http://www.frontiersin.org/Journal/10.3389/fncom.2013.00160/full?
Jaap van Pelt
Arjen van Ooyen- Computational Neuroscience Group, Department of Integrative Neurophysiology, Center for Neurogenomics and Cognitive Research, VU University Amsterdam, Amsterdam, Netherlands
Introduction
Because synapses can form only when axons and dendrites
are in close proximity, the connectivity in neuronal networks strongly
depends on the three-dimensional morphology of the constituting neurons.
Neuronal morphology varies greatly, and the substantial variability in
neuronal morphologies will consequently also produce large variability
in their connections with other neurons. An additional factor
determining connectivity is the spatial position of neurons, leading to
widely varying distances between neurons pairs. The morphology of
neurons is complex, with branches of varying orientations and diameters
bifurcating at different lengths. In reconstructions this complex
morphology is usually approximated in a piece-wise linear fashion, i.e.,
by a number of line pieces or cylinders (the latter when the diameter
is also measured). These reconstructions in continuous space preserve
the details of the arbor structures of the neurons. Another way of
characterizing the spatial structure of neurons is by discretizing space
by means of a grid of voxels and defining in each voxel the neuronal
“mass” (i.e., the length or the volume of a branch in that voxel). When
the mass in each voxel is divided by the voxel volume, this description
results in a neuronal “mass” density field (in short called density
field). Clearly, the density field of a single neuron fully reflects the
arbor structure of the neuron, with non-zero densities in voxels
occupied by arbors and zero densities elsewhere.
When an average density field is obtained from a number
of neurons (after alignment of the somata), the individual arbor
structures get lost, and the number of non-zero voxel densities
increases because of the large variations in neuronal morphologies. Only
for very high neuron numbers will a stable estimate of the population
mean density field be obtained. Although the level of smoothness of the
population mean density field may be high in areas near the soma, it
will remain low in remote areas, which are visited only by spurious
branches of individual neurons. The smoothness of a density field may be
enhanced if certain symmetries can be assumed in the averaged
morphology of cells. For instance, when neurons grow out without any
orientation preference, a spherical symmetry in the density field may be
assumed. In that case, the total mass at a certain radial distance from
the soma can be smeared out uniformly over the sphere with that radius.
Similarly, when rotation invariance around a central axis can be
assumed, the total mass at a certain radial distance from, and a certain
height at the axis, can be smeared out uniformly over the circle with
that radius and at that height. Stable estimates of the population mean
density fields of neurons reflect shape characteristics that are typical
for a given cell type. Therefore, these estimates can be regarded as
powerful statistical descriptors of the neurons' spatial innervation
patterns and, as such, as templates for various neuronal cell types.
Synaptic contacts may occur when axonal and dendritic
elements are very close in space, i.e., within a few microns, a
condition usually referred to as Peters' rule (Peters, 1979). Binzegger et al. (2004)
use another interpretation of Peters' rule in that axons connect in
direct proportion to the occurrence of the synaptic target structures in
the neuropil. Locations where candidate synapses can be formed can be
found by testing the proximity of any pair of line pieces of the axonal
and dendritic arborizations of neuronal reconstructions. Recently, we
developed a new method for finding candidate synaptic locations in areas
innervated by both axonal and dendritic arborizations. The method
defines the precise locations of the candidate synaptic contact points
on the axonal and dendritic segments. The term candidate is used because
it refers to the minimal geometric requirement for a synapse. Whether
in neuronal tissue functional synapses will actually develop at the
locations of candidate synapses depends on other factors as well. When
we use the word synapse in the following, it is meant to mean candidate
synapse. The method is based on proximity and crossing of axonal and
dendritic line pieces (van Pelt et al., 2010).
By varying the positions of the somata of the pre- and post-synaptic
neurons, one can obtain the number of synaptic contacts as a function of
neuron positions. Repeating this process for many neuron pairs of a
population of reconstructed neurons yields a statistical estimate of the
number of synaptic contacts vs. soma positions. From these outcomes,
one can also derive an estimate of the connection probability (the
probability that an arbitrary neuron pair is connected, i.e., has at
least one synaptic contact) as well as the mean number of synaptic
contacts per connected neuron pair.
The question whether connectivity can also be derived
from the overlap of dendritic and axonal density fields has been
addressed by Liley and Wright (1994), based on the work of Uttley (1955).
They derived an analytical expression for the expected number of
synapses between two neurons at given positions. They assumed spherical
symmetry in the axonal and dendritic density fields and used exponential
decaying radial functions. Their analytical approach in continuous
space required smooth density functions. Kalisman et al. (2003)
constructed averaged templates of axonal and dendritic fluxes in 3D
space (preserving spatial and directional information) for calculating
the expected number of contacts. They found a good agreement with the
actual number of autapses in reconstructed rat cortical layer 5
pyramidal neurons. Stepanyants and Chklovskii (2005) calculated neurite segment length density functions from reconstructed neurons and applied the formalism of Liley and Wright (1994)
to study the relation between neurogeometry and potential synaptic
connectivity. In order to obtain spatially smooth density fields, they
convolved the skeleton densities with a Gaussian function with a typical
standard deviation of 10–30 μm. Recently, McAssey et al. (in revision)
used the Liley and Wright method to investigate the propagation of
individual neuron variability via the density fields into variability in
the estimated number of contacts. Using sets of generated neuron
morphologies of different sizes, they showed how the standard deviation
in the estimated number of contacts decreases with increasing size of
the data set used for calculating the density fields. Instead of
deriving connectivity from density fields, Cuntz (2012)
followed the reverse way by using a minimal spanning tree approach to
derive the dendritic density fields from the spatial distribution of
contacts points between the neurons.
Important for the validity of the methodology developed
for deriving connectivity from density fields is that the estimated
connectivity from overlapping density fields is consistent with the
connectivity derived from the actual arborizations. To our knowledge
such a rigorous validation has never been carried out before.
The objectives of this paper are (i) to derive
connectivity from overlapping density fields using our recently
developed criterion for the formation of synaptic contacts (van Pelt et al., 2010);
(ii) to develop a method that is also applicable for highly irregular
density fields (such as those of individual neurons) and that is thus
not dependent on any smoothness requirement of the density fields; (iii)
to validate the neuronal connectivity estimates from the overlapping
density fields with the actual connectivity derived from mutually
innervating axonal and dendritic arborizations.
The method developed here is based on a discretization
of space by a grid of voxels of a given size (here set to 1 μm), with
each voxel having a certain dendritic and/or axonal mass density. These
densities are then used to calculate the probabilities of finding axonal
and dendritic line pieces in the voxels. Assuming uniform random
orientations of these line pieces in each voxel, we then apply the above
mentioned proximity/crossing criterion to axonal and dendritic line
pieces (van Pelt et al., 2010)
in the same or within different voxels. The connectivity measures
between two neurons at given positions are obtained by evaluating all
voxel pairs of the axonal and dendritic density fields. The new method
is used to make predictions of the expected number of contacts, the
connection probability between a pre-synaptic and post-synaptic neuron,
and the number of contacts between a connected pre- and post-synaptic
neuron pair, using their density fields. In addition, the mean
connection probability and the Euclidean distances of synapses to their
pre- and post-synaptic somata are estimated in a given network of
neurons represented by their density fields.
The data set of neuronal arborizations used for the
calculation of the density fields and the actual connectivity between
the individual neuronal arborizations (validation) was obtained using
our simulator NETMORPH (Koene et al., 2009).
A number of 50 random neuron morphologies were generated with growth
parameters optimized on a set of rat cortical L2/3 pyramidal neurons,
reconstructed by Svoboda (Shepherd and Svoboda, 2005) and made available by the NeuroMorpho.org data base (Ascoli, 2006).
