plosPLoS Comput BiolploscompPLoS Computational Biology1553-734X1553-7358Public Library of ScienceSan Francisco, USA08-PLCB-RA-0793R210.1371/journal.pcbi.1000251Research ArticleComputational Biology/Computational NeuroscienceNeuroscience/NeurodevelopmentNeuroscience/Theoretical NeuroscienceThe Morphological Identity of Insect DendritesThe Morphological Identity of Insect
DendritesCuntzHermann12*ForstnerFriedrich2HaagJuergen2BorstAlexander2Wolfson Institute for Biomedical Research and Department of Physiology,
University College London, London, United KingdomDepartment of Systems and Computational Neurobiology, Max-Planck
Institute for Neurobiology, Martinsried, GermanyFristonKarl J.EditorUniversity College London, United Kingdom* E-mail: h.cuntz@ucl.ac.uk
Conceived and designed the experiments: HC FF AB. Performed the experiments:
FF JH. Analyzed the data: HC FF. Wrote the paper: HC FF AB.
The authors have declared that no competing interests exist.
12200826122008412e10002511592008121120082008Cuntz et alThis is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and
source are credited.
Dendrite morphology, a neuron's anatomical fingerprint, is a
neuroscientist's asset in unveiling organizational principles in the
brain. However, the genetic program encoding the morphological identity of a
single dendrite remains a mystery. In order to obtain a formal understanding of
dendritic branching, we studied distributions of morphological parameters in a
group of four individually identifiable neurons of the fly visual system. We
found that parameters relating to the branching topology were similar throughout
all cells. Only parameters relating to the area covered by the dendrite were
cell type specific. With these areas, artificial dendrites were grown based on
optimization principles minimizing the amount of wiring and maximizing synaptic
democracy. Although the same branching rule was used for all cells, this yielded
dendritic structures virtually indistinguishable from their real counterparts.
From these principles we derived a fully-automated model-based neuron
reconstruction procedure validating the artificial branching rule. In
conclusion, we suggest that the genetic program implementing neuronal branching
could be constant in all cells whereas the one responsible for the dendrite
spanning field should be cell specific.
Author Summary
Neural computation has been shown to be heavily dependent not only on the
connectivity of single neurons but also on their specific dendritic
shape—often used as a key feature for their classification. Still,
very little is known about the constraints determining a neuron's
morphological identity. In particular, one would like to understand what cells
with the same or similar function share anatomically, what renders them
different from others, and whether one can formalize this difference
objectively. A large number of approaches have been proposed, trying to put
dendritic morphology in a parametric frame. A central problem lies in the wide
variety and variability of dendritic branching and function even within one
narrow cell class. We addressed this problem by investigating functionally and
anatomically highly conserved neurons in the fly brain, where each neuron can
easily be individually identified in different animals. Our analysis shows that
the pattern of dendritic branching is not unique in any particular cell, only
the features of the area that the dendrites cover allow a clear classification.
This leads to the conclusion that all fly dendrites share the same growth
program but a neuron's dendritic field shape, its “anatomical
receptive field”, is key to its specific identity.
HC, FF, JH and AB were supported by the Max-Planck Society.Introduction
Dendrite morphology is the most prominent feature of nerve cells, typically used by
neuroanatomists to discriminate and classify them [1]. These tree-like
ramifications represent the input region of the neurons and fulfil the role of a
complex computational unit [2]–[5]. Typically, dendritic
arborizations are analyzed in a descriptive way, e.g. by enumerating local and
global branching parameters [6]–[8]. Very little is known
about the general rule leading to their distinct appearance partly due to the wide
variety among different neurons. In insects, same neurons across individuals are
rather invariant in their anatomy and constant in their function. Lobula plate
tangential cells (LPTCs) of the fly visual system [9] are uniquely
identifiable and are therefore ideal subjects for investigating the basic rule
constraining dendrite formation. They integrate local motion information over an
array of retinotopically arranged columnar elements [10]. Accordingly, their
planar dendritic trees cover the area corresponding to their distinct primary
receptive fields. In this paper we isolate potential fundamental principles which
may lead to the morphological identity of individual LPTCs.
Results
We studied inter-individual constancy and variability in four members of the LPTC
group: the equatorial and the northern cell of the horizontal system (HSE and HSN,
Figure 1A), and two members
of the vertical system (VS2 and VS4, Figure 1B), each of them represented by at least ten individuals from
different flies. Two-photon image stacks were acquired from cells filled with
fluorescent dye in the living blowfly, Calliphora vicina.
Subsequently, the anatomy of each neuron was manually traced and described by a set
of connected cylinders (see detailed explanation on the reconstruction procedures in
the Methods section). The idea was, in
concordance with previous publications [6]–[8], to use
statistical distributions over morphological parameters thereby isolating key
features of dendritic branching. Next to classical branching parameters on the
“topological points” (branching and termination points in the
tree) such as branching order and path lengths to the root, we parameterized the
area covered by the dendritic tree, the so-called “dendrite spanning
field” [11]. We defined spanning field by drawing a contour
around the dendrite at a distance of 25 µm after orienting the
reconstructed neuron along its axonal axis (Figure 1C and 1D).
10.1371/journal.pcbi.1000251.g001
Dendrite morphological statistics.
(A,B), Sketches showing HSE and HSN (A) and VS2 and VS4 (B) in the context of
the lobula plate (gray). (C,D), Superimposed full anatomies of all
individual cells sorted according to their respective cell type. Cells were
aligned along their axonal axis (red lines). To the right, the corresponding
dendrite spanning fields are outlined. (E–K) Statistics
specifically related to dendrite branching. Statistics are represented as
superimposed distribution histograms, filled squares show mean values and
error bars correspond to standard deviation between individual dendrites:
(E) path length to root values for all topological points; (F) ratios
between direct and path distances from each topological point to the
dendrite root; (G) topological point branching order values, a measure for
the topological distance from the dendrite root; (H) length values of branch
pieces between topological points; (I) branching angle values at all
branching points between the two direct daughter branches within the plane
in which they lay; (K) surface area values assigned to each topological
point after Voronoi segmentation indicating topological point density and
distribution homogeneity. (L) Sholl intersection plots: number of
intersections of each tree with circles with increasing diameter.
(M–R) Statistics describing the dendrite spanning field: (M) total
surface value of spanning field; (N) percentage of the spanning field below
the axonal axis; (O) convexity index of the spanning field; (P) ratio of
width against height of the spanning field; (Q and R) horizontal and
vertical coordinates of centre of mass of the dendrite spanning field.
Regarding branching-specific statistics (Figure 1E–K), qualitative distinction was possible only by
detailed examination of distributions of topological point density, path length to
the root and branch order. Ratios between direct and path distances of the root
(Figure 1F) followed a
narrow distribution close to 1 in all cases for all topological points. Path length
histograms (Figure 1E) therefore
corresponded to the Sholl intersection diagram (Figure 1L), a measure typically used to describe
branching topology. On the other hand, parameters relating to the spanning field
plainly reflected cell type specific differences: All four cells could be readily
discriminated by eye by their relative position and the shape of their dendrite
spanning fields (Figure 1C and
1D, parameters see Figure
1M–R). Those differences were in conformity with the respective
primary receptive field locations in the retinotopic arrangement. HS and VS spanning
fields were easily distinguished by either their convexity index (Figure 1O) or the ratio of width
against height (Figure 1P).
Finer differentiation of HSE against HSN and VS2 against VS4 was provided directly
by their relative location to the axonal axis (Figure 1N), and accordingly by their centre of
mass (Figure 1Q and 1R). We
investigated the descriptive power of spanning field parameters versus branching
parameters in a quantitative way (Figure 2). Spanning field related parameters readily grouped individual
cells into their respective cell types as shown simply by plotting convexity index
values against the contextual relative location off the axonal axis (Figure 2A). On the other hand,
even a highly-dimensional clustering analysis on the basis of parameterized shape
fits of the distributions in Figure
1E–K (see Figure S2) or subsets of these did not allow the
separation of the real cells into their respective groups. Best clustering was
obtained using path length, density and branching order distributions which
separated HS from VS cells but not the members of the two families (Figure 2B). In accordance to these
findings we postulated that if the spanning area best determines neuronal
appearance, the particularities in branching parameter distributions might be merely
a consequence of the neuronal target zone.
10.1371/journal.pcbi.1000251.g002
Cluster analysis.
(A) Dendrite spanning fields are readily separable into the individual cell
types at the example here of two parameters only: the convexity and the
relative location to the axonal axis (B) Cluster analysis using three
parameters of a generalized extreme value distribution fits for branching
properties from Figure 1E, 1G,
and 1K.
In order to identify the critical impact of spanning field shape on branching
parameters, artificial dendrites were constructed covering the same region. Inside
the contours of the original cells, random points were distributed following their
respective density map. An iterative greedy algorithm was launched starting at the
coordinates of the real dendrite root. At each step, a connection was added from the
existing tree to one of the unconnected random points according to a cost function
which kept house of both total amount of wiring and total path length from the root
to each point [12]. The number of random points was set to match the
resulting number of topological points with the original dendrites. Improved
appearance and overall path distance to the root was achieved by a subsequent
smoothing step along primary branches (see Methods section). This resulted in artificial dendrites confined to the same
area as the corresponding in-vivo dendrite reconstructions which
were virtually indistinguishable from their real counterpart (Figure 3A; see Figure S3 for a
full overview and Video S1 depicting the constructing process). Interestingly, artificial
dendrites also yielded quantitatively similar parameter distributions in all cases
(Figure 3B, compare with
Figure 1E–L). The
exact same branching rule can therefore account for all individual morphologies
after constraining the spanning field shape alone.
10.1371/journal.pcbi.1000251.g003
Artificial dendrites grown in real dendrite spanning fields.
(A) Artificial dendrites: two examples of each cell type. Upper row: real
dendrites. Lower row, marked by preceding “m”:
artificial dendrites corresponding to each of the spanning field. (B)
Artificial dendrite parameter distributions as in Figure 1E–K showing the
similarity to their real counterparts.
Consequently, one could consider that original raw fluorescent images containing a
labelled neuron would correspond to a distribution of interconnected points within a
spanning field. Then, if our assumptions about the branching rule are correct, one
should be able to apply it to obtain the branching model directly from the image
material. We therefore applied the same greedy algorithm describing our branching
rule for artificial dendrites on structural points extracted from the raw data via
image skeletonization. The results of such an attempt are shown at the examples of
an HSE dendrite (Figure 4A and
4B) and a full VS2 cell (Figure
4C and 4D). Faithful cylinder models of almost all branches could be
retrieved in a fully automatic way from the image material after simply assigning
manually a starting location at the dendrite root (see detailed description of the
procedure in the Methods section).
10.1371/journal.pcbi.1000251.g004
Model-based reconstruction of neuronal branching from 3D two-photon image
stacks.
Depicted at the example of an HSE dendrite (A,B) and of a VS2 cell (C,D).
Left, maximum intensity projections of the image stacks containing
fluorescent cells. Right, overlaid reconstructed branching in red.
Discussion
In summary, we claim that all cells analysed here follow the same branching rule, and
that their morphological identifier is the shape of their dendrite spanning fields.
This claim is supported by the presented branching statistics, the previously
proposed branching rule [12] and its reapplication in a heuristic
reconstruction algorithm. Early approaches to describing and reconstructing dendrite
branching in general had failed to take into account a major functional constraint
governing dendrites: their need to reach specific input locations. More recent
attempts to constructing dendrite morphology in relation to their function and the
location of their inputs had led to dendrite structures of low complexity and
accuracy in spite of high computational costs [13],[14]. However, circuitry
and connectivity as well as simple wire packing issues are known to be determinants
of dendrite morphology [15],[16]. In addition, the
specific organization and architecture of many parts of the brain helps to reduce
wiring costs for the circuitry [17],[18]. It is therefore
not surprising that such constraints can be used to describe dendrite branching in
LPTCs and other cells. Other planar space-filling cells, the cerebellar Purkinje
cells, certainly follow a similar rule [19]. However, the suggested approach is not
restricted to planar dendrites and future analysis will cover all different neuron
arborizations to clarify the ubiquity of the suggested branching rule. At the
example of LPTCs, the usefulness of the approach presented here can be put forward:
LPTC electrophysiology was studied in great depth e.g. [20] and precise models,
so-called compartmental models, including the detailed anatomical structure were
designed and are continuously being improved [21]–[24].
Understanding LPTC branching, these constraints can be directly put in relation with
the optic flow processing occurring within their circuitry [20],[23]. Assuming generality of
principles, even the function of cells, which have not yet been reconstructed, can
be inferred based on the contours of their dendrites alone. Moreover, the fly is the
model animal in which the molecular components that determine neural growth are
currently being unveiled, mainly through genetic tools [25],[26]. Our
framework therefore allows a quantitative study of the impact of gene modifications
far beyond basic statistics. In particular, molecular principles guiding neuronal
self-avoidance during development [27] and others can now be put in relation with the
branching constraints presented here. Eventually, studying molecular factors shaping
dendritic spanning fields separately from a specific branching rule within should
elucidate a fundamental organizational element in the brain, i.e. the
neuron's branching structure.
MethodsReconstructions
Female blowflies (C. vicina) were dissected as described in
[28]. In each subject either one or two different HS
or VS cells were filled with a fluorescent dye (Alexa 488). Flies were viewed
under a custom built two-photon microscope [29], orienting the planar
cells as orthogonal as possible in respect to the laser beam to minimize the
amount of images in the Z-direction. In order to capture the entire expansion of
the cells, 6 to 15 adjacent stacks (210 µm×210 µm
area in XY x ∼30 in 2 µm Z-steps) were taken from different
XYZ positions with an overlap of about 10 percent (Figure
S1A). The image stacks were then transferred to Matlab (Mathworks,
Natick, MA) and all further analysis was performed there in custom written
software. Manual fine tuning of the original coordinates from the individual
stacks was usually necessary to obtain a precise alignment in three dimensions.
Maps of maximum intensity and corresponding depth were computed along the
Z-axis. This reduction from 3D-data to two 2D images was sensible as there were
no or very few 3D crossings of branches and all cells were planar. Based on
these images cylinder models of the branching structure were obtained in a
semi-automated way: interactive software allowed switched viewing of either
Z-projection or an individual slice of an image stack (Figure
S1B). The widths of 2D rectangles connecting the end points were fitted
by gauss functions to suggest a diameter for the cylinders (Figure
S1D). Z-values were attributed to each cylinder directly from the
depth-map according to their 2D location. Quick tracing results (30 min) were
achievable working with maximum Z-projections alone, although slight movements
of the living fly compromised the accuracy of the projection image (Figure
S1C). In order to achieve a higher accuracy, some manual corrections
based on individual slices were necessary in all reconstruction steps. Taking
advantage of the planar cell morphology allowed quicker reconstructions compared
to other approaches [30]: detailed cell models with about 700 to 1600
compartments were obtained typically within around 2 hours. Jumps in the Z-axis
were smoothed by use of linear interpolation. Internally and externally, the
models were stored in the SWC format [31]. The
reconstructions can be downloaded at: (http://www.neuro.mpg.de/english/rd/scn/research/ModelFly_Project/downloads/)
Dendrite Statistics
For simplification, the resulting generic directed graphs were transformed into
strict binary trees by substituting multifurcations with several bifurcations
after minimally shifting the branches on their parent cylinder. Region indices
[32] (soma (1), axon (2) or dendrite (3)) were
manually attributed written to the SWC file. The somata in all cells consisted
of a clearly separated bag-like structure that branched from the axon or
dendrite. The last branch point (very short branches were ignored) before the
soma was chosen to be the end of the dendrite and the beginning of the axon. The
dendrite root was set to the primary branching point. Axonal parameters showed
no trend to classify the cells (data not shown). There was no obvious
correlation between axonal and dendrite length measures. Hence, size
normalization of the cells was discarded. Dendrite flattening was performed as a
morphometric transform [33] (Figure S1E). A distance isoline to any point
on the dendrite was drawn at a 25 µm threshold to determine the
dendrite spanning fields (Figure S1F). This corresponds to performing a
morphological dilation on the same points with a 25 µm radius disc.
For most statistics, only the branching and termination points
( = topological points) were selected as the
carrier points for the topological complexity. A Voronoi segmentation was
performed on these points in order to express space-filling density distribution
(Figure
S1G, used in Figures
1K and 3B). The
density value therefore describes the area in vicinity of a specific branching
or termination point. All LPTC reconstructions were rotated in order for both
the dendrite root and the furthest axon terminal tip to lie on the horizontal
line building the axonal axis. In order to obtain a measure for the convexity of
dendrites, the convex hull was drawn around all dendrite nodes. The surface
ratio between the dendrite spanning field (see above) and this convex hull was
chosen as a characteristic spanning field parameter, the convexity index (Figure 1O). Centre of mass was
calculated by taking the mean horizontal and vertical values of the line
surrounding the dendrite spanning field (Figure 1Q and 1R).
Cluster Analysis
Clustering (Figure 2) was
done on the three parameters which enabled a by eye discrimination of VS and HS
cells in Figure 1: the
branching order, the path length and the density. Their histograms were
collapsed to three values (mean, standard deviation and shape parameter) by
fitting them to a generalized extreme value distribution (Figure S2).
After normalizing to weigh parameters equally, Euclidean distances between the
different dendrites in the resulting 9 dimensional parameter space were
clustered hierarchically using the single linkage algorithm and displayed as
dendrograms. As an alternative, the principal components of the matrix
containing the normalized scalar parameters for each tree were obtained and the
trees observed in the corresponding reduced dimensionality plot: no better
grouping was possible with this method (data not shown).
Artificial Dendrites
Boundary-corrected density maps of dendrite topological points were derived from
real cell dendrites (Figure S1H–L). Random points were
distributed according to the obtained density maps (Figure
S1M). An extended greedy minimum spanning tree algorithm [12] was
applied on these points starting at the root point of the original dendrite
(Figure
S1N). The number of random points was increased until the resulting
number of topological points in the artificial dendrites matched the original
dendrites. XY-coordinates of points on longer branches were smoothed by Spline
interpolation to result in realistic dendrites (Figure
S1O). Similar conclusions would arise if artificial dendrites were
constructed on random points distributed entirely homogeneously (data not
shown).
Automatic Reconstruction
3D image stacks from one HSE dendrite and a full VS2 cell were submitted to 2D
anisotropic filtering, morphological closure and subsequent brightness level
thresholding. After 3D skeletonization and sparsening the carrier points, the
remaining points were submitted to the same greedy algorithm (started at a user
defined dendrite root location) as used for obtaining artificial dendrites
Quadratic diameter decay was mapped on the resulting trees [12] (see Figure
S1P).
Supporting Information
Sketches describing the manual cell reconstruction process and the subsequent
handling of dendrite morphology. (A) Assembled maximum Z-Projection of an
HSN with ten overlapping image stacks. (B) Example of a reconstructed
sub-tree of an HSN cell superimposed on a single slice from one image stack.
(C) Compromising effects of the maximum Z-Projection (right) compared to the
original slice (left, arrows show loss of branches). (D) Examples of
automatic diameter approximations. Normalized positions 0.25, 0.5 and 0.75
on the midline and 40 half pixels in orthogonal direction were used to
construct a sampling grid that covers a branch's thickness (first
panel). The average over the resulting sampling matrix was convolved with
the first derivative of a Gaussian distribution (little box) to emphasize
brightness changes. The diameter was obtained by the distance from the
centre of the maximum plateau in the mean signal to the null in the
derivative of the convolved signal. (E) Planar dendrites were mapped
entirely to two dimensional space (black original, red flattened dendrite).
(F) The dendrite spanning fields were determined by drawing a region at 25
μm away from any point on the dendrite. (G) Topological point density
distribution was obtained by Voronoi segmentation (green borders) with a
dendrite spanning field boundary. Shaded gray scale indicates surface area
of Voronoi pieces. Overlaid dendrite in red. (H–P) Steps in the
creation of artificial dendrites: (H) dendrite topological points were
morphologically closed (dilation followed by erosion) with a 25 μm
radius disc and the resulting binary image smoothened with a Gaussian filter
of 25 μm variance; (I) This was then cut out by the boundaries of the
closed image, representing for each location in the dendrite spanning field
the error made when smoothly averaging the density; (K) density estimation
of topological points by Gaussian filtering with a 25 μm variance.
(L) the density map in (K) was normalized by the estimation error obtained
in (I); (M) random points (green) were distributed according to the
corrected density distribution with sharp boundaries; (N) preliminary
artificial dendrite following the iterative greedy algorithm presented
previously [12] on green points in (M); (O) Artificial
dendrite after smoothing along heavier branches; (P) quadratic diameter
decay was mapped on the resulting dendritic structure according to an
optimization of synaptic democracy [12]. The resulting
artificial dendrite shows similarity with the original tree in (F). In
(H–L) the flattened original dendrite from (F) was overlaid on top
of the respective gray-scale maps.
(4.55 MB TIF)
Supplementary information on cluster analysis. (A) Generalized extreme value
fits for the distributions shown in Figure 1E–K. This approach
allowed obtaining three parameters for each distribution. These were used
for the cluster analysis in Figure 2B and here. (B) Full cluster analysis for all models
presented in the article. The method applied corresponds to Figure 2B of the article.
Dendrites of individual cells were tagged by index numbers. Additionally,
artificial dendrites from Figure 3 marked with a preceding “m” and
lighter colours were included. Artificial dendrites mingled with their
corresponding real counterparts indicating that they were similar to real
cells in respect to their branching rule. Automatically reconstructed
dendrites from Figure 4
were included marked by a preceding “--------r”. (C)
Spanning field parameters as in Figure 2A but including the automatically reconstructed VS2 and
HSN cells marked by a star. Line connects the automatically reconstructed
dendrites with the corresponding manual reconstructions.
(1.20 MB TIF)
Overview of all 45 reconstructed LPTC dendrites and their artificial
correlates. (A) Real manual dendrite reconstructions. (B) All constructed
artificial LPTC dendrites. The model dendrites were grown in the spanning
fields displayed in (A) in the same order. Diameter tapering was mapped here
onto the branching structures for visual aesthetic purposes [12].
The artificial dendrites are hard to distinguish from their biological
counterparts. (C, D) Overview of all dendrograms: comparison between
reconstructed and artificial dendrites same colours as used in the main
article. Dendrograms were sorted to put heavier trees (with larger
sub-trees) on the left side. Although, taken one by one, dendrograms of
artificial dendrites would not perfectly reproduce the corresponding partner
(since they originated from random distributions of points), a trend of
similarity is evident. This particularly illustrates how the differences in
branching between HS and VS cells relates to the spanning fields of their
dendrites since all artificial dendrites originate from the same branching
rule. This is strongly in favour of a common branching rule for all cells.
And this common rule is most likely very similar to the one applied to
obtain the artificial dendrites.
(2.02 MB TIF)
Demonstration of the artificial growth process. Dark red axonal arborizations
are randomly distributed and correspond to target points. Iteratively,
unconnected points are added to the tree (green). At each time step, for
visual purposes, diameter tapering was mapped onto the tree as in Figure
S1P and the existing tree was smoothed. These two steps were really
performed after the entire growth in the artificial dendrites used in this
paper. The artificial dendrite shown here is based on the HSN cell of Figure
S1.
(2.77 MB AVI)
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