Gliotransmitter Exocytosis and Its Consequences on Synaptic Transmission

Abstract

Calcium-dependent exocytosis of glutamate and purines from astrocytes is the mechanism of gliotransmission that has best been characterized up-to-date, but its putative functional consequences remain elusive. Here we review and expand a mathematical modeling framework originally introduced by De Pittà et al. (2011) to study how gliotransmission could affect synaptic coding and mechanisms of short-term plasticity. Consideration of analytical solutions for rate-based, mean field dynamics of gliotransmission-mediated synaptic neurotransmitter release provides a testable rationale to record functional modulations of synaptic transmission by gliotransmitters in experiments. At the same time, we present theoretical arguments that reveal how functional gliotransmission is a complex phenomenon that depends on the nature of structural and functional coupling between astrocytic and synaptic elements.

Appendices

Appendix 1 Derivation of Analytical Results

1.1 Appendix 1.1 Time-Dependent Solution of Mean Field Synaptic Dynamics

Equations 10.34 and 10.35 can conveniently be rewritten in the form

$$\begin{aligned} \frac{\mathrm {d}U_S(t)}{\mathrm {d}t}&= \left( \Omega _f+\nu _S(t)\right) U_0(t) - \left( \Omega _f + U_0(t)\nu _S(t)\right) U_S(t) = \left( \Omega _f+\nu _S(t)\right) U_0(t) - \Omega _u(t)U_S(t)\end{aligned}$$

(10.53)

$$\begin{aligned} \frac{\mathrm {d}X_S(t)}{\mathrm {d}t}&= \Omega _d - \left( \Omega _d + U_S(t)\nu _S(t)\right) X_S(t) = \Omega _d - \Omega _x(t)X_S(t) \end{aligned}$$

(10.54)

where $\Omega _u(t),\,\Omega _x(t)$ are, respectively, defined by Eqs. 10.17 and 10.18. The equations for $U_S,\,X_S$ are thus in the general form

$$\begin{aligned} \frac{\mathrm {d}x}{\mathrm {d}t} = -a(t) \cdot x + b(t) \end{aligned}$$

(10.55)

so that their solutions are of the type

$$\begin{aligned} x(t) = h(t)\,e^{-\int _0^t a(t\,')\mathrm {d}t\,'} \end{aligned}$$

(10.56)

In particular, an expression for h(t) in the above equation can be obtained replacing the generic solution x(t) in Eq. 10.55 and reads:

$$\begin{aligned} \frac{\mathrm {d}h(t)}{\mathrm {d}t} \mathrm {e}^{-\int _0^t a(t\,')\mathrm {d}t\,'} -a(t)x(t)&= b(t) -a(t)x(t) \nonumber \\&\Rightarrow h(t)=\int _{-\infty }^t b(t\,')\,\mathrm {e}^{\int _0^{t\,'} a(t\,'')\mathrm {d}t\,''}\mathrm {d}t\,' \end{aligned}$$

(10.57)

Accordingly,

$$\begin{aligned} U_S(t)&= \int _{-\infty }^t U_0(t\,') \left( \Omega _f + \nu _S(t\,')\right) \,\mathrm {e}^{-\int _{t\,'}^t \Omega _u(t\,'')\mathrm {d}t\,''}\mathrm {d}t\,'\nonumber \\&= \int _{-\infty }^t U_0(t\,') \left( \Omega _f + \nu _S(t\,')\right) \,\mathrm {e}^{-\int _{t\,'}^t \left( \Omega _f + U_0(t\,'')\nu _S(t\,'')\right) \mathrm {d}t\,''}\mathrm {d}t\,' \end{aligned}$$

(10.58)

$$\begin{aligned} X_S(t)&= \Omega _d \int _{-\infty }^t \mathrm {e}^{-\int _{t\,'}^t \Omega _x(t\,'')\mathrm {d}t\,''}\mathrm {d}t\,' = \Omega _d \int _{-\infty }^t \mathrm {e}^{-\int _{t\,'}^t \left( \Omega _d + \nu _S(t\,'')\right) \mathrm {d}t\,''}\mathrm {d}t\,' \end{aligned}$$

(10.59)

