A comparative computer simulation of dendritic morphology

Abstract

Computational modeling of neuronal morphology is a powerful tool for understanding developmental processes and structure-function relationships. We present a multifaceted approach based on stochastic sampling of morphological measures from digital reconstructions of real cells. We examined how dendritic elongation, branching, and taper are controlled by three morphometric determinants: Branch Order, Radius, and Path Distance from the soma. Virtual dendrites were simulated starting from 3,715 neuronal trees reconstructed in 16 different laboratories, including morphological classes as diverse as spinal motoneurons and dentate granule cells. Several emergent morphometrics were used to compare real and virtual trees. Relating model parameters to Branch Order best constrained the number of terminations for most morphological classes, except pyramidal cell apical trees, which were better described by a dependence on Path Distance. In contrast, bifurcation asymmetry was best constrained by Radius for apical, but Path Distance for basal trees. All determinants showed similar performance in capturing total surface area, while surface area asymmetry was best determined by Path Distance. Grouping by other characteristics, such as size, asymmetry, arborizations, or animal species, showed smaller differences than observed between apical and basal, pointing to the biological importance of this separation. Hybrid models using combinations of the determinants confirmed these trends and allowed a detailed characterization of morphological relations. The differential findings between morphological groups suggest different underlying developmental mechanisms. By comparing the effects of several morphometric determinants on the simulation of different neuronal classes, this approach sheds light on possible growth mechanism variations responsible for the observed neuronal diversity.

Figure 1. Dendritic diversity.

Sample cells showing the variety of tree morphologies used as data for this study.

Figure 2. Model parameters and flow.

(A) Flowchart showing how virtual trees are created from sampled basic parameters. (B) Depiction of the basic parameters. (C) Depiction of the three fundamental determinants which constrain the sampling of the basic parameters. (D) Morphometrics which are neither fundamental nor basic are emergent to the model and can be employed to compare the real and virtual trees.

Figure 3. Basic parameter distributions.

Example basic parameter (daughter diameter ratio) distributed irrespective of fundamental parameters (inset), as used in previous studies, and the same parameter binned by Path Distance (main plot: columns and error bars are means and standard deviations, respectively). Both the main graphs and the inset only include daughter ratio values greater than one. The solid line (secondary axis in the main plot) shows the percentage of unitary values in each bin. The dotted line represents the overall percentage of unitary daughter ratios.

Figure 4. Differential ability of the three individual models to capture four emergent morphometrics.

The upper portion of each panel shows the average relative difference between the means of virtual trees and those of real trees for each fundamental determinant (RAD = Radius, PD = Path Distance, BO = Branch Order). The lower portions show the proportion of times each model is the best (as measured by the distance metric) at determining the emergent morphometrics for the 68 tree groups. The fundamental determinants differ in their relative ability to capture each of the emergent morphometrics. The asterisk (*) signifies p<.05 for all figures as determined by the Mann-Whitney U non-parametric test. All error bars show standard error. (A) Number of bifurcations is best captured by Branch Order. (B) Branch Order is significantly worse than Radius or Path Distance at capturing bifurcation asymmetry. (C) Surface area is captured equally well by all three fundamental determinants. (D) Path Distance best captures surface area asymmetry.

Figure 5. Ability of the models to capture apical and basal number of bifurcations.

(A) Apical trees have their number of bifurcations best captured by Path Distance (RAD = Radius, PD = Path Distance, BO = Branch Order). (B) Basal and (C) non-pyramidal trees have their bifurcation numbers best determined by Branch Order. This may point to different underlying developmental mechanisms between apical and other tree types. (D, E) Sholl-like plots showing bifurcation number as a function of branch order for sample apical (D) and basal (E) groups of cortical pyramidal cells (Markram layer 4, N = 24). Path Distance better captures apical bifurcations while Branch Order better captures basal arbors.

Figure 6. Ability of the models to capture apical and basal bifurcation asymmetry.

(A) Apical trees have their bifurcation asymmetry best determined by Radius (RAD = Radius, PD = Path Distance, BO = Branch Order). (B) Basal trees have their bifurcation asymmetry best determined by Path Distance, which wins over the other two models two-thirds of the time. (C) Non-pyramidal trees lie somewhere in the middle, with neither Path Distance nor Radius giving better bifurcation asymmetry results. (D, E) The values of bifurcation asymmetry vary as a function of branch order in representative apical (D) and basal (E) groups (Amaral CA1, N = 23). Path Distance better captures the basal pattern, while the Radius model better captures apical asymmetry.

Figure 7. Relative magnitude of apical-basal divide.

(A) Number of bifurcations is better captured by both Radius and Branch Order in basal than in apical trees. (B) Conversely, bifurcation asymmetry is better captured by both models for apical trees. In either case, non-pyramidal trees tend to lie in between apical and basal trees. (C) The relative ability of the individual models to differentiate apical from basal trees is greater than for other tree divisions. The Performance Ratio is the absolute value of the log of the ratio between the two tree types of the mean differences between real and virtual trees. Number of bifurcations is shown as positive bars (black), bifurcation asymmetry as negative bars (gray). With models based on Branch Order and Radius, the apical-basal divide shows the largest performance ratios for both bifurcation number and asymmetry. The numbers above the Radius columns represent the count of tree groups for the corresponding divisions.

Figure 8. Mix model results.

(A) The ability of the different model variants to capture the emergent morphometrics. The best individual (BI) and percent mixing (% Mix) were repeated with different random number seeds until they produced 243 virtual tree groups for every real one to match the number produced in the determinant mixing paradigm (243 Mix). The determinant mixing paradigm, where the sampling of each basic parameter could be controlled by a separate fundamental determinant, was significantly better at capturing bifurcation asymmetry and total surface area. Both mixing paradigms were better than the best individual models at capturing surface area asymmetry. (B) Sample real and virtual dendrograms using the determinant mixing paradigm. Scale bars are the same for each real-virtual pair.

Figure 9. Relative contributions of the three fundamental determinants to the best models.

The top row compares the percentage of winning best individual models (BI) to the relative contribution of the three fundamental determinants (RAD = Radius, PD = Path Distance, BO = Branch Order) to the winning models in the percent mixing (% Mix) and determinant mixing (243 Mix) paradigms for (A) number of bifurcations, (B) bifurcation asymmetry, (C) surface area, and (D) surface area asymmetry. The bottom row shows how the fundamental determinant contribution to the winning 243 Mix model breaks down by basic parameter (DR = daughter-ratio, PDR = parent-daughter-ratio, TR = taper rate, BPL = branch path length, BIF = bifurcation probability). The overall trend in the determinant mixing paradigm is for a more even distribution of fundamental determinant influence than seen in the best individual and percent mixing paradigms. The basic parameters with fundamental determinant weights close to those seen in the best individual model are likely the strongest drivers in the best individual model selection.