Modeling of Astrocyte Networks: Toward Realistic Topology and Dynamics

Abstract

Neuronal firing and neuron-to-neuron synaptic wiring are currently widely described as orchestrated by astrocytes-elaborately ramified glial cells tiling the cortical and hippocampal space into non-overlapping domains, each covering hundreds of individual dendrites and hundreds thousands synapses. A key component to astrocytic signaling is the dynamics of cytosolic Ca²⁺ which displays multiscale spatiotemporal patterns from short confined elemental Ca²⁺ events (puffs) to Ca²⁺ waves expanding through many cells. Here, we synthesize the current understanding of astrocyte morphology, coupling local synaptic activity to astrocytic Ca²⁺ in perisynaptic astrocytic processes and morphology-defined mechanisms of Ca²⁺ regulation in a distributed model. To this end, we build simplified realistic data-driven spatial network templates and compile model equations as defined by local cell morphology. The input to the model is spatially uncorrelated stochastic synaptic activity. The proposed modeling approach is validated by statistics of simulated Ca²⁺ transients at a single cell level. In multicellular templates we observe regular sequences of cell entrainment in Ca²⁺ waves, as a result of interplay between stochastic input and morphology variability between individual astrocytes. Our approach adds spatial dimension to the existing astrocyte models by employment of realistic morphology while retaining enough flexibility and scalability to be embedded in multiscale heterocellular models of neural tissue. We conclude that the proposed approach provides a useful description of neuron-driven Ca²⁺-activity in the astrocyte syncytium.

Keywords: astrocytes; calcium signaling; cell morphology; modeling; noise-driven dynamics.

Figure 1

Model structure and molecular mechanisms. (A) Astrocytic network is segmented in three spatial compartments: I—cell bodies and thick branches; II—the mesh of thin branches; III—extracellular space. (B) Model variables (in colored ovals) and main regulatory pathways of intra-astrocyte calcium dynamics.

Figure 2

Algorithm to create surrogate templates of astrocyte network. First, a set of seeding points on a regular grid (light-gray) is perturbed with random shifts (dark-gray points). Voronoi diagram is then drawn for these points (blue lines). Each patch in the Voronoi partitioning is then filled with the best shape-matching template from an augmented collection of astrocyte images. The lookup collection is created from a set of experimental images taken from CCDB (Martone et al., 2002, 2008) by applying multiple different random rotations and shears to each experimental image.

Figure 3

Example of AVF color-coding. (A) Maximal projection of a confocal image of a cortical astrocyte. (B) color-coded template ready for simulation; regions with non-zero blue channel delineate astrocyte domain, while intensity of the red channel encodes AVF. (C) Link between color-coded AVF and SVR parameters.

Figure 4

Simulations of single-astrocyte spatial templates. (A) top row: Ca²⁺ transient initiation sites (red) and maximum span contours at low (left, p_syn = 0.005 Hz) and high (right, p_syn = 0.01 Hz) synaptic drive, in both cases 25 largest events are shown, at high drive all such events span the whole template; bottom row: snapshots of instantaneous extracellular glutamate concentrations at low and high synaptic drive parameters. Scale bar: 25 μm. (B) Effect of synaptic drive on Ca²⁺ transient frequency and sizes (n = 27 templates, simulation time 2,500 s after burn-in period of 2,000 s); top row: number of events covering more than 25% of cell area (“large” events) increases with excitation strength (left), number of events covering less than 25% of cell area (“small” events) decreases (right); bottom row: average areas of both “large” (left) and “small” (right) events increases with excitation strength in most templates. (C) Distribution of baseline [Ca²⁺]_i levels with AVF parameter at low and high stimulation drives (left) and an example of spatial distribution (right). Each transparent line corresponds to one template, thick lines: average. (D) Same as in (C), but for [IP₃]_i . (E) Evolution of a single Ca²⁺ transient starting in top-right corner of the template; top row: [Ca²⁺]_i , middle row: [IP₃]_i , bottom row: relative change in [IP₃]_i .

Figure 5

Ca²⁺ dynamics in single-cell templates self-organizes in repeatable spatiotemporal patterns. (A) A spatial template for simulation and 2 path-scans (#1, purple and #2, yellow) used for rasters in (B); (B) Rasters of [Ca²⁺]_i and relative change in [IP₃]_i along paths; path starting points are shown as magenta squares. (C) Transient-triggered averages, linked to the [Ca²⁺]_i peaks at the origin of path #1 (left) and to the [Ca²⁺]_i peaks at the origin of path #2 (right); initiation of a [Ca²⁺]_i peak at the origin of path #1 typically leads to a full-cell [Ca²⁺]_i wave, while [Ca²⁺]_i transients emerging at the origin of path #2 remained localized. (D) Left: clustered initiation points (red) and contours of 25 biggest [Ca²⁺]_i waves; right: 5 “large” [Ca²⁺]_i waves with color-coded delay before reaching the peak in [Ca²⁺]_i show a tendency to start in the same area and spread with similar spatiotemporal profiles. (E) Examples of repeated spatiotemporal patterns in 4 other spatial templates: in some cells there was more than one preferred initiation site, the full delay before initiation and waning of the wave also varied.

Figure 6

Complementary cumulative distribution functions for areas (left) and durations (right) of Ca²⁺ events in all single-cell templates. Red lines—low excitation (p_syn = 0.005 Hz), blue lines—high excitation (p_syn = 0.01 Hz). Event area CCDFs: slopes of the fits for the low and high excitations are 3.0 and 2.3, correspondingly; the bend at the large areas corresponds to transition to whole-cell activation. Event duration CCDFs: slopes of the fits are 4.1 and 3.5 for the low and high excitations, correspondingly.

Figure 7

Ca²⁺ activity in multicellular template. (A) Layout of individual spatial domains, each cell is color-coded. (B) Activity level is not uniform: color-coded percentage of time that each pixel had [Ca²⁺]_i above 0.1μM. Domain periphery is more active than somatic regions. (C) Example of spatiotemporal evolution of two co-occurring waves. Contours are separated by 1 s, time delay since the first contour is color-coded. (D) Cellwise dynamics: Ca²⁺ and IP₃ values averaged over individual cell domains, line color corresponds to the map in (A), visible are single-cell events as well as packed Ca²⁺ transients representing multi-cellular waves. (E,F) Repeated patterns of cell activation. (E) Spike-triggered averages of cellwise [Ca²⁺]_i profiles initiated by the cells indicated as #, *, and & in (A,F); activity of the first cell is time-locked to activation of a single other cell, Ca²⁺ transient the second cell consistently leads to activation of several cells, while the third cell does not participate in repeatable patterns. (F) Local score of activation pattern repeatability based on approach shown in (E) (see text for more details); activation of a subpopulation of cells leads to repeated activation of its neighbors; red regions denote wave initiation sites as in Figure 4A.

Figure 8

Modification of spatial templates changes activation patterns. Color-coding: repeatability score, red: wave initiation sites. (A,B) Unlinking single cells from the neighbors; Blue contours—isolated cells, red regions denote wave initiation sites as in Figure 4A. Isolation of the cell often activated the cell denoted “*” in Figure 7 lowers repeatability of the latter. (B) Isolation effectively silences the cell denoted “*” in Figure 7, which was a center of repeated patterns before. (C,D) Modifications of wave initiation sites. (C) Averaging AVF values within the activation sites increases repeatability of the waves, starting at the cells, isolated in (A,B) and one other cell. (D) Removing cell content from the spatial template within active initiation sites leads to a lower number of wave initiation sites and increased repeatability in several cells.