Under the assumptions that $\nu _S(t)$ is continuous and differentiable, its value at $t\,''$ can conveniently be expressed by the mean value theorem as $\nu _S(t\,'') = \nu _S(t) - \nu _S\,'(t)(t-t\,')$. Substituting this latter in the above equations, we obtain

$$\begin{aligned} U_S(t)&\approx \int _{-\infty }^t U_0(t\,') \left( \Omega _f + \nu _S(t\,')\right) \,\mathrm {e}^{-\Omega _f(t-t\,') - U_0(t)\nu _S(t)(t-t\,') + U_0(t)\nu _S\,'(t)\frac{(t-t\,')^2}{2}}\mathrm {d}t\,'\nonumber \\&=\int _{-\infty }^t U_0(t\,')\left( \Omega _f + \nu _S(t\,')\right) \,\mathrm {e}^{-\Omega _u(t)(t-t\,')}\cdot \mathrm {e}^{U_0(t)\nu _S\,'(t)\frac{(t-t\,')^2}{2}}\mathrm {d}t\,'\nonumber \\&=\int _{-\infty }^t U_0(t\,')\left( \Omega _f + \nu _S(t\,')\right) \,\mathrm {e}^{-\Omega _u(t)(t-t\,')}\left( 1+U_0(t)\nu _S\,'(t)\frac{(t-t\,')^2}{2}\right) \mathrm {d}t\,'\nonumber \\ X_S(t)&\approx \Omega _d \int _{-\infty }^t \mathrm {e}^{-\Omega _d (t-t\,') - U_S(t)\nu _S(t)(t-t\,') + U_S(t)\nu _S\,'(t)\frac{(t-t\,')^2}{2}}\mathrm {d}t\,'\nonumber \\&=\Omega _d \int _{-\infty }^t \mathrm {e}^{-\Omega _x(t)(t-t\,')}\cdot \mathrm {e}^{U_S(t)\nu _S\,'(t)\frac{(t-t\,')^2}{2}}\mathrm {d}t\,'\nonumber \\&=\Omega _d \int _{-\infty }^t \mathrm {e}^{-\Omega _x(t)(t-t\,')}\left( 1+U_S(t)\nu _S\,'(t)\frac{(t-t\,')^2}{2}\right) \mathrm {d}t\,'\nonumber \end{aligned}$$

where the rightmost terms in parentheses follow by first-order Taylor series expansion of the last exponential by the assumption of gradual rate changes such that $\nu _S\,'/\nu _S \ll \nu _S$. The expressions for $U_S(t)$ and $X_S(t)$ obtained in this fashion coincide with those reported in Eqs. 10.15 and 10.16 and may then be used to study mean field synaptic transmission for different ranges of $U_0(t)$ values. In particular, in the likely scenario of slowly decaying gliotransmitter modulation of synaptic release (Appendix 2), $U_0(t)$ can be assumed almost constant during variations of $\nu _S(t)$, i.e. $U_0(t) \approx U_0 = \mathrm {const}$, so that

$$\begin{aligned} U_S(t) \approx U_0 \int _{-\infty }^t \left( \Omega _f + \nu _S(t\,')\right) \,\mathrm {e}^{-\Omega _u(t)(t-t\,')}\left( 1+U_0\nu _S\,'(t)\frac{(t-t\,')^2}{2}\right) \mathrm {d}t\,' \end{aligned}$$

(10.60)

which, in the release-decreasing regime of gliotransmission such that $0 < U_0 \ll 1$, provides

$$\begin{aligned} U_S(t)&\approx U_0 \int _{-\infty }^t \left( \Omega _f + \nu _S(t\,')\right) \,\mathrm {e}^{-\Omega _f (t-t\,')}\mathrm {d}t\,' = U_0 + U_0\int _{-\infty }^t \nu _S(t\,')\,\mathrm {e}^{-\Omega _f (t-t\,')}\mathrm {d}t\,'\nonumber \\&= U_0 + U_0\,\nu _S(t)*\mathrm {e}^{-\Omega _f\,t} \end{aligned}$$

(10.61)

in line with Eq. 10.21. In the case of $X_S(t)$ instead, one may notice that the integral can be solved regardless of the functional form of $\nu _S(t)$ and $U_S(t)$, since the variable of integration is $t\,'$ and not t, so that

$$\begin{aligned} X_S(t) = \frac{\Omega _d}{\Omega _x(t)} + \frac{\Omega _d U_S(t)\nu _S(t)}{\Omega _x^3(t)} \end{aligned}$$

(10.62)

which accounts for Eq. 10.19 in the main text.

