Clustering of large cell populations: method and application to the basal forebrain cholinergic system

Abstract

Functionally related groups of neurons spatially cluster together in the brain. To detect groups of functionally related neurons from 3D histological data, we developed an objective clustering method that provides a description of detected cell clusters that is quantitative and amenable to visual exploration. This method is based on bubble clustering (Gupta and Ghosh, 2008). Our implementation consists of three steps: (i) an initial data exploration for scanning the clustering parameter space; (ii) determination of the optimal clustering parameters; and (iii) final clustering. We designed this algorithm to flexibly detect clusters without assumptions about the underlying cell distribution within a cluster or the number and sizes of clusters. We implemented the clustering function as an integral part of the neuroanatomical data visualization software Virtual RatBrain (http://www.virtualratbrain.org). We applied this algorithm to the basal forebrain cholinergic system, which consists of a diffuse but inhomogeneous population of neurons (Zaborszky, 1992). With this clustering method, we confirmed the inhomogeneity in this system, defined cell clusters, quantified and localized them, and determined the cell density within clusters. Furthermore, by applying the clustering method to multiple specimens from both rat and monkey, we found that cholinergic clusters display remarkable cross-species preservation of cell density within clusters. This method is efficient not only for clustering cell body distributions but may also be used to study other distributed neuronal structural elements, including synapses, receptors, dendritic spines and molecular markers.

Figure 1

Two-dimensional illustration differentiating between random and clustered distribution. The upper panel represents the space filled with randomly distributed cell bodies. Lower panel represents the clustered distribution. To quantify the difference between the two distributions we align a bubble of increasing diameter to each neuron (sub-panels from left to right) and we delete all bubbles which contain less than 8 cells bodies (n=8). After the deletion, the overlapping bubbles are merged and the number of isolated bubbles is determined. In the random distribution, as the diameter of bubbles increases, the number of cell bodies contained by bubbles increases exponentially while the number of bubbles after merging remains only one. In contrast, when we apply the same method for the clustered distribution (lower panel), the bubbles containing at least 8 cells quickly segregate into 3 bubble clusters. These clusters remain segregated after merging the bubbles despite the increasing diameter. The clusters only fuse when the bubble diameter exceeds the inter-cluster distance. Under the panels are the number of clusters and the number of cells contained by the clusters. Filled circles represent cells that are surrounded by 8 cells in the specified diameter (seeds). Small empty circles represent cell bodies for which this assumption is not fulfilled.

Figure 2

Quantification of “clusteredness” on real and homogeneous random data. (A) The number of cell bodies within clusters (ordinate: % total) is a sigmoid function of the bubble diameter (abscissa). The cell count starts to increase from 0.1 mm diameter and reaches near maximum at d=0.5 mm. The characteristic cluster size d= 0.2–0.3 mm is the diameter at the steepest increase of cell count, containing approximately 40–70% of neurons. (B) Comparing the observed cell distribution with the homogeneous random distribution. Ordinate is the number of cells divided by the bubble diameter. Abscissa is the bubble diameter. As the bubble diameter increases the number of cells per diameter increases exponentially while the observed distribution from real data reaches a maximum at 0.2 mm, marking the putative average cluster size. This analysis is from a gapless series of 100 µm horizontal sections (n=34) stained for choline acetyltransferase containing about 15,700 cholinergic cell bodies (case 95124).

Figure 3

The parameter space reveals the characteristic cluster size. The basal forebrain cholinergic system of four rats (A–D) and two monkeys (E–F) were mapped and an initial cluster parameter scan was performed on the 3D cell body distribution. Each histogram depicts the number of clusters c as a function of bubble diameter d (ordinate), and minimum number of nearest neighbors n (abscissa). As the predefined bubble size d increases from 100 µm the number of clusters also increases at a broad range of cell density n. However, the number of clusters starts decreasing beyond a critical bubble size d. The characteristic bubble diameter corresponds the maximum d (thin dashed lines) at which the number of clusters c forms a plateau (thick dashed line) within a range of n-s (white arrows). (A) rat #95124 d=150 n=11 c=22; (B) rat #96001R d=200 n=10 c=9; (C) rat #04012 (VRB) d=300 n=10 c=21; (D) rat #96002R d=200 n=10 c= 8; (E) monkey #1 d=200 n=7 c=26; (F) monkey #2 d=200 n=8 c=5.

Figure 4

Distribution of cholinergic cells (red) from three plotted sections (A) and clusters as identified in the cluster program (B). For clarity cholinergic cells forming seeds in (B) are labeled in black all other cholinergic cells shown in white. Case #95124, sections 7–9. (C) Montage from one of the sections to compare the computed clusters with the location of high density cell aggregates (arrows). acp= anterior commissure; cp= cerebral peduncle; opt=optic tract; 3V= third ventricle HDB= horizontal limb of the diagonal band; SI=substantia innominata; LV=lateral ventricle. Two parallel arrows point to clusters in the HDB. Single arrow points a large cluster in SI. Bar scale: 1 mm.

Figure 5

Screenshot of the VRB program during clustering. The tree displays the various clusters identified by their seed numbers. Arrow point to cluster with 207 seeds. The bottom form on the left displays the parameter used for clustering. The right window shows a dorsal view of the cholinergic cells, case #95124. The colors of the various clusters correspond to the colors of clusters as identified in the tree. Cluster with 207 seeds is marked with an arrow in the window. The parameters for cluster identification (lower left window) were: n=11; d=150 µm; c=22. Outlines of individual sections are in white. Bar scale: 2.5 mm

Figure 6

Distribution of cholinergic cells (unassigned in white) and clusters (in various colors) viewed from approximately the same angle in four different rats. Case 95124 were cut in horizontal plane; the three other brains were cut coronal. Cluster parameters are summarized in Table 1. The location of the septum (S), horizontal limb of the diagonal band (HDB) and globus pallidus (GP) are indicated. For better view of the clusters, section outlines are omitted. Scale in case #04012: 0.85 mm; case 96002: 1mm; case 96001: 1.5 mm.

Figure 7

Distribution of clusters from monkey #1 using increasing cluster diameter (from 100 µm to 700 µm) while keeping n=8. Medial is septum, lateral is globus pallidus. Scale: 3.0 mm