Astrocyte Networks and Intercellular Calcium Propagation

Abstract

Astrocytes organize in complex networks through connections by gap junction channels that are regulated by extra- and intracellular signals. Calcium signals generated in individual cells can propagate across these networks in the form of intercellular calcium waves, mediated by diffusion of second messengers molecules such as inositol 1,4,5-trisphosphate. The mechanisms underpinning the large variety of spatiotemporal patterns of propagation of astrocytic calcium waves, however, remains a matter of investigation. In the last decade, awareness has grown on the morphological diversity of astrocytes as well as their connections in networks, which seem dependent on the brain area, developmental stage, and the ultrastructure of the associated neuropile. It is speculated that this diversity underpins an equal functional variety, but the current experimental techniques are limited in supporting this hypothesis because they do not allow to resolve the exact connectivity of astrocyte networks in the brain. With this aim, we present a general framework to model intercellular calcium wave propagation in astrocyte networks and use it to specifically investigate how different network topologies could influence shape, frequency, and propagation of these waves.

Appendices

Appendix 1 Simulations of 3D Astrocytic Networks

1.1 Construction of Networks with Different Topology

The five different topologies for 3D astrocytic networks considered in this chapter were constructed as following detailed (see also Lallouette et al. 2014).

• Link-radius networks were constructed connecting each astrocyte i to all cells contained in a sphere of radius d centered on i. The degree distribution of these networks displays some variability around the mean degree $\langle k \rangle$, due to preliminary jitter of astrocyte locations in the absence of highly connected cells.

• Regular-degree networks were developed connecting each astrocyte to its k nearest neighbors while forbidding links longer than $d_{max}={150}\,{\upmu {\text {m}}}$. In doing so, connections were established in $k_{reg}$ iterations to avoid directional biases. Namely, all nodes were randomly taken once per iteration m and linked to the nearest node i with degree $k_i < m \le k_{reg}$ and located within $d_{max}$ from the selected node.

• Shortcut networks were constructed in a way similar to small-world networks (Watts 1999). We started by positioning astrocytes on a cubic lattice with internode distance a, linking each cell to its nearest neighbors at distances that were multiples of a up to l times. We then rewired each connection with probability $p_s$ thereby randomly assigning one of its endpoint. Finally, we jittered the nodes positions as explained in the main text.

• Spatial scale-free networks were incrementally built by spatially constrained preferential attachment (Barthélemy 2010). Briefly, astrocytes were progressively included in the network, one by one, and connected with $m_{sf}$ cells. Each connection between a new astrocyte i and a target cell j was established with probability $p_{i\rightarrow j} \propto k_j \exp (-d_{ij}/r_c)$, where $k_j$ is the degree of the target cell j, $d_{ij}$ represents the Euclidean distance between astrocytes i and j, and $r_c$ sets the range of interaction between cells in the space. Small values of $r_c$ result in connections between astrocytes that are limited to their neighbors, while large $r_c$ values allow establishing long-distance connections. Spatially constrained preferential attachment may also produce some highly connected astrocytes or “hubs.”

• Erdős-Rényi networks were built by linking each astrocyte pair with probability p, independently of their distance and existing degree. These networks are the only ones in our analysis that do not bear any spatial constraint.

Depending on whether $r_c$ (respectively $p_s$) is large or not, spatial scale-free networks (respectively shortcut networks) can be regarded either as spatially constrained networks or as spatially unconstrained networks. Due to random wiring, some of the above procedures could result in disconnected networks. To minimize this scenario, we iterated the wiring procedure to ensure that, in our networks, disconnected node pairs accounted for ${<}2$% of all possible node pairs. Parameters used to build the different networks in the simulations discussed in Sect. 7.3.2 are detailed in Table 7.2.

1.2 Numerical Procedures

Each network model consisted of $3N=3993$ ODEs which we numerically solved by fourth-order Runge–Kutta integration with a time step of 0.01 s. The extent of ICW propagation ($N_{act}$) was quantified by the number of astrocytes that were activated at least once, where an astrocyte was considered to be activated whenever its Ca$^{2+}$ concentration exceeded 0.7 ${\upmu }$M. Each network model was produced into $n=20$ different realizations, and mean degree ($\langle k \rangle$) and mean shortest path length (L) of each network model were averaged over realizations.

To generate ICWs, we stimulated the cell whose location was the closest to, if not coincident with the center of the 3D cubic lattice containing the network. Stimulation was delivered for $0\le t \le {200}\,{\text {s}}$ connecting an IP$_3$ reservoir of ${2}\,{\upmu }$M to the central cell and allowing IP$_3$ diffusion into that cell according to Eq. 7.5.

In networks with UAR astrocytes, we considered step increases of time by $\Delta t = {0.1}\,{\text {s}}$, simulating a transition from a state x to a state y (with rate $k_{x\rightarrow y}$ and $x,y=\mathrm {U,\,A,\,R}$), every time that a random number r drawn from a uniform distribution in [0, 1] at each $\Delta t$ was such that $r\le k_{x\rightarrow y}\cdot \Delta t$. In those networks, stimulation of the central cell was deployed forcing activation of its connected neighbors, since this was observed to be case in the majority of networks with biophysically modeled astrocytes.

Appendix 2 Supplementary Online Material and Software

Detailed derivation of the shell model (Sect. 7.4.2) can be downloaded from https://github.com/mdepitta/comp-glia-book/tree/master/Ch7.Lallouette. The file is provided along with the original LaTeX files within the folder associated with this chapter. In the same folder, the WxMaxima file is also provided. This file was used to analytically solve the ODE system at the core of the derivation of the shell model (Eqs. 1–3 in the supplementary online text).

Within the same repository, the code used for simulations of astrocyte networks presented in this chapter is also provided. The core source code is implemented in C++ and is located in folder. This code must preliminarily be compiled by from this directory. The Python script, relies on the compiled source code to generate all data sets to plot the figures of this chapter. Depending on the hardware configuration, it might take up to a day to complete all the simulations involved. By default, the software will attempt using all available cores on the local machine. Individual figures can be generated by for Fig. 7.3c, d; for Fig. 7.5c, d; and for Fig. 7.6e, f (Table 7.4).