1.2 Appendix 1.2 Validity of the Mean Field Description

The error introduced by the approximation of statistical independence between $u_S$ and $x_S$ in the derivation of Eqs. 10.34 and 10.35 may be estimated by the Cauchy–Schwarz inequality of probability theory whereby (Tsodyks et al. 1998):

$$\begin{aligned} \frac{\left| \langle \bar{u}_{S}\bar{x}_{S} \rangle -\langle \bar{u}_{S} \rangle \langle \bar{x}_{S} \rangle \right| }{\langle \bar{u}_{S} \rangle \langle \bar{x}_{S} \rangle } \le \chi _{u_S}\cdot \chi _{x_S} \end{aligned}$$

(10.63)

where $\chi _{u_S}$ and $\chi _{x_S}$ stand for the coefficients of variation of the random variables $u_{S}$, $x_{S}$ and, in the approximation of slow $U_0(t)$ dynamics (i.e. $\Omega _G \ll \Omega _d,\,\Omega _f$), they are equal to (Tsodyks et al. 1998; De Pittà et al. 2011):

$$\begin{aligned} \chi _{u_S}^{2}&= \frac{\Omega _f (1-U_0)^2 \nu _S}{(\Omega _f + \nu _S)\left( 2\Omega _f + U_0(2-U_0)\nu _S\right) }\end{aligned}$$

(10.64)

$$\begin{aligned} \chi _{x_S}^{2}&= \frac{U_S^2\nu _S}{2\Omega _d + (2-U_S)U_S\nu _S} \end{aligned}$$

(10.65)

Only if $\chi _{u_S}\chi _{x_S}<0.1$, and Eqs. 10.34 and 10.35 provide a realistic mean field description of synaptic dynamics (Tsodyks et al. 1998; Tsodyks 2005). In the simulations presented here, as well as for the majority of values of synaptic parameters within their estimated physiological range (Appendix 2), this constraint on the error is generally satisfied, resulting in an error that overall is less than 8%, thereby validating our mean field description of synaptic dynamics.

Analogous considerations can also be made for the derivation of Eq. 10.40. In that equation, the error introduced by the approximation of statistical independence between $x_A$ (via $G_A$) and $\Gamma _S$ is upper bounded by the Cauchy–Schwarz inequality

$$\begin{aligned} \frac{\langle x_A\,\Gamma _S \rangle -\langle x_A \rangle \langle \Gamma _S \rangle }{\langle x_A \rangle \langle \Gamma _S \rangle } \le \chi _{x_A}\cdot \chi _{\Gamma _S} \end{aligned}$$

(10.66)

where $\chi _{x_A},\,\chi _{\Gamma _S}$ denote the coefficients of variation of $x_A$ and $\Gamma _S$ for stochastic gliotransmitter release. Derivation of $\chi _{x_A}$ directly follows from the above formula for $\chi _{x_S}^2$ replacing $u_S,\,\Omega _d$, and $\nu _S$ by $U_A,\,\Omega _A$, and $\nu _A$, thus providing

$$\begin{aligned} \chi _{x_A}^{2} = \frac{U_A^{2}\,\nu _A}{2\Omega _A + (2-U_A)U_A\,\nu _A} \end{aligned}$$

(10.67)

More involved is instead the estimation of $\chi _{\Gamma _S}$. For this latter, we start from a recursive expression of the peak value of activated presynaptic receptors ($\Gamma _{S_n}$) as a function of the timing of gliotransmitter release ($\tau _n$). With this regard, the rate of GREs ($\nu _A$) can be assumed much lower than the recovery rate of released gliotransmitter resources, i.e. $\nu _A\ll \Omega _A$ (Montana et al. 2006; Bowser and Khakh 2007), so that, at every release event it is $r_{A_n}=U_{A}\,x_{A_n}\approx U_A$. In this fashion, each release event increases gliotransmitter concentration in the ECS by (Eq. 10.7):

$$\begin{aligned} G_{rel_n}=\rho _e G_T r_{A_n}\approx \rho _e G_T U_A = \hat{G}_A \end{aligned}$$

