Three-dimensional distribution of cortical synapses: a replicated point pattern-based analysis

Abstract

The biggest problem when analyzing the brain is that its synaptic connections are extremely complex. Generally, the billions of neurons making up the brain exchange information through two types of highly specialized structures: chemical synapses (the vast majority) and so-called gap junctions (a substrate of one class of electrical synapse). Here we are interested in exploring the three-dimensional spatial distribution of chemical synapses in the cerebral cortex. Recent research has showed that the three-dimensional spatial distribution of synapses in layer III of the neocortex can be modeled by a random sequential adsorption (RSA) point process, i.e., synapses are distributed in space almost randomly, with the only constraint that they cannot overlap. In this study we hypothesize that RSA processes can also explain the distribution of synapses in all cortical layers. We also investigate whether there are differences in both the synaptic density and spatial distribution of synapses between layers. Using combined focused ion beam milling and scanning electron microscopy (FIB/SEM), we obtained three-dimensional samples from the six layers of the rat somatosensory cortex and identified and reconstructed the synaptic junctions. A total volume of tissue of approximately 4500μm(3) and around 4000 synapses from three different animals were analyzed. Different samples, layers and/or animals were aggregated and compared using RSA replicated spatial point processes. The results showed no significant differences in the synaptic distribution across the different rats used in the study. We found that RSA processes described the spatial distribution of synapses in all samples of each layer. We also found that the synaptic distribution in layers II to VI conforms to a common underlying RSA process with different densities per layer. Interestingly, the results showed that synapses in layer I had a slightly different spatial distribution from the other layers.

Keywords: 3D Ripley's K function; Besag's L function; FIB/SEM; dual-beam electron microscopy; neocortex; random sequential adsorption; replicated spatial point patterns; spatial distribution of synapses.

Figure 1

Diagram of data extraction to analyze whether the synaptic densities of cortical layers are significantly different. The figure shows how we randomly selected a sample from layer III, then we extracted, also randomly, a box inside this sample and counted the number of synaptic junctions in the box. We repeated this process 50 times for each layer. The dimensions of the box were the same for all layers, and it had the maximum volume that could be extracted from all the samples, i.e., it had the minimum length in each dimension (x, y, z) considering all samples.

Figure 2

Layer I, Sample 1. An example of K and L functions for CSR and RSA processes. K (left) and L (right) functions of the experimentally observed data (blue) along with the theoretical CSR (red) and the average of 99 RSA process simulations fitted for sample 1 (green). The K functions of the sample, CSR and RSA processes are very similar. The L functions of the RSA and the experimentally observed sample are positioned well below the diagonal (CSR) for short distances and are fairly close to the diagonal for larger distances.

Figure 3

Analysis of spatial patterns using global envelopes (sample 1 for each layer of the somatosensory cortex). The L functions of the experimentally observed samples are shown in blue, and the averages of 99 RSA simulations are shown in green. The shaded area represents the envelopes of values calculated from a separate set of 99 RSA simulations. We do not reject the RSA null hypothesis for any sample because no observed L function lies outside the envelope for any value of distance d. The results for all samples in the study were the same (see Supplementary material). Dashed red lines show the theoretical value for CSR (for the purpose of visual comparison only).

Figure 4

Diagram of the random thinning process for three groups of replicated point patterns, A, B, and C, for which the Diggle test did not find significant differences. Our goal is to check if these groups are differentially thinned versions of a common underlying RSA process. Random thinning of dense simulations is performed for each experimentally observed sample j in each group i (test sample, shown in blue). Random thinning continues until we reach the intensity $\widehat{\lambda }$_ij, estimated from all samples in group i excluding sample j. Then, for each experimentally observed sample j in each group i, we used simulation-based envelopes to test for differences in the spatial distributions of the thinned RSA simulations and the sample (we used 99 thinned simulations to estimate the L function for the RSA_ij model and the other 99 to calculate the maximum deviation necessary to build the envelope).

Figure 5

(Left) Mean synaptic density of the six layers of the somatosensory cortex. The synaptic density of the six layers is significantly different. However, we found no significant differences between the densities of layers I vs. V or between the densities of layers II vs. III. (Right) Mean distance to nearest synapse for each layer. Nearest synapse distances are significantly different in the six layers of the somatosensory cortex, but we found no significant differences between distances of layers I vs. V, I vs. VI, II vs. III, and III vs. V.

Figure 6

Aggregated K and L functions for each animal. The Diggle test found no significant differences between the three animals used in the study.

Figure 7

For each layer, aggregated L function (dark blue) of experimentally observed data (dashed blue) along with the average of 99 RSA simulations (green) fitting the model for all samples of the layer. This figure shows the envelope obtained using a separate set of 99 RSA simulations. We do not reject the RSA model for any layer of the somatosensory cortex because all the aggregated L functions were within the boundaries of the envelopes. We added the theoretical L function for a random pattern (dashed red diagonal) for the purpose of visual comparison.

Figure 8

Aggregated K and L functions for each layer. The Diggle test found no significant differences between K functions of layers II, III, IV, V, and VI (shown in different shades of violet). Layer I (green) is significantly different from other layers.

Figure 9

(A) RSA simulation with λ = 1.4 for the group of layers II, III, IV, V, and VI. (B) Thinned RSA simulation, λ = 0.932, for sample 10 of layer III. λ estimated from the remaining nine samples of layer III. (C) Thinned RSA simulation, λ = 0.457, for sample 1 of layer VI. λ estimated from the remaining three samples of layer VI.