Note on statistical analysis and outcomes

Due to the large n available within this dataset, standard practices relying on reporting p-values are not suitable here (See Materials and Methods, Statistical Analysis for further discussion). As such, we focus on reporting effect sizes in order to emphasise the magnitude of differences over statistical significance alone [17,18]. For main effects and interactions we use partial eta squared , In order to define a meaningfully large effect size we use guidelines from [19]. These are as follows: “small” effect size as , a “medium” effect size as , and a “large” effect size as . We use these as a cut off, and require that a reported meaningfully large effect size, and the confidence interval around it, is typically above the “small” cut-off.

Similarly, for post hoc comparisons, we report (“small”: , “medium”: , “large”: ) with the same criteria.

Dendrite spatial embedding

We first turn to understanding the spatial structure and embedding of T4 and T5 dendrites in order to understand differences in the manner in which these dendrites occupy their densely overlapping environment. Here we focus on the shape and volume of the spanned region of T4 and T5 dendrites, the manner in which these dendrites innervate across the depth axis of their respective neuropil layers, and the spatial relationships between the first branching point of different dendrites.

Dendrite shape.

T4 and T5 dendrites are often characterised by their asymmetrical orientation [3], whereby the direction of dendrite outgrowth is inverse to the direction of functionally encoded motion (Fig 1C). In order to visualise the occupied space of dendrites by subtypes we align individual dendrite in a manner which allows us to overlay them within subtypes. For each dendrite we specify its internal axis using Eigendecomposition of the underlying point cloud representation of each dendrite. As the covariance matrix of 3-dimensional coordinates is a square symmetric matrix, the eigenvectors are orthogonal to each other, and relate directly to vectors in space for each dendrite - their internal axis. We can then order axis based on the magnitude of their corresponding eigenvalues. We term the axis corresponding to the largest eigenvalue as the ’longest’ internal axis, as this is the axis which explains the greatest variance within the data. As these data are points in 3-dimensional space this will be the axis of longest extent spatially. We orient each ensuring that the longest axis aligns with the y-axis basis , and scale by the mean standard deviation along the y-axis within each subtype (See Materials and Methods, Alignment and Scaling). By then centring individual dendrites on the root of their dendritic branching (referred to as the dendrite root point) we can overlay all dendrites within subtypes (Fig 2A). Additionally, as the spatial coordinates are scaled to one standard deviation along the y-axis within each population, we can overlay both hemispheres within the same space. This shows us a clear elliptical spatial occupancy shared across all dendrite subtypes. Having centred dendrite subtype populations on their dendrite root point, we also observe clear offsets from the centre of each elliptical shape for each subtype (Fig 2A; Red points), with the exception of T5a subtypes, where the dendrite root point appears more centrally positioned. Additionally, in comparison to T4 c and d subtypes, the dendrite root point of T5 c and d subtypes appears additionally offset towards the anterior. When this alignment is performed separately within each hemispheres dataset (S1 Fig), the same offset patterns in root positions are observed.

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Fig 2. Spatial Embedding and Volume of T4 and T5 Dendrites.

a) Scaled and locally aligned subtype specific dendrite populations, pooled over hemispheres. Each individual dendrite is aligned to its internal axis and scaled within each type specific population (T4 and T5) to one standard deviation of the population dispersion along the longest axis. Individual dendrites are centred on the root of the dendritic tree (red points). KDE’s show point density for each subtype. b) Schematic illustrating angular deviation between a dendrites longest internal axis (blue line) and the global Dorsal-Ventral (DV) axis (red dashed line). Red point shows the dendrite root. c) Distribution of angles between DV axis and dendrites longest internal axis for T4 and T5 dendrites d) Principal component analysis variance explained along each principle component (PC) for each subtype, illustrating dispersion of points along each internal axis per dendrite. e) Relationship between variance explained between first and second principle components for T4 and T5 dendrites. f) Distribution of raw convex hull volumes for T4 and T5 dendrites. Black horizontal lines denotes difference in means. Gray horizontal lines denote difference in medians. g) Distribution of convex hull volumes scaled by Medulla layer 10 volume (T4) and Lobula layer 1 volume (T5). Black horizontal lines denotes difference in means. Gray horizontal lines denote difference in medians. h) Spatial distribution of T4 and T5 convex hull volumes within their respective neuropil (T4, Medulla, left; T5, Lobula, right) on the globally aligned DV / AP plane collapsed together for both hemispheres.

https://doi.org/10.1371/journal.pcbi.1014657.g002

Given these dendrites’ characteristic asymmetry, it is unclear from this alone if the longest internal axis, and the extension of these dendrites in space, is shared along either the dorsal-ventral (DV) or anterior-posterior (AP) global axis across all subtypes, or differs between horizontally oriented (a, b) and vertically oriented (c, d) subtypes. In order to investigate this we measure the angle between each dendrites’ longest internal axis and the global DV axis (Fig 2B). We find that the longest internal axis of all neuron types and subtypes in both hemispheres is aligned well with the global DV axis (Fig 2B). There is no meaningfully large difference between types, subtypes, or hemispheres . This elongation is in line with the observed elongation of columns along the DV axis in both Medulla layer 10 and Lobula layer 1 [4].

We turn next to understand if these dendrites differ in their spread along the DV, AP, and depth axis. We extend the Eigendecomposition utilised for alignment to principal component analysis (PCA). As in standard PCA, the variance explained by each principal component is given by dividing each eigenvalue by the sum of eigenvalues. This tells us the amount of dispersion within the point cloud representation of dendrites along each of its internal axes. As we have shown already that the longest internal axis aligns well with the global DV axis, it follows that the central internal axis aligns with the AP axis, and the shortest internal axis with the depth axis globally.

The variance explained by PC1 we find no meaningfully large difference between neuron types or hemispheres . Between subtypes we find a medium effect size . Post hoc pairwise comparisons find all comparisons are meaningfully large . Considering the mean differences between subtypes, there is a clear structuring in this result. The variance explained along the first axis, which translates to the DV axis, is ordered in a subtype-specific manner following d > c > b > a.

Similarly, for variance explained by PC2 we again find no meaningfully large difference between neuron types or hemispheres . We again observe a medium effect size between subtypes . Post hoc comparisons are all have a meaningfully large effect size, with the exception of a vs b subtypes, where the tail of the confidence interval falls bellow the small effect size threshold . Interestingly however, the previously observed ordering between subtype means is reversed (d < c < a < b).

Within PC3 we find a slightly different result. Here, we observe a small effect of neuron type . post hoc comparisons show that T4 < T5 with a medium effect size . This difference may be a result of the two different depths of the respective neuropil layers being innervated by T4 and T5. No meaningfully large differences are observed between neuron subtypes or hemispheres.

Given the orderings observed in variance explained by PC1 and PC2, we fit an ANCOVA model for PC2 with PC1 as a covariate (Fig 2E). Here we find a large main effect indicating PC1 as a strong predictor of PC2 , with a slope of . Here, no meaningfully large differences are observed between neuron types, subtypes, hemispheres, or interactions.

Together, these results tell us that all T4 and T5 dendrites elongate primarily along the DV axis in both the Medulla and Lobula. However, the extent of this elongation is subtype-specific, although in the same manner between T4 and T5. Interestingly, the observed negative correlation between the variance explained along PC1 and PC2, which relate to the DV and AP axis, indicates that the greater the extent of outgrowth along the DV axis, the lesser the extent along the AP axis in a proportional manner.

Dendrite depth.

We have focused thus far on the manner by which T4 and T5 dendrites fill space along the DV and AP axes. We turn next to understanding how individual dendrites occupy space along the remaining depth axis. In order to do so, we calculate the normalized neuropil layer depth of dendrite nodes (Fig 3A; Materials and Methods, Surface Fitting, Volume, and Depth Measurement). When globally aligning each population of dendrites, we use a shared origin determined by the centre of mass of a sphere fitted to each population. Given the surface mesh of each neuropil layer, we can determine the inner and outer surface of each mesh based on the dot product of face normals within the mesh to a ray cast from this shared origin, with values less than -0.8 giving faces on the inner surface, and greater than 0.8 providing the outer surface. In the case of the Medulla, the outer surface corresponds to the boundary between Medulla layer 10 and Medulla layer 9, and the inner surface the outer boundary of the Medulla and the surface through which the T4 dendrites innervate. This is inverted within T5, where the inner surface corresponds to the boundary between Lobula layer 1 and Lobula layer 2, and the outer surface to the boundary of the Lobula, and the surface through which T5 dendrites innervate the neuropil. This inversion is why depth is inverted within T5. As a result of the mesh fitting procedure, an offset along the depth axis is introduced between the two hemispheres (See Materials and Methods, Surface Fitting, Volume and Depth Measurement). As a result, analysis on the left and right hemisphere datasets are treated independently here. This is due to the fact that this offset dominates the results showing a misleading difference in hemispheres, however the observations of interest remain consistent between the two hemispheres.