(10.68)

Hence, following the nth release event occurring at $t=\tau _n$, the time course of gliotransmitter in the ECS is (Eq. 10.9):

$$\begin{aligned} G_{A}(t\ge \tau _n) = \hat{G}_A\,\mathrm {e}^{-\Omega _e\,t} \end{aligned}$$

(10.69)

Experimental evidence hints that onset of modulation of synaptic release by gliotransmitters and gliotransmitter clearance from the ECS is generally much faster than recovery of synaptic dynamics from gliotransmitter-mediated modulation, that is $\Omega _G \ll \Omega _e,\,O_{G}G_A$ (Appendix 2). Therefore, it may be assumed that, between the onset of modulation of synaptic release by gliotransmitters (at $t=\tau _n$) and the peak of this modulation (at $t=\hat{t}_n$), the effect of modulation on synaptic release evolves according to Eq. 10.14 as:

$$\begin{aligned} \frac{\mathrm {d}\Gamma _S}{\mathrm {d}t}&\simeq O_G\,G_A \,(1-\Gamma _S)&\qquad \, \text {for}\,\tau _n\le t < \hat{t}_{n} \end{aligned}$$

(10.70)

where $G_A$ is given by Eq. 10.69. Accordingly, the general solution of Eq. 10.70 reads:

$$\begin{aligned} \Gamma _S(t)&= 1+\left( \Gamma _S(\tau _n)-1\right) \beta ^{1-\mathrm {e}^{-\Omega _e t}}&\text {for}\,\tau _n\le t < \hat{t}_{n} \end{aligned}$$

(10.71)

with $\beta = \exp (-J_S U_A)$, and asymptotically reaches a maximum value of $\hat{\Gamma }_{S_n} =\Gamma _S(t\rightarrow \infty ) = 1 + \left( \Gamma _S(\tau _n)-1\right) \beta$.

Following fast gliotransmitter clearance from the extracellular space instead, the effect of modulation of synaptic release essentially decays from its peak value (at $t=\hat{t}_n$) at rate $\Omega _G$ till the next event of gliotransmitter release from the astrocyte (at $t=\tau _n + \Delta t = \tau _{n+1}$). That is:

$$\begin{aligned} \frac{\mathrm {d}\Gamma _S}{\mathrm {d}t}&\simeq -\Omega _G\, \Gamma _S&\qquad \qquad \qquad \text {for}\,\hat{t}_n\le t < \tau _n + \Delta t \end{aligned}$$

(10.72)

so that

$$\begin{aligned} \Gamma _S(t)&= \Gamma _S(\hat{t}_n) \mathrm {e}^{-\Omega _G\,t}&\qquad \qquad \text {for}\,\hat{t}_n\le t < \tau _n + \Delta t \end{aligned}$$

(10.73)

Then, assuming that $\Omega _e \gtrsim O_G\, G_A$, we can make the approximation that $\Gamma _S(\hat{t}_n) \approx \hat{\Gamma }_{S_n}$. Furthermore, since $\nu _A \ll \Omega _A \ll \Omega _e,\,O_G\, G_A$ (Appendix 2), we may assume that the quasi-totality of gliotransmitter released by the nth GRE is cleared from the ECS before the following release event, so that upon any GRE, $\Gamma _S(\tau _n)$ in Eq. 10.71 is given by solution 10.73 for $t\rightarrow \tau _n$. In this fashion, posing $\Gamma _{S_n} \equiv \Gamma _S(\tau _n)$, we obtain the recursive expression

$$\begin{aligned} \Gamma _{S_{n+1}} \approx \hat{\Gamma }_{S_n} \mathrm {e}^{-\Omega _G\, \Delta t} \approx \left( 1 - \beta (1 - \Gamma _{S_n})\right) \mathrm {e}^{-\Omega _G t} \end{aligned}$$

(10.74)