Summary of Findings
An exact expression was derived for the expected number
of contacts between two neurons based on the overlap of their axonal and
dendritic density fields. This density-field based estimate of the
number of contacts turned out to be fully consistent with the number of
contacts calculated directly from the actual arborizations. The method
is applicable to any arbitrary filling of space with density values,
thus also to “fields” obtained from single dendritic or axonal
arborizations. No assumptions were needed for the “smoothness” of the
density fields. A significant reduction in computational load was
achieved when local uniformity of axonal densities in the neighborhood
of dendritic densities could be assumed. This approximated expression
was consistent with the expression derived by Liley and Wright (1994),
using analytical methods. The accuracy of the approximated expression
was quantified. Our attempt to estimate the connection probability and
the expected number of contacts per connection (connected neuron pair)
from the density fields failed because the fields do not carry anymore
the underlying correlative structure in the spatial distribution of
arbors and synapses. Using empirical mapping functions, however, we
could well estimate both connectivity measures from the expected number
of contacts. For a network of spatially distributed neurons the average
connection probabilities between neuron pairs vs. their intersoma
distance were calculated from their population mean density fields. We
showed how Euclidean distances of synapses to their pre- and
post-synaptic somata can be estimated from the density fields, and how
these distances for a centrally located neuron in a network depend on
the spatial distribution of the other neurons.
The paper is organized as follows. The Materials and
Methods section gives a brief summary of the developed methodology; the
developed methodology is fully described in the Appendix (see
Supplementary Material). The Results section includes an application
part with the estimation of connectivity measures between two neurons
based on their density fields, a validation part in which the
density-field estimates are compared with the estimates based on the
original arborizations, and an application part with the estimation of
averaged connectivities between neurons in a network. The findings are
discussed in the Discussion section.
Materials and Methods
Axonal and Dendritic Mass Distributions in a Spatial Grid of Voxels
Axonal and dendritic arborizations innervate space in a
manner that is determined by their morphological characteristics. Like
the morphology of neurons, the spatial innervation patterns of neurons
may vary considerably between neurons. To quantify these spatial
patterns, we discretize space by a cubic three-dimensional grid, with
volume elements (voxels) of size sv and volume s3v μm3 (Figure 1).
FIGURE 1
Figure 1. Discretization of space by means of a three-dimensional grid of voxels. Red, voxels occupied by the branching structure.
A single arborization will
intersect only a fraction of the voxels in the 3D grid, and within each
such voxel it will do so with a certain “mass.” “Mass” in this context
refers to the volume or to the length of the arbor structure. In this
study we will use the length of the part of the arborization that lies
in the voxel, thus ignoring the diameters of the arborizations. For a
large number of arborizations aligned according to their somata, many
more voxels will be intersected depending on the variability of the
arborizations. The summed “mass” per voxel is then a measure for the
total mass of the population of arborizations at that location in space.
Dividing the summed “mass” per voxel by the number of arborizations
gives an estimate for the population mean mass mv of a single arborization per voxel, or for its density ρ in the case of a unit voxel (sv
= 1 μm). Voxel densities, calculated separately for axonal and
dendritic arborizations, result in so-called (population mean) axonal
and dendritic density fields. The mass per voxel is then obtained via
indicating the expected length of an axonal or dendritic arborization in that particular voxel.
Scale of the 3D Grid
The scale of the grid is defined by the size of the individual voxels sv.
Evidently, this size determines the level of fine structure that is
preserved in the density fields. Neuronal branches contain branch points
and their branches may be curved. Coarse grid scales do not capture
these finer details and integrate all length within a voxel. Finer grid
scales increasingly capture more linear parts of the branches. A voxel
size of 1 μm is considered to be appropriate in capturing the branching
structure in all its relevant details. In addition, the intersections of
the branches with voxels of this size can be expected to deviate little
from straight lines. For fine grid scales, single axonal or dendritic
trees will intersect only a small fraction of the total number of
voxels. A large number of trees is therefore needed to obtain
statistically sufficiently stable density fields of axonal and dendritic
arbors.
Estimation of Density Fields
A dendritic arbor of a cortical L2/3 pyramidal neuron
may fill voxels up to distances of about 400 μm from the soma. With a 1
μm voxel size, there are already 4π * 4002 = 2.010.619 voxels
at that distance in 3D space (i.e., the surface of the sphere with a
radius of 400 μm). When an individual dendrite reaches such distances
with one branch, then only single voxels are intersected at these
distances. If one wants a population sum with all voxels at that
distance intersected by at least one branch, a total number of about 2 *
106 dendrites is needed. To obtain stable statistical
averages per voxel, one needs a multitude of this number, say at least
20 * 106 dendrites. Axonal fields extend over larger
distances of, say, 1000 μm for local arborizations. The number of voxels
at this distance from the soma is 4π * 10002 = 12.566.371 and for stable density field estimates in peripheral areas one needs a number of at least 1.2 * 108
axonal arborizations. Evidently, these are unrealistically high numbers
if experimental reconstructed neurons need to be used for building
density fields. Neural simulators could possibly do the job, but the
numbers are still huge.
The estimation of (smooth) density fields becomes more
tractable when the density fields can be assumed to have some symmetry.
For instance, if the arborizations invade space without any preferred
direction, then spherical symmetry may be assumed. Under these
conditions it is sufficient to have a stable estimate of the radial
distribution of dendritic mass Md(r) and axonal mass Ma(r) vs. distance r from the soma. The spatial densities ρ per unit volume are then obtained via
When spherical symmetry cannot be assumed, the
arborization may show axial symmetry around a central axis (i.e., being
invariant for rotations around the axis). Axial symmetry may be present
in cortical pyramidal neurons, with the apical dendritic main stem as
the axis of symmetry. Axial symmetry was implicitly assumed in the
so-called fan-in projection method by Glaser and McMullen (1984). With axial symmetry it is sufficient to have stable estimates of the mass distribution at different heights z and distances rp perpendicular to the central axis, Md(z, rp) and Ma(z, rp). The spatial densities per unit volume are then obtained via
The estimation of density fields becomes even more
tractable without any requirement on smoothness or complete filling of
space. In this study 50 neurons are used to construct a population mean
axonal and dendritic density field.
Connectivity and Axonal and Dendritic Density Fields
Axons can make synaptic connections with dendrites when their branches are sufficiently close to each other (Peters, 1979).
Given reconstructed axonal and dendritic arborizations one can search
the whole space for locations of sufficient proximity. With
arborizations approximated by series of line pieces, one needs to test
all combinations of axonal and dendritic line pieces. An algorithm for
such a search has recently been developed by van Pelt et al. (2010).
The algorithm is based on the requirement that pairs of axonal and
dendritic line pieces cross with a crossing distance smaller than a
given criterion distance. In density fields, however, the individual
branch structure is lost and replaced by the probability of a finding a
certain mass in the individual voxels. The question then becomes how
these densities can be used in estimating the connectivity between axons
and dendrites. We propose an answer to this question by the following
method.
Voxel mean intersection length, densities, and hit probabilities
A line intersecting a voxel has intersecting points with
two voxel planes. The line piece between these intersecting points,
called the intersecting line piece (or intersection), has a certain
length lint. For a voxel of size s, lint
can be as small as 0 μm when the line is intersecting a corner of the
voxel and as long as the diagonal in the voxel, thus having a range of lint(s)∈[0,s3√]μm . Intersecting a voxel of size s
by a large number of randomly oriented lines gives a characteristic
distribution of intersection lengths (see Appendix section A1) with a
mean of
and a standard deviation of
When a randomly oriented line is drawn in a space larger
than the voxel, the line may or may not intersect the voxel; that is,
in a statistical sense, the voxel will be hit with a certain probability
phitvoxel(s). When there are N randomly oriented lines in that space, the voxel will be hit by an expected number of E{nhitv} = N × phitv(s). The total length of the intersecting line pieces Ltotint(s) (total mass) then becomes
Rewriting this equation as
gives us the expected number of intersecting line pieces
in a voxel, expressed in terms of the total “mass” in the voxel and the
mean intersection length. When the probability of hitting a voxel is
very low, this equation applies to the hit probability itself with
which gives us the probability that a voxel is hit by a
random line in the surrounding space, expressed in terms of the total
“mass” in the voxel and the mean intersection length. Let the density ρ
denotes the mass per unit voxel (i.e., with s = 1 μm), then the mass per voxel of size s becomes ρ × s3. Dendritic mass mvd and axonal mass mva in a voxel v can now be related to the probability phitvd that a voxel v is intersected by a dendritic branch and the probability phitva that the voxel is intersected by an axonal branch, respectively:
(see also Appendix section A2).
Crossing line pieces, crossing probabilities, and crossing distances
An estimate can now be made of the connectivity between
axonal and dendritic arborizations when they are expressed in terms of
density fields (see also Appendix section A3). Two infinite lines in
space are at their shortest distance at the site where they are
crossing. At this site a connection line can be drawn orthogonal to both
infinite lines, with a length called crossing distance. Although two
infinite lines will cross with certainty (except when they are parallel
or coincide), two line pieces with finite length may or may not cross in
space. This principle is used for defining possible unique synaptic
locations between dendritic and axonal arborizations, with the
additional requirement that in the case of crossing the crossing
distance should not be larger than a given distance criterion (van Pelt et al., 2010).
The crossing of random intersections in a single voxel or
in different voxels is described in Appendix section A3. The results
are briefly summarized here. The probability pcross that a pair of random line pieces in a single voxel cross is equal to
which is independent of the size of the voxel. In
contrast, crossing distances between crossing line pieces in a single
voxel do scale linearly with the size s of the voxel and are given by their mean and standard deviation
(Figure A5). For a pair of voxels v and w at a given distance dv, w
from each other, the crossing probability of random line pieces in both
voxels is dependent on the voxel distance, as shown in the graph of Figure A6. A best fit through the data points was given by Equation A16
Conditional crossing probabilities
When a distance criterion of δ μ m is set to the
crossing distance between crossing line pieces the conditional crossing
probability
becomes dependent on δ and on the size of the voxel (see
also Appendix section A4). For two random lines in a single voxel the
conditional crossing probability pcrossv, v(s|δ) is shown in Figure A7 of Appendix section A4.1. For two random lines in different voxels v and w at a distance dv, w from each other, the conditional crossing probability pcrossv, w(s, dv, w|δ) is shown in Figure A8 of Appendix section A4.2 (for the unconditional values, see Figure A6
of the Appendix section A3.2). The figures illustrate how the crossing
probability decreases with increasing distance between the voxels
particularly when this distance is near the criterion value (Figure A8).
Note that the distance between voxels is taken as the distance between
corresponding voxel corners (or centra). The crossing distances of
crossing line pieces in voxel pairs are in the range of [dv,w−s3√;dv,w+s3√] .
Density-weighted conditional crossing probabilities
In the foregoing the crossing probabilities were
determined on the basis of the presence of a random line piece in a
voxel. When the presence of a line piece is a stochastic event then the
crossing probabilities need to be multiplied with the probabilities that
the line pieces are present (see Appendix section A5). In that case,
the crossing probability of line pieces in two voxels v and w at a given distance dv, w from each other is given by
In the overlap area of a dendritic density field D and an axonal density field A,
each voxel has a dendritic and an axonal mass that determines the
probability of finding a dendritic or an axonal line piece in these
voxels, which is dependent on the size of the voxels. The probability
that a dendritic line piece in voxel v and an axonal line piece in voxel w cross is now given by
with ρvD the dendritic density in voxel v and ρ wA the axonal density in voxel w.
Expected number of synapses in overlapping axonal and dendritic density fields
The expected number of crossing line pieces of the
axonal and the dendritic field in the overlap area can now be obtained
by calculating the expected number of crossing axonal and dendritic line
pieces in all the pairs of axon and dendrite voxels in the overlap area
that meet the distance criterion.
Assuming that a synaptic connection may be present at
locations where axonal and dendritic line pieces cross each other at
sufficient small crossing distances, we now have an expression for the
expected number of synaptic contacts in the overlap area of axonal and
dendritic density fields, given by
The double summation in Equation 17 runs over all voxel pairs (v, w) in the given space. However, for each dendritic voxel v only the axonal voxels w within the criterion distance δ contribute to the sum. The second summation over the axonal voxels w can therefore be restricted to the ones in the local environment venv (see Equation A23) of the dendritic voxel v:
Approximation of the expected number of synapses—local uniformity in axonal densities
If it can be assumed that the axonal densities ρwA in the local environment of a dendritic voxel v are not very different from the axonal density ρvA in voxel v itself, Equation 19 can be simplified into
The second summation now runs over all voxels in the local environment of a given voxel v but does not depend on the position of voxel v anymore. The outcome that we will call the local environment crossing factor fenv(s, δ) now becomes a fixed number that is only dependent on the size of the voxels s and the distance criterion δ (see Appendix section A5.2):
The values of the local environment crossing factor fenv(s, δ) are shown in Table A1 of Appendix section A5.2.1. The local environment crossing factor fenv(s, δ) can be approximated by a linear dependence on the criterion δ as (for s = 1) fenv(s = 1, δ)≅ 0.69822 × δ (Equation A38). Then, Equation 20 simplifies into
with IDA denoting the overlap sum IDA=∑vspaceρvD×ρvA .
Connection probability and number of contacts per connection (connected neuron pair)
The connection probability of two neurons denotes
the probability that they are connected, i.e., have at least one
synaptic contact. The question whether and how the connection
probability for a neuron pair can be estimated from their population
mean density fields, can be answered as follows: Let E{nsynv} denotes the expected number of synapses in voxel v. Because the voxel size is small, this expected number will be much smaller than one and can be interpreted as the probability psynv of finding a synapse in that voxel. The probability of no-synapse in that voxel pnosynv is then given by pnosynv = 1 − psynv. The product of the no-synapse probabilities of all voxels in area A, assuming independency, then yields the probability of no-synapse in the overlap space, pnosynA = ∏i(1 − psynvi). The connection probability pconA, i.e., the probability of at least one contact in the overlap space, is then given by pconA = 1 − pnosynA.
A basic assumption in this approach is that the synapse
probabilities of all the voxels are independent of each other. As will
be shown in the Results section, this approach gave inconsistent
outcomes, indicating that the basic assumption of independency is not
justified. Alternatively, the connection probability was estimated from
the expected number of contacts by using a mapping function derived from
the connectivity between the actual arborizations. Also for the
estimation of the number of contacts per connected neuron pair
from the expected number of contacts a mapping function was used that
was derived from the connectivity between the actual arborizations.
Euclidean distances of synapses to their pre- and post-synaptic somata
Euclidean distance distributions of synapses to their
pre- and post-synaptic somata can also be obtained from the overlapping
density fields. For a given neuron pair the probability of finding a
synaptic contact is calculated in each voxel of the overlap space. With
the Euclidean distance of this voxel to the pre- and post-synaptic
somata, the probability of the synaptic contact is then accumulated to
the pre- and post-synaptic Euclidean distance probability distribution,
respectively. Summing over all voxels then yields the distance
distributions for a single neuron pair. In an area with many neurons
this procedure must be repeated for all neurons pairs. The final pre-
and post-synaptic Euclidean distance distributions, averaged over all
neuron pairs, thus depend on the number and positions of all the somata.
Results
Estimation of the Connectivity between an Axonal and a Dendritic Neuron using Population Mean Density Fields
For the application of the method the morphologies of a
number of 50 neurons were generated with the simulator NETMORPH, using a
parameter set optimized on a set of rat layer 2/3 pyramidal cells
obtained from the Svoboda data set in the NeuroMorpho.org data base
(Figure 2).
FIGURE 2
Figure 2. Display of the set of 50 random neuronal
morphologies with their axonal (green), basal (red), and apical (blue)
dendritic arborizations, generated with the NETMORPH simulator using a
parameter set optimized on a set of rat cortical layer 2/3 pyramidal
neurons from the NeuroMorpho.org database. The neurons are aligned according to their apical dendrites.Density Fields with Axial Symmetry
An example of density field calculations based on axial symmetry is given in Figure 3.
