1004 VOLUME 20 | NUMBER 7 | JULY 2017 nature neurOSCIenCe

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The organization of synaptic connections constrains the dynamics of a neural circuit, shaping the repertoire of collective activities that gives rise to neural functions or dysfunctions1–3. A growing body of data on the anatomical microcircuitry at the cellular and synaptic level, ‘microconnectomes’1, has shed light on information processing in specific neural systems4,5. However, primarily due to the enormous technological challenge involved in mapping large-scale circuits at the synaptic level6,7, these microscale wiring diagrams are still rare, with only a few seminal exemplars8,9.

For the mammalian neocortex, which is related to many high-level computational processes such as sensory processing, planning, motor control, perception and language, detailed microconnectomes may still be years away. A prevailing alternative approach has been to sta-tistically characterize connectivity patterns between different pairs of cell types10–13. However, remarkably few studies have been devoted to higher-order connectivity patterns14–16; these have uncovered traces of complex network topologies. For instance, particular three-neuron network motifs were found to be overrepresented in the neocortex compared to in corresponding random network models14,16. To better understand how random or specific these patterns are, their emer-gence in the neocortical microcircuit and their functional implica-tions, a comprehensive wiring diagram of this circuit is required.

The lack of detailed wiring diagrams for cortical circuits has prompted experimental efforts to reconstruct the neocortical micro-connectome using electron microscopy. To date, these studies have provided highly detailed anatomical reconstructions (at cellular and subcellular resolutions) within volumes of a few individual cells7.

In parallel, complementary modeling efforts, based on biologically constrained algorithms, have started to provide computer-generated predictive wiring diagrams of the prototypical neocortical micro-circuit (NMC) at the scale of thousands of cells17,18. These in silico models integrate sparse (anatomical and physiological) experimental data and utilize interdependencies between the properties of the neo-cortical microcircuitry to digitally reconstruct a dense map of all syn-aptic connections in the NMC (including those that were not directly measured experimentally). Anatomical data, such as cell types and their related densities at various cortical layers, dendritic and axonal geometries and the density of axonal boutons, are used to reconstruct biologically constrained wiring diagrams. In addition, incorporation of measurements from in vitro electrophysiological recordings has extended the structural connectivity with biophysical properties of neurons and individual synapses10, both of which are essential to further reconstruct the functional dynamics of the neural circuit.

Acquiring these huge volumes of big data describing the micro-connectome of a large neuronal circuit is only the beginning of the scientific challenges involved. The extreme magnitude and complex-ity of brain connectomes defeat traditional approaches for unraveling wiring specificities based on manual inspection of individual neu-rons and tabulation of their connections. It calls for a new approach for concisely describing the topological structures of such large net-works and, consequently, illuminating how these structures affect cortical dynamics.

This study describes a framework for the systematic analysis of such large-scale cellular-level connectomics information. We used

1Edmond and Lily Safra Center for Brain Sciences, The Hebrew University, Jerusalem, Israel. 2Department of Neurobiology, The Hebrew University, Jerusalem, Israel. 3The Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel. 4Sagol School of Neuroscience, Tel Aviv University, Tel Aviv, Israel. 5Blue Brain Project, École Polytechnique Fédérale de Lausanne (EPFL) Biotech Campus, Geneva, Switzerland. Correspondence should be addressed to E.G. (eyal.gal@mail.huji.ac.il) or I.S. (idan@lobster.ls.huji.ac.il).

Received 9 March; accepted 3 May; published online 5 June 2017; corrected online 14 June 2017 (details online); doi:10.1038/nn.4576

Rich cell-type-specific network topology in neocortical microcircuitryEyal Gal1,2 , Michael London1,2, Amir Globerson3,4, Srikanth Ramaswamy5, Michael W Reimann5, Eilif Muller5, Henry Markram5 & Idan Segev1,2

Uncovering structural regularities and architectural topologies of cortical circuitry is vital for understanding neural computations. Recently, an experimentally constrained algorithm generated a dense network reconstruction of a ~0.3-mm3 volume from juvenile rat somatosensory neocortex, comprising ~31,000 cells and ~36 million synapses. Using this reconstruction, we found a small-world topology with an average of 2.5 synapses separating any two cells and multiple cell-type-specific wiring features. Amounts of excitatory and inhibitory innervations varied across cells, yet pyramidal neurons maintained relatively constant excitation/inhibition ratios. The circuit contained highly connected hub neurons belonging to a small subset of cell types and forming an interconnected cell-type-specific rich club. Certain three-neuron motifs were overrepresented, matching recent experimental results. Cell-type-specific network properties were even more striking when synaptic strength and sign were considered in generating a functional topology. Our systematic approach enables interpretation of microconnectomics ‘big data’ and provides several experimentally testable predictions.

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the most comprehensive and biologically constrained in silico recon-structed microcircuit available, which was recently obtained from the somatosensory neocortex of young (P14) rats18. The volume of this circuit is around ~0.3 mm3 and contains 31,346 cells (Fig. 1a) that are interconnected by ~7.8 million multisynaptic connections, form-ing ~36 million synapses. This framework employs network-science methodologies to define multiple functionally meaningful statistical and topological features of neocortical connectivity.

RESULTSThe in silico neocortical networkThe in silico neocortical microcircuit used in the present study and its consistency with the integrated experimental observations is described in Figure 1. The 31,346 neurons belong to 55 morphological cell types (13 excitatory and 42 inhibitory; Supplementary Fig. 1) and are distributed across the five cortical layers (layers 2 and 3 were uni-fied; Fig. 1a). Briefly, the process of reconstructing the in silico NMC algorithmically started with deciding on the total number of neurons in the selected volume, and then the various morphological types (Fig. 1b) were positioned according to their distribution at a given position (Fig. 1c). Next, putative synapses were identified based on the proximity between dendritic and axonal arbors. Axon-to-dendrite proximity was a necessary but insufficient condition for defining a functional synapse. Therefore, further experimental observations of axonal boutons densities (Fig. 1d) and numbers and strength of syn-apses between specific connected cell types (Fig. 1e,f) were applied to constrain and reconstruct synaptic connectivity. Integrating this set of

local observations was sufficient to provide comprehensive and bio-logically consistent maps of the connections between all neurons in the NMC that, when used in numerical simulations, also reproduced mul-tiple in vitro and in vivo experiments without any parameter tuning18. The reconstruction process is fully described in ref. 18 and data are available online at http://bbpteam.epfl.ch/nmc-portal.

We analyzed seven NMC instances; they were spatially tiled such that one central NMC was surrounded by six neighboring NMCs. Each instance defined a network of n = 31,346 neurons (nodes) that were sparsely interconnected by 7.8 ± 0.1 million connections (mean ± s.d., N = 7 throughout this work, unless stated otherwise). The overall sparsity was 0.8%; the connection probability is much larger for intersomatic distances ≤ 100 µm; Supplementary Fig. 2). On average, a connection between two neurons was formed by 4.7 synapses (contacts), with an average of 13.9 contacts per connec-tion for inhibitory and 3.7 contacts per connection for excitatory synapses, respectively. Altogether, the circuit consisted of 36.5 ± 0.6 million contacts.

For each instance of NMC, three different n × n adjacency matrices were created. In matrix A, the element in the ith row and jth column, aij, denotes whether there is a synaptic connection from neuron i to neuron j (a binary matrix). In matrices B and C, the elements rep-resent the number of synaptic contacts per connection and the total conductance impinging on a neuron via all contacts in that connec-tion, respectively (weighted connectivity matrices). Matrices A and B depict strict anatomical connectivity (structural connectivity map), whereas matrix C reflects the effective (functional) connectivity.

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Figure 1 The present in silico model of neocortical microcircuitry (NMC) and its consistency with experiments. (a) Left: blue stripe in rat cartoon indicates the sagittal plane of the neocortex used to obtain somatosensory slices; black lines in brain image indicate location of the modeled NMC. Right: dimensions and number of cells per layer in the seven instances of the model (mean ± s.d., N = 7). The total number of neurons in this circuit is depicted at the top. (b) Example of reconstructions of two morphological cell types (layer (L) 2/3 ChC (chandelier) inhibitory interneuron, left, and L5 TTPC2 (thick-tufted pyramidal) neuron, right) out of the 55 neuron types used in this study (Supplementary Fig. 1). Red, soma and dendrites; magenta, apical dendrites for the pyramidal cell; blue, axonal arbor. (c–f) In silico synaptic connectivity consistency with four biological in vitro statistics. (c) Densities of various types of interneurons at equidistant bins throughout the height of the column. (d) Density of axonal boutons per cell type. (e) Number of postsynaptic contacts per connection for various cell-type to cell-type connections. (f) Amplitude of postsynaptic potentials for various cell-type to cell-type connections. Data in c–f are mean ± s.e.m. Circles and triangles, mean values of corresponding cell-types (presynaptic cell types in e and f), averaged over biological data (x axis) and over in silico instances (y axis). Pearson correlation coefficients are depicted (P < 0.001 for all). Color code is displayed at top right; Inh, inhibitory; Exc, excitatory; L, layer. Black diagonal lines indicate the identity line. (All data from Markram et al.18.)

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1006 VOLUME 20 | NUMBER 7 | JULY 2017 nature neurOSCIenCe

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Comparison of the matrices could thus shed light on the rela-tionship between structural and functional connectivity in the studied circuit.

Emergence of excitation–inhibition balanceThe NMC circuit is composed of two populations of neurons: excita-tory (E), comprising 85% of the total neurons, and inhibitory (I), com-prising the remaining 15%. To quantify the level of structural balance between excitation and inhibition, we inspected the connectivity within and between the populations (considering only synapses aris-ing from within the NMC). We found a highly asymmetrical distribu-tion across the populations, with an excess of excitatory connections (90% of the total number of connections) in the microcircuit (Fig. 2a, Supplementary Table 1 and Online Methods). About 86% were E-to-E and 4% were E-to-I. The inhibitory connections comprised 10% of connections; 9% were I-to-E and I-to-I accounted for only about 1%. This imbalance is likely to be even higher in vivo, given the estimation that about 80% of excitatory synapses are formed by afferent fibers originating from outside the NMC19.

At the level of individual cells, the number of incoming excita-tory and inhibitory connections converging on an excitatory neuron ranged between 0 and 945 and between 0 and 141, respectively; yet these two types of input sources were correlated (Fig. 2b; Pearson

correlation coefficient r = 0.63 ± 0, P < 0.001 for all). The E/I con-nection ratio was very high, 13.8 ± 13.6 (averaged across excitatory neurons in all the seven networks, N = 185,355; Fig. 2c). For inhibi-tory neurons, the average E/I ratio was lower, 6.6 ± 7.0 (N = 32,814; Fig. 2c); however, the variance across individual neurons was higher (lower correlation between the number of E and I connections: r = 0.30 ± 0.01, P < 0.001 within a network; Fig. 2b).

The ratios discussed above characterize the structural balance of inputs based on the organization of ‘binary’ connections, while neglect-ing their efficacies. As the number of synaptic contacts and their con-ductances vary among individual connections and affect the effective balance impinging a cell, we computed the E/I ratios in terms of number of contacts and conductances. When considering contacts (Fig. 2d–f), the ratio of E/I inputs became more balanced for both populations (aver-age E/I contacts ratio of 3.5 ± 4.0 for excitatory neurons and 3.7 ± 4.6 for inhibitory neurons), due to the higher average number of contacts per inhibitory connection (13.9) than per excitatory connection (3.7; Fig. 2f). When conductance was also considered, the E/I ratio was reduced slightly further (3.4 ± 3.5 for E and 2.3 ± 3.4 for I; Fig. 2g–i). In addition, the E/I correlations in the excitatory neurons increased (r = 0.75 ± 0 for contacts and r = 0.77 ± 0 for conductance), in agreement with experimental functional findings20. In contrast, the E/I correla-tions per inhibitory neurons were still low (Fig. 2e,h).

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Figure 2 Cellular-level E/I balance is emphasized when considering synaptic conductance. (a) The percentage of E (arrow head) and I (circular head) pathways in terms of the total number of connections (7,824,436 ± 104,092). Line thickness illustrates the percentage of connections in the corresponding pathway. (b) Incoming E/I connections for excitatory (left, nE = 26,567) and inhibitory (right, nI = 4,779) cells. Data depict a single exemplar NMC instance (darker colors indicate deeper layers, as in Figure 1; black diagonal lines indicate the identity line; white crosses indicate the mean). (c) E/I connections ratio distribution among excitatory (red) and inhibitory (cyan) population (lines indicate the average over all seven instances and shading indicates the s.d.; the mean E/I ratio per curve is indicated by triangle markers). (d–i) As in a–c but in terms of (d–f) total number of synaptic contacts (synapses: 36,471,080 ± 554,503) and (g–i) total peak synaptic conductance (30.74 ± 0.47 mS). Note the excess excitation vs. inhibition in all three cases. The shifting of the bottom histograms toward 1 implies a higher conductance-based E/I balance. P values of Pearson correlations coefficients are < 0.001 for all six scatterplots.

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nature neurOSCIenCe VOLUME 20 | NUMBER 7 | JULY 2017 1007

Cell-type-specific in-hub and out-hub neurons forming a rich clubThe convergence (number of presynaptic cells) and divergence (number of postsynaptic cells) of a neuron profoundly influences its impact in information flow across the network. We computed for each neuron its in-degree, din, a property representing the number of presynaptic cells, and its out-degree, dout, representing the number of postsynaptic target cells. We found that in- and out-degrees varied from 0 to 1,033 (r = 0.27 ± 0.01; P ≤ 0.001 within a network). This wide heterogeneity of the in-degrees is already implied in Figure 2b. The mean in- and out-degree averaged over all neurons were din = 249.61 ± 176.19 and dout = 249.61 ± 152.10 (N = 31,346 × 7). In other words, the number of presynaptic or postsynaptic cells arising from within the NMC to/from a given neuron was ~250 on average.

Since both in- and out-degrees span three orders of magnitude, the distributions of degrees, p(din) and p(dout), are much more informative

than the mean. Figure 3 shows that both in-degrees and out-degrees displayed skewed multimodal distributions. The long tail indicates the existence of hub neurons with exceptionally high numbers of incoming or outgoing connections. The tails of both distributions decayed more slowly than the exponential distribution but faster than the power-law distribution p(d) ∝ d−γ (Online Methods). Indeed, it is unlikely that we would find a pure power-law degree distribution (i.e., a scale-free network21) in a spatially bounded system in which the formation of synaptic connections involves a cost22,23 (for example, a metabolic cost).

An examination of the identity of hub neurons showed that they were restricted to a small subset of cell types. Notably, hub neurons with a high number of incoming connections (in-hubs) were generally not the same neurons as those that had a high number of outgoing con-nections (out-hubs). For example, the top half-percentile of the in-hubs

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)Figure 3 In-hub and out-hub neurons belonging to a small subset of cell types and forming a rich club. (a) Long-tail distributions of in-degrees (number of presynaptic cells per neuron) in the seven NMC instances. The top 157 (0.5%) in-degrees (in-hubs) arise primarily from only four cell types residing in deep layers (pie chart). (b) Long-tail distribution of out-degrees (number of postsynaptic cells per neuron); the top 157 out-degrees (out-hubs) arise from multiple cell types, the majority of which are pyramidal and Martinotti cells from intermediate layers (pie chart). In a and b, lines and shading indicate the mean and s.d. (N = 7), respectively. Gray areas are degree distributions expected from ER (dark) and ER-mixture (light) null random networks, presented for reference. (c) Within the central NMC, a neuron’s inclusive in-degree (when taking the extrinsic connections from surrounding NMCs into account) is correlated with its total dendritic length. (d) A neuron’s inclusive out-degree is less correlated with its total axonal length. P values of Pearson correlations coefficients, r, are both < 0.001 (N = 31,346 neurons). (e,f) As in c and d but for thick-tufted pyramidal neurons from layer 5 (L5-TTPC1) only (N = 2,403). (g) Correlations between inclusive in-degrees and total dendritic lengths for all 55 cell types. (h) Correlations between inclusive out-degrees and total axonal lengths for all 55 cell types. (i) Ratios of the number of connections among NMC neurons whose total degree (in-degree + out-degree) > d to the number of connections expected from random networks with matching degree sequence (namely, in which each neuron maintains its original degree values) is plotted for all d (N = 1,000; s.d. is depicted in gray). This ratio is > 1 for high-degree neurons (P < 0.001, Monte Carlo), reflecting the presence of a rich club24.

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(157 neurons), which were highly connected (836–1,010 connections), were pyramidal cells from deeper layers (Fig. 3a), whereas the top 157 out-hubs (with 767–1,026 outgoing connections) were mostly pyrami-dal and Martinotti cells from the intermediate layers (Fig. 3b).

A reconstruction of a bounded tissue volume might exhibit a ten-dency for high-degree neurons in the center and low-degree neurons near the edges merely because, in the latter case, connections to neurons that are outside of the modeled volume are not included. To overcome this edge effect, the extrinsic connections from surrounding NMCs to the central NMC were also considered (Supplementary Fig. 3). It was found that the long-tail distribution and cell-type-specificity of hubs remained even for these inclusive degree measures (Supplementary Fig. 4a,b), but now the less dominant cell-types serving as hubs disap-peared (compare Fig. 3a,b to Supplementary Fig. 4c).

The statistical significance of the hub neurons was evaluated by comparing the NMC network to various reference random networks (Online Methods), including random networks with uniform connec-tion probabilities (Erdős–Rényi (ER) random networks; Fig. 3a,b), random networks that preserve the probabilities of connection within and between the different cell-type populations (ER-mixture random networks; Fig. 3a,b) and random networks that preserve the nonuni-form dependency of connection probability on intersomatic distance as observed in the NMC (distance-dependent (dd) random networks; Supplementary Fig. 2). In all cases, NMC hubs were statistically highly significant (P < 0.001, Monte Carlo; Online Methods).

To investigate the origins of the observed degree heterogeneity, we analyzed the relationship between the degrees of a given neuron and its morphological properties. Figure 3c demonstrates that the inclu-sive in-degree of a neuron is largely explained by the total length of its dendrites (r = 0.88). The correlation between the inclusive out-degree and the total axonal length was smaller (r = 0.52; Fig. 3d). However, when each cell-type was inspected independently, both correlations were generally higher (Fig. 3e,f). Low correlations were observed only in populations with small numbers of cells (Fig. 3g,h), suggesting that such low correlations could be attributed to noise due to small sample size. Indeed, we found that 92% of the inclusive in-degree variance was explained by dendritic lengths and 83% of the inclusive out-degree vari-ance was explained by axonal lengths (P < 0.001 for both, linear mixed-effect models conditioned on cell-type group; Online Methods).

Hubs in a network permit higher resilience of signal flow to random loss of cells. On the other hand, hub-specific damage (for example, in particular neurodegenerative diseases) is likely to have a dramatic effect on network dynamics, since these cells are most intensely connected to the rest of the network. An optional strategy to alleviate this vulner-ability is to ensure that hubs are densely interconnected among them-selves, forming ‘rich clubs’24. We found that NMC hub neurons indeed tended to be connected among themselves ~1.4 times more frequently than expected from their degrees (P < 0.001, Monte Carlo; Online Methods). In further systematic analysis of this over-connectivity ratio, we measured it among all neurons whose total degree (in-degree + out-degree) was > d, for every possible d. The ratio was significantly larger than 1 for larger d values (Fig. 3i), indicating the emergence of a rich-club phenomenon24.

The standard concept of connection-based hubs as analyzed above was extended to the weighted (contact-based or conductance-based) adjacency matrices. In both weighted matrices we found that the top in-hubs were excitatory (L5 pyramidal cells), consistent with the fact that connection-based in-hubs were also mostly excitatory. However, the top-weighted out-hubs belonged mostly to inhibitory cells-types (Supplementary Fig. 4d,e), in contrast to the excitatory-connection-based out-hubs (Fig. 3a,b). This indicates that, whereas the out-hub

neurons with the largest number of postsynaptic target cells are excita-tory, the most effective out-hub neurons, in terms of their total synaptic strength, are in fact inhibitory. In other words, excitatory hubs distrib-ute their impacts to many target cells while making only a few contacts per target cell, whereas inhibitory hubs connect to a fewer target cells but via more reliable connections (more contacts per target cell).

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Figure 4 Reciprocity of cortical networks under ER and dd assumptions. (a) Scatterplot of the relationship between sparsity p and reciprocity precip for the entire microcircuit (cross), excitatory (triangles) and inhibitory (circles) network-wide subnetworks (open triangle and circle, respectively), and within each layer. The values of p and precip are averaged over the seven NMC instances (mean ± s.d.; error bars smaller than markers are hidden). The dashed line depicts the cubic relationship y = x2 as expected from the ER random network. (b) Hyper-reciprocity levels (precip/p2) from the seven NMCs (colored boxplots) with respect to matching ER random networks (gray boxplots around 1; Monte Carlo with N = 100, P = 0.0099 (left) and N = 1,000, P < 0.001 (right); Online Methods), for all networks depicted in a. Note that most of the subnetworks display high hyper-reciprocity. (c) As in b but with respect to corresponding dd-random networks (note the change in vertical scale). Under the distance-dependence assumption, hyper-reciprocity is significantly reduced and excitatory subnetworks in particular display under-reciprocity within layers (left: P = 0.0099; right: P = 0.195 for L1I, P = 0.043 for L6I; P < 0.001 otherwise). In each boxplot, center lines show the medians, box limits indicate the first and third quartiles, whiskers extend 1.5× the interquartile range from the first and third quartiles, and outliers are represented by crosses. *P < 0.05, **P < 0.01, ***P < 0.001.

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Experimental demonstration of anatomical hub neurons in the neo-cortex is currently lacking (however, functional hub neurons have been identified in hippocampal networks25). Thus, by using the net-work embedding of neurons to predict their functional impact and by uncovering the hub-specific cell types, this study can serve as an experimental guidance toward targeted perturbations (e.g., stimula-tion, silencing or knockout) that are expected to have a profound influence on network dynamics. Rich clubs in the nervous system were first observed in macroscale human brain networks26 and recently, at the cellular scale, in functional networks of slice cultures of mouse somatosensory cortex27. In addition, anatomically detected rich clubs in C. elegans were found to evolve early in development and play a functional role in coordinated movement of the nematode28. In this light, our anatomical- and cellular-scale analysis of the mammalian neocortex coincides with the hypothesis that rich-club organization represents a cross-scale and cross-species invariant in the architecture of neural networks29.

Cell-type-specific triad motifsThe measures employed above analyzed connectivity properties of individual neurons. We also analyzed the wiring specificities within small groups of either two or three neurons and identified patterns that were significantly more frequent in the NMC than expected by chance (network motifs30). Such pair motifs or triad motifs have been previously studied experimentally in the neocortex for particular cell types14,16.

A pair of two indistinct nodes has three possible configurations of connectivity: no connection at all, a single unidirectional connec-tion from one node to the other or a reciprocal connection between the two nodes. The number of reciprocal connections in the NMC was tabulated and compared to that expected from matching random networks. In simple ER random networks having a connection prob-ability p, the resulting probability of finding a reciprocal connection is quadratic, p2 (Fig. 4a). In the NMC, the reciprocal connections were 3.04 ± 0.03 times more prevalent than expected from ER random networks (Fig. 4b). Specifically, the hyper-reciprocity was larger for inhibitory subnetworks than for excitatory subnetworks; this was also the case within individual layers (Fig. 4b).

Could this hyper-reciprocity be captured by a higher-order random process? When dd-random networks were used for comparison, our results showed that the distance-dependent connectivity was sufficient to reproduce about 86 ± 1% of the observed reciprocal connections. In other words, with respect to dd-random connectivity, the net-work-wide hyper-reciprocity (1.17 ± 0.01; Fig. 4c) was substantially lower compared to that in ER networks. In particular, inhibitory subnetworks displayed small, but significant, hyper-reciprocity with respect to dd-random networks, whereas excitatory subnetworks mostly exhibited under-reciprocity (Fig. 4c).

To further evaluate how well the distance-dependent connectiv-ity reproduced higher-order connectivity patterns, we studied the representations of triads. A triad has thirteen possible configurations in which all three neurons are connected (Fig. 5). The z-score was used to quantify the statistical significance of the number of appear-ances of each connected triad i:

ZN N

Ni

i i

i=

−

( )( ) ( )

( )( )

NMC dd

dds1

where Ni( )NMC is the number of times the triad appeared in the

NMC network (averaged over the seven instances); Ni( )dd and

(1)(1)

s Ni( )dd( ) are, respectively, the mean and the s.d. of its appearances

in the ensemble of corresponding dd-random networks (Online Methods). It was found that triads having a connection between all three pairs of neurons (triads #4 and #9–13) were overrepresented in the NMC, compared to the matching dd-random networks (Fig. 5a). These triads were obviously more overrepresented when compared to the ER random network (Fig. 5a) as well as to random networks that matched the prevalence of reciprocal connections (Fig. 5a).

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Figure 5 Local wiring-specificities within the NMC show overrepresented three-neuron network motifs. (a) Significance level (z-score) for all 13 triads for all cells in the NMC with respect to three types of random networks (Monte Carlo with N = 100; Online Methods). ER random networks with matching numbers of overall connections, light gray with squares; ER with additional matching of reciprocal connections, dark gray with diamonds; dd-matched random networks, black with stars. (b) Normalized z-scores for all 13 triads in both excitatory (black) and inhibitory (gray) subnetworks with respect to the dd-matched random networks (Monte Carlo with N = 100). Normalization factors are 1,752 for the E network and 207 for the I network. (c,d) As in b but for cell-type-specific subnetworks (c) of all 13 excitatory neurons (each layer is depicted by a different color) and for (d) the 11 largest inhibitory cell type populations (total number of cells > 150 cells per type). Color codes correspond to specific layers, as in Figure 4. Red motifs in c were found to be overrepresented in experiments performed on L5–L5 thick-tufted pyramidal cells by Perin et al.16, and green motifs in d were found in cerebellar inhibitory cells by Rieubland et al.31. Both of these motifs were found in this study to be common to all excitatory and inhibitory cell types. Monte Carlo with N = 1,000 was used in c and d.

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Although the standard graph-theoretical analysis employed above is useful for discovering patterns of anatomical connections, it ignores the heterogeneity in cell types. Next, we further examined motifs in the excitatory and inhibitory subnetworks (Fig. 5b). To compare the two subnetworks, which differ in size, the z-score vector of each one was normalized to 1 (Online Methods). In both the E and I subnet-works the convergent triad (#2), the divergent triad (#3) and all closed triads (#9–#13), except the loop triad (#7) were overrepresented with respect to the dd-matched networks (Fig. 5b).

Very few experimental studies have attempted to quantify network motifs of cortical microcircuitry at the cellular level. These studies esti-mated the representation of triads among particular cell types (layer 5 thick-tufted excitatory neurons in the neocortex14,16 and GABAergic interneurons in the cerebellar cortex31) using samples of four14,31 and twelve16 simultaneously recorded neurons in vitro. Our results were found to be consistent with the experimental observations when motifs were measured in layer 5 thick-tufted subnetworks (Fig. 5c) and in subnetworks of inhibitory cell-types (Fig. 5d). Extending the analysis to the various excitatory cell-types showed that, collectively, all excitatory subnetworks in the NMC exhibited overrepresentation of common motifs (Fig. 5c). These are mainly ‘transitive’ triads (in which a connection from cell i to cell j and from cell j to cell k also implies a direct connection from i to k; triads #4, #7 and #11) and the ‘divergent’ triad (#3). In contrast to the prototypical behavior of E–E motifs, inhibitory subnetworks showed higher variability in represen-tation levels for most triads, except for #3 and #4, which are common triads to all inhibitory subnetworks (Fig. 5d).

Structural and functional small-world topologyProperties characterizing the efficiency of communication and com-putation across a whole network were studied next. Small-world topol-ogy describes network architectures that simultaneously reconcile two opposing demands in sparse networks: a short path length between any pair of nodes and clustered interconnectivity within groups of nodes32. Intuitively, in a neural network, the ‘small-world-ness’ of its wiring reflects the degree to which it balances the needs for global integration and local segregation of neural information33.

In the NMC, the mean shortest path length32 (characteristic path length) averaged over all pairs of neurons was l = 2.48 ± 0.01, implying that, on average, 2.48 synapses separated any two cells in the NMC (Fig. 6a and Online Methods). The clustering coefficient of a node in directed network, ci, indicating the tendency of the node’s neighbors

to cluster together, was defined as the ratio of the number of existing triangles among the node and its neighbors to the number of all pos-sible triangles34 (Fig. 6b and Online Methods). Theoretically the value of ci ranges from 0 (none of the neighbors are connected to each other) to 1 (all neighbors are mutually connected). In the NMC, the mean clustering coefficient averaged over all neurons, c, was 0.029.

To evaluate the significance of the observed characteristic path length and clustering coefficient, they were compared against that expected from dd-random networks. The matching dd-random net-works had a characteristic path length, l(dd) = 2.20 ± 0.01, and a clus-tering coefficient of c(dd) = 0.012 (Fig. 6a,b). The NMC network thus displayed clusters of nodes that were about 2.4× denser (0.029 versus 0.012) than what was expected from corresponding random networks while preserving the short characteristic path length of comparable random networks (2.48 versus 2.2). The tradeoff between the high clustering coefficient and the short characteristic path length was quantified by the small-world index33

sic c

l l=

/

/( )

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Namely, si is defined as the ratio of the clustering coefficient to the characteristic path length, after normalizing both by the corresponding values from dd-random networks. Thus, values of si > 1 imply a small-world topology. In the NMC network, si = 2.08 ± 0.01 (Fig. 6c).

Next the small-world indices of the E and I subnetworks were meas-ured independently. The characteristic path-lengths were lE = 2.43 ± 0.01 for the excitatory subnetwork and lI = 3.84 ± 0.03 for the inhibitory subnetwork. For comparison, the matching dd-random networks had shorter characteristic path-lengths of about lE

( ) .dd = 2 14 and lI( ) .dd = 3 53,

respectively (Fig. 6a). The clustering coefficient of both the E and I net-works was cE = cI = 0.029, whereas the clustering coefficients expected from the corresponding dd-random networks were cE

( ) .dd = 0 014 and cI

( ) .dd = 0 010 (Fig. 6b). This implies that although I-to-I connec-tions were sparser than E-to-E connections, they were wired much more efficiently to form clusters. Hence, the small-world index was higher for the inhibitory subnetwork, siI = 2.80 ± 0.06 versus siE = 1.82 ± 0.01 (Fig. 6c).

Experimental measurement of small-world organization in cortical cellular-level microcircuits is difficult because of the need for a full map of all connections between all the neurons involved. A few stud-ies, however, have approximated it using samples for the functional

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Figure 6 Small-world organization of neocortical networks. Entire network and E/I subnetworks, both at the network-wide level and within layers demonstrate (a) a characteristic path length, l, that is comparable to that expected from the corresponding dd-random networks l(dd); and (b) a clustering coefficient, c, that is higher than expected based on those of the corresponding random networks c(dd) (scatterplot data are mean ± s.d.); and (c) subnetworks display small-world wiring topology si > 1 (colored boxplots depict all seven NMC instances) with respect to dd-random networks (gray boxplots around 1; Monte Carlo with N = 100, P = 0.0099 (left) and N = 1,000, P < 0.001 (right); Online Methods). In each boxplot, center lines show the medians, box limits indicate the first and third quartiles, whiskers extend 1.5× the interquartile range from the first and third quartiles, and outliers are represented by cross.

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connectivity estimated from in vitro35 and in vivo36 recordings from several dozen of neurons. These studies also reported low, but sig-nificant, degrees of small-world architecture. Taken together with our in silico findings, it seems that the small-world topology is apparent at multiple anatomical and functional levels in cortical microcircuitry (Fig. 6a–c; and see Supplementary Table 2 for the levels of individual layers). The functional implications of this observation are elaborated on in the Discussion section below.

We next extended the above structure-based small-world analysis to consider the strengths and signs of synaptic connections (‘functional small-world’ analysis). Given a pair of neurons connected by a chain of synapses, the challenge is to quantify the distance between neurons in terms of the strength and sign of synapses along the path rather than in terms of the mere number intermediate synapses. To this end, we relied on a recent experimental study demonstrating that the probability of an extra (or an abolished) spike in a cell is linear with the total charge arriving at the cell’s soma37. We derived a general-ized path-length measure that both incorporates synaptic strengths and retains the E and I (sign) information; it essentially measures the effect of activity in the first neuron on the activity in the last neuron. It can be shown that given a path between two neurons, i and j, the probability that a spike in neuron i will initiate an extra (or abol-ished) spike in neuron j is proportional to the product of all synaptic weights (and signs) along that path (termed the ‘path strength’; Online Methods). We defined sij as the strength of the strongest path among all paths with minimal distance between the two cells. Their weighted path length was defined accordingly as l sij ij

W =1/ . The characteristic weighted path length, lW, is the average over all lij

W.To further define a weighted small-world index (wsi) we used lW

together with a recent generalization of the clustering coefficient, cW, which takes into account weighted connectivity34. The weighted small-world index was defined as

wsic c

l l l l=

W

W/

( ) (| |)/ (| |)( )

W(dd)

W(dd) W W(dd)sgn log log3

(this equation should be compared to equation (2) above; see details in Online Methods).

We found that the wsi for the NMC increased from 2.06 ± 0.01 for the connection-based matrix to 3.30 ± 0.18 for the (signed) conductance-based matrix (P = 0.0180, two-tailed Wilcoxon signed-rank test, n = 7). This implies that synaptic physiology boosts the structurally embed-ded small-world topology of the NMC. The increase in value arises from larger ratios between the NMC weighted clustering-coefficient and that of the corresponding random network (Supplementary Table 3). The positive sign of wsi indicates that the majority of the strongest paths (among the shortest paths) are effectively excitatory (encouraging the initiation of a spike in the last neuron in the chain due to a spike in the first neuron).

DISCUSSIONWe presented a systematic network analysis of a dense, biologically constrained, computer-generated (in silico) NMC. This circuit, which contains 31,000 cells interconnected by 36 million synapses, is the most complete biologically constrained model of neocortical cellu-lar-level connectivity available today. Having such comprehensive wiring diagrams enabled us to identify key emergent network topolo-gies, some of which cannot be experimentally unraveled given the as-yet-absent dense biological maps at the above scale. We demon-strated that, despite the large heterogeneity in the numbers of E and

(3)(3)

I connections per cell, the E/I ratios are fairly constant for excitatory cells (Fig. 2). The circuit contains a rich-club core of interconnected hub neurons belonging to a surprisingly small number of cell-type sub-classes. Moreover, hubs with the largest spread of outputs (binary out-hubs) were mainly excitatory, whereas hubs with the strongest spread (weighted out-hubs) were inhibitory (Fig. 3 and Supplementary Fig. 4). The circuit also contains overrepresented transitive local triad motifs, which are typical for all excitatory cell-types, whereas inhibitory cell types exhibited a diverse motif profile (Fig. 5). At the global level, the structural connectivity demonstrated small-world topology with an average synaptic distance of about 2.5 between any two neurons; this topology was enhanced when the synaptic properties (strength and sign) were considered (Fig. 6). The connectivity backbone underpinning the NMC, as it emerged here, is summarized schemati-cally in Figure 7.

The structural topologies identified here were shown to play an important role in dynamics of complex networks in general38 and neuronal networks in particular. The tendency to have a fixed E/I ratio for neurons may promote a balanced state of the network39. Long-tail distributions and hubs strongly impact network-wide collective behaviors, such as oscillatory dynamics and synchronization25,40. Rich-club organization could facilitate efficient communication across the network with short synaptic delays and low noise27,28. Small-world topology was theoretically shown to enhance both responsiveness and

L1L2

/3L4

L5L6

Figure 7 A schematic diagram of the key topological features of the neocortical microcircuit. Hubs: among the 55 morphological cell types, in-hubs emerge from a small subset of types. All are excitatory and reside in L5 and L6 (three examples are depicted by the pyramidal cells (triangle shapes) with incoming arrows). Out-hubs emerge from another small subset of excitatory and inhibitory (round shapes) cells residing in L4–L6 (five examples are depicted by cells with outgoing connections). Hubs are defined as neurons receiving or making ~1,000 synaptic connections within the NMC; the average number of connections per cell is ~250. Rich club: hub neurons tend to connect to each other more than would be expected if connections were random (colored lines connecting hubs). Small-world wiring topology: the network demonstrates a combination of short average distances (~2.5 synapses) between any two neurons (gray path) and high local clustering of neurons (not shown). Motifs: E–E transitive triad motifs are overrepresented (colored triplets of triangle cells). I–I divergent and feedforward triplet motifs, as well as reciprocal connections, are overrepresented (colored triplet and pairs of round cells). Other NMC features, such as a relatively constant E/I conductance ratio for excitatory cells, are not shown in this schematic illustration.

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synchronization in neuronal networks41. Thus, the present study pro-vides guidance for focusing experiments toward anatomical structures of key functional roles. For example, targeted stimulation or silencing of candidate hub neurons (Fig. 3) is expected to profoundly influence the network dynamics.

We demonstrated that network-wide degree distributions are mostly explained by the total lengths of the dendrites and axons (Fig. 3). Using multiple models of random networks, we found that the short characteristic path-length in the NMC is merely a consequence of connections density (captured by Erdős–Rényi random networks; Supplementary Fig. 5), whereas the prevalence of reciprocal con-nections is primarily a consequence of the spatial embedding of the neurons in 3D space and their distance-dependent connectivity42 (captured by dd-random networks; Fig. 4). Triads and clustering coef-ficients were not fully captured by distance dependence, implying the need for a higher-order generative model; for example, consider-ing the polarity of neurons (axon versus dendrite). These assertions require further analysis.

We emphasize that emergent large scale properties analyzed here are the result of the NMC model and the way it was algorithmically recon-structed. The reconstruction is subjected to several limitations. For instance, the extent of axons might not have been fully captured from cortical slices; synaptic conductance among most types of neurons was inferred from a limited number of pair recordings from a subset of cell types; and there are probably connectivity rules, including structural plasticity rules, that were not taken into account when building the in silico circuit. Still, we believe that our findings reveal general net-work topologies of cortical circuits, given their corroboration by currently available empirical results, the large set of solid biological observations constraining the reconstruction and the extensive in vitro and in vivo experimental results, which the circuit reproduced without any parameter tuning18. In particular, several recent results from the first-ever electron-micrograph reconstruction of an 80,000-µm3 corti-cal volume7, such as the existence of ‘redundant’ dendritic synapses arising from the same presynaptic neuron, are compatible with our NMC model. We therefore view the NMC as an important reference for suggesting a set of large scale network predictions that can now be empirically tested, as well as for developing methods to concisely describe the extraordinary complexity of neuronal microcircuits.

We note that the circuit studied here is of young rats (P14), in which synaptic connectivity and intrinsic excitability have not matured43,44, implying that the connectivity found hereby may be subjected to ongo-ing developmental changes. Nevertheless, anatomical studies demon-strate that the morphological properties of dendrites have already largely matured to the adult level45,46. The network topology described here may be a transient manifestation reflecting the immaturity of the circuit. If indeed some topological features (for example, hubs) appear at P14 (as our model predicts) and then disappear at a mature age, then the functional importance of these features to development, learning and brain disorders3 would be uncovered by comparative studies. It is worth noting that several network properties that were found in the present study were also evident in more mature neocortices14 (motifs) as well as throughout development (hub neurons and rich clubs) and in late adolescence (hub regions) of different species at different scales28,47.

In addition, the unique complexities of the nervous system, particu-larly at the synaptic level, present network science with new challenges. For instance, communication between neurons goes beyond the sim-ple signal flow along single shortest path and involves integration of multiple—possibly longer—paths with different time delays. The physiological properties of neurons and their synapses should be taken into account to constrain network dynamics more faithfully48.

Diverse types of neurons might represent fundamentally different processes for network nodes (for example, excitatory versus inhibi-tory). The nodes themselves might be effectively transient (for exam-ple, refractory period) and the synaptic connections might undergo dynamic changes (for example, short-term plasticity). These domain-specific complexities necessitate extensions of current standard net-work measures (for example, the signed-and-weighted characteristic path length that we introduced in equation (3)). Such newly extended measures, although derived primarily to tackle neuroscience data, may also have broader applications to complex systems in diverse scientific disciplines (for example, signed social networks). We expect that such advanced network science49, together with forthcoming microconnectomics data and models, will play a central role in uncov-ering hidden organization principles of neuronal networks50, as well as in elucidating the much-needed link between anatomical structure and computational functions in the nervous system.

METhODSMethods, including statements of data availability and any associated accession codes and references, are available in the online version of the paper.

Note: Any Supplementary Information and Source Data files are available in the online version of the paper.

AcknowledgmenTSWe thank the members of the Segev Lab for helpful discussions related to this project. We also thank I. Nadav for the image processing. This work was supported by the Gatsby Charitable Foundation and the EPFL-Hebrew University Collaborative Grant, the EPFL support to the Laboratory of Neural Microcircuitry (LNMC), the ETH Domain for the Blue Brain Project (BBP), the Human Brain Project through the European Union Seventh Framework Program (FP7/2007-2013) under grant agreement no. 604102 (HBP) and from the European Union H2020 FET program through grant agreement no. 720270 (HBP SGA1), the Brain Science grant of the Sachs Family and by the ISF Centers of Excellence grants 1789/11 and 2180/15.

AUTHoR conTRIBUTIonSE.G. and I.S. conceived the study and wrote the manuscript. E.G. carried out the analysis. M.L. helped in developing the functional small-world analysis. A.G. and M.L. participated in discussions. S.R., M.W.R., E.M. and H.M. developed the in silico microcircuit and provided the respective data.

comPeTIng FInAncIAl InTeReSTSThe authors declare no competing financial interests.

Reprints and permissions information is available online at http://www.nature.com/reprints/index.html. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

1. Sporns, O., Tononi, G. & Kötter, R. The human connectome: a structural description of the human brain. PLoS Comput. Biol. 1, e42 (2005).

2. Seung, H.S. Reading the book of memory: sparse sampling versus dense mapping of connectomes. Neuron 62, 17–29 (2009).

3. Fornito, A., Zalesky, A. & Breakspear, M. The connectomics of brain disorders. Nat. Rev. Neurosci. 16, 159–172 (2015).

4. Varshney, L.R., Chen, B.L., Paniagua, E., Hall, D.H. & Chklovskii, D.B. Structural properties of the Caenorhabditis elegans neuronal network. PLoS Comput. Biol. 7, e1001066 (2011).

5. Briggman, K.L., Helmstaedter, M. & Denk, W. Wiring specificity in the direction-selectivity circuit of the retina. Nature 471, 183–188 (2011).

6. Chklovskii, D.B., Vitaladevuni, S. & Scheffer, L.K. Semi-automated reconstruction of neural circuits using electron microscopy. Curr. Opin. Neurobiol. 20, 667–675 (2010).

7. Kasthuri, N. et al. Saturated reconstruction of a volume of neocortex. Cell 162, 648–661 (2015).

8. Helmstaedter, M. et al. Connectomic reconstruction of the inner plexiform layer in the mouse retina. Nature 500, 168–174 (2013).

9. White, J.G., Southgate, E., Thomson, J.N. & Brenner, S. The structure of the nervous system of the nematode Caenorhabditis elegans. Phil. Trans. R. Soc. Lond. B 314, 1–340 (1986).

10. Thomson, A.M. & Lamy, C. Functional maps of neocortical local circuitry. Front. Neurosci. 1, 19–42 (2007).

© 2

017

Nat

ure

Am

eric

a, In

c., p

art

of

Sp

rin

ger

Nat

ure

. All

rig

hts

res

erve

d.

a r t I C l e S

nature neurOSCIenCe VOLUME 20 | NUMBER 7 | JULY 2017 1013

11. Barbour, D.L. & Callaway, E.M. Excitatory local connections of superficial neurons in rat auditory cortex. J. Neurosci. 28, 11174–11185 (2008).

12. Lefort, S., Tomm, C., Floyd Sarria, J.C. & Petersen, C.C. The excitatory neuronal network of the C2 barrel column in mouse primary somatosensory cortex. Neuron 61, 301–316 (2009).

13. Pfeffer, C.K., Xue, M., He, M., Huang, Z.J. & Scanziani, M. Inhibition of inhibition in visual cortex: the logic of connections between molecularly distinct interneurons. Nat. Neurosci. 16, 1068–1076 (2013).

14. Song, S., Sjöström, P.J., Reigl, M., Nelson, S. & Chklovskii, D.B. Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS Biol. 3, e68 (2005).

15. Silberberg, G. & Markram, H. Disynaptic inhibition between neocortical pyramidal cells mediated by Martinotti cells. Neuron 53, 735–746 (2007).

16. Perin, R., Berger, T.K. & Markram, H. A synaptic organizing principle for cortical neuronal groups. Proc. Natl. Acad. Sci. USA 108, 5419–5424 (2011).

17. Egger, R., Dercksen, V.J., Udvary, D., Hege, H.-C. & Oberlaender, M. Generation of dense statistical connectomes from sparse morphological data. Front. Neuroanat. 8, 129 (2014).

18. Markram, H. et al. Reconstruction and simulation of neocortical microcircuitry. Cell 163, 456–492 (2015).

19. Stepanyants, A., Martinez, L.M., Ferecskó, A.S. & Kisvárday, Z.F. The fractions of short- and long-range connections in the visual cortex. Proc. Natl. Acad. Sci. USA 106, 3555–3560 (2009).

20. Xue, M., Atallah, B.V. & Scanziani, M. Equalizing excitation-inhibition ratios across visual cortical neurons. Nature 511, 596–600 (2014).

21. Barabasi, A.L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).

22. Amaral, L.A., Scala, A., Barthelemy, M. & Stanley, H.E. Classes of small-world networks. Proc. Natl. Acad. Sci. USA 97, 11149–11152 (2000).

23. Clauset, A., Shalizi, C.R. & Newman, M.E.J. Power-law distributions in empirical data. SIAM Rev. Soc. Ind. Appl. Math. 51, 661–703 (2009).

24. Colizza, V., Flammini, A., Serrano, M.A. & Vespignani, A. Detecting rich-club ordering in complex networks. Nat. Phys. 2, 110–115 (2006).

25. Bonifazi, P. et al. GABAergic hub neurons orchestrate synchrony in developing hippocampal networks. Science 326, 1419–1424 (2009).

26. van den Heuvel, M.P. & Sporns, O. Rich-club organization of the human connectome. J. Neurosci. 31, 15775–15786 (2011).

27. Nigam, S. et al. Rich-club organization in effective connectivity among cortical neurons. J. Neurosci. 36, 670–684 (2016).

28. Towlson, E.K., Vértes, P.E., Ahnert, S.E., Schafer, W.R. & Bullmore, E.T. The rich club of the C. elegans neuronal connectome. J. Neurosci. 33, 6380–6387 (2013).

29. van den Heuvel, M.P., Bullmore, E.T. & Sporns, O. Comparative connectomics. Trends Cogn. Sci. 20, 345–361 (2016).

30. Milo, R. et al. Network motifs: simple building blocks of complex networks. Science 298, 824–827 (2002).

31. Rieubland, S., Roth, A. & Häusser, M. Structured connectivity in cerebellar inhibitory networks. Neuron 81, 913–929 (2014).

32. Watts, D.J. & Strogatz, S.H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

33. Rubinov, M. & Sporns, O. Complex network measures of brain connectivity: uses and interpretations. Neuroimage 52, 1059–1069 (2010).

34. Fagiolo, G. Clustering in complex directed networks. Phys. Rev. E 76, 026107 (2007).

35. Bettencourt, L.M., Stephens, G.J., Ham, M.I. & Gross, G.W. Functional structure of cortical neuronal networks grown in vitro. Phys. Rev. E 75, 021915 (2007).

36. Yu, S., Huang, D., Singer, W. & Nikolic, D. A small world of neuronal synchrony. Cereb. Cortex 18, 2891–2901 (2008).

37. London, M., Roth, A., Beeren, L., Häusser, M. & Latham, P.E. Sensitivity to perturbations in vivo implies high noise and suggests rate coding in cortex. Nature 466, 123–127 (2010).

38. Newman, M.E.J. The structure and function of complex networks. SIAM Rev. Soc. Ind. Appl. Math. 45, 167–256 (2003).

39. Landau, I.D., Egger, R., Dercksen, V.J., Oberlaender, M. & Sompolinsky, H. The impact of structural heterogeneity on excitation-inhibition balance in cortical networks. Neuron 92, 1106–1121 (2016).

40. Roxin, A. The role of degree distribution in shaping the dynamics in networks of sparsely connected spiking neurons. Front. Comput. Neurosci. 5, 8 (2011).

41. Lago-Fernández, L.F., Huerta, R., Corbacho, F. & Sigüenza, J.A. Fast response and temporal coherent oscillations in small-world networks. Phys. Rev. Lett. 84, 2758–2761 (2000).

42. Barthélemy, M. Spatial networks. Phys. Rep. 499, 1–101 (2011).43. Atkinson, S.E. & Williams, S.R. Postnatal development of dendritic synaptic

integration in rat neocortical pyramidal neurons. J. Neurophysiol. 102, 735–751 (2009).

44. Lorenzon, N.M. & Foehring, R.C. The ontogeny of repetitive firing and its modulation by norepinephrine in rat neocortical neurons. Brain Res. Dev. Brain Res. 73, 213–223 (1993).

45. Larkman, A.U. Dendritic morphology of pyramidal neurones of the visual cortex of the rat: I. Branching patterns. J. Comp. Neurol. 306, 307–319 (1991).

46. Romand, S., Wang, Y., Toledo-Rodriguez, M. & Markram, H. Morphological development of thick-tufted layer v pyramidal cells in the rat somatosensory cortex. Front. Neuroanat. 5, 5 (2011).

47. Baker, S.T. et al. Developmental changes in brain network hub connectivity in late adolescence. J. Neurosci. 35, 9078–9087 (2015).

48. Kopelowitz, E., Abeles, M., Cohen, D. & Kanter, I. Sensitivity of global network dynamics to local parameters versus motif structure in a cortexlike neuronal model. Phys. Rev. E 85, 051902 (2012).

49. Bassett, D.S. & Sporns, O. Network neuroscience. Nat. Neurosci. 20, 353–364 (2017).

50. Schröter, M., Paulsen, O. & Bullmore, E.T. Micro-connectomics: probing the organization of neuronal networks at the cellular scale. Nat. Rev. Neurosci. 18, 131–146 (2017).

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ONLINE METhODSStatistics. The statistical significance of the differences between structural prop-erties of the NMC and random connectivity models was assessed using the Monte Carlo method. Therefore, there were no assumptions made about the data distribu-tion. For each test statistic, Monte Carlo P values were calculated as (r + 1)/(N + 1), where N is the number of generated random networks and r is the number of random networks that produced a test statistic with deviation (from the random ensemble mean) greater than or equal to that calculated for the NMC (a two-tailed test). When the statistical analysis involved measurements from the seven NMC instances, the mean value was used as the statistic (and sets of 7 correspond-ing random networks were generated N times). We used N = 100 randomizations for evaluating the whole NMC network and the E/I subnetworks and N = 1,000 for evaluating layer-specific and cell-type-specific subnetworks.

The linear mixed-effects analysis of the relationship between in-degrees/out-degrees and dendritic/axonal lengths was performed in Matlab. As a fixed effect, the model included the dendritic/axonal length. As random effects, the model included intercepts for the length, as well as by-cell-type random slope for the effect of length. P values were obtained by likelihood ratio tests of the full model with the by-cell-type random effect in against the model without this effect (χ2

3 = 30,539 for the in-degree/dendrite model and χ2

3 = 44,815 for the out-degree/axon model, P < 0.001 for both; note that this χ2 distribution assumption is based on the convergence of the log-likelihood ratio statistic to this distribution). To reduce heteroscedasticity, we considered a log-transform of the dependent variable and also took into account the effect of cell depth in the volume (y coordinate). Namely, we extended the model with an additional fixed effect for the depth as well as the by-cell-type random slope of the depth. These analyses demonstrated even higher coefficients of determination (R2). Assumptions of the two-tailed Wilcoxon signed-rank test were verified by quantifying the symmetry of a set around its median (n = 7, W = 0). For the Pearson correlation test, data distribu-tion was assumed to be normal, but this was not formally tested.

The seven NMC reconstructions that we analyzed here are referred to as the BioM microcircuits in Markram et al.18. The seven instances of the NMC were generated so that the central (hexagonal) circuit was tiled with six surrounding additional circuits to overcome boundary effects. In principle, it is possible to reproduce additional instances, but the effect is already evident for this number (they produce reasonably similar statistics). No further statistical methods were used to predetermine sample sizes. Randomization was not applicable in this study. Data collection and analysis were not performed blind to the conditions of the experiments. The reconstruction algorithm is fully described in ref. 51. Data is available online at the neocortical microcircuit collaboration portal52 (http://bbpteam.epfl.ch/nmc-portal).

Significantly asymmetric e/I organization. To evaluate the significance of the excitatory–inhibitory asymmetric organization (Fig. 1a), it was compared to that expected from a random network with number of nodes (nNMC = 31,346) and sparsity of connections (pNMC = 0.008) similar to the NMC. The Erdős–Rényi (ER) model for the generation of random networks53 assumes that the probability of forming a connection between any pair of nodes is same for all pairs and that connections are formed independently. Thus, the model requires matching the nNMC and pNMC parameters alone. In the ER random case, the edges within and between two subpopulations of similar sizes, nE and nI, are expected to be propor-tional to the number of available pairs within and between these subpopulations. Specifically, when 85% of the nodes are excitatory (nE = 26,568 ± 3) and 15% are inhibitory (nI = 4,778 ± 3) as in the NMC, the random network is expected to have approximately 72% ((nE/n)2 = 0.72) of its connections within the excitatory population. The actual percentage of E-to-E connections in the NMC was 86%.

Note that the number of E-to-E connections in the random network con-sisting of nE nodes with connection probability of pNMC follows a binomial distribution, K B n n pEE

ERE E NMC

( ) ~ ( , )( )− 1 . With pNMC = 0.008 and nE as above, the mean number of edges is equal to E K n n p[ ] , ,( )( )

EEER

E E NMC= − =1 5 646 656 and the s.d. is s[ ] ( )( ) ( )( ) /K n n p pEE

ERE E NMC NMC= − −1 1 1 2 = 2,367. Thus, the

number of excitatory edges in the NMC, kEE = 6,730,851 ± 86,461, was approxi-mately 460 s.d. above the mean, which implies that the probability of achieving this in the random model was essentially zero. This could also be computed by approximating the number of edges drawn from the normal distribution K NEE

ER( ) ~ ( , , , , )m s= =5 646 656 2 3672 2 and using its cumulative distribution func-tion to compute the probability P K k( )( )

EEER

EE≥ .

The computed probability of finding this excitatory density is an upper limit on the probability of finding the overall E/I connections organization as found in the NMC. Thus, the significant asymmetrical organization of excitation– inhibition connectivity in the NMC cannot be accounted for merely by the number of E and I neurons.

degree distributions do not follow power law. Detection and accurate fitting of power-law distribution, p(d) ∝ d−γ, to empirical data requires careful analysis, given the noisiness of values sampled from its long tail. In particular, it requires an estimate of potentially infinite s.d. (if the exponent γ < 3) from a finite sample set. Using a recent approach, based on maximum-likelihood fitting of the exponent and goodness-of-fit based on the Kolmogorov–Smirnov (KS) statistic23,54, the supposition that the in-degree or out-degree distributions in the NMC obeyed a power-law was ruled out (neither their tails alone).

eR random networks with reciprocity match. To generate random networks that matched both total number of connections and the number of reciprocal connections, the ER random model was generalized as follows. Given the required sparsity, p, and reciprocity, r, we were interested in the ensemble of matrices (Aij) constrained by

E A n n pi j

ij≠∑

= −( )1

and

E A A n n ri j

ij ji>∑

= −0 5 1. ( )

where the expectation E[·] is over the whole ensemble of n × n binary matrices. This was done by defining the connectivity between all pairs of neurons (i,j) as independent random variables, whereby the probability for reciprocal connection is r, that of a unidirectional connection from i to j or from j to i is each (p − r), and the probability for no connection is z = 1 − 2p + r.

degree distributions when extrinsic connections are taken into account. We verified that the considerable variability in the in- and out-degrees was not merely a boundary effect due to cutting neuronal processes at the edge of the NMC. The results indeed showed a negative correlation between a neu-ron’s degrees and its horizontal distance from the NMC central vertical axis (Supplementary Fig. 3a,b). To rule out this effect, we computed the total in- and out-degrees for neurons in the centrally located NMC (which is surrounded by six other NMCs), taking into account the extrinsic connections arising from the surrounding NMCs. This substantially reduced the boundary artifacts (Supplementary Fig. 3c,d); however, the long-tail distributions and the hub neurons remained (Supplementary Fig. 4). In this case, the newly identified hub neurons were not necessarily the same neurons as those that were previously observed (about a 26% overlap). Specifically, the dominant cell types compos-ing the hubs remained the same and less-dominant ones disappeared (compare Fig. 3 to Supplementary Fig. 4).

The statistical significance of nmc hub neurons. To evaluate the significance of NMC hubs, they were compared to the highest connected nodes expected from various random networks. The simplest comparable random networks that match the size (nNMC = 31,346) and the sparsity (pNMC = 0.008) of the NMC network but otherwise maximize the connectivity randomness are the ER random networks. The degrees expected from matching ER random networks followed a Poisson distribution, which have a sharp peak around the mean degree λ = 249.61 ± 0.38 (Fig. 3a,b and see below). Thus, it did not reproduce the observed long tail and, therefore, the existence of hubs with degrees of ~1,000 was highly significant (the existence of any cell with > 500 connections was significant, P < 0.001, Monte Carlo with N = 1,000).

While the use of ER random networks for comparison is very attractive because of its simplicity and its widespread use in general network-theory studies, the analysis above has shown that sparsity alone is a very poor statistic for captur-ing the NMC hubs. Thus, two more highly tuned, biologically driven random networks were considered.

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The neurons composing the NMC belong to 55 different cell types. This heterogeneity could have given rise to the large variance in degrees observed among the NMC neurons. To account for this, random networks composed of 55 subpopulations with matching sizes and sparsities between and within them (552 values; ER-mixture networks) were used. This mixture model predicted the existence of neurons with degrees of 550 connections (Fig. 3a,b). However, these were still far from the magnitude of hubs that were observed in the NMC (in the ER-mixture, the probability of obtaining a neuron with more than 650 connections is less than 10−4).

Both the standard ER and the ER-mixture random models used above essen-tially assume that neurons are dimensionless points. However, the ability of two neurons to connect is physically constrained by their geometric locations and by the overlap between their axonal and dendritic arbors. This was reflected in the NMC network, whereby the connection probability depends on the interso-matic distance between the two neurons (Supplementary Fig. 2). To assess the impact of the observed distance-dependence on the emergence of hubs, we used random networks (dd-random networks) that preserve the original positions of all neurons and the particular distance dependence computed for the NMC (see below). The results showed that these dd-random networks exhibited a sharply decaying Poisson degree-distribution similar to that in ER random networks (not shown). This implies that the presence of observed hubs was highly significant (P < 0.001, Monte Carlo with N = 1,000), even when the distance-dependence in the NMC was taken into account.

Poisson degree-distributions of eR random networks. Given an ER network with n nodes and edge probability p, the degree (in-degree or out-degree) of a given node ν follows a binomial distribution B(n − 1,p):

Pr( ( ) ) ( )deg v knk p pk n k= =−

− − −11 1

In the limit of large n, it converges to a Poisson distribution with parameter λ = (n − 1)p:

Pr( ( ) )!

deg v k ek

k= = −l l

Thus, the degree distribution of the complete ER random network is a Poisson distribution.

Random networks preserving the nmc distance-dependence. The first step was to compute the probability for a connection between pairs of neurons as a function of their intersomatic distance. This probability, pNMC(x), was com-puted as the ratio of the number of observed connections at a given distance (Supplementary Fig. 2) to the number of overall existing pairs at this distance. Fifty bins were used to describe the distance domain (0–2,090 µm).

Next, we generated random networks with a distance-dependent connectiv-ity profile that matched the NMC profile (dd-random networks). The original coordinates of all neurons were preserved, and the connections were defined as independent binomial random variables Xij, with a success probability pij = pNMC(dij) that corresponded to their intersomatic distance dij.

To generate distance-dependent random networks corresponding to weighted networks, the connections of the original network were divided into bins as above, and the weights were reshuffled with each bin (wdd-random networks).

Rich-club organization. To verify that the high interconnectivity between the hubs was not merely a consequence of their high degree, it was compared to that expected from random networks (N = 1,000) with matching degree sequence (using the Configuration model55, we generated random networks that approxi-mately preserve the original in- and out-degrees for each node). Thus, these ran-dom networks consisted of hubs with similar degrees, but their interconnectivity was at chance level. It was found that hub interconnectivity in the NMC is 1.44 times higher than observed in these corresponding random networks.

To systematically quantify a rich-club organization24 we first defined the rich-club coefficient Ø(d) as the ratio between the number of connection among all neurons whose total degree (in-degree + out-degree) is ≥ d and the maximum number of possible connections among them. Then we measured it in the NMC, ØNMC(d), and compared it to that expected from the degree-matched random networks, Øran(d). We