In this fashion, in the hypothesis of steady-state gliotransmitter release (i.e. $\Gamma _{S_{n+1}}=\Gamma _{S_n}$), Eq. 10.74 can be used to estimate $\langle \Gamma _S \rangle$ and $\langle \Gamma _S^2 \rangle$ to eventually obtain an analytical expression for $\chi _{\Gamma _S}$. With this purpose, it may be noted that the average value of the exponential decay factor in Eq. 10.74 is the integral over all positive $\Delta t$ values of $\mathrm {e}^{-\Omega _G\,\Delta t}$ multiplied by the probability density of GREs in the time interval $\Delta t$. Under the assumption that such GREs are Poisson distributed (Skupin and Falcke 2010) and occur at an average frequency $\nu _A$ (Eq. 10.36), the probability of an inter-event interval of duration $\Delta t$ is then $f(\Delta t) = \nu _A(\Delta t)\,\mathrm {e}^{-\nu _A(\Delta t)\,\Delta t}$; that is, $\Delta t$ is exponentially distributed. Accordingly, denoting by $\left( \mathcal {L}f\right) (s)$ the Laplace transform of $f(\Delta t)$, the average exponential decay reads

$$\begin{aligned} \langle \mathrm {e}^{-\Omega _G\, \Delta t} \rangle = \int _{0}^{\infty } \mathrm {e}^{-\Omega _G\, \Delta t}f(\Delta t) = \left( \mathcal {L}f\right) (\Omega _G) \underset{\lim \xi \rightarrow 0}{\simeq } \frac{\nu _{A}}{\Omega _G + \nu _{A}} \end{aligned}$$

(10.75)

where, for simplicity of notation, we redefined $\nu _A \leftarrow \nu _{A,0}$ (Eq. 10.36) in the last equality. Thus, at steady state and in the limit of $\xi \rightarrow 0$ (i.e. homogeneous Poisson process at constant rate $\nu _A$), it follows that (Eq. 10.74):

$$\begin{aligned} \langle \Gamma _S \rangle&= \langle \Gamma _{S_{n+1}} \rangle = \langle \Gamma _{S_n} \rangle \nonumber \\&= \left( 1 - \beta (1 - \langle \Gamma _S \rangle )\right) \langle \mathrm {e}^{-\Omega _G\, \Delta t} \rangle \nonumber \\&\Rightarrow \langle \Gamma _S \rangle = \frac{(1-\beta )\nu _A}{\Omega _G + (1-\beta )\nu _A} \end{aligned}$$

(10.76)

In a similar fashion, averaging over the square of $\Gamma _{S_n}$ given by Eq. 10.74 provides $\langle \Gamma _S^{2} \rangle$, which reads

$$\begin{aligned} \langle \Gamma _S^2 \rangle&= \langle \Gamma _{S_{n+1}}^2 \rangle = \langle \Gamma _{S_{n}}^2 \rangle \nonumber \\&= \left( \left( 1-\beta \right) ^2 + \beta ^2 \Gamma _S^2 + 2\beta (1-\beta ) \langle \Gamma _S \rangle \right) \langle \mathrm {e}^{-2\Omega _G t} \rangle \nonumber \\&\Rightarrow \langle \Gamma _S^2 \rangle = \frac{(1-\beta )^2\left( \Omega _G + (1+\beta )\nu _A\right) \nu _A}{\left( \Omega _G + (1-\beta )\nu _A\right) \left( 2\Omega _G + (1-\beta ^2)\nu _A\right) } \end{aligned}$$

(10.77)

Accordingly, the coefficient of variation of $\Gamma _S$ ($\chi _{\Gamma _S}$) satisfies the equation:

$$\begin{aligned} \chi _{\Gamma _S}^{2} = \frac{\langle \Gamma _S^2 \rangle -\langle \Gamma _S \rangle ^2}{\langle \Gamma _S \rangle ^2} = \frac{\Omega _G^2}{\left( \Omega _G + (1-\beta )\nu _A\right) \left( \Omega _G + (1+\beta )\nu _A\right) } \end{aligned}$$

(10.78)

Comparison of Eq. 10.76 with Eq. 10.49 shows that the error introduced by the above rationale in the computation of $\chi _{\Gamma _S}$ roughly reaches $\sim$10% within the frequency range of ${\text {Ca}}^{2+}$ oscillations $\nu _A=0.01-1\,{\text {Hz}}$ considered in this chapter (Fig. 10.9). Therefore, Eq. 10.78 can be taken as an acceptable estimation of the $\chi _{\Gamma _S}$ and be used in the evaluation of the upper boundary of the error introduced by the mean field description (Eq. 10.66). Computation of this boundary for different values of gliotransmission parameters across their estimated physiological range reveals that this boundary is generally <7%, implying that our description of mean field gliotransmission is generally realistic and introduces a negligible error.

Fig. 10.9

a Approximation of $\langle \Gamma _{S_n} \rangle$. Comparison between exact steady state, mean field solution of $\langle \Gamma _S \rangle$ (Eq. 10.40, red curve), and the approximated solution in Eq. 10.76 (gray curve). b Percentage error of the approximating solution with respect to the exact mean field solution. Model parameters as in Table 10.1

Appendix 2 Parameter Estimation

An in-depth estimation of parameters of gliotransmitter-regulated synapses was produced by De Pittà and Brunel (2016), and the reader is referred to that study for details on the derivation of most of the parameter ranges reported in Table 10.1. In addition, we consider new estimations on rise ($\tau _r$) and decay time constants ($\tau _d$) of gliotransmitter modulation of synaptic release, that can be obtained by manual fitting of experimental data by difference of exponentials

$$\begin{aligned} y(t) = C\left( \mathrm {e}^{-\frac{t}{\tau _d}}-\mathrm {e}^{-\frac{t}{\tau _r}}\right) \end{aligned}$$

(10.79)

where C is a normalization factor equal to

$$\begin{aligned} C = \left( \left( \frac{\tau _r}{\tau _d}\right) ^{\frac{\tau _x}{\tau _d}}-\left( \frac{\tau _r}{\tau _d}\right) ^{\frac{\tau _x}{\tau _r}}\right) ^{-1} \end{aligned}$$

(10.80)

with $\tau _x = \tau _r\tau _d/(\tau _d - \tau _r)$. Limiting our analysis, for simplicity, to release-increasing effects of glutamatergic gliotransmission, both of short- (Covelo and Araque 2018) and long-term nature (Perea and Araque 2007), and short-term release-decreasing effect mediated by purinergic gliotransmission (Covelo and Araque 2018), we obtain: for glutamatergic gliotransmission $\tau _r>0.05-0.2\,{\text {s}}, \, \tau _d=5.9-6.7\,{\text {s}}$ ($n=2$, short-term); $\tau _r=6.2-28.3\,{\text {s}}, \, \tau _d=18.7-95.7\,{\text {s}}$ ($n=3$, long-term); and for purinergic gliotransmission ($n=2$): $\tau _r=1.6-8.2\,{\text {s}}, \, \tau _d=8.2-12.0\,{\text {s}}$. These values may be translated into equivalent ranges for $O_G$ and $\Omega _G$ considering $O_G G_A \approx 1/\tau _r$ and $\Omega _G \approx 1/\tau _d$, whereby for an average gliotransmitter concentration in the ECS of 100–300 $\upmu {\textsc {m}}$, we get $O_G=1.2\times 10^{-4}-0.2\,\upmu {\textsc {m}}^{-1}{\text {s}}^{-1}$ and $\Omega _G=0.01-0.17\,{\text {s}}^{-1}$.

Concerning gliotransmitter uptake, because the inclusion of $J_u$ (Eq. 10.10) in the time course of gliotransmitter in the ECS was only marginally considered in this study, mostly for the sake of completeness, we provide here only estimates for glutamate uptake by astrocytic EAATs, leaving more detailed investigations on the effect of gliotransmitter uptake on functional gliotransmission to future studies. In this spirit, the kinetic of glutamate (Glu) uptake by astrocytic transporters (T) may be approximated by the Michaelis–Menten reaction

$$\begin{aligned} {\text {Glu}}_e + {\text {T}} \mathop {\rightleftharpoons }^{O_b}_{\Omega _b} {\text {Glu}}-{\text {T}} \mathop {\longrightarrow }^{\Omega _t} {\text {Glu}}_i + {\text {T}} \end{aligned}$$

