Topological characterization of neuronal arbor morphology via sequence representation: I--motif analysis

Abstract

Background: The morphology of neurons offers many insights into developmental processes and signal processing. Numerous reports have focused on metrics at the level of individual branches or whole arbors; however, no studies have attempted to quantify repeated morphological patterns within neuronal trees. We introduce a novel sequential encoding of neurite branching suitable to explore topological patterns.

Results: Using all possible branching topologies for comparison we show that the relative abundance of short patterns of up to three bifurcations, together with overall tree size, effectively capture the local branching patterns of neurons. Dendrites and axons display broadly similar topological motifs (over-represented patterns) and anti-motifs (under-represented patterns), differing most in their proportions of bifurcations with one terminal branch and in select sub-sequences of three bifurcations. In addition, pyramidal apical dendrites reveal a distinct motif profile.

Conclusions: The quantitative characterization of topological motifs in neuronal arbors provides a thorough description of local features and detailed boundaries for growth mechanisms and hypothesized computational functions.

Figure 1

Converting tree to sequence. a. Bifurcation nodes are encoded as characters based on whether their child branches bifurcate or terminate. Arborization (A) nodes have two bifurcating children. Continuation (C) nodes have one bifurcating and one terminating child. Termination (T) nodes have two terminating children. b. Nodes are traversed depth-first starting from the smaller side which optimally preserves locality. c. Hippocampal pyramidal cell apical (green) and basal (blue) dendrograms and morphologies are shown (NMO_00191 from [62]), with enlargement of a portion of the apical dendrite (right) and coloring in the sequence. Node types are colored and numbered by their order in the sequence starting with the first node in the subtree. d. The entire pyramidal cell morphology is shown (top), with dendrogram (bottom) and sequence representations (background) of the axonal arbor (magenta) (NMO_07897 from [63]).

Figure 2

Measuring k-mers. a. Highlighted dimers in a portion of a fly tangential cell [64] and its associated sequence. b. Dimer schematics displaying the possible configurations. Triangles represent subtrees of unspecified size and shape. Bold segments indicate branches leading to the larger side subtree of the parent A node. Given the (Smaller then Larger) traversal method, the TT dimer must be preceded by an A. TA and TC schematics are examples, as additional bifurcations could be found between the parent A and small-side T nodes. c. Number of k-mers (with examples) by k shows an approximately exponential rise. d. Calculating the percentile rank of a k-mer given the distribution of k-mer counts in the source sequence’s surrogate population. An example apical dendrite (NMO_02582 from [65]), dendrogram, and sequence are shown along with cumulative distribution of k-mer counts for k-mers AT (red) and AC (green). Below: Six out of 100 node-type-constrained surrogates are shown. The example k-mers are highlighted and their counts compose the distributions. Colored dots show the respective percentile ranks of the apical dendrite k-mer counts, with AT being above nearly the entire surrogate distribution (thus constituting a motif) and AC being “captured” inside the middle 95% of its surrogate distribution.

Figure 3

Tree size and complexity. a. Complexity of trees is limited by tree size. Here are shown the set of possible tree shapes for trees with 1 to 6 bifurcations. Additionally, the number of T nodes (red dots in sample trees) is always 1 more than A nodes (green dots). Thus, size and number or percent of C nodes (yellow dots) fully captures node-type statistics. b. Number of tree shapes for tree size (in # bifurcations, or sequence length). The relationship is approximately exponential, though the number is smaller than for the set of all possible sequences of the same size and alphabet unconstrained by “treeness”. Green lines indicate the 11 topologies of trees with 6 bifurcations displayed in (a) and the 680,000 tree shapes with 20 bifurcations which serves as the minimum complexity cutoff for the analysis.

Figure 4

Neurite size and node type features. a. Example morphologies along with node-type proportions (%C) illustrate how difficult it is to estimate topological patterns by visual inspection of full morphologies [65-67]. b. Percent C distributions of axons (magenta) and dendrites (blue) overlap but are clearly distinct. Inset: Pyramidal cell basal apical dendrites (green) fall between basal dendrites and axons, which respectively are similarly distributed to the (non-apical) dendrites and axons of all (non-pyramidal) neurons. Biased terminal (dark gray) and segmental (light gray) growth bound the neurite populations, with the unbiased distribution of tree shapes (black) falling in between. c. Schematics of terminal (top) and interstitial (bottom) growth starting from a representative seed tree shape with sequence CCT. Colored dots represent potential bifurcation points given the growth mechanism, with their respective resulting branches seen in the trees surrounding the initial tree. Segmentally grown trees contain more C bifurcations than terminally grown trees on average, though the percentages stabilize at lower values (seen in panel b) at around 15 bifurcations. d. Percent C versus sequence length for axons and dendrites compared to the tree shapes baseline, segmentally grown trees, and terminally grown trees with low-order bias. Axons fall in between segmental and terminal growth while the bulk of dendrites display terminal growth followed by a possibly segmental growth-based rise in %C with larger sequence lengths. Below: Distribution of sequence lengths for axons and dendrites. e. Percent C as a function of relative position within sequences for dendrites of several sequence size groups along with segmental and terminal growth. The increase in %C with sequence length is sequence-wide and not specific to distal portions of trees. The initially low %C and rise to stability is similar to that displayed by terminally grown trees.

Figure 5

Dimer analysis reflects terminal growth effects. a. Average proportions of captured k-mers (percentile rank between 2.5^th and 97.5^th) at each k for all neurites, with a break between 0 and 80%. Over 95% of trimers are captured, as are over 98% of tetramers, suggesting that most analyses should focus up to dimers, and possibly trimers. Darker descending bars represent the baseline, or chance level based on bootstrapping of surrogates. Numbers between or above bars signify the gap in percent captured between real neurites and the baseline (i.e. statistical equivalence). b. All axon and dendrite dimers except CC are motifs or anti-motifs. Colored lines below dimer schematics show associations between dimers. When one dimer proportion increases (and in turn its percentile rank) another must decrease. c. Growth processes produce specific topological patterns. Starting with a tree of a single bifurcation, the tree grows one bifurcation at a time from terminal nodes. After two bifurcation steps the CT to TT ratio is 2:1 as there are twice as many ways a CCT shape can emerge compared to an ATT shape. After a third bifurcation (at right), the dimer ratio is 5:1, and still 3:1 controlling for %C. d and e. CT and TT motifs and anti-motifs shown in an exemplar interneuron’s dendritic and axonal arbors (NMO_00340 from [68]) in sequence and dendrogram (d), and full morphology (e). Sequence and dendrogram highlighting indicate CT dimers in the axon (pink) and dendrite (blue). In the morphology, the darker color indicates the CT dimers. Asterisks (*) indicate the TT dimer in both representations. All error bars are standard error of the mean.

Figure 6

Distinguishing trimers of axons and dendrites. Axons and dendrites share some motifs consistent with dimers and their interpretations, but they differ on a small related set of k-mers. A rat hippocampal CA3 interneuron dendrite (left; NMO_00837 from [69]) displays a relative abundance of ACC trimers (asterisks) but only one ACT occurrence (dark blue). A rat cortical basket cell axon (right; NMO_07461 from [70]) has 9 occurrences of ACT (magenta) but only 2 of ACC (asterisks). Graph Inset: The ACT k-mer is a motif for axons (magenta bars) and a slight anti-motif for dendrites (blue ovals), while ACC is an anti-motif for axons and neither motif nor anti-motif for dendrites.