Assuming that the axial symmetry axis coincides with the apical main
stem of the neuron, we calculated the axonal and dendritic mass of 50
NETMORPH-generated neurons as a function of the position along the
symmetry axis (height, also referred to as Z-axis) and the radial
distance (radius, i.e., orthogonal distance to the symmetry axis. To
this end, each neuron was first soma-centered at the origin and aligned
according to its apical main stem, and then “sliced” into layers of 1
micron thick. Subsequently, the axonal and dendritic intersections per
layer were analyzed for their radial mass distribution. The axonal and
dendritic density fields are calculated by dividing the “mass” at a
given height and radius r from the symmetry axis by the perimeter (2π r) of the circle with radius r,
under the assumption that the mass is distributed uniformly over the
circle centered at the symmetry axis. These density fields are shown in
Figure 3.
Because of the large perimeters of the circles, the densities decrease
rapidly with increasing radius down to very low levels at large radial
distances, as shown in the logarithmic plot for the density field. These
plots also show the ranges over which the axons and dendrites send
their branches. The population mean density fields clearly show the
non-smoothness due to the isolated branches in remote areas from the
soma center.
FIGURE 3
Figure 3. Population mean density distributions of (left)
dendrites and (right) axons of 50 aligned neurons, plotted as function
of the axial (height) and radial positions. The color-coded
log10-density scales run from the values indicated at the left of the
color bars. The solid dots along the height axes indicate the position
of the cell body. Note that a number of −9 was assigned to voxels
whenever their original density was zero.
Finally, for a given spatial positioning of the two cell bodies, the overlap sum IDA
(Equation 22) of the axonal and dendritic density fields was determined
by calculating for each voxel the density product of both fields and
summing these products over all voxels in the overlap area.
Subsequently, Equation 22 is used to calculate the expected number of
contacts between both neurons for various values of the proximity
criterion δ. The outcomes are given in Figure 4 as contour plots (panel A) and axial-radial curves (panel B),
which show how the expected number of synaptic contacts decrease
monotonically with increasing distance between the cell bodies. Note
that the expected number of contacts has its maximum when the
pre-synaptic neuron is positioned about 50 μm above the post-synaptic
neuron.
FIGURE 4
Figure 4. Expected number of contacts between two neurons shown in (A) contour and (B) axial-radial plots.
The neurons are aligned according to their apical main stem. In all the
plots the dendritic neuron is soma-centered at the origin. In (A)
the position of the soma of the axonal neuron is given by the
coordinate axes in the plot. The contours are labeled by the respective
values of the expected number of contacts (as a multiple of the
criterion value δ, the inner contours maintain the stepwise increase of
0.2δ). In (B) the radial position of the axonal soma is
given by the abscissa coordinate, while each curve is labeled with the
positive (upper panel) and negative (lower panel) displacement along the
Z-axis (ΔZ) of the axonal soma relative to the dendritic soma. The ordinate scale is normalized for δ = 1 μ m.Validation of the Density-Field Estimated Number of Contacts between Two Neurons
The number of contacts estimated from overlapping axonal
and dendritic density fields is validated by comparison with the number
of contacts between the actual 3D arborizations of the same data set of
simulated neurons. The actual number of contacts was determined for all
the 50 * 49 = 2450 neuron pairs with the soma of the dendritic neuron
centered at the origin and the soma of the axonal neuron positioned at a
given axial and radial distance. The number of contacts was determined
by assessing, for all the pairs of dendritic and axonal line pieces,
whether they were crossing and whether the crossing distance was smaller
than or equal to the given proximity criterion (van Pelt et al., 2010).
The mean number of contacts for all the neuron pairs, and the mean
number of contacts for all the connected neuron pairs, were determined
for a number of different axial and radial positions of the axonal cell
bodies. The results are shown in Figure 5.
The solid curves indicate the expected number of contacts from the
density fields; these curves are identical to the ones in Figure 4. The individual data points show the mean and standard error in the mean (sem) (n
= 2450) of the number of contacts actually determined from the
overlapping axonal and dendritic arbors between all neuron pairs. An
excellent agreement was found between the density-based expectations and
the arbor-based calculations, even within the small standard error
values. A similar agreement was found for the distance criteria δ = 2
and δ = 3 (not shown in Figure 5).
Although the actual number of contacts is highly variable between
neuron pairs, as reflected in the standard deviation in the distribution
of data points (Figure 10),
it is because of the large number of 2450 data points that the sem
values become very small. This agreement thus validates the density
field approach for estimating the number of contacts between neuron
pairs.
FIGURE 5
Figure 5. Comparison of the expected number of contacts
predicted by the population mean density-field approach (solid curves),
and obtained directly from the axonal and dendritic arbors of the
aligned neurons [individual data points with mn(sem) values]. Each curve is labeled with the positive (Upper panels) and negative (Lower panels) displacement along the Z-axis (ΔZ) of the axonal soma relative to the dendritic soma. Shown are the validations for criterion values of δ = 1 (Left column) and δ = 4 (Right column).
The validations for δ = 2 and δ = 3 showed a similar agreement between
density-based and arbor-based calculations (not shown in figure).Estimation of the Connection Probability from the Expected Number of Contacts
The connection probability between two neurons was
calculated from their population mean density fields according to the
approach described in the Materials and Methods section. For validation,
the connection probability was also calculated from the actual
arborizations as the ratio of the number of connected neuron pairs (with
at least one contact) and the total number of 2450 neuron pairs. Both
approaches turned out to give inconsistent results. The density-field
expected values were significantly larger than the arbor-based data
points. A generalization of the approach in the Materials and Methods
section is further described in Appendix section A7, where it is
explained how the connection probability between two neurons can be
estimated from the expected number of contacts when independency is
assumed for the spatial distribution of synapses. This resulted in a
“theoretical” mapping curve, which is shown in Figure A13 and in Figure 6
(solid curve). The relation between the connection probability and the
expected number of contacts, estimated from the population mean density
fields, was found to exactly match this theoretical mapping curve.
However, the density-field estimated connection probability was
inconsistent with the arbor-based connection probability. This thus
implicated that the theoretical mapping function was not appropriate.
For validation, it was compared with an empirical mapping function,
derived from the arbor based calculated number of contacts and
connection probability. To this end, for a given spacing of the cell
bodies, both the mean number of contacts and the connection probability
for all the 2450 neuron pairs were determined from the actual
arborizations. By varying the spacing for x-shifts of (0, 20, 50, 100,…, 500 μm) and y-shifts of (−300, −200,…, 500 μm) one obtains 12 * 9 = 108 data points, as shown in the scatterplots of Figure 6.
The actual data points indeed show significantly lower connection
probabilities than those predicted by the theoretical curve (upper solid
curve). For low number of contacts the data points are very close to
but do not exceed the theoretical curve. Apparently, the theoretical
curve, derived from the expected number of contacts, provides an upper
limit for the connection probability. Figure 6 includes best-fitting regression functions of the type f(x) = a(1 − ebxc)
through the data points. The method for calculating the connection
probability (see Materials and Methods section) and its generalization
in A7 are based on the assumption that the expected number of contacts
in the voxels in the overlap space are independent of each other (see
also Equation A49). The incorrectness of this assumption is likely
caused by the fact that synapses are restricted in their positions to
the axonal and dendritic arborizations, which provide an underlying
correlative structure to the synapse positions that is not reflected
anymore in the density fields.
FIGURE 6
Figure 6. Scattergram of the mean connection probability vs.
the mean number of contacts (obtained from the actual arbors of all the
2450 neuron pairs in the validation set). Each panel is labeled by the used distance criterion δ and includes the theoretical mapping function (solid line; see also Figure A13), a best-fitting regression function (dashed line) through the data points of the type f(x) = a(1 ‒ ebxc), and the values of the optimized parameters. The data points are labeled by their z-shift values (see symbols).
An explanation for the
overestimation of the connection probability can be given by referring
to the procedure in Section Connection Probability and Number of
Contacts per Connection (Connected Neuron Pair). Because actual synapses
are restricted to the arbor subspace they are spatially correlated. In
other words, finding an actual synapse implicates a higher probability
of finding another actual synapse in that subspace. Similarly, not
finding an actual synapse at a given location implicates a high
probability to be not at the arbor subspace and also implicates a higher
probability of not finding an actual synapse nearby. In the density
field approach the probability of finding or not finding a synapse at a
given location (voxel) is assumed to be independent of the probability
of finding or not finding a synapse elsewhere, respectively. The product
of the probabilities of not finding a synapse in the different
locations in the overlap area is thus higher in the actual case than in
the density field case. Consequently, the probability of at least one
contact will be lower in the actual case than in the density field case.
Thus, the density field approach overestimates the connection
probability between two neurons.
Because the connection probability could not be estimated
from the density fields, we alternatively estimated it from the
(correct) density-field estimated number of contacts using the empirical
mapping functions. The results, shown in Figure 7
for distance criterion values of δ = 1 μ m and δ = 4 μ m, are now in
good agreement with the validation data, and for several cell positions
the expected curves even agree within the sem values of the validation
data. However, for other cell-cell positions the validation data lie
somewhat above or below the expected curve. Figure 6
shows that the scatter of the validation data around the best-fitting
curve is not random but mainly positive or negative for the different
cell-cell positions. For instance, the ΔZ = 0 data points are lower than the curve, whereas the ΔZ
= 200 data points lie above the curve. This structure in the variation
of the data translates also directly into the deviations shown in Figure
7.
The small deviations between the expected and validation data can
therefore be explained by the structure in the variations in the
validation data, which appeared to depend on the cell-cell positions.
Thus, the connection probabilities can be well-estimated from the
density-field expected number of contacts using the empirical mapping
function.
FIGURE 7
Figure 7. (Solid lines) Connection probabilities estimated
from the expected number of contacts using the best-fitting mapping
functions shown in Figure 6. Each curve is labeled with the positive (Upper panels) and negative (Lower panels) displacement along the Z-axis (ΔZ)
of the axonal soma relative to the dendritic soma. Individual data
points are the arbor-based results. Results are shown for distance
criterion values of δ = 1 μ m (Left column) and δ = 4 μ m (Right column).
For δ = 2 μ m and δ = 3 μ m a similar agreement between density-based
estimations and arbor-based calculations was obtained (not shown).Estimation of the Expected Number of Contacts per Connected Neuron Pair
The expected number of contacts per connection between
two neurons is defined as the mean of the number of contacts in a
connected neuron pair, averaged over all the connected neuron pairs in
the data set. This number is equal to the ratio of the expected number
of contacts and the connection probability (Equation A50 in Appendix
section A7). But similarly to the connection probability, the
density-field expected values were significantly different from the
validation data. These deviations can be seen in Figure 8
by comparing the relationship between the number of contacts per
connection vs. the number of contacts as predicted from the
density-field approach (thick solid line) and calculated from the actual
arborizations (individual data points). The empirical mapping functions
(dashed lines in Figure 8) were obtained by regressing the data points with a function of the form f(x) = a + bx + cedx. The theoretical mapping curve (solid line in Figure 8) is also shown in Appendix section A7 (Figure A13).
FIGURE 8
Figure 8. Scattergram of the mean number of contacts per
connection vs. the mean number of contacts (obtained from the arbors of
all the 2450 neuron pairs in the validation set). Each panel is labeled by the distance criterion δ and includes the theoretical mapping function (solid line and see Figure A13), a best-fitting regression function (dashed line) through the data points of the type f(x) = a + bx + cedx, and the values of the optimized parameters.
Because the number of contacts per
connection also could not be estimated from the density fields, we
alternatively estimated it from the (correct) density-field estimated
number of contacts using the empirical mapping functions. The results
for δ = 1 μ m and δ = 4 μ m are shown as solid curves in Figure 9.
The estimated values appear to be in very good agreement with the
validation data for several cell-cell positions, even within the sem
values of the data points. But for other cell-cell positions the
deviations show the same systematic structure as in the scatterplots of
Figure 6,
indicating that they originate from the variability structure in the
validation data for the different cell-cell positions. Thus, also the
number of contacts per connection can be well-estimated from the
density-field expected number of contacts using the empirical mapping
function.
FIGURE 9
Figure 9. (Solid lines) Number of contacts in connected
neuron pairs, estimated from the expected number of contacts using the
best-fitting mapping function shown in Figure 8. Each curve is labeled with the positive (Upper panels) and negative (Lower panels) displacement along the Z-axis (ΔZ)
of the axonal soma relative to the dendritic soma. Individual data
points are the arbor-based results. Results are shown for distance
criterion values of δ = 1 μ m (Left column) and δ = 4 μ m (Right column).
For δ = 2 μ m and δ = 3 μ m a similar agreement between density-based
estimations and arbor-based calculations was obtained (not shown).Density Fields of Individual Neurons—Validation of Equation A24
In Equation A24 it was shown that the expected number of
contacts obtained from the overlap of population mean density fields is
equal to that obtained from the sum of the overlap of individual neuron
density fields. To test this equality, we estimated the expected number
of contacts in a neuron pair from the overlap between the axonal and
dendritic density fields of the individual neurons at given spatial
locations by means of the exact expression (A24). Next, the outcomes
were averaged over all the 2450 neuron pairs. The calculations were
repeated for a range of mutual locations of the neuron pairs. The
distributions for the averaged expected number of contacts between
individual neuron density fields turned out to match exactly the ones
obtained from the population mean density fields as shown in Figure 4.
This agreement thus validates Equation A24 and also demonstrates that
connectivity estimates based on density fields of individual neurons
give consistent results, irrespective of the irregularities of the
individual neuron fields.
Density Fields of Individual Neurons—Variability in the Connectivity between Neurons
Measures of connectivity between individual neuron pairs
show large variations. As illustration, connectivity measures were
calculated for all the 2450 neuron pairs, with the axonal neuron placed
at an x-shift of 100 μm and a z-shift of 100 μm relative
to the dendritic neuron. Again the exact expression (A24) was used. The
distributions of these measures are shown in Figure 10.
FIGURE 10
Figure 10. Distributions of connectivity measures of the 2450 pairs of individual neurons, with the axonal neuron placed at an x-shift of 100 μm and a z-shift of 100 μm. Shown are the distributions of (A) the density-product values, (C,F) the expected number of contacts, (D,G) the connection probability, and (E,H)
the expected number of contacts per connected neuron pair. These
measures were calculated for proximity criteria of δ = 1 μ m (2nd row)
and δ = 4 μ m (3rd row). The top-right panel (B) shows
the mapping functions used for calculating the connection probability
(left ordinate) and the expected number of contacts per connection
(right ordinate) from the expected number of contacts. The curves are
labeled with the value of the proximity criterion δ, with T denoting the
theoretical mapping curve.Density Fields of Individual Neurons—Validation of the Local Uniformity Assumption in the Axonal Density in the Calculation of the Density Fields Overlap
Thus far, all the calculations involving the population
mean density fields used the approximated expression in Equation 22,
which is based on the assumption that the axonal densities in the local
environment of a dendritic voxel do not differ much from the axonal
density in the dendritic voxel itself. For a density field that is
calculated as the mean of a large population of neurons, this is a
reasonable assumption. For density fields of individual neurons,
however, this may not be a good assumption, as the density field then
reflects the individual arbors, which are not filling space in a smooth
manner. This is also the case when the density field is obtained by
spreading arbor mass in an axial symmetric way. Therefore, also the
approximated expression of Equation 22 needs to be validated. To this
end, the number of contacts between two neurons is calculated using (1)
the approximated expression of Equation 22, (2) the exact expression in
Equation 19, and (3) the actual contacts points between the arbors
themselves. The results for the 2450 neuron pairs, with the axonal soma
shifted −100 μm in the Z-direction and 150 μm in the X-direction
relative to the dendritic soma, and with δ = 4μ m, are displayed in
Figure 11.
When the approximated expectations are plotted vs. the exact
expectations for all the 2450 neuron pairs, they show a clear diagonal
pattern (Figure 11A).
When the relative difference between the approximated and the exact
expectations are plotted vs. the exact expectations, the data points
show a jitter around zero, with larger fluctuations for smaller values
of the exact expectations (Figure 11C).
For very small values of the exact expectations, the approximated
expectations are systematically smaller than the exact expectations
(Figure 11D).
These findings can be understood by realizing that the approximated
expectations are based on the product of axonal and dendritic densities
per voxel. In the case of a positive dendritic density but zero axonal
density, the product will be zero. For the exact expectation, however,
also the axonal densities in the environment of the dendritic voxel
contribute to the density product sum, implicating that even when the
axonal density in the dendritic voxel is zero its environment may
contribute positively. Thus, for small values of the expected number of
contacts, the approximated expectation as given by Equation 22
underestimates this number. Figure 11B
shows the comparison of the exact expectation of the number of contacts
with the actual number of contacts between the overlapping axonal and
dendritic arbors of all the 2450 neuron pairs. It is clear from Figure 11B
that even if the actual arbors do not have contacts, the expected
number of contacts can be positive. Also, for a given value of the
expected number of contacts the actual numbers of contacts can range
between zero and 20, a range also shown in Figure 10F.
FIGURE 11
Figure 11. (A) Expected number of contacts between
two neurons calculated with the approximated voxel-voxel overlap
approach (ordinate) vs. the expected number of contacts calculated with
the exact voxel-environment overlap approach (abscissa). (B)
Actual number of contacts between two neurons vs. the expected number
of contacts calculated with the voxel-environment overlap approach. Note
that many data points are plotted over, but this information is not
essential because the figure only aims at illustrating the range of
actual values underlying a given expected value. (C)
Relative difference between expected number of contacts according to the
voxel–voxel and voxel-environment approach vs. expected number of
contacts according to the voxel-environment approach. (D) Similar to (C) but with finer abscissa scale.
For the expected mean number of contacts, averaged over all the 2450 neuron pairs, with z-shift = −100 μm and x-shift
= 150 μm, the relative differences between the exact and approximated
expectations were −0.003% (δ = 1μ m), −0.005% (δ = 2 μ m), 0.037% (δ = 3
μ m) and 0.016% (δ = 4 μ m). Compared over a large number of soma-soma
positions, the mean value of the expected number of contacts between two
neurons calculated with the exact and the approximated expression (both
averaged over all 2450 neuron pairs) showed a relative difference of
less than 0.05% (δ = 1 μ m), 0.1% (δ = 2 μ m), 0.2% (δ = 3 μ m), and
0.2% (δ = 4 μ m).
Network Applications of Density Fields
Thus far, the focus was on using density fields for
estimating the connectivity between two neurons at given positions in
space (see Figures 5, 7, 9).
In a network, however, neurons all take their individual positions.
Network connectivity is therefore determined by the mean of the
connectivities between all pairs of neurons. Evidently, this network
connectivity is highy dependent on the actual positions of the neurons.
An example is given in section Estimation of the Connection Probability
in Neuronal Networks for the averaged connection probability in a
network. Density fields can also be used for estimating the Euclidean
distance distributions of synapses to their pre- and post-synaptic
somata. The method and example results are explained and discussed in
section Estimation of Euclidean Distances of Synapses to their Pre- and
Postsynaptic Somata.
Estimation of the connection probability in neuronal networks
For deriving the mean connection probability in a network
one needs to average over all the different mutual positions of the
neuron pairs. This can be done by calculating the distance distributions
of all the neuron pairs in the network and convoluting the
distributions with the expected connection probabilities, as shown in
Figure 7. An example is given in Figure 12
for a network composed of 2000 neurons, all represented by the same
population mean density field, obtained from the data set of 50 neurons
(see Figure 2).
The somata of the 2000 neurons were uniform randomly distributed in a
cylindrical space with a height of 360 μm and a diameter of 1000 μm.
FIGURE 12
Figure 12. Network connection probabilities, averaged over
all neuron pairs in the network, as a function of their Euclidean
intersoma distance. A number of 2000 somata were uniform randomly distributed in a cylinder with a height of 360 μm and a diameter of 1000 μm.Estimation of euclidean distances of synapses to their pre- and post-synaptic somata
Density fields can also be used to derive the Euclidean
distance distributions of synapses to their pre- and post-synaptic
somata. To this end, the probability of finding a synapse is determined
in each voxel in the overlap area as well as the voxel's Euclidean
distance to pre- and post-synaptic somata. The distance distributions
are then constructed by summing the probabilities sorted by their
distances. Evidently, pathlength distributions of synapses to their
pre-and post-synaptic somata cannot be determined, as the arbor
structure is lost in creating the density fields.
Synapses can occur only where axons and dendrites overlap
in space. These overlap areas are determined by the positions of the
somata and the extents of their arbors. When a dendrite overlaps only
with remote areas of an axonal field, the possible synaptic locations
will have large Euclidean distances to their pre-synaptic somata.
Alternatively, when a dendrite overlaps with central areas of an axonal
field, possible synaptic locations will have short Euclidean distances
to their pre-synaptic somata. When synapses are distributed
homogeneously over the axonal and dendritic arborizations, their
Euclidean distance distributions reflect the axonal and dendritic mass
distributions vs. Euclidean distance to their somata (Figure 13).
FIGURE 13
Figure 13. (A) Axonal and (B)
dendritic mass distributions vs. Euclidean distance to somata. The tail
in the dendritic mass distributions originates from the apical dendrite
and its apical tuft. (C) Pre-synaptic and (D)
post-synaptic Euclidean distance distributions of a centrally located
neuron, as calculated from its connections with all other neurons. The
total number of 5000 neurons are uniform randomly distributed in a
cylinder of height 360 μm and radius 1000 μm. The pre-synaptic distance
distribution has a mean(sd) value of 216 (120) μ m and the post-synaptic distance distribution a mean(sd) of 91(63) μ m.
To illustrate the effect of
spatial boundaries, we calculated the pre- and post-synaptic distances
for a centrally located neuron in the cylindrical space of height 360 μm
and diameter 2000 μm, with a total number of 5000 neurons (density of
4421 neurons/mm3) that are uniform randomly distributed in the cylindrical space (Figures 13C,D).
Note that the expected number of synapses between two neurons vs. their
intersoma distance follows the patterns as shown in Figure 4. Although the pre-synaptic distribution (Figure 13C) has a rough resemblance with the axonal mass distribution in Figure 13A,
it differs from that, with a mean distance of 281 μm for the mass
distribution and a mean distance of 216 μm for the pre-synaptic
distribution. Also the post-synaptic distribution (Figure 13D) has a rough resemblance with the dendritic mass distribution in Figure 13B,
but differs in particular in the tail of the distribution, with a mean
distance of 112 μm for the mass distribution and a mean distance of 91
μm for the post-synaptic distribution.
These differences can be understood from a cartoon
drawing illustrating the dimensions of the dendritic and axonal density
fields and the cylindrical space. Figure 14A
shows an axonal density field of a centrally located neuron in the
cylindrical space and three dendritic density fields of neurons at
nearby and remote locations. The figure illustrates that a large part of
the axonal field of the central neuron cannot be overlapped by
dendritic fields of the other neurons due to their spatial constraints
within the cylinder. At low cell densities, the shape of the
pre-synaptic distance distribution becomes in a sensitive way dependent
on the particular locations of the dendritic density fields (not shown
here). This was not so much the case for the post-synaptic
distributions. Because of the size of the axonal fields the central
dendritic density field will have overlap with many more axonal fields
(Figure 14B),
even at low cell densities. Because of the spatial constraint, the
apical part of the dendritic density field will generally be overlapped
by a less dense part of the axonal density fields; this explains why the
tail in the post-synaptic distributions differs from that in the
dendritic mass distribution.