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Fig 3. Dendrite neuropil layer specific depth.

a) Schematic illustrating Neuropil specific depth calculation within T4 and T5. Layer surface meshes are generated using all dendrite points, and inner and outer faces estimated using the dot product of a ray from the coordinate origin and the normal of each surface face (See Materials and Methods, Surface Fitting and Volume). The inner surface within T4 dendrites corresponds to the boundary of the Medulla, and the surface through which T4 dendrites innervate their neuropil. The outer surface is the boundary between Medulla layer 10 and Medulla layer 9. Within T5 dendrites, this is inverted, so the inner surface represents the boundary between Lobula layer 1 and Lobula layer 2, and the outer surface the boundary of the Lobula through which T5 dendrites innervate their neuropil. As T5 dendrites innervate from the side of the outer surface, and T4 from the inner, T5 normalized depths are inverted. b) Distribution of T4 dendrite point depths within the left hemisphere. Solid lines indicate dendrite root points, with a bootstrapped confidence interval around the probability mass function. Dashed lines represent the distribution of non-root points within dendrites, shaded region show asymmetric MAD within the probability mass function across individuals. c - e) As in b, for T4 dendrites within the right hemisphere (c), T5 dendrites within the left hemisphere (d), and T5 dendrites within the right hemisphere (e). f) Multiple Regression showing subtype specific slopes within T4 dendrites within the left hemisphere illustrating the relationship between normalized euclidean distance from root (normalised by the maximum per dendrite) and normalized point depth within Medulla layer 10. Root and furthest points within each dendrite (x = 0 and x = 1) are excluded. g-i) As in f, but for T4 dendrites within the right hemisphere (g), T5 dendrites within the left hemisphere (h), and T5 dendrites within the right hemisphere (i).

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We first investigate the depth of the dendrite root point of individual dendrites (Dendrite root, Fig 3B–3E; solid lines). Within the right hemisphere, we observe a small main effect of neuron subtype . This same small main effect is observed within the left hemisphere . Post hoc comparisons of both hemispheres reveal this is caused by the ’a’ dendrite subtype root positions being positioned marginally closer to the neuropil layer surface in comparison to other subtypes. Within the right hemisphere, post hoc comparisons show the ’a’ subtype is closer to the neuropil surface in comparison to all other subtypes with a small to medium effect size . Within the left hemisphere, this difference is only observed between the ’a’ and ’d’ subtypes with a small effect size . We note that within the left hemisphere, the depth of T4 dendrite root points is significantly more varied and noisy than in the right hemisphere, or T5 dendrites (Fig 3B). This is unlikely to be a result of the dendrite extraction procedure as all dendrite root points are reviewed manually in a random order across the whole dataset. It may be a result of varying reconstruction quality of T4 neurons within the left hemisphere of the FAFB-FlyWire dataset, but it is unclear if this is the case or this is a biological difference at this stage.

When considering the depth of non-dendrite root points we find no differences between type, subtype, or interaction within both hemispheres (Fig 3B–Fig 3E; dashed lines). This indicates that individual dendrites, excluding the start of dendritic branching, occupy the same depth within the neuropil layer regardless of type and subtype, or direction of innervation within the neuropil layer itself.

We next consider the manner by which T4 and T5 dendrites project across their respective neuropil layer. The previously mentioned offset introduced within the layer meshes is most obvious here in the shift along the y-axis (Fig 3F–3I). As both Medulla layer 10 and Lobula layer 1 are well approximated by a curved surface, two possibilities exist. Either, dendrites could follow this curvature as they fill space, or they could project across this curvature. In the latter case, we would expect each dendrite to exist primarily on a 2D plane, and fill space at an angle across the neuropil layer curvature. Given the proximity of the dendrite root point to their respective neuropil layer surface we would expect dendrite points further from this point to be closer to the opposing layer surface. We include the normalised euclidean distance from the dendrite root point, normalised by the maximum within each dendrite (referred to a root distance), as a covariate predictor for normalised depth within each hemisphere. Within both hemispheres, we find a small main effect of root distance as a covariate predictor of neuropil depth. It is noted that within the right hemisphere, the confidence interval around this effect size drops marginally below our small effect size cut off , with a slope of . Within the left hemisphere, this effect is more pronounced and a slope of . With no meaningful effects of neuron type or subtype, this indicates that all T4 and T5 dendrites project across the neuropil layer at a slight angle (Fig 3F–3I), with increasing depth in the neuropil layer away from the dendrite root. We note however that these data are exceptionally noisy and caution is taken with interpretation.

Taken together our analysis shows that the depth at which dendrite subtypes begin arbourising may differ by subtype, more noticeably for T4 than T5 subtypes. The depth of the dendritic arbour itself, excluding the dendritic root point, does not differ between types and subtypes. Finally, we find that these dendrites project across their respective neuropils, rather than growing along the curved space.

Dendrite initial positions.

A single column within either Medulla layer 10 or Lobula layer 1 contains the initial innervation of a full complement of all four subtypes of T4 or T5 dendrites [1]. With the absence of neuropil specific spatial column information, we investigate the pairwise nearest-neighbour relationships between dendrite root points. This informs us about any possible spatial structure in the initial starting positions of dendritic arbourisation. We find both the subtype-specific nearest-neighbour pairings for dendrite root points, and additionally the global subtype-agnostic unique pair assignment which minimises the total distance between pairs of dendrite root points (Fig 4A; Materials and Methods, Dendrite Root Nearest Neighbour Matching). Alongside subtype labels, we additionally group data into “same” (e.g., aa), “opposite” (e.g., ab), or “orthogonal” (e.g., ac and ad) groups.

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Fig 4. Dendrite root nearest neighbour relationships.

a) Illustration of nearest neighbour calculations 20μm from a section of dendrite root positions within the central medulla. (left) shows subtype specific nearest neighbours from T4d dendrites to T4 a,b, and c subtypes (T4d→T4d are omitted from the schematic for simplicity). (right) Global subtype agnostic nearest neighbour calculations determined by maximum cardinality matching in order to find the unique set of nearest neighbour pairs which minimises the global distance between pairs of dendrite root points. b) Nearest neighbour distance of dendrite root points within T4 dendrites, using matching as in (a, left), grouped into horizontal (a, b subtypes, solid lines) and vertical (c, d subtypes, dashed lines). “Opposite” distributions denote opposingly oriented subtypes (e.g., a→b). “Same” subtypes denote same subtype combinations (e.g., a→a). “Orthogonal” denotes orthogonally oriented subtypes (e.g., a→c and a→d). c) Probability of the nearest neighbour being of each of the “Opposite”, “Same”, and “Orthogonal”, inferred using nearest neighbour matching derived from (a, right) within T4 dendrites. Dashed lines denote approximate chance probability for each group (0.25 for “Opposite” and “Same” groups, 0.5 for “Orthogonal” groups). d) Subtype specific nearest neighbour probabilities inferred from nearest neighbour matching derived from (a, right) within T4 dendrites. e-g) as in b-d but for T5 dendrites.

https://doi.org/10.1371/journal.pcbi.1014657.g004

When looking at the subtype specific distribution of nearest-neighbours, we find that in all cases the “opposite” subtype is the closest nearest neighbour for any given dendrite’s root point, and the “same” subtype is always the furthest, with “orthogonal” subtypes occupying the middle distance, with no separation between which of the “orthogonal” subtypes is closer (Fig 4B and 4E). Additionally, we observe that within T4 but not T5 dendrites, the horizontal (a,b; Fig 4B, solid lines) and vertical (c,d; Fig 4B dashed lines) subtype distributions differ. The vertical subtypes show a greater distance between their nearest neighbours in relation to the horizontal, which is not the case in T5 dendrites (Fig 4E). One possibility is that this difference within T4 is a result of differing neuropil layer depths of the dendrite root points between subtypes. In order to asses this, we repeat this analysis on dendrite root points projected onto the surface of a fitted sphere, fit individually to each neuron type population within each hemisphere (See Supplementary S1 Text, Materials and Methods, Spherical Surface Projection). In doing so, all points are projected to the same depth, removing this as a possible confound. We find no difference in our observed results (S4 Fig).