(10.81)

where $O_b=10^{4}\,{\textsc {m}}^{-1} {\text {ms}}^{-1}$, $\Omega _b=0.2\,{\text {ms}}^{-1}$, and $\Omega _t = 0.1\,{\text {ms}}^{-1}$ (Rusakov 2001). In this fashion $K_u=(\Omega _b + \Omega _t)/O_b = 30\,\upmu {\textsc {m}}$ which is in agreement with independent estimates of EAAT binding affinity in the range of $\sim$10–20 $\upmu {\textsc {m}}$ (Tzingounis and Wadiche 2007). More problematic is instead the estimation of the maximal rate of glutamate uptake $O_u$ which depends on the expression of available transporters. With this regard, reported values of transporter concentration in the perisynaptic volume occupied by astrocytic transporters are comprised in a wide range of values, i.e. $[T]_{0} =$ 30–1300 $\upmu {\textsc {m}}$, depending on how this volume is defined (Lehre and Danbolt 1998). These values were, however, estimated assuming uniform transporter distribution, which is likely not the case at perisynaptic astrocytic processes, where transporters are known to cluster in proximity of synaptic interface, and be instead poorly expressed away from it (Danbolt et al. 2002). Consequently, because gliotransmitter release sites are located away from synaptic interfaces as they rather face extrasynaptic targets (Jourdain et al. 2007), it is plausible to consider smaller values for $[T]_0$. In this fashion, it follows that $O_u=\Omega _t \cdot [T]_0 < 3\,{\text {m}}{\textsc {m}}{\text {s}}^{-1}$.

Appendix 3 Software

The Python file used to generate the figures of this chapter can be downloaded from the online book repository at https://github.com/mdepitta/comp-glia-book/tree/master/Ch10.DePitta. The software for this chapter is organized in two folders. The folder contains experimental data from Perea and Araque (2007) and Covelo and Araque (2018) used for parameter estimations in Appendix 2. Experimental data from those studies were extracted by WebPlotDigitizer 4.0 (https://automeris.io/WebPlotDigitizer). The same folder also contains the Jupyter notebook used to fit those data by Eq. 10.79.

The folder contains instead all the routines (including ) used for the simulations of this chapter. The files contain the classes that implement spiking and mean field models for gliotransmitter-regulated synapses introduced here. The core model is implemented in C/C++11 ( and files), whereas the file provides a user-friendly Python interface for simulations.

Numerical solutions of the spiking model were computed by a hybrid integration scheme that combined an event-based strategy to compute exact solutions of synaptic variables ($u_S,\,x_S,\,r_S$, and I) and releasable gliotransmitter resources ($x_A$), with a backward Gaussian second-order Runge–Kutta integration scheme for $G_A$ and $\Gamma _S$. Integration of the stochastic Li–Rinzel model in Figs. 10.2b and 10.3 was performed by a Milstein integration scheme for Stratonovich formulation of multiplicative noise (Milshtejn 1975). On the contrary, numerical solutions of mean field model in Fig. 10.6 were computed by an implicit Bulirsch–Stoer method suitable for stiff problems like ours, for the existence of largely separated timescales between synaptic dynamics and gliotransmitter-mediated modulations.

Appendix 4 Model Parameters Used in Simulations

Parameter of the stochastic Li–Rinzel model used to simulate astrocytic calcium dynamics in Figs. 10.2 and 10.3 is as following (see also Chap. 4): $d_1=0.1\,\upmu {\textsc {m}}$, $O_2=0.4\,\upmu {\textsc {m}}^{-1}{\text {s}}^{-1}$, $d_2=2.1\,\upmu {\textsc {m}}$, $d_3=0.9967\,\upmu {\textsc {m}}$, $d_5=0.2\,\upmu {\textsc {m}}$, $C_T=4\,\upmu {\textsc {m}}$, $\rho _A=0.4$, $\Omega _C=7\,{\text {s}}^{-1}$, $\Omega _L=0.05\,{\text {s}}^{-1}$, $O_P=0.1\,\upmu {\textsc {m}}{\text {s}}^{-1}$, $K_P=0.1\,\upmu {\textsc {m}}$. To induce oscillations, we set intracellular ${\text {IP}}_{3}$ at $0.4\,\upmu {\textsc {m}}$ starting from initial conditions $C(0)=0.05\,\,\upmu {\textsc {m}}$ and $h(0)=0.9$.