Figure 1.
The topological and electrotonic identity of a neuronal tree.
(A) The tree consists of cylinders or frusta (red) connecting each two nodes along the directed edges (away from the root node, arrows). Branch points and termination points represent the topology (topological points). A branch is a set of continuation points between two topological points. The labelling of the nodes is unique following three principles: hierarchical sorting, continuous labelling preserving sub-tree consistency and topological sorting (see text). (B) Rearrangement of node locations on a sample tree. Examples of equidistant node redistribution resulting in 10 or 20 µm resampling and a 20 µm resampling including length conservation (see text and “resampling” section of Methods). (C–E) Unique representations of topology and electrotonic properties from sample tree from (B). (C) Applying topological sorting, a unique electrotonic equivalent tree can be constructed by mapping node label hierarchy on the branch angle (equivalent tree). (D) The adjacency matrix depicts the connectivity between the nodes of a tree. The corresponding electrotonic signature (current transfer from a node to another, i.e. the potential difference measured in one node as a result of a current injection into another) describes the dendritic compartmentalization (see text). The electrotonic signature corresponding to the 20 µm resampled tree preserves the compartmentalization of the original tree. (E) A one-dimensional string fully describes the topology once the nodes of a tree are sorted topologically. Green pieces represent branches ending with a branch point while black pieces end with a termination point. Branch lengths correspond to real metric length and their order follows the node label sorting. Because all representations observe the same continuous labelling, they preserve the sub-tree structure (a red transparent patch highlights one such sub-tree throughout all representations in (C–E)).
Figure 2.
Generating neuronal branching structures using optimized graphs.
(A) The growth described by an extended minimum spanning tree algorithm (see text). Unconnected carrier points (red) are connected one by one to the nodes of a tree (black). Red dashed lines indicate three sample Euclidean distances to the nodes of the tree for sample point P. (B) Example trees grown on homogeneously distributed random carrier points in a circular hull starting from a root located at its centre (see top). Plotted as a function of the balancing factor bf, the trees range from perfect minimum spanning trees (left) to almost direct connections from the root to any point (right).
Figure 3.
Generating dendritic structures by constructing geometric spanning fields: I. the retinal starburst amacrine cell.
(A) Reconstruction of a starburst amacrine cell in the inner plexiform layer of the rabbit retina (data from [24]). (B) Synthetic starburst amacrine cell morphologies can be best obtained by distributing random carrier points along a density ring limited by a circular hull. (C) An example tree grown on random carrier points distributed according to B following the algorithm described in Figure 2. Spatial jitter was added to reproduce the wriggliness of the original structure. (D) A tree grown on exactly the same points as (C) with a lower balancing factor. (E) The number of randomly distributed carrier points and the balancing factor bf determine the synthetically generated morphology. Here, the areas are plotted in which the synthetic trees match the original according to certain criteria (blue: total cable length ±200 µm; red: total number of branch points ±5; green: mean path length to the root ±3 µm). The area of overlap corresponds to a reasonable parameter set for the synthetic trees. (F–H) Branch order distribution, path length distribution and Sholl intersections are compared for the original tree (red) and for one sample synthetic tree (grey).
Figure 4.
Generating dendritic structures by constructing geometric spanning fields: II. the hippocampal dentate gyrus granule cell.
(A) Reconstructions of four sample hippocampal granule cells (data from [28]). (B) After centring, rotating and scaling all cells adequately, the 50 µm iso-distance volume hulls (black lines) around the set of all topological points (black dots) overlap in all dimensions. Left, xy-projection; Middle, xz-projection; Right; yz-projection. Overlay colours represent local density with same colormap as in Figure 3. (C) Examples of synthetically generated granule cells (based on the data in AB) with bf = 0.85. (D) Third cell from the left in C was grown on the same carrier points with different balancing factors to show the effect of bf here. (E–G) Overlaid branch order distributions, path length distributions and Sholl intersections for original trees (red) and for synthetic trees with suitable parameter bf = 0.85 (black).
Figure 5.
A general strategy for generating synthetic morphologies: Cortical pyramidal cells.
(A) After rotating rat somatosensory cortex layer 2/3, 4 and 5 pyramidal cells to overlap, the limits of their individual regions were extracted: black shaded boxes show the mean limits in XY for the apical region; the black empty boxes delineate one standard deviation away from the mean. Corresponding red boxes duplicate this procedure for the basal dendrites. Cells are then scaled region-by-region to the mean limits of each region. Overlay colours describe local density (colormap see Figure 2D) of lumped topological points of scaled trees. (B) Same procedure for three groups of cortical pyramidal cells during development. (C) Construction stages of a sample layer 5 pyramidal cell according to spanning fields described in A. First the apical tuft is constructed, then oblique dendrites and finally the basal dendrite. Spatial jitter and diameter values are added subsequently.
Figure 6.
Sample cells grown using the general strategy.
(A), (B) and (C) show sample synthetically generated model cells of layer 2/3, layer 4 and layer 5 cortical pyramidal cells respectively, all grown using the general strategy described in Figure 5. In A and B, all dendrites were thickened by 1 µm over all cells for clarity purposes. (D) When data from developing neurons was binned into three groups (P0–5, P8–12, P36–44), synthetically generated cortical pyramidal cells could be generated for the different developmental stages. Vertical or horizontal locations of the cells are purely for layout purposes in all cases.
Figure 7.
Validating synthetic branching structures of pyramidal cells.
While branching statistics of starburst amacrine cells and hippocampal granule cells were moderately homogeneous, pyramidal cells exhibited stronger variations. Balancing factors leading to reasonable branching statistics ranged from 0.4 to 0.7. In the following we compare branching parameter distributions as in Figure 4 for synthetic (black) and original dendrites (red) of layer 2/3 (A–C), layer 4 (D–F) and layer 5 (G–I) pyramidal cells grown with a balancing factor bf = 0.7, 0.6 and 0.5 respectively. (K) Representative layer 5 pyramidal cells grown with different balancing factors bf = 0, bf = 0.2, bf = 0.7. (L) Representative electrotonic signatures of these synthetically generated dendrites and of one original layer 5 pyramidal cell for comparison. (M) Simple relationship between electrotonic compartmentalization and balancing factor. Straight line connects averages of each 100 model dendrites at different balancing factor values. Dashed line shows average compartment size of real reconstructions.
Figure 8.
The interactions between neuronal branching and the network context.
(A) Nine synthetic neuronal trees grown competitively on a sample square substrate of homogeneously distributed random carrier points: the competitive greedy growth results automatically in tiling of the available space. (B) Three out of 16 neuronal trees grown competitively on random carrier points distributed on a ring: this simulates well the sharp borders of Purkinje cells in the cerebellum. Whether Purkinje cell dendrites actually tile in sagittal planes of the cerebellum remains to be determined. (C) Hippocampal granule cells from Figure 4 were scaled and positioned along the contours of a human dentate gyrus obtained from a sketch by Camillo Golgi [31]. Growing synthetic CA3 hippocampal pyramidal cells competitively with the limits from the template resulted in realistic hippocampal pyramidal cells affected by mutual avoidance. Synthetic dendrites were overlaid on the background of the original sketch. (D) Bipolar cells (black) in the retina were grown competitively to connect an array of photoreceptors (yellow) to an array of starburst amacrine cells (green, obtained using the algorithm in Figure 3). In such a case the full morphology of bipolar cells is determined by the context of the circuitry, after prescribing soma locations of the bipolar cells. For all panels of Figure 8 precise scale bars would depend on the details of the preparations and were therefore omitted.
Figure 9.
Automated reconstruction of multiple cells using the greedy algorithm.
(A) Example of an additional application of the algorithm: automated model-based tree reconstruction from image stacks. Maximum intensity projection of tiled image stacks containing a sample sub-tree of a fluorescently labelled neuronal tree. Blue overlay in top panel corresponds to the output of a non-linear thresholding. The resulting binary matrix is then reduced to single points in space (green dots) via a skeletonization procedure. After a distance graph is obtained which describes the probability of a connection between these points due to the image data the points are used as carrier points for the growth algorithm to obtain the corresponding tree using the distance graph as an additional cost factor. After unlikely branches are removed the underlying tree structure is captured (green tree structure in the lower panel, see text for more detail; note absence of scale bar since this a sample image stack). (B) Maximum intensity projections of tiled 2-photon fluorescent image stacks acquired at 820 nm from primary visual cortex of a p13 JAX transgenic mouse (strain #007677, [38]) expressing GFP in parvalbumin interneurons, of which one is present. Three further layer 5 pyramidal neurons are also imaged; all cells were filled with a fluorescent dye Alexa 594 via whole cell patch-clamp recording. Data courtesy of Kate Buchanan and Jesper Sjöström. (C) Corresponding reconstructions (with the interneuron in green) grown in a competitive manner on the image stacks after manual post-processing.