Based on the subtype agnostic nearest-neighbour pairing, we can additionally calculate the probability of a given nearest neighbour being of the “Opposite,” “Same” or “Orthogonal” classification (Fig 4C and 4F). We find that in both T4 and T5 the nearest-neighbour root node is most likely to be of the “Opposite” classification in a manner which is well above chance. This pattern also extends to subtype-specific pairings (Fig 4D and 4G). Here, for both T4 and T5, we see that the most probable nearest neighbour is of the opposite subtype (a to b, c to d), and the same subtype the least likely (Fig 4D and 4G; diagonal), with the orthogonally oriented subtypes positioned between the opposite and same subtypes.

This indicates a clear spatial organisation in the dendrite root node positions, highlighting the potential importance of the initial pattern of innervation before dendritic arbourisation. How this result relates to the spatial position of dendrite root nodes within optic lobe columns remains to be determined however.

Dendrite orientation and geometry

The asymmetric orientation of T4 and T5 dendrites has been well established [1,2,4,10,20]. Beyond this orientation however, the high resolution of these data allows for the analysis of the orientation of individual dendritic sections, and the geometry of bifurcations. We have already observed that all dendrites, regardless of subtype, occupy an elliptical space oriented along the DV axis. This alone does not show their characteristic directionality however. The manner by which individual sections within each dendrite are oriented tells us how this asymmetry may emerge, and if outgrowth during development is coordinated along the DV/AP axes, or independently.

Dendrite orientation.

For each dendrite we can calculate the mean angular orientation of individual sections within the dendrite, which shows a clear organisation into the four known subtypes in both T4 and T5 (Fig 5A and 5B). This dendrite-section-based analysis of orientation is in line with similar analyses conducted on the orientation of differing Strahler order sections [10]. Interestingly here the distribution of T5a dendrite orientations is, although still oriented, far less clear than all other subtypes (Fig 5B). This may relate to the previously noted more centralised dendrite root positions (Figs 2A and S1).

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Fig 5. Dendrite orientation and Geometry.

a) Distribution of mean section angles for each dendrites relative to the anterior-posterior axis, coloured by subtype. b) As in a, for T5 dendrites. c) Schematic illustration of radial angle calculation. Note that denotes a section perfectly aligned with the ray cast from the dendrite root (gray dashed arrow). As such, segments where the source node is the dendrite root point have and are excluded from analysis. Plane normal orientations used for calculation of the sign of are oriented in the same direction to ensure accurate signs for . d) Radar plot showing distribution of section radial angles within T4 subtypes. Solid lines show internal dendrite sections, and dashed lines denote external leaf sections. e) As in d, but for T5 dendrites. f) Schematic illustration indicating notation of bifurcation angular components. The red arrow denotes the parent section, and blue and green indicate the two child sections. and show the angles from the parent section to either of the two child sections. denotes the angle between the two child sections. g) Distribution of parent to child angles and within T4 and T5 dendrites. and are collapsed together as they are determined arbitrarily. Shaded region show asymmetric MAD within the probability mass function across individuals. h) Distribution of child angle for T4 and T5 dendrites. Line colours are the same as in g. i) Joint probability distribution illustrating inverse relationship between parent to child angles and . Data for type and subtype are collapsed together. j) as in i, showing the relationship between parent to child angles and with child angle . k) Distribution of the sum of bifurcation angular components (, and ) for T4 and T5 dendrite bifurcations. Gray dashed (T4) and dotted (T5) lines show bifurcation angular component sums for randomly generated bifurcations for comparison (See Materials and Method, Geometry Analysis). l) Distributions of Dihedral angle , used as a measure of 3D bifurcation flatness for T4 and T5 dendrites. Gray dotted and dashed lines facilitate comparison against randomly generated bifurcations as in (k). Inset shows a schematic illustration of the calculation of Dihedral angle for a single bifurcation.

https://doi.org/10.1371/journal.pcbi.1014657.g005

We next analyse the radial orientation sections within the dendrite. In order to do so we calculate the signed radial angle of each dendrite section [21] (Fig 5C; Materials and Methods Geometry Analysis). Given the dominant elongation along the DV axis observed, coupled with four orientations of subtypes, we wish to know if either, dendrite sections align with the DV and AP axis, or alternatively, if sections orient radially outwards from the dendrite root point.

We find a tightly symmetrical distribution of radial section angles with no difference between subtypes (Fig 5D (T4) and 5E (T5)) oriented around . We have classified individual sections as “internal” or “external” within this analysis (See Materials and Methods, Neurons as Tree Graphs). This denotes whether a section is contained within the dendrite, starting at the dendrite root point or at a branch point, and ending at a branch point (internal), or if a section terminates at its end point (external).

We note that T5 dendrites show a broader set of radial orientations than T4 (Fig 5E in comparison to 5D), likely caused by growth in differing neuropils and different local spatial constraints. We observe that radial orientation of internal and external sections differs, with internal sections (Fig 5D and 5E; solid lines) showing tighter radial angles closer to , more closely aligned with the vector from the dendrite root in comparison to external sections (Fig 5D and 5E; dashed lines).

Although previous work has identified a radial angle distribution centred around [21], we find that within T4 and T5 dendrites, sections are still oriented consistently outward from the start of branching, however distributed symmetrically around at . This shows that individual dendrite sections, rather than being oriented along or across the DV or AP axis, are oriented outwards from the initial branching point in a similar manner for both T4 and T5 dendrites as well as their subtypes. We also note that, although still oriented outwards, external sections show a broader distribution and are less constrained by the radial orientation away from the definite root node in comparison to internal sections (Fig 5D and 5E; solid lines compared to dashed). This indicates that the manner in which these dendrites orient individual sections, and by extension their asymmetric outgrowth, is to project outwards from the start of dendrite branching in a systematic and conserved manner across subtypes, as opposed to orienting along the DV or AP axis.

Dendrite topology

Our final section of analysis considers the topological and graph-theoretical structure of T4 and T5 dendrites. Analysis at this level, which is concerned with the underlying structure of dendrites, is less influenced by external factors, such as mechanistic forces dictated by their external environment. As such, we can infer from these metrics details of the internal developmental process for each dendrite, independent of external signals. Conservation of structure at this level is seen as further evidence in favour of a shared developmental mechanism across types and subtypes, independent of variability introduced between neuropils.

Dendrite graph structure.

We first determine the nature of the underlying tree-graph for T4 and T5 dendrites. At one extreme, a star graph contains a single internal node and all other nodes are external, leaf nodes. For such a graph the number of branching nodes (I) is equal to 1. At the other extreme, a full binary tree is a tree-graph whereby all internal nodes have an out-degree of two. Within the reduced tree-graphs used here where all nodes with a single parent edge and a single child edge are removed, a full binary tree will have for n total nodes (see Materials and Methods, Neurons as Tree Graphs). When plotting the relationship between total number of dendrite nodes and total number of branching nodes (Fig 6A) we find that all T4 and T5 dendrites are slightly beneath this upper bound for a full binary tree. Statistical analysis identifies the total number of nodes as a predictor of total number of branch nodes with a large effect size with a slope of . We find no meaningfully large differences between other main effects or interactions .

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Fig 6. Graphical Structure of Dendrites.

a) Scatter plot showing the relationship between total number of nodes and number of branching (internal) nodes within a dendrite tree graph. Gray filled in space denotes the region of possible tree-graphs, where by outside this region a given graph will no longer be a tree-graph. Black dashed line denotes a fully binary tree. b) Distribution of node out-degree for T4 and T5 dendrites. Nodes with out-degree of 0 are leaf nodes, and and out-degree of 2 show bifurcations. No nodes with an out-degree of 1 exist in the reduced tree-graph representations used (See Materials and Methods, Neurons as Tree Graphs). Error bars show asymmetric MAD of probability mass functions across individual dendrites. c) Distributions of weighted partition asymmetry across T4 and T5 dendrites (See Materials and Methods, Branching Structure and Symmetry Quantification). Values of 0 show symmetric branching and 1 show asymmetry. Error bars show asymmetric MAD of probability mass functions across individual dendrites. d) Distribution of external (leaf node) counts at increasing dendrite tree-graph depths. Error bars show asymmetric MAD of probability mass functions across individual dendrites. e) Relationship between number of dendrite sections and total dendrite cable length in T4 and T5 dendrites. f) Section length with increasing tree-graph depth for both T4 and T5 dendrites in internal (left) and external (right) sections. Error bars show asymmetric MAD of probability mass functions across individual dendrites. g) Schematic illustration showing curvature of individual segments around the euclidean vector between source and target nodes of a section. “Raw” dendrite skeletons are sampled at , so a single dendrite tree-graph section is comprised of multiple edges, and the euclidean distance between the source and target of dendrite sections is less than the sum of edge weights unless the section comprises of a single edge, in which case they will be equal. h) (left) Distribution of the mean angle of edges within a section relative to the euclidean vector between source and target within the same section. Error bars show asymmetric MAD of probability mass functions across individual dendrites. (right) Angular variance of edges within a section relative to the euclidean vector between the source and target point of the same section. Error bars show asymmetric MAD of probability mass functions across individual dendrites.

https://doi.org/10.1371/journal.pcbi.1014657.g006

When observing the out-degree distribution of dendrite tree-graphs we see that these dendrites are dominated by nodes with an out-degree of 0 (all leaf nodes) and 2 (bifurcating branch nodes). Interestingly, there is a highly consistent but small number of trifurcations within the dataset ( per dendrite). We note that these trifurcations could be a result of the skeletonization process, although their consistency leads us to believe otherwise. We find no meaningfully large differences between neuron types, subtypes, hemispheres, or interactions ().

