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Edited by: Patrik Krieger, Ruhr University Bochum, Germany

Reviewed by: Jochen Ferdinand Staiger, University Medicine Goettingen, Germany; Sean L. Hill, International Neuroinformatics Coordinating Facility, Sweden; Jaap Van Pelt, VU University Amsterdam, Netherlands

*Correspondence: Marcel Oberlaender, Computational Neuroanatomy Group, Max Planck Institute for Biological Cybernetics, Spemannstraße 38-44, Tuebingen 72076, Germany e-mail:

This article was submitted to the journal Frontiers in Neuroanatomy.

†These authors have contributed equally to this work.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Sensory-evoked signal flow, at cellular and network levels, is primarily determined by the synaptic wiring of the underlying neuronal circuitry. Measurements of synaptic innervation, connection probabilities and subcellular organization of synaptic inputs are thus among the most active fields of research in contemporary neuroscience. Methods to measure these quantities range from electrophysiological recordings over reconstructions of dendrite-axon overlap at light-microscopic levels to dense circuit reconstructions of small volumes at electron-microscopic resolution. However, quantitative and complete measurements at subcellular resolution and mesoscopic scales to obtain all local and long-range synaptic in/outputs for any neuron within an entire brain region are beyond present methodological limits. Here, we present a novel concept, implemented within an interactive software environment called

One of the major challenges in neuroscience is to relate results from structural and functional measurements across multiple spatial scales. Current anatomical approaches either provide information of synaptic connectivity at macroscopic, i.e., between brain regions (e.g., using bulk injections of retro/anterograde agents, Oh et al.,

At present, methods that allow for measurements of synaptic connectivity at sufficiently high resolution (i.e., (sub)cellular levels) can be grouped into three main categories: First, electrophysiological approaches determine connectivity between pairs (or small numbers) of neurons using simultaneous patch-clamp recordings (e.g., Feldmeyer et al.,

Second, electron-microscopic approaches, such as serial block face scanning (SBFSEM, Denk and Horstmann, ^{3} (Helmstaedter,

Third, statistical approaches allow to determine cell type- and/or location-specific connectivity patterns by measuring structural overlap between reconstructed axons and dendrites of individual (Lubke et al.,

Here, we present a novel approach, implemented within an interactive software environment called

We illustrate our approach using the vibrissal part of rat primary somatosensory cortex (i.e., barrel cortex, vS1), present the required anatomical data and compare our

The interactive software environment _{ij}

All routines of

Mandatory anatomical input data to

The most important prerequisite to assemble average dense models of the neuronal circuitry is the definition of a standardized 3D reference frame that allows integration of anatomical data obtained from many animals. In general, the reference frame describes the 3D geometry of the brain region(s) of interest in terms of anatomical landmarks. Further, it specifies the variability of these landmarks across animals, which serves as a resolution limit of the average circuit model. More specifically, the 3D reference frame has to describe (i) the boundaries of the brain region(s) of interest, (ii) anatomical substructures within these regions, and (iii) a global and/or multiple local coordinate systems. The latter reflects the general scenario that brain areas have irregular and/or curved boundaries and sub-structures.

In case of rat vS1, the 3D reference frame has been generated by reconstructing the pial surface of entire rat cortex, the white matter tract (WM) and the circumferences of 24 cortical barrel columns (i.e., each representing one of the large facial whiskers on the animal's snout, Woolsey and Van der Loos,

^{*}) in

Because the pial and WM surfaces are curved, the orientation of each barrel column is tilted with respect to the (D2) z-axis. Therefore, we determined 23 additional local coordinate systems (i.e., for each barrel column), using the same approach used to determine the global D2 coordinate system. The final standardized reference frame of rat vS1 thus comprises the average pial and WM surfaces, 24 column center coordinates and diameters with respect to the global D2 coordinate system and 24 z-axes, representing local coordinate systems that define the orientation of each barrel column within the curved cortex. We further determined the variability of these anatomical landmarks across animals. The standard deviations (SDs) of the column center locations were on average ~90 μm, of the pia-WM distances ~100 μm and of the column orientations ~4.5 degrees (Egger et al., ^{3} voxels and a local z-axis was calculated for each voxel by interpolating from the respective nearest barrel column axes.

The 3D reference frame of rat vS1 is presented to

The second anatomical prerequisite to generate an average dense model of the neuronal circuitry are measurements of the number and 3D distribution of excitatory and inhibitory neuron somata for the entire brain region(s) of interest. These distributions have to be obtained with respect to, and at the resolution of, the anatomical reference frame. In case of rat vS1, we stained 50 μm thick histological sections, cut tangentially to the D2 barrel column axis from the pia toward the WM, for NeuN (Mullen et al., ^{3} per mm^{3}). The two average soma density fields are provided to

The third prerequisite to generate an average dense model of the neuronal circuitry are reconstructions of complete 3D soma/dendrite/axon morphologies. The morphological dataset has to be representative for the brain region, fulfilling two criteria: (1) objective classification approaches should reveal all axo-dendritic cell types (i.e., dendrite as well as axon projection patterns are similar within, but significantly different between cell types) reported for the brain region(s) of interest (see, Narayanan et al., under review, for excitatory cell types in rat vS1), and (2) spatial sampling of neurons should be performed at the resolution of the anatomical reference frame (i.e., revealing location-dependent differences in morphology, spatial distribution and overlap of different cell types). For each cell type, a number of properties is defined using a spreadsheet (

In case of rat vS1, we labeled individual neurons with Biocytin using cell-attached recordings

The final anatomical prerequisite to generate an average dense model of the neuronal circuitry is measurements of the density of postsynaptic target sites (PSTs), i.e., spines along dendrites of excitatory neurons and surface areas of somata and dendrites of excitatory/inhibitory neurons for all cell types present within the brain region(s) of interest. 3D reconstruction of soma and dendrite diameters of excitatory and inhibitory neurons was performed manually using ^{3} voxel size) along skeleton tracings of

Connections between cell types are specified in ^{2} area, and/or per μm branch length is defined, based on measured values (using the methods stated above) for each cell type and substructure (soma, apical dendrite, or basal dendrite). This meta-connectivity list thus specifies general knowledge of whether two cell types can in principle connect to each other and at which substructures. For example, inhibitory interneurons may specifically innervate somata and dendritic shafts of excitatory neurons. Thus, connections from interneuron to excitatory cell types can be specified in the meta-connectivity list such that PSTs are exclusively calculated by the surface areas of the excitatory somata and dendrites (i.e., soma/dendrite surface-specific PSTs). In contrast, connections from excitatory to excitatory cell types may be specified in the meta-connectivity list such that PSTs are calculated exclusively by the spine densities (i.e., dendrite length-specific PSTs).

Upon availability of the above described anatomical data in appropriate formats, ^{3} μm^{3}) and rounding to the nearest integer. 3D soma locations within a voxel are drawn from a uniform distribution. Based on the 3D location, each soma is further assigned to a unique substructure (barrel column) and cell type (Figure

The dense statistical connectome _{ij}_{spines}_{surface}

Here, _{j,L}_{T(i),T(j)}(^{−1}), as provided by spine density measurements and specified in the meta-connectivity spreadsheet. _{j,L}^{2}). α_{T(i),T(j)}(^{2} soma surface) for connections from neurons of type ^{−2}). Whereas spine and bouton distributions can be measured (e.g., using the methods stated above), we derived surface PST densities by assuming that the total number of boutons _{all}

Reducing this equation to 1 dimension (i.e., collapsing the 3D densities to the z-axis), we fit the respective surface PST density values α_{T(i),T(j)} using standard least squares algorithms (see fitting result in the online meta-connectivity list).

Third, the precision (across animal variability) of the geometrical reference frame determines the voxel resolution, i.e., the smallest scale at which axo-dendritic overlap can be calculated between morphologies obtained in different animals. Thus, locations of somata/dendrites/axons within a voxel cannot be further resolved and proximity of boutons and PSTs within a voxel cannot be used to estimate synaptic innervation. Instead, we assume that all PSTs within a voxel are equally likely to receive any bouton in the same voxel (i.e., independent synapse formation at resolutions smaller than the accuracy of the reference frame). The probability that neuron

Here, _{all}

If _{i}

Average values for _{i}_{j}^{1})-^{2}) and ^{−3}), respectively. Given the ~5 orders of magnitude differences between _{i}_{j}_{i}_{j}

Here, we defined the average innervation _{ij}

The connectivity statistics between any two neurons _{ij}

Because we assume that synapses in different voxels are formed independently of another, the total probability of finding a connection between two neurons

Here, _{ij}

Using the innervation matrix _{ij}_{b}_{a}_{AB}_{ij}_{ij}

Here, 〈···〉_{a ϵ A} is the ensemble average across all neurons _{AB}_{ij}_{ij}

Based on the anatomical input data (Figure ^{3} (Egger et al.,

First, the average 3D distributions of excitatory and inhibitory somata were registered to the reference frame and somata were placed and assigned to cell types (Figure ^{10} μm^{2}. The total number of spines was 5.2 × 10^{9}, and the total number of boutons was 6.4 × 10^{9}.

The average bouton (synapse) density across entire rat vS1 was 1 bouton per μm^{3}, which matches previous measurements (0.94 ± 0.12 synapses per μm^{3}) of synapse densities using electron-microscopic tomography on small tissue (~200 μm^{3}) volumes of rat vS1 (Merchan-Perez et al.,

Within the dense model of rat vS1, we used

_{ij}_{ij}_{ij}_{ij}_{ij}

However, within the overlap volume, dendritic spines originating from other excitatory neurons are present, rendering as equally likely contact sites for the VPM boutons as the spines of the exemplary L4ss neuron. The total number of spines within the BB of the overlap volume was 2.1 × 10^{7}, with a maximum of 130,000 spines per voxel. Furthermore, VPM axons could also target somata and/or dendritic shafts of inhibitory interneurons (Staiger et al., ^{6} PSTs on inhibitory surfaces are present within the BB of the overlap volume, with a maximum of 13,500 surface PSTs per voxel. Consequently, the 3D innervation field _{ij}^{7}) was four orders of magnitude larger than the number of spines/boutons from the individual neurons, justifying the approximation of the binomial connection probability by a Poisson distribution.

The resultant 3D innervation field _{ij}_{ij}_{ij}_{ij}

In consequence, we argue that structural axo-dendritic overlap should never be calculated from sparse morphological data alone and that connectivity measurements by Peters' rule should not be presented in a binary fashion (i.e., overlap equals connectivity, no overlap equals no connectivity). Instead, structural overlap in the present form results in innervation measurements at subcellular (reference frame) resolution, which can be converted into pairwise connection probabilities and a range of putative synapse numbers. In case of the present example, the overlap between 2964 VPM boutons with 4640 L4ss spines did thus not result in a connection probability of 1, but instead, the probability that the two neurons were unconnected was 52%, that they were connected by a single synapse was 34%, and by two or three synapses was 12% and 2%, respectively (Figure

In the following, we compare our

_{ij}_{ij}_{ij}_{ij}

The D2 column comprised 17810 excitatory neurons including 4657 neurons of L4 cell types (2480 L4ss; 1707 L4sp; 470 L4py), 1386 L5st, 1103 L5tt, 1391 L6cc, 767 L6inv, and 4048 L6ct neurons. Further, the D2 column model contained 2545 inhibitory neurons and 311 thalamocortical axons originating in the D2 barreloid (Meyer et al., _{ij}_{ij}_{AB}

Finally, the range of putative synapses per connection for L4ss-to-L4ss connections was 1–5 (

Because the average dense model of rat vS1 resembles the structural organization of this neuronal tissue at meso-, micro- and nanoscopic scales (see Application example 1) and structural overlap measurements within the model reproduced cell type-specific pairwise connectivity measurements (see Application example 3), we investigated whether the resultant dense statistical connectome can be used to investigate higher-order connectivity patterns beyond pairwise measurements.

The simplest higher-order pattern to be investigated is connectivity between three neurons (Sporns and Kotter, _{ij}_{ij}_{12} = 0.68, which corresponds to a pairwise connection probability of _{12} = 0.49. This can be interpreted as the probability that the triplet motif contains an edge from node 1 to node 2. Conversely, the probability that this particular edge is missing is 1-_{12} = 0.51.

_{ij}

In general, three nodes can be connected by 64 different motifs of bidirectional edges. However, multiple motifs are redundant (e.g., 1 connected to 2 and no other edge present is the same motif as 2 connected to 3 and no other edge is present). Thus, the 64 triplet motifs can be reduced to 16, of which 7 contain three edges (three-connected), 6 contain two edges (two-connected, 2 contain one edge (one-connected) and 1 motif (no edges) represents the absence of any connections between the three neurons (Figures _{12}, _{21}, _{13}, _{31}, _{23}, _{32}) allows computing the probability of finding each triplet motif by multiplying the probability of finding/not finding all six possible edges. For example, the probability that the three neurons are connected according to motif 7 (i.e., three-connected by unidirectional edges) is computed as follows:

There are five other possibilities of arranging connections between these three neurons that result in the same triplet motif. Thus, the total probability of finding this triplet motif among these three neurons is the sum over these six individual connection arrangements, resulting in a total probability of _{123} = 0.146 (Figure

In the same way, we calculated the probability of occurrence for each of the 16 possible non-redundant triplet motifs, illustrated as a motif spectrum (Sporns and Kotter,

The deviations between the “uniform” spectra of triplet motifs from those predicted by the dense statistical connectome were substantial (Figure

In the present study, we introduced a novel quantitative approach for measuring synaptic connectivity at subcellular resolution and mesoscopic scales. The measurements are based on sparse morphological datasets, integrated into a common anatomical reference frame that allows up-scaling to an average dense model of the neuronal circuitry and determining axo-dendritic overlap between any two neurons in the model. Illustrating our approach for excitatory thalamo- and intracortical circuits in rat vS1, we (i) defined the mandatory anatomical information required to generate average dense circuit models, (ii) introduced the interactive software environment

In recent years, multiple approaches began integrating morphological data to generate anatomically well-constrained neuronal network models. However, compared to

Therefore, we argue that our approach can be regarded as more general for investigating structural organization principles of the neuronal circuitry. First, the present concept relies on definition of a standardized 3D reference frame that describes the average geometry of the brain structure (and substructures) of interest. Consequently, no assumptions about the mesoscopic organization of neuronal circuits are required. For example, in case of rat vS1, we previously reported that each cortical barrel column has a specific diameter, height and orientation, and barrel columns representing whiskers located within different rows along the animals' snout have substantially deviating volumes (Egger et al.,

Second, the up-scaling to a dense average circuit model is based on measured 3D distributions of excitatory and inhibitory neurons. Consequently, no assumptions about the microscopic (i.e., cellular) organization of the neuronal circuits are required. For example, in case of rat vS1, we previously reported that separation between individual barrel columns is only present within the distribution of excitatory neurons in L4, where neuron densities are significantly lower in the septum, compared to densities in barrel columns (Meyer et al.,

Finally, connectivity measurements are based upon complete 3D reconstructions of

In summary, because organizational principles of the neuronal circuitry are generally influenced by brain region- and species-specific mesoscopic, cellular and subcellular quantities, generation of well-constrained network models should not be based on assumptions, but on measurements of these quantities instead. Assessments of these quantities provide information about the respective variability across animals, allowing to determine (i) the appropriate resolution for connectivity measurements within an average representation of the neuronal circuitry and (ii) how representative the average model is (i.e., in terms of SDs of (sub)cellular properties).

The validity of measuring synaptic innervation by structural overlap between dendrites and axons has been discussed controversially (Stepanyants and Chklovskii,

However, to date, neither the appropriate spatial resolution to apply Peters' rule, nor a coherent framework to obtain structural overlap in terms of connection probabilities with respect to all neurons projecting dendrites into the overlapping volume existed. We provide both. First, the resolution for determining structural overlap within an average network model (i.e., integration of morphological data from different animals) is defined by the inter-animal variability of the geometrical reference frame used to integrate the data. Increasing the voxel size will provide less accurate connectivity estimates (i.e., cells or cell types that do not overlap at 50 μm resolution may overlap at 100 μm scales). In contrast, decreasing the voxel size below the precision of the registration framework would imply inappropriate accuracy. Hence, implications of synaptic innervation below the resolution limit, or even at submicron resolution, are beyond the limits of Peters' rule. Instead, measurements of subcellular synapse locations remain exclusive to reconstructions at electron-microscopic levels (but see, Druckmann et al.,

Second, we illustrate that in general, millions of potential postsynaptic target sites (PSTs) from unstained neurons are present within the overlap volume of two stained neurons. Hence, when normalizing innervation by the total number of PSTs, the resultant innervation and pairwise connection probabilities are small. In case of the exemplary calculation between the dendrites of one L4ss and one thalamocortical VPM axon in rat vS1, overlap between ~4500 spines and ~3000 boutons did not result in a connection probability of one, but instead there is a 52% chance that the two neurons are unconnected. Hence, connectivity measurements by structural overlap have to be performed with respect to

In addition to illustrating that pairwise connection probabilities determined by structural overlap are in line with measurements using conventional recording/reconstruction techniques, we provide a strategy that allows investigation of higher-order connectivity patterns within dense statistical connectomes. On the example of the population of L4ss neurons located within a barrel of rat vS1, we determined the probabilities of obtaining all possible three-neuron (triplet) motifs and compared the resultant motif spectra with those to be expected from randomly connected networks that have the same average pairwise connection probability. Interestingly, we found that the two spectra displayed significant deviations. For example, unidirectional triplets (i.e., recurrent loops) are much less likely to occur within the L4ss population compared to randomly connected networks. In contrast, other triplet configurations were significantly more likely. Arguably such deviations can be considered as evidence for specificity in the organization of the neuronal circuitry, for example caused by inhomogeneous distributions of somata (e.g., excitatory soma density decreases from the barrel center toward the borders), dendrites and axons (e.g., polar dendrite morphologies pointing toward the barrel center).

Hence, we suggest using statistical spectra of higher-order motifs as a definition of cell type-specific “structural fingerprints” for the respective neuronal circuits. Comparing these fingerprints with dense connectomes obtained at electron-microscopic resolution, will indicate whether such cell type-specific higher-order patterns can be explained by the meso- and microscopic organization of the network, or whether additional specificity originates at nanoscopic scales. In consequence, not the absence of synapses between touching dendrites/axons, but deviations of higher-order connectivity patterns observed in statistical and electron-microscopic dense connectomes should be considered as evidence for violations of statistical network organization.

We present a novel concept for measuring pairwise and high-order connectivity patterns at subcellular resolution and mesoscopic scales. We provide the required software to generate average dense circuit models, to calculate structural overlap, and to convert these measurements into dense statistical connectomes. Further, we describe the anatomical data necessary to assess structural organizational principles of the neuronal circuitry without assumptions about homogeneity at meso/microscopic and subcellular scales. Given that the required anatomical data is available, we consider our approach as generalizable to other brain structures and species. This sets the stage to generate well-constrained network models that allow simulating sensory-evoked signal flow to provide unprecedented insight into the interplay between the structural organization and function of the respective local and long-range neuronal circuits.

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 Bert Sakmann for discussions and his generous support. Funding was provided by the Max Planck Florida Institute for Neuroscience (Marcel Oberlaender), the Studienstiftung des deutschen Volkes (Robert Egger), the Bernstein Center for Computational Neuroscience, funded by German Federal Ministry of Education and Research Grant BMBF/FKZ 01GQ1002 (Daniel Udvary, Robert Egger and Marcel Oberlaender), the Max Planck Institute for Biological Cybernetics (Daniel Udvary, Robert Egger and Marcel Oberlaender), the Werner Reichardt Center for Integrative Neuroscience (Marcel Oberlaender), the Max Planck Institute of Neurobiology (Vincent J. Dercksen) and the Zuse Institute Berlin (Vincent J. Dercksen and Hans-Christian Hege).