FIGURE 14
Figure 14. Cartoon drawing of axonal (green) and dendritic
(red) density fields with their somata in a cylindrical space (blue
rectangle) of height 360 μm and diameter 2000 μm. (A) An axonal field with its soma centered, and three dendritic fields at various locations in the space. (B) A dendritic field with its soma centered, and three axonal fields at various locations in the space.
The bounded area of the cylinder also puts constraints on the intersoma distance distribution, as shown in Figure 15. The rather linear pattern differs significantly from a quadratic pattern expected in unconstrained space.
FIGURE 15
Figure 15. Frequency distribution of distances between a
central soma and 5000 other somata uniform randomly distributed in a
cylindrical space of height 360 μm and diameter 2000 μm (density of
4421/mm3).Discussion
Rationale and Summary
Neuronal density fields are statistical descriptors of
the spatial innervation of axonal and dendritic arborizations. They were
used in several studies to estimate neuronal connectivity (Uttley, 1955; Liley and Wright, 1994; Kalisman et al., 2003; Stepanyants and Chklovskii, 2005).
Recently, we developed a new criterion for determining the location of
synaptic contacts in areas innervated by both dendritic and axonal
arborizations (van Pelt et al., 2010).
In order to apply this criterion to connectivity studies based on
density fields, we needed to develop new methodology. A second objective
of the present study was to validate the connectivity estimates based
on the density field approach with the connectivity data derived from
the actual arborizations.
Our recently developed method for finding synaptic
locations is based on crossing dendritic and axonal line pieces in
combination with a distance criterion (van Pelt et al., 2010).
The application of this criterion to density fields required an
investigation into the statistical geometry of intersections of lines
and voxels (Appendix section A1). First we needed to obtain
intersections of randomly oriented lines with cubic voxels, a procedure
that turned out to be not trivial. The intuitive procedure of first
selecting a uniform random point within the cube through which a uniform
oriented line is drawn was incorrect. Essential is that first a uniform
random orientation is selected followed by the selection of a uniform
random point in space (thus not restricted by the cube) through which
the line is drawn. The length distributions of the intersections were
highly irregular. Their orientations (in terms of azimuth and elevation
angle distributions) were significantly different from those expected
for random oriented lines (showing uniform and cosine distributions,
respectively). We were not able to trace earlier literature on these
topics; thus to our knowledge these findings are new. For sake of
completeness, the 2D case for intersections of random lines with a
square in a plane has been included in Appendix section A1.
Knowing the mean intersection length makes it possible
to relate the density in a voxel to the probability of an intersection.
By taking a random “dendritic” intersection in a given voxel and a
random “axonal” intersection in another voxel, we were able to apply the
crossing/proximity criterion. If both line pieces cross and the
crossing distance between the line pieces was within the distance
criterion, a synaptic connection was identified. Repeating this
procedure many times yielded the probability of a synaptic connection,
weighted by the intersection probabilities for this voxel pair. The
connectivity of a given “dendritic” voxel could be obtained by pairing
it with all “axonal” voxels in its close environment. The total sum for
all dendritic voxels in the overlap area of the axonal and dendritic
density fields resulted in the expected number of contacts between the
“axonal” and “dendritic” neuron.
The summation over all local “axonal voxels” around a
dendritic voxel can be simplified if the local axonal densities do not
vary much. Then the summation can be replaced by the product of the
axonal and dendritic density in the dendritic voxel only, multiplied
with a local environment crossing factor that integrates the
crossing properties of random dendritic line pieces in the dendritic
voxel and random axonal line pieces in the local environment. This
factor is independent of the density fields, and thus can be obtained
once and applied to all voxel pairs. For smooth axonal density fields
without strong gradients, this assumption is warranted, but for
individual neuron density fields it may not. To test the error made in
such conditions, we calculated the expected number of contacts between
neurons using their individual density fields with the exact procedure
and the approximated one. Both procedures yielded similar results as
long as the expected number of contacts was not too small. For very
small values, however, for instance in the case of large intersoma
distances, the approximation procedure underestimated the number of
contacts compared with the exact procedure, down to even 100% (Figure 11). However, averaged over neuron pairs for a range of intersoma distances, the relative difference was less than about 0.2%.
With the approximation expression, the expected number of
synaptic contacts between two neurons reduces to a simple summation
over all the voxels in the overlap area of the axonal and dendritic
density product per voxel, multiplied with Icoef (which includes the local environment crossing factor) (Equation A28, and Table A1). The expression derived by Liley and Wright (1994)
had a similar structure but with an integral of density products,
because of the formulation in continuous space. The coefficient in their
expression was equal to πε2 , with ε denoting the distance criterion. This coefficient turned out to be equal to our coefficient Icoef (at least up to the 3rd decimal, Table A4, and see Equation A39). This proves the consistency between the two fully independent and different approaches.
Estimation of Connectivity Measures
The calculations were based on a data set of 50 neurons generated with the simulator NETMORPH (Koene et al., 2009).
The growth rules were optimized on a data set of rat cortical L2/3
pyramidal cells from the NeuroMorpho.org database. For the calculation
of the density fields the neurons were aligned according to their apical
dendrites, and axial symmetry was assumed. Although the population mean
density fields were far from smooth (particularly in remote areas; see
Figure 3), accurate estimates could be obtained for the connectivity measures between neuron pairs at varying locations of their somata.
An important objective of this study was the validation
of the density-field based connectivity expectations with the data
obtained from the actual arborizations.
Validation of the estimation of the number of contacts
As shown in Figure 5,
the agreement between both approaches for calculating the number of
contacts was extremely good, even within the small standard error in the
mean of the actual arborizations (because of averaging over 2450 neuron
pairs). This implies that the number of contacts estimated using
population mean density fields is a full alternative to the averaging
over the number of contacts between the actual neuronal arborizations.
Estimation of the connection probability
An attempt was made to estimate the connection
probability from the density fields. A basic assumption in the approach
used was that synaptic contacts are independently distributed in 3D
space. The incorrect outcomes made clear that this assumption was not
valid. Actually, it emphasizes the correlative structure in the spatial
distribution of synapses, which may not be surprising as synapses are
distributed along neuronal arborizations. These correlative structures
are not preserved in the population mean density fields, making density
fields not suitable for predicting connection probabilities.
Alternatively, we estimated the connection probabilities from the
correct expected number of contacts by using empirical mapping
functions, which produced outcomes that agreed very well with the
validation data.
Estimation of the number of contacts per connection
Because this connectivity measure is calculated as the
ratio of the expected number of contacts and the connection probability,
it cannot be estimated from the density fields either. Alternatively,
we estimated the number of contacts per connected neuron pair from the
correct expected number of contacts by using empirical mapping
functions, which produced outcomes that agreed very well with the
validation data.
Empirical mapping functions
The empirical mapping functions for both the connection
probability and the number of contacts per connection were dependent on
the distance criterion for synaptic contacts. Whether these mapping
functions are also dependent on the morphology of the cell types is
still unknown. If not, the mapping functions could have a general
validity. Investigation of this question was considered to be outside
the scope of this paper.
Distinction between basal and apical dendrites of pyramidal cells
In the calculation of the dendritic density fields, no
distinction was made between basal and apical dendrites. When such a
distinction is made, the connectivity measures can be estimated for
basal and apical dendritic connectivity separately.
Comparison of Present Findings with Other Connectivity Studies
Number of contacts between two neurons
Hellwig (2000)
estimated computationally the number of contacts between eight
experimentally reconstructed rat cortical L2/3 pyramidal neurons by
placing them at several distances from each other. For two groups of
four neurons each, the number of contacts as a function of the cell
separation was determined using a distance criterion of 1 μm. Our
results (Figure 5)
compare well with the two regression curves of Hellwig. The difference
between the two regression curves of Hellwig illustrates the effect of
small sample sizes (four) when the number of contacts between individual
neuron pairs may vary as strongly as shown in Figure 10 (see also McAssey et al., in revision).
Connection probabilities in neuronal networks
In neuronal networks neurons take different positions.
To derive connectivity estimates for the whole network, one needs to
average over all the different relative positions of the neuron pairs.
This can be done by calculating the distance distributions of all the
neuron pairs and convoluting the distributions with the expected
connectivity data (see Figures 5, 7, 9). An example of this procedure for calculating the averaged connection probability is given in Figure 12.
For a distance criterion of 1 μm, the connection probability shows a
monotone decreasing pattern, from a value of about 0.7 at very short
intersoma distances down to about 0.04 at an intersoma distance of 500
μm. For larger distance criteria the connection probabilities slightly
increase, while the intersoma distance dependency becomes more linear.
Experimental data on connection probabilities of rat layer 2/3 rat
pyramidal neurons have been collected by Holmgren et al. (2003).
In paired electrophysiological recordings, they found connection
probabilities of about 0.09 at intersoma distances of 0–25 μm,
decreasing down to about 0.01 at intersoma distances of 100–200 μm.
Using multipatch experiments on a large set of thick-tufted layer 5
pyramidal neurons in rat cortical somatosensory slices, Perin et al. (2011)
estimated the mean (functional) connection probability as a function of
intersoma distance. The connection probabilities for this type of
neuron showed a similar dependence on distance but with values about a
factor of 3–4 lower than our outcomes.
In general, our estimates are substantially higher than
the experimental estimates. Several notes need to be made. Our estimates
are based solely on geometrical considerations and mark only possible
candidate synaptic locations. Whether at these locations actual synapses
are present and whether they are functional and measurable in
electrophysiological experiments are open questions. It is notoriously
hard to collect experimentally reliable estimates of connectivities in
neuronal networks, an effort that is hampered by issues such as cutting
effects in slices, unbiased sampling of patched neurons, and measuring
resolutions. The computational predictions strongly depend on the chosen
distance criterion for synapse formation and, although a criterion of
about 4 μm seems plausible in view of the local geometry, it still has
to be validated. A larger uncertainty is the probability that a
candidate synapse location really represents a functional synapse. With a
computational estimate of about 0.9 for the connection probability at
very short distances and an experimental estimate of about 0.09 (Holmgren et al., 2003), there is still a factor of 10 difference to be explained.
An interesting finding from the present study is that the
expected number of contacts was highest when the pre-synaptic neuron
was placed about 50 μm above the post-synaptic neuron (Figure 4). Kalisman et al. (2003) reported a similar observation (with a maximal number of contacts at a displacement of 100 μm for layer 5 pyramidal cells).
Pre- and post-synaptic euclidean distance distributions
The probability of having a synapse at a particular
location in space directly depends on the local values of the axonal and
dendritic densities. The distribution of synapses on the axonal and
dendritic arborizations is thus determined by the overlap profile of the
density fields, which depends on the locations of the somata. An
example is given for a number of neurons with their somata uniform
randomly distributed in a cylindrical space (Figure 13).
In the case of uniform density fields and unrestricted space, one would
expect pre- and post-synaptic distances to be equal to the radial mass
distributions of the axons and dendrites. The comparison thus shows the
effect of inhomogeneous density fields and restricted space on the
spatial distribution of synapses.
Feldmeyer et al. (2002)
studied connectivity between layer 4 spiny neurons and the dendrites of
layer 2/3 pyramidal cells in the rat barrel cortex by means of paired
recording and reconstruction techniques. The distribution of the
post-synaptic Euclidean distances of synapses on the pyramidal dendrites
turned out to correspond quite well with our predictions, although the
limited number of their observations (59 synapses in 13 neuron pairs)
prevented a detailed shape comparison. Data on pre-synaptic Euclidean
distance distributions appears to be absent in the literature, probably
because of the experimental challenges involved in reconstructing full
axonal arbors of neurons that project to a given target neuron.
Experimental Challenges in Measuring Network Connectivity
Helmstaedter (2013)
recently evaluated the state-of-the art of experimental techniques for
resolving the connectivity matrix in neuronal circuits (connectomics).
As structures involved in connections (axonal diameters, spine necks)
have minimal dimensions of less than about 50 nm, the minimal required
imaging resolution must be less than about 30 nm. Present electron
microscopy techniques meet these resolution requirements but are limited
in the volume that they can image. While these reconstructions are time
consuming, the time needed for segmentation and determining the wiring
exceeds these imaging times by factors. Therefore, computational
approaches may provide valuable alternative approaches for studying
connectivity at a cellular level in neuronal networks.
Future Challenges
Density fields calculated from experimentally reconstructed neurons
The present study was based on a set of neuronal morphologies produced by the simulator NETMORPH (Koene et al., 2009),
but could equally well have been based on a set of experimentally
reconstructed neurons. A set of simulated rather than experimentally
reconstructed neurons was chosen because it puts no restriction on the
number of neurons and because simulated neurons do not suffer from
incompleteness caused by tissue sectioning, a problem that affects many
sets of experimentally reconstructed neurons.
If a sufficiently large set of experimentally fully
reconstructed neurons became available, the axonal and dendritic density
fields derived from these neurons would provide powerful statistical
representations of their spatial innervation patterns. These density
fields can replace actual arborizations when one wants to build networks
of these neurons. The limited availability of actual neuronal
reconstructions is then no longer restricting the size of the network.
The connectivities emerging in such networks can then be reliably
estimated from the overlap of the neurons' density fields, as has been
shown in this study. For building cortical networks, one needs density
fields of a variety of neuron types. These neuron-specific density field
templates are not yet available, and constructing them would be an
interesting challenge for the future.
Variability in neuronal morphologies and density fields
Neurons vary substantially in their morphologies. Density
fields based on different data sets from the same neuron population
will also show variations, which inevitably propagate to variations in
the estimated connectivity values. This issue has recently been
addressed by McAssey et al. (in revision). They show how the variation
in the estimated number of contacts between two neurons decreases with
increasing size of the data set used for calculating the density fields.
They advocate the use of neuronal simulators, because simulators enable
the generation of any desired number of morphologies so that the
density fields can be estimated with any desired level of statistical
stability. Essential is that the simulated neurons are realistic in all
relevant aspects of their morphology.
Density field completion of sectioned incomplete neurons
Many neuronal reconstructions for a variety of cell types
and species that are made available through the NeuroMorpho.Org data
base (Ascoli, 2006)
are, unfortunately, incomplete and not directly suitable for
constructing density fields. However, when parts of the density fields
within the spatial constraint of a section can be reliably estimated
from incomplete neuronal reconstructions, it should be possible to make
the density field complete provided axial or spherical symmetry can be
assumed.
Density fields of neuronal populations at various developmental stages
During neuronal development, axonal and dendritic arbors
increase their spatial innervation area by neurite elongation and
branching. Connectivity studies on developing networks critically rely
on the availability of reconstructed neurons at different developmental
stages, but such morphological time series are unfortunately scarce.
Density fields of outgrowing neurons will also change with developmental
stage and presumably according to a particular growth pattern. If
density fields can be determined for a number of developmental stages,
such growth patterns could possibly be described in terms of a density
field growth function. These density field growth functions could then,
for example, be used for (i) interpolating or extrapolating to
developmental stages for which experimental data is not available, and
(ii) studying connectivity in developing neuronal networks.
Conclusion
Determining the connectivity between neurons requires
knowledge about their innervation of space. Neurons can be represented
by their actual arborizations, but also by their density fields. In this
paper, we have shown that the number of contacts between neurons
estimated from their population mean density fields is fully consistent
with the number of contacts calculated from their actual arborizations.
However, the connection probability and the number of contacts per
connection cannot be reliably estimated from the density fields.
Alternatively, they can be estimated from the expected number of
contacts by using empirical mapping functions. The population mean
density fields are powerful representations of the mean axonal and
dendritic spatial innervation patterns of a given cell type. These
density fields can be used in neuronal network studies to obtain
statistical connectivity estimates by representing each neuron by the
population mean density field of its cell type. The large variation
between individual neurons is then already expressed in the density
field itself.
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