^{*}

Edited by: Rava A. Da Silveira, Ecole Normale Supérieure, France

Reviewed by: Markus Diesmann, Jülich Research Centre and JARA, Germany; Giorgio Ascoli, George Mason University, USA

*Correspondence: Jaap van Pelt, Computational Neuroscience Group, Department of Integrative Neurophysiology, Center for Neurogenomics and Cognitive Research, VU University Amsterdam, De Boelelaan 1085, 1081 HV Amsterdam, Netherlands e-mail:

This article was submitted to the journal Frontiers in Computational Neuroscience.

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Neurons innervate space by extending axonal and dendritic arborizations. When axons and dendrites come in close proximity of each other, synapses between neurons can be formed. Neurons vary greatly in their morphologies and synaptic connections with other neurons. The size and shape of the arborizations determine the way neurons innervate space. A neuron may therefore be characterized by the spatial distribution of its axonal and dendritic “mass.” A population mean “mass” density field of a particular neuron type can be obtained by averaging over the individual variations in neuron geometries. Connectivity in terms of candidate synaptic contacts between neurons can be determined directly on the basis of their arborizations but also indirectly on the basis of their density fields. To decide when a candidate synapse can be formed, we previously developed a criterion defining that axonal and dendritic line pieces should cross in 3D and have an orthogonal distance less than a threshold value. In this paper, we developed new methodology for applying this criterion to density fields. We show that estimates of the number of contacts between neuron pairs calculated from their density fields are fully consistent with the number of contacts calculated from the actual arborizations. However, the estimation of the connection probability and the expected number of contacts per connection cannot be calculated directly from density fields, because density fields do not carry anymore the correlative structure in the spatial distribution of synaptic contacts. Alternatively, these two connectivity measures can be estimated from the expected number of contacts by using empirical mapping functions. The neurons used for the validation studies were generated by our neuron simulator NETMORPH. An example is given of the estimation of average connectivity and Euclidean pre- and postsynaptic distance distributions in a network of neurons represented by their population mean density fields.

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,

The question whether connectivity can also be derived from the overlap of dendritic and axonal density fields has been addressed by Liley and Wright (

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.,

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.,

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.,

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 (

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.

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 _{v}^{3}_{v}^{3} (Figure

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 _{v}_{v}

indicating the expected length of an axonal or dendritic arborization in that particular voxel.

The scale of the grid is defined by the size of the individual voxels _{v}

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^{*}400^{2} = 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^{*}10^{6} dendrites is needed. To obtain stable statistical averages per voxel, one needs a multitude of this number, say at least 20 ^{*} 10^{6} 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 ^{*} 1000^{2} = 12.566.371 and for stable density field estimates in peripheral areas one needs a number of at least 1.2^{*}10^{8} 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 _{d}_{a}

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 (_{p}_{d}_{p}_{a}_{p}

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.

Axons can make synaptic connections with dendrites when their branches are sufficiently close to each other (Peters,

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 _{int}. For a voxel of size _{int} 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

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 ^{hit}_{voxel}(^{hit}_{v}^{hit}_{v}^{tot}_{int}(

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 ^{3}. Dendritic mass _{vd}_{va}^{hit}_{vd}^{hit}_{va}

(see also Appendix section A2).

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.,

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 ^{cross} 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

(_{v, w}

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 ^{cross}_{v, v}_{v, w}^{cross}_{v, w}_{v, w}

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, w}

In the overlap area of a dendritic density field

with ρ_{vD}_{wA}

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 (_{env} (see Equation A23) of the dendritic voxel

If it can be assumed that the axonal densities ρ_{wA}_{vA}

The second summation now runs over all voxels in the local environment of a given voxel ^{env}(

The values of the ^{env}(^{env}(^{env}(

with _{DA}

The ^{syn}_{v}^{syn}_{v}^{nosyn}_{v}^{nosyn}_{v}^{syn}_{v}^{nosyn}_{i}^{syn}_{vi}^{con}_{A}^{con}_{A}^{nosyn}_{A}

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

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.

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

An example of density field calculations based on axial symmetry is given in Figure

Finally, for a given spatial positioning of the two cell bodies, the overlap sum _{DA}

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.,

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 ^{*}9 = 108 data points, as shown in the scatterplots of Figure ^{bxc}

^{bxc}

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

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 ^{dx}

^{dx}

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

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

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

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

For the expected mean number of contacts, averaged over all the 2450 neuron pairs, with

Thus far, the focus was on using density fields for estimating the connectivity between two neurons at given positions in space (see Figures

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

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

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/mm^{3}) that are uniform randomly distributed in the cylindrical space (Figures

These differences can be understood from a cartoon drawing illustrating the dimensions of the dendritic and axonal density fields and the cylindrical space. Figure

The bounded area of the cylinder also puts constraints on the intersoma distance distribution, as shown in Figure

^{3})

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,

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.,

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

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 _{coef} (which includes the _{coef} (at least up to the 3rd decimal,

The calculations were based on a data set of 50 neurons generated with the simulator NETMORPH (Koene et al.,

An important objective of this study was the validation of the density-field based connectivity expectations with the data obtained from the actual arborizations.

As shown in Figure

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.

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.

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.

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.

Hellwig (

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

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.,

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

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

Feldmeyer et al. (

Helmstaedter (

The present study was based on a set of neuronal morphologies produced by the simulator NETMORPH (Koene et al.,

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.

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.

Many neuronal reconstructions for a variety of cell types and species that are made available through the NeuroMorpho.Org data base (Ascoli,

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.

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.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Ir. H. van Pelt for support with one of the figures, and Prof H. B. M. Uylings for critical comments on the manuscript.

The Appendix of this article is supplied as Supplemental Data (Data Sheet.pdf) and can be found online at: