\mathrm{{[Mg^{2+}]_e}}) O_e^\alpha (G_0^\alpha ) (v_\alpha -E_e)\end{aligned}$$

(13.29)

$$\begin{aligned} O_e^\alpha (G_0^\alpha )&= 0.397\, (G_0^\alpha )^{1.5} \end{aligned}$$

(13.30)

1.4 Background Current

All neurons in the network are stimulated by a conductance-based background current that comprises both excitatory and inhibitory contributions, i.e.,

$$\begin{aligned} i_\mathrm {X}^\alpha = g_{\alpha \mathrm {E}}^\mathrm {X}(t)(v_\alpha - E_\mathrm {E}^\mathrm {X}) - g_{\alpha \mathrm {I}}^\mathrm {X}(t) (v_\alpha - E_\mathrm {I}^\mathrm {X}). \end{aligned}$$

(13.31)

where conductances $g_{\alpha \beta }^\mathrm {X}$ follow an Ornstein-Uhlenbeck process, i.e.,

$$\begin{aligned} \frac{\mathrm {d}g_{\alpha \beta }^\mathrm {X}}{\mathrm {d}t} = \frac{1}{\tau _\alpha ^\mathrm {X}}\left( g_{\alpha \beta }^\mathrm {X}(t) - g_{\alpha \beta }^\mathrm {X}\right) + \sigma _\beta ^\mathrm {X} \Gamma (t) \end{aligned}$$

(13.32)

with $\Gamma (t)$ denoting white Gaussian noise.

1.5 Network Architecture

The V1 network considered in this chapter was constructed following the procedure originally outlined Stimberg et al. (2009), and the reader may refer to that study for implementation details on pinwheel domain-organized, two-dimensional orientation-selective cortical maps, such as that considered in Fig. 13.6. In our model, excitatory neurons are located on regular grid of $N_\mathrm {E} \times N_\mathrm {E}$ nodes. Inhibitory neurons are instead randomly positioned on 1/3 of all available nodes. All neurons receive $N_{\alpha X}$ afferent inputs in addition to a fixed number of recurrent (lateral) excitatory ($N_{\alpha \mathrm {E}}$) and inhibitory inputs ($N_{\alpha \mathrm {I}}$). Recurrent connections were randomly drawn from a cell-centered, two-dimensional Gaussian distance distribution with standard deviation $\sigma _\alpha$ according to Efraimidis and Spirakis (2008), so that nearest-neighbor connections are more likely. Synaptic delays (not shown in the model equations) were drawn from a gamma distribution $\Gamma _d^\alpha (k_\alpha ,\theta _\alpha )$. To minimize finite-size artifacts, periodic boundary conditions were adopted accordingly. Stimulation was modeled by Poisson-distributed spike trains delivered by afferent inputs at orientation-dependent rate given by

$$\begin{aligned} f(\vartheta ) = \bar{\nu }\left( a_0 + (1-a_0)\exp \left( -\frac{\left( \vartheta - \theta (x,y)\right) ^2}{4w}\right) \right) \end{aligned}$$

(13.33)

where $\vartheta$ is the orientation of the visual stimulus, $\bar{\nu }$ is the maximum firing rate attained at preferred afferent orientation, $a_0$ denotes the part of firing rate that is independent of stimulation, $\theta (x,y)$ is the spatial map of preferred orientations in the network, and w stands of the afferent input tuning width.

1.6 Numerical Methods

The network model was implemented in Python-based Brian 2.0 simulating environment (Chap. 18) using a standard Euler integration scheme with 0.01 ms time step. Simulations were run for 2 s of simulated time, discarding the first 0.4 s.