We next consider the symmetry of dendrite tree-graphs. Tree asymmetry has long been used as a metric in the analysis of neuron morphology [26–28]. Symmetry here refers to the symmetry of branching through the dendritic tree and is calculated at each branch node. As an example, if one pictures a bifurcation whereby the left downstream node is a termination, and the right contains the entirety of the downstream subtree which follows this same right-dominated rule, this bifurcation would be asymmetric and as such have a tree asymmetry value of 1. On the other hand, if all branching nodes downstream from a bifurcation are also bifurcations, this subtree would be symmetric, and as such have a tree asymmetry value of 0. We implement here a weighted tree asymmetry metric, similar to [28] (See Materials and Methods, Branching Structure and Symmetry Quantification). Rather than using a single summary statistic per dendrite, we show the full distribution of scores across a single dendrite, showing the clear symmetric nature of T4 and T5 dendrites (values close to 0 are closer to symmetric, Fig 6C). This is in line with previous observations [26,28,29] of tree symmetry in neuron morphologies. Again, no meaningfully large differences are found for neuron type, subtype, hemispheres or interactions ().

We note that external leaf nodes are distributed throughout tree graph depths (Fig 6D), again with no differences observed between type and subtypes . Much like the relationship between total dendrite cable length and dendrite convex hull volume, and between dendrite convex hull volume and the number of sections within a dendrite (S2C and S2F Fig), there is a strong relationship between the total dendritic cable length and the number of individual segments within each dendrite (Fig 6E). With the number of segments as a covariate of total cable length we find segment count is a strong predictor of total cable length . With a slope of this tell shows that all dendrite sections are, on average, in length.

Together, these results speak of a highly conserved topological structure for all T4 and T5 dendrites, as well as their subtypes. We do not find any distinguishable characteristic within this structure which would be in favour of arguing for differences between subtypes mediated by differing orientations, or neuropil specific differences between T4 and T5.

Dendrite section lengths.

We next turn to investigate any differences in the individual sections between T4 and T5 and their subtypes. As was done previously when considering the radial orientation of individual sections, we again include the internal or external nature of individual sections within our analysis. We first investigate the decay of section lengths as a function of tree depth (Fig 6F). Tree depth here refers to the number of (unweighted) steps taken from the root of the dendritic tree. We find no meaningfully large differences between hemispheres, neuron type, subtype, source, or interactions (all ). We note, however, that sections closer to the dendrite root (with lower depth), particularly within internal sections, appear marginally longer before the section lengths stabilise. This observation of an initial decay followed by stabilization has been observed previously in mouse Purkinje cells [30], and in line with an optimal wiring and conduction time trade-off, which would imply proximal longer runs and distal finer segmentation [31]. T4 internal section lengths also appear qualitatively greater than T5 across all tree depths (Fig 4F; left).

The individual section within a dendrite are a collection of short edges which curve around the individual dendrite root point, branch and leaf nodes within the tree graph (Fig 6G). We would like to understand how much deviation there is from the shortest vector between points as the dendrite section projects between them. As such, we measure for each section the mean angular deviation between individual tree graph edges and the euclidean vector between the source and target point of the section, and the angular variance of the individual edges. We find no differences between types and subtypes , showing the curvature within individual sections is tightly conserved again across hemispheres, neuron types, subtypes, and sources. Additionally, individual sections show very little deviation, indicating that outgrowth is essentially straight for within all dendrites between branching and terminal events.

We next investigate the distribution of individual section lengths across T4 and T5 and their subtypes. As well as “raw” section lengths, we additionally scale individual dendrites along their internal axis to have equal variance, creating uniform spherical dendrites without changing the topological structure. The intention of this scaling is to remove the influence of the local dendrite environment, such as external mechanistic forces, or the confines of the neuropil layer from the length of dendrite sections, without altering the topological structure. Ideally, this results in a dendrite representation independent of external influence (Fig 7A and 7B).

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Fig 7. Dendrite Section length Distribution and Explorative Distribution Fitting.

a) Distribution of dendrite section lengths for T4 and T5 dendrites, with dashed lines showing internal sections and solid lines showing external leaf sections. Schematic (insert) shows example neuron with internal (cyan) and external (orange) sections coloured for illustration, along with first two internal axis. Errors show asymmetric MAD of probability mass functions across individual dendrites. b) Distribution of section length as in a, but for variance scaled dendrites (See Materials and Methods, Alignment and Scaling). Dendrites are individually scaled to have equal variance along each internal axis in order to account for neuropil specific differences. Schematic (insert) illustrated variance scaling along internal axis. This results in a spherical dendrite. Errors show asymmetric MAD of probability mass functions across individual dendrites. c - f) Best fitting probability distribution for pooled section lengths within each subtype. Left hand plots show unscaled section lengths (c and e) and right hand plots show variance scaled section lengths (d and f). The top row shows internal sections (c and d) and bottom external sections (e and f). Both unscaled and variance scaled internal section lengths are best fit by a gamma distribution (c and d). Both unscaled and variance scaled external section lengths are best fit by a log-normal distribution (e and f). Gray histograms show the probability density for all pooled data.

https://doi.org/10.1371/journal.pcbi.1014657.g007

Interestingly, in both original unscaled and scaled datasets, we find no meaningful differences between hemispheres, neuron types, subtypes, source, or their interactions (). Due to the high skew observed in these data, we also test log-transformed values, which have the same result (). Our analysis here again shows a highly conserved structure with no meaningfully large differences between T4 and T5 dendrites and their subtypes.

Dendrites are structurally homogeneous

Given the distinct characteristic orientation of T4 and T5 dendrite subtypes [3], one may assume that this would lead to morphological differences within the dendrite branching structure. We, however, observe consistent homogeneity in all geometric, and graph-theoretical, measures across all T4 and T5 dendrites, as well as their subtypes, across both hemispheres.

The dendrites of T4 and T5 neurons, and their subtypes, are all well characterised by the same close to binary and symmetric tree-graph with little variability across hemispheres, neuron types, and subtypes (Fig 6A–6C). Interestingly, the deviation from a binary tree manifests in a small but consistent proportion of trifurcations (Fig 6B). Their consistency leads us to believe that they are less likely to be a direct result of the skeletonization procedure. This may indicate that they are biologically meaningful, for example, being Steiner points which minimize the total dendritic cable [39], however further analysis is needed in order to determine if this is the case.

We also observe a striking similarity in the geometry of dendritic branching, even though individual subtypes are globally oriented (Fig 5). The angular components of all dendrite bifurcations are indistinguishable, as is their planarity, illustrating that dendritic branching itself is indistinguishable between hemispheres, neuron type, and subtype. The implication here being that the mechanism of branching within these dendrites is conserved and, regardless of the ultimate directionality of the individual dendrite, or the neuropil which it exists in, the result of a consistent mechanism.

The observed planarity is potentially the result of additional constraints on branching geometry. Previous work has related the flatness of bifurcations with both the stochastic process of connecting to the closest point in space randomly [25], and being an emergent property of the local 3D elastic tensions in neurites, where distal parts of branches develop force equilibrium when anchored to the substrate [22]. Either mechanism is plausible here, although we cannot disregard that the observed flatness is a result of the relative planarity of the occupied space of these dendrites.

Further, our results indicate a simple rule whereby the two child sections are maximally separated from each other. This follows from the relative planarity of these bifurcations, coupled with the negative correlation between both parent and child branching angles. This maximal partitioning may relate to self-avoidance mechanisms, as well as optimal spatial embedding.

These dendrites also show similar spatial occupancy, shown in their elliptical structure (Figs 2A and S1), their consistent elongation along the Dorsal-Ventral axis regardless of subtype (Fig 2C), as well as neuropil layer depth innervation (Fig 3). Once appropriate scaling is applied, the mean volume of these dendrites additionally does not differ (Fig 2G), in line with the importance of the spatial occupancy of columns and necessities of motion detection to integrate across multiple adjacent visual points [2,4]. Independent of this spatial scaling, the total cable length and number of sections within the dendrite tree-graph also does not differ markedly between hemispheres, neuron types, or subtypes (S2 Fig, S1 Text: Statistical Analysis of Dendrite Size Metrics).

Together, these analysis highlight that the spatial occupancy, geometry and topology of these dendrites is, essentially, identical. From this, it seems unlikely that differing mechanisms of branching, or space filling, are employed during the emergence of these dendrites asymmetry.

Observed differences are exclusively constrained to spatial embedding, not structure

Although the structure of individual dendrites is highly conserved across types and subtypes, there remain some insightful differences particularly focused on elements of their individual spatial embedding. Of particular interest is the differing distribution shape and spatial organisation in all size based metrics. Specifically, the population of T4 dendrites shows a heavy tail not present in the comparably symmetric T5 distribution (Figs 2F, S2A and S2D). The tail of larger T4 dendrites is focused within the anterior-ventral region of Medulla layer 10 (Figs 2H, S2B and S2E). There are two opposing explanations to this observation. Either, the visual columns within Medulla layer 10 are physically larger, or column size is uniform and these larger dendrites, by extension, sample from a greater proportion of the visual space. Interestingly, it has been previously observed that within the compound eye, across Drosophilidae species, larger lenses are present within the anterior - ventral region [40–42]. Strikingly, we find that the column space within Medulla layer 10, which determines a T4 dendrites sampling of visual space, is uniform (S3 Fig). This indicates that individual T4 dendrites within the anterior-ventral region span across a larger portion of the columnar grid, and as such seem to integrate visual inputs from a larger portion of visual space. It is important to note that this this only relates to the spatial span across the columnar space in relation to Mi1 morphology, and does directly not relate to difference in features such as input density, or the spatial distribution of synapses. That this region of enlarged dendrites correlates with larger lens sizes on the retina seems unlikely to be a coincidence. Importantly, future functional analyses will be needed to determine if there is any implication to this observation in synaptic integration of receptive field size, or if this is merely a quirk of morphology.

Interestingly, we also find evidence of differences in the positions of the initial point of branching within these dendrites. Within both T4 and T5 neurons, there is some evidence in favour of subtype specific variability in the depth of dendritic root points within their neuropil layer. Statistically, this is dominated by the ’a’ subtypes being closer to the layer surface. Qualitatively, particularly within T4 dendrites, our analysis indicates a possible split between horizontal and vertical subtypes when considering layer depth for dendritic root points (Fig 3C). Interestingly, within the left hemisphere, T4 dendrite root point depths are significantly more varied in comparison to the T4 dendrites within the right hemisphere, or T5 dendrites in general. As individual dendrites are reviewed in a random order across the mix of all subtypes and hemispheres, this is unlikely to be a systematic bias introduced within our pre-processing procedure. We cannot rule out that this is some artefact within the reconstructions of T4 dendrites within the left hemisphere, but these have also been independently proofread within the FAFB-FlyWire dataset.

We also observe differences between horizontal and vertical subtypes in terms of the dendritic root point. We see a clear difference between nearest-neighbour partners in T4 horizontal and vertical subtypes when considering oppositely oriented nearest-neighbours, whereby horizontal subtypes are closer nearest-neighbours than vertical (Fig 4B). This trend is not present in T5 neurons (Fig 4E), and is not a result of the observed differences in the depth of dendrite root points (S4 Fig). This observed difference may be a result of the columnar structure itself, as noted in [4], whereby the columns within both layers are elongated along the DV axis, however within Lobula layer 1 they are additionally shortened along the AP axis, resulting in a triangular columnar array as opposed to hexagonal. When we consider the aspect ratio (AR) between AP and DV axis between neuropils (Simply AP / DV extent within each neuropil) we find the Lobula is, indeed, proportionally narrower (AR: Medulla right = 0.66, Medulla left = 0.72, Lobula right = 0.55, Lobula left = 0.58). How dendritic root point innervations into their target neuropils map to the columnar lattice is unclear, for example it is unclear if the dendrite root point of each subtype within a column are all centrally located, or some sit nearer the boundary. The narrower columns along the AP axis within the Lobula may serve to constrain horizontal and vertical subtypes, forcing them closer together in comparison to horizontal and vertical subtypes within Medulla layer 10. Additionally, as the first NB division during development separates horizontal and vertical subtypes, the manner of targeting used to innervate their eventual neuropil prior to dendritic outgrowth may differ, facilitating the observed difference. As a final consideration, Medulla layer 10 and Lobula layer 1 also differ in some manner in terms of their global structure. Medulla layer 10 is somewhat closer to a slice taken from the surface of a sphere, whereas Lobula layer 1 shows greater variability and deviations from this, although still well approximated by a section taken from a spherical surface. These differing geometries neuropil can also have an impact on our observations of differing elements of spatial embedding within these dendrites.

Of additional interest is the observed subtype, but not type, specific relationship in the extent of dendrite innervation along the dorsal-ventral (DV) and anterior-posterior (AP) axes. Vertical (c,d) subtypes show a greater elongation along the DV axis, and narrower width along the AP axis, a trend which is smaller within the horizontal (a,b) subtypes (Fig 2D). Interestingly, however, the difference between elongation along the DV and AP axis is negatively correlated (Fig 2E), indicating that as each dendrite extends along the DV axis, it extends less along the AP axis proportionally. This could be evidence of a cost-constrained mechanism limiting the total size of dendrites, coupled with dominant, subtype-specific, elongation along the DV axis. The general elongation along the DV axis is in line with observed column elongation along this same axis [4], however how or why the extent of this elongation is subtype specific is unclear.

These results highlight subtle differences between the T4 and T5 dendrites. Notably, in terms of the initial innervation pattern, we see evidence of a horizontal / vertical subtype split within T4 and the Medulla which is not evident within T5 and the Lobula. It is interesting that we only see this separation between horizontal and vertical subtypes in relation to the point at which the dendrite starts its branching. Given that these dendrites first innervate their entire target neuropil, whilst remaining unicolumnar, before asymmetry emerges within the dendrites, how this may play a role in the emergence of dendritic orientation, and what impact the horizontal / vertical distinction may have in T4 but not T5 is unclear. What is clear, however, is that regardless of the starting position of dendritic branching, the resulting structure of the differing dendrites for both T4 and T5, and their subtypes, is conserved.

Second, the observation of larger T4 dendrites alongside a uniform columnar space, does raise the interesting question of how this may impacts function. We have shown that within the region where lenses on the retina are larger, T4 dendrites additionally appear to span a larger area, even though the columnar grid through which they sample their visual inputs, is uniform. This directly implies that T4 dendrites within the anterior-ventral region integrate motion over a marginally larger region of visual space, and as such would be expected to posses a larger receptive field. We are not aware, however, of any functional studies which show this. On the other hand, given the fundamental necessity of motion detection within visual processing, this may well be an artefact of these dendrites development which is modulated by functional tuning, leading to a morphological, but not functional, difference.

The emergence of orientation

Given the homogeneity of dendrite structure across subtypes and types, it is important to pinpoint exactly what is meant by their characteristic orientation. At some stage between 36 and 72 hours APF [8], the well documented orientation of these dendrites must emerge [1,4,10]. We have shown that these dendrites, regardless of their spatial orientation, share a common structure. Their spatial embedding, although more variable, follows the principle of elliptical space filling elongated along the DV axis. Additionally, we have shown that, although the directed subtypes can be distinguished by the mean orientation of section angles relative to the start of dendritic branching, these sections are consistently oriented away from this point. This appears indicative of a shared space filling algorithm across these dendrites regardless of subtype specific orientation [43,44].

The characteristic orientation of these dendrites then seems to be determined by the centrally offset root position of the dendritic branching process in relation to the centre of the occupied elliptical space (Figs 2A and S1). How then, given that individual sections are oriented radially away from the dendrite root point (Fig 5D and 5E), does the mean direction of these sections result in orientation? To illustrate a process by which orientation emerges as a result of offsets in the position of initial branching (the dendrite root point) relative to the final filled elliptical space, consider an ellipse in which a point is randomly placed at an off-centre location. If short lines a drawn within the ellipse, oriented radially away from the point but directed towards the boundary, the average angle of these lines relative to a global reference will be oriented, as is the case between subtypes here. It is evident that the initial branching positions for all subtypes exhibit a consistent offset relative to the occupied space. Specifically, dendrite roots in the vertical subtype are positioned centrally within the lower and upper regions of their spanned area, whereas horizontal subtype roots are located within the central left or right regions (Figs 2A and S1). This configuration enables dendritic orientation within the occupied space, while preserving the structural homogeneity independent of overall orientation.

T5a dendrites, however, show a markedly more central dendrite root point (Figs 2A and S1). Interestingly, the orientation of dendrites from the mean angle of dendrite sections for T5a is still present, however it possesses a somewhat broader and less directed distribution in comparison to all other subtypes (Fig 5B). While this observation is in line with the above described mechanism: a more central initial position would result in reduced directionality, it is unclear why this is the case within T5a dendrites. Similarly, within T5c dendrites we see a shift in their orientation toward the posterior direction (Fig 5B), in line with the observed shift of the dendrite root point towards the anterior (Figs 2A and S1). This is not observed however within the T5d subtype, although we would expect it to be. Although we observe this result separately within both hemispheres, we are cautious to not rule out the possibility of artefacts within either the FAFB-FlyWire dataset itself or our processing pipeline, although these seem unlikely. Future work should seek to validate, or refute, this observation in a separate dataset before speculating about the broader impact of this observation.

Insights into dendrite development

We have set out our analysis in order to quantify the morphology of all dendrites within these closely related populations of neurons with the aim of better understanding how the characteristic subtype specific orientation may develop between 36 and 72 hours APF [8]. Our analysis has shown that these dendrites are structurally homogeneous, which we take as strong evidence of a shared algorithm of dendritic outgrowth within this period. Our analysis indicates a developmental process whereby new dendrite arbours orient outwards from the initial branching point to occupy space in an optimal manner at equivalent rates (Fig 6) and with shared geometry within both T4 and T5, and their subtypes (Fig 5). This, alone, would produce a circular dendrite, and it is unclear how such outgrowth facilitates symmetry breaking within this period.

A likely contender for the emergence of orientation during development is that subtypes make use of chemical gradients. The identification of chemical gradients within the Medulla and Lobula during development which may be related to T4 and T5 development have remained elusive however. One contender would be the identified DWnt4 and DWnt10 A/P gradient identified within the Medulla during development [45]. However this gradient is observed at 25 hours APF, and not present at 45hrs APF. This is an issue, as the dendritic orientation of T4 and T5 dendrite subtypes does not start to emerge until later than 36hrs APF [8].

Given the absence of a definitive chemical gradient, we propose two potential local cues which could influence dendritic orientation in light of our analysis which would maintain structural conservation. Our analysis shows that dendrites innervate their target neuropil in a subtype specific manner, with opposingly oriented dendrites positioned as nearest neighbours (Figs 4 and S4). Previous work suggests that these nearest neighbour partners likely originate from the same neuroblast (NB) [7]. Additionally neurons from the same NB have been shown to be able to distinguish themselves from other neurons from the same lineage [46,47]. During the initial dendritic outgrowth phase where dendrites remain confined within a single column [8] dendrites originating from the same NB may provide attraction or avoidance cues determining later orientation and outgrowth. Second, each dendrite densely overlaps in space. During development, neurons will form synapses promiscuously and, taken out of their developmental context, form any synapses they can [48]. This implies that dendrites will compete spatially for synapses. For two dendrites which initiate branching close together, this competition will be reduced in the opposing direction, facilitating directed outgrowth.

While such promiscuity could disrupt the precise synaptic compartmentalisation observed within these dendrites [4], we have shown that they orient across, not along, the neuropil depth (Fig 3D and 3E). If input partners are stratified within the neuropil layer, or if their innervation times differ, spatial compartmentalisation of synapses may, in part, be a result of the spatial or temporal availability of these partners during development. This allows for promiscuity during development, while maintaining the spatial compartmentalisation of synapses onto these dendrites.

Although we invite caution in putting to much emphasis on the specific distributions identified within our explorative analysis of section lengths, the observed distributions do offer some interesting interpretations to development. We observe a gamma distribution as the consistently best fitting distribution to internal dendrite section lengths, effectively the inter branch interval. The gamma distribution relates to a process involving the necessity for multiple events to occur before a branching event takes place. Giving a more concrete example, this implies that a branching event will occur during growth after n sub-events, which are exponentially distributed, have first occurred [49]. When branching is treated as a Poisson process, and multiple events are required before a stable branch occurs, we would expect to observe a gamma distribution as a result [50]. Empirically fit models which treat branching in this manner have previously been used to accurately capture the observed distribution in real data [51–53]. This assumes a stable dendrite branch requires some number of molecular sub-events before formation. Actin nucleation, specifically transient, localized Arp2/3 recruitment, precedes branch formation in Drosophila DA neurons [54], as an example of one such potential molecular sub-event. Further, microtubules have been shown to transiently invade branch precursors, providing a pathway for additional material required for stabilisation, initiating changes in dendrite spine morphology [55]. MAP2c in cultured hippocampal neurons has also been shown to promote stabilisation of neurites, through the coupling of F-actin and microtubules, consolidating stabilisation [56]. Our data is indicative of a complex multi-event mechanism which governs branching within these dendrites.

Within external dendrite sections, a log-normal distribution is the consistently best fitting distribution to the data. A log-normal distribution has been previously related to dendrite spine outgrowth and lengths [36], as well as axonal tree section lengths [28]. The log-normal distribution is a result of a central limit theorem-like process resulting from the product, rather than the sum, of multiple samples [57,58]. Dendritic spine lengths have previously been observed to follow this type of distribution, both experimentally and within modelling studies [36,59,60]. How this may relate mechanistically to the outgrowth of dendrite external sections is not as clear however. Live imaging has shown stochastic switching between growth, paused, and shrinking states in an episodic manner within dendrite tips [61]. This dynamic state switching, coupled with the available resources in the dendrite tips, could, in principle, generate the multiplicative effects required to produce the observed log-normal distribution [28].

Future analysis and perspectives

Multiple avenues of future work open up in order to better understand T4 and T5 dendrites from connectomics data. The most obvious is to investigate comparative differences in T4 and T5 dendrites between datasets, utilising the available male optic lobe connectome which has been recently released [62]. This would allow us to better capture variability across individuals, and also allow for sex-specific comparisons relating to differing size, shape, and orientation of Drosophila compound eyes between males and females across species [41].

Second, it is important to understand the synaptic distributions within T4 and T5. Synaptic inputs are vital to dendritic development [63–65], and given the fundamental role of motion detection within visual processing, it seems likely that the functional output of T4 and T5 dendrites is tightly regulated. Further, the origin of the distinct spatial compartmentalisation of synapses onto T4 and T5 dendrites is not well understood. Extending our analysis to included synaptic information, as well as additional functional imaging or modelling with the available highly detailed dendritic morphology, will be insightful.

Finally, we have limited out analysis of columns to only the size of columns as defined by Mi1 neurons. As we consider the optic lobe column structure to be the map of visual space through which these dendrites perform motion detection, it will be important to extend these analysis to better understand the relationship between visual columns and the spatial embedding of these dendrites, particularly in light of the observed size differences of T4 dendrites within the anterior-ventral region. However, see [10,66] for analysis of T4 and T5 neurons which includes visual space mapping. Of additional interest will be understanding how the structural relationship between the start of dendritic branching is related to the column map, and if this relates to emergent orientation of these dendrites through the selection of columns which they ultimately sample from.

Limitations

The data presented here are from a single adult female Drosophila, offering a view of a single snapshot of T4 and T5 dendrite morphology. Although thought of as complete, and offering a new gold standard in data quality, some noise is undoubtedly introduced from the complex processing steps needed to generate a usable dataset from the EM image stack.

Before consideration for inclusion within this study, we ensure that all dendrites are labelled as complete and proofread within the FAFB-FlyWire dataset, ensuring they have already been independently reviewed before our use. We additionally manually verifying the accuracy of each individual dendrite. However, particularly small branches are likely missing from the morphological reconstructions of neurons. It is worth noting however that previous work using manually reconstructed dendrites, the gold standard in morphological reconstruction, shows that such small branches are not directed and typically concentrated on connecting a small number of synapses to the dendrite [10], and a such minimally impact the dendritic morphology.

In addition to this we would argue that as this error is likely missing at random, and as the n of T4 and T5 neurons is sufficiently large, this error rate is likely uniform across the dataset and is unlikely to influence our core findings. As these data also come from a single Drosophila, it is also possible that different individuals show differing results, keeping in mind however that our results are consistent across both hemispheres. Given the fundamental nature of T4 and T5 neurons in early visual processing and motion detection, we feel this is unlikely to be the case. The question of individual variability between datasets is an intriguing one which requires further work and comparative analysis between datasets in order to be fully understood.

T4 and T5 dendrite annotation and pre-processing

The FAFB-FlyWire dataset comprises of fully reconstructed neurons from a single resolution electron microscopy (EM) image stack [13–16,67]. Neuron type annotations within the right optic lobe of this dataset are available through [9], and include annotated T4 and T5 neurons. These annotations have been updated since the publication of [9] to include both optic lobes bringing the total number of annotated T4 and T5 neurons to . Neuron type annotations use the current annotation state within FAFB-FlyWire as of April, 2026 [9,14–16].

Each neuron is represented as a 3D surface mesh. Skeletonization, the method of extracting tree-graph representations from neuron surface meshes, is implemented using a standard procedure [68,69]. The correct rooting of each neuron at the soma is manually verified and corrected if needed. All reconstructions have been annotated as complete within the FAFB-FlyWire dataset. After skeletonization, all neurons are converted to from nm, and resampled to . Table A in S1 Text gives counts by subtype and hemisphere for available and included dendrites.

With skeletonization, each neuron is modelled as a tree-graph (See Materials and Methods, Neurons as Tree Graphs). We further require the accurate annotation of the dendritic compartment of each neuron tree-graph in order to extract and analyse dendrite morphology (See Materials and Methods, Dendrite Extraction and Pre-processing).

Neurons as tree graphs

Each neuron represented as a tree-graph is rooted at the soma, in the case of complete neurons, or the first branching point of nodes within the dendritic subtree (referred to as the dendrite root point), and all edges are oriented away from this root node. For completeness, we provide a brief overview of graph-theory specific to directed tree-graphs here and the components relevant to this manuscript.

A neuron is modelled as a directed tree:

(1)

where V is the set of 3-D coordinates defined by the skeletonization algorithm and the set of oriented edges. r denotes the root node of the tree. denotes an edge from u to v, where u is the source node, and v the target.

Each edge carries the Euclidean length . The tree weight (total cable length) is:

(2)

For :

(3)

Where deg⁺ denotes the out-degree of a node, deg^- denotes the in-degree, and deg(v) the total degree. As edges point away from the root, , and for every . branch nodes have and leaf nodes . An edge is external if is a leaf, otherwise internal.

For any two nodes there exists a unique undirected path

(4)

with k = d(u,v) steps.

The corresponding edge sequence:

(5)

lists the k oriented edges traversed in order; note that orientation may be reversed relative to E when the walk proceeds toward the root.

Unweighted and weighted path lengths are

(6)

Node depth is its unweighted distance to the root, , and an edge inherits the depth of its distal node.

Every maximal path of intermediate nodes with is contracted into a single edge whose weight is the sum of the collapsed weights. The reduced graph is . Such reduced trees dramatically minimise the number of nodes and edges within each neuron skeleton.

For any node we define the subtree rooted at v as:

(7)

where means that v lies on the unique root–to–x directed path in T. Thus contains v and all its descendants together with the edges that connect them, and is itself a rooted, oriented tree with root v.

Dendrite extraction

Previously, separation of neuron morphologies into biologically relevant subtrees, such as the axon and dendrite, has been done manually [70] or derived from synapse data [71,72]. Within this dataset of neurons, manual annotation of each dendrite is not practical. Methods based on the input-to-output ratios of synapses [71,72] assume clearly defined information flow between compartments, which does not always hold [70], and require additional synapse annotations. We instead implement a novel, semi-automated method to annotate the dendritic compartment of T4 and T5 neurons based on tree structure alone.

Given a tree-graph correctly rooted at the soma, we then identify the subtree in T which maximises the number of leaves within the subtree whilst minimizing the sum of edge weights using a simple metric, :

(8)

Where is the set of branch nodes within T. Although often the identified subtree will be the dendrite within T4 and T5 neurons, due to either routing errors resulting from the surface mesh and skeletonization procedure, or instances where no single node isolates the root of the dendritic subtree, we manually verify and modify the identified dendritic compartment for all neurons, and exclude those with an inaccurate or visibly incomplete dendritic tree.

This manual verification step is completed using the GUI available within the NeuRosetta toolbox (See Materials and Methods, Data and Code Availability). This GUI allows for the re-rooting of neurons, manual subtree annotation from a single point (alongside automated identification using the above metric) and flagging of neurons for exclusions from further analysis. This manual review step can be completed using this GUI in a relatively short time by a single expert reviewer.

Global population dendrite alignment

As T4 and T5 dendrites exist in two separate neuropils, across two hemispheres, we perform a global alignment step in order to ensure each dendrite population is aligned with the dorsal - ventral and anterior - posterior axis in visual column space. This process is performed by taking the following steps, where a dendrite population refers to each individual population of dendrites within each hemisphere, so for example all T4 dendrites within the right hemisphere.

First, we subtract the mean coordinate within each population, and perform a rotation. The aim of this rotation is to ensure that the point cloud is oriented in a way which ensures that the longest extent of dispersion is along the y-axis, the second along the x-axis, and third along the z-axis in euclidean space. This is done by performing eigendecomposition on the covariance matrix of the point cloud coordinates, providing three eigenvectors and three eigenvalues. Our rotation then aligns the eigenvector corresponding to the highest eigenvalue to the y-axis basis vector and the eigenvector corresponding to the second highest highest eigenvalue to the x-axis basis vector . As the covariance matrix is square and symmetric, the eigenvectors are guaranteed to be orthogonal and as such by definition the final eigenvector is aligned with the final basis .

The second step taken is to fit a sphere to each population and subtract the centre of mass of this fitted sphere to provide a shared, central, coordinates origin for all dendrite populations regardless of neuropil. This works well, as both Medulla layer 10 and Lobula layer 1 are well approximated by a section on the surface of a sphere. As a result of this, the greatest extent of each populations points is aligned with the polar axis, and the y-axis, and the second with the equatorial axis, and the x-axis.

As a final step, we ensure that the anatomically dorsal population for both T4 and T5 is positioned within the northern hemisphere to ensure that both T4 and T5 dendrite populations are aligned with the same dorsal-ventral and anterior- posterior space. This final rotation is completed with reference to the column annotations within FAFB-FlyWire. This is done by identifying example dorsal, ventral, anterior and posterior columns as well as the T4 and T5 neurons associated with them and using these to ensure the correct global orientation. These available annotations assign neurons to a column in visual space, but do not include neuropil-specific spatial columns, necessitating this final step in order to facilitate the shared DV/AP alignment of the T4 and T5 dendrite populations.

These steps results in all T4 and T5 dendrites within both hemispheres being aligned so that the y-axis follows the Dorsal - Ventral axis and x-axis the Anterior-Posterior.

Individual dendrite alignment and scaling

We utilise a similar procedure to align and scale individual dendrites where applicable. When aligning individual dendrites, we calculate the covariance matrix using a robust M-estimation procedure based on a Huber-type influence function. This is done in order to minimise the influence of node outliers on eigenvectors as a single long outreaching sections may impact the eigenvector orientation within single dendrites. After Eigendecomposition of the covariance matrix, eigenvectors are aligned similarly to the global procedure so the primary eigen-axis aligns with the y-axis, the secondary eigen-axis to the x-axis and the third to the z-axis. Before calculating angles of rotation we ensure that each eigenvector is oriented along it’s positive target basis vector, in order to minimise rotation.

In order to scale point node positions we scale coordinates along their eigenvectors by s. Within Fig 2A, these eignevectors are the basis vectors in euclidean space due to the above described alignment. Again for Fig 2A, s is defined as:

(9)

where c is the node coordinates of a single dendrite, is the mean coordinate, and is the mean standard deviation along the y-axis of the current population. This scales all dendrites along each axis so that , placing the coordinates in standard deviation units. Alternatively, for unit variance scaling (Fig 7B, 7D and 7F) s is a vector equal to where is the variance along each eigenvector, e. This scales dendrite to have equal variance along each axis.

Branching structure and symmetry quantification

Considering a tree graph with no nodes where , as is the case for the reduced neuron tree-graph representations used here, let n = |V| and L be the number of leaves. Internal node number . As (i) a star tree has one branch point and , and (ii) a full binary tree satisfies L = I + 1, hence , thus for any tree graph we have an upper bound:

(10)

This provides limits between maximally condensed and maximally branching tree-graphs, and no tree-graph can exist outside of these bounds. The closer I is to , the closer to a full binary tree.

Tree asymmetry for a single node, , as defined in [26,27], is extended similarly to [28] in order to weight the metric by the weight of the subtree defined at v with child nodes. In the case of deg⁺(v) > 2, we use a mean of all child pairs (p, q):

(11)

lies in [0,1], where 0 indicates full symmetry in the branching structure.

Surface fitting, volume, and depth measurement

In order to find the spanning volume of single dendrites, a convex hull is fit to the node coordinates of each dendrite using scipy.spatial.ConvexHull [73]. We reconstruct surfaces meshes of Medulla layer 10 and Lobula layer 1 via voxelization and marching cubes of all node coordinates in T4 (Medulla layer 10) or T5 (Lobula layer 1) dendrites. Step by step, we first compute the Axis-aligned bounding box for the given point cloud, extended by a fixed padding factor to ensure a spatial margin. This is then discretized into voxels and each point is assigned to a voxel. 1 iteration of morphological dilation is applied in order to fill small gaps in the set of occupied voxels, and then unoccupied voxels are removed, retaining only the largest connected region of voxels. This binary grid is then converted to a continuous surface using the marching cubes algorithm as implemented in scikit-image. The resulting mesh is then cleaned by removing unreferenced nodes, duplicated or degenerate faces, filling holes in the mesh surface, and correcting face normals to ensure they are outwardly oriented. Surface meshes are smoothed using the Windowed Sinc method and a band pass filter with a width of , as implemented in [74].

In order to calculate the depth of any given point within a neuropil layer mesh, we first define two subsets of faces on the surface mesh which represent the two sides of the neuropil we are measuring depth across. We can define these sub-surfaces base on the orientation of face normals in relation to the shared coordinate origin. This is done by taking the dot product of each face normal with a vector from the coordinate origin. Faces with values are treated as belonging to the outer mesh surface, and values as the inner surface. A KDTree is calculated for both the inner and outer surface. For any given point, the nearest neighbour distance is calculated to both surfaces. We then calculate a normalized distance for the point from the inner surface. As T5 dendrites innervate Lobula layer 1 from the opposite side of curvature in comparison to T4 dendrites innervating Medulla layer 10, we invert T5 depths. Depth is calculated as follows:

(12)

When generating neuropil surface meshes, a dilation step which fills in missing voxels is required in order ensure a single continuous surface is recovered. Due to the curvature of these neuropils, this results in a degree of padding within the concave region which is empty of dendrites. The extent of this padding differs between the two hemispheres, resulting in an offset when estimating neuropil layer depth. As such, we treat analysis of depth in the two hemispheres separately.

Dendrite root point nearest neighbour matching

In order to identify dendrite root point nearest neighbours we first assign subtype specific nearest neighbours using KDTrees within each type - subtype combination, allowing us to identify the distance between each T4a, for example, and the closest T4a (not including itself), T4b, T4c, and T4d dendrite root.

In order to determine the probability of a nearest neighbour being of either a specific classification (“opposite,” “same,” or “orthogonal”), or specific subtype, we calculate an optimal pairing within all T4 and all T5 dendrite root positions using a maximum-cardinality matching approach as implemented within the graph-tool python toolbox [75]. This method aims to match as many points as possible, maximising the number of pairs while simultaneously minimising the total euclidean distance between the matched pairs.

In order to account for depth as a possible confound within this procedure we project each set of points to the surface of their fitted sphere prior to calculating nearest neighbour distances as described above (S3 Fig).

Geometry analysis

When calculating angles between vectors in we project vectors to the plane orthogonal to the normal vector, n, by taking the rejection of each vector from n:

(13)

We then calculate the signed angle, , between v₁ and v₂ in the plane as:

(14)

Where is the unit normal of the basis vector, is the cosine of the angle. gives the signed sine, where the sign is determined by the right hand rule with respect to n. returns the signed angle .

As defined in [76], the circular mean is calculated as:

(15)

and Variance as:

(16)

The Dihedral angle , is the “the angle between the daughters’ half-plane and the plane formed by the parent segment and the line perpendicular to daughters’ bisection through the bifurcation point in the daughters’ plane” [22] (Fig 5L, insert). In order to calculate this for a set of parent and two child vectors centred on their branching point, we first find the bisector, b, of c₁ and c₂ as:

(17)

We then calculate the angle between b and p from the perspective of the plane defined by the normal vector, n, given by:

(18)

This results in the Dihedral angle which is between 0 and . At both extremes, this value represents bifurcations sitting on the same plane, however at 0 child vectors are on the plane oriented in the same direction as the parent relative to the bifurcation points, and at they will be oriented away.

In order to generate bifurcation random null-models we randomly generate 1,000,000 sets of three 3-dimensional unit vectors. We ensure that one vector is oriented towards the negative x-axis to represent the parent section, and the other two are oriented along the positive x-axis. This ensures an approximation of the outward facing branching structure observed. Second, sets of bifurcation vectors are then scaled along their z-axis separately for T4 and T5 by the cube root of the mean variance explained by the third principle component, in order to mimic the natural flatness of the occupied space of T4 and T5 dendrites.

Explorative distribution fitting

We divide dendrite populations by type (T4, T5), subtype (a,b,c,d) and section type (internal, external). We then fit using maximum likelihood, for each group, six test distributions in order to compare goodness of fit. The distributions used are as follows:

Wald (Inverse Gaussian) Distribution, with mean and shape :

(19)

Log-normal Distribution, with log-mean and log-standard deviation :

(20)

Gamma Distribution, with shape k > 0 and scale :

(21)

where is the gamma function.

Exponential Distribution (Gamma distribution special case with k = 1) with scale :

(22)

Log-logistic Distribution with shape c > 0 and scale s > 0:

(23)

Minimum Weibull Distribution with shape c > 0 and scale :

(24)

For each fitted distribution we calculate the Bayesian Information Criterion (BIC) as:

(25)

where k is the number of free parameters (2 for Wald, log-normal, gamma, log-logistic, and Weibull Distributions; 1 for exponential). n is the number of observations within the sample and is the maximised log-likelihood. Lower BIC indicates a better trade off between model fit and parsimony.

Statistical analysis

We test for hemisphere (Right, Left), neuron type (T4, T5) and subtype (a,b,c, and d) effects, as well as all interactions, on different measured variables using ordinary least squares (OLS) regression and analysis of variance (ANOVA) / covariance (ANCOVA). Models are fit with a full factorial structure, and sum of squares were evaluated using Type III tests. We additionally, for each effect (main effects and interaction terms), calculate the partial eta-squared as an effect size, which quantifies the proportion of variance uniquely attributed to each factor given the model structure [18]:

(26)

where SS is the sum of squares. We additionally report slopes obtained from the regression model fit for meaningfully large main effects, as well as group specific slopes and their differences where applicable.

We additionally obtain, for each effect size, a non-parametric confidence interval obtained through stratified bootstrap resampling (1000 samples). Post hoc pairwise comparisons were conducted on marginal means, both at the main effect level and within interactions, and Cohen’s d was calculated for effect sizes:

(27)

with

(28)

where is the group mean and the group standard deviation. Again, non-parametric confidence intervals using 1000 bootstrap resamples are calculated.

Plotting conventions

We make liberal use of the empirical probability mass functions (epmf) for plotting distributions throughout, without smoothing. To allow for comparison within plots, binning along the x-axis is the same for all epmf’s shown within a plot. Bin counts and ranges are adjusted manually. Error bars are presented in two forms, as specified in a plot by plot basis. In cases where a single observation is available per individual dendrite, a bootstrapped confidence interval around the mean in each bin is shown (1000 bootstrap replicates). Alternatively, when a repeated measure is available within each dendrite, a single epmf is generated for each individual dendrite and, in order to give an idea of individual variability across the population, error bars within each bin ar calculated as the asymmetric median absolute deviation :

(29)

where is the median within the respective bin. This illustrate variance in each bin across neurons. An Asymmetric MAD allows for the possibility that individual variability within each bin is skewed, and can be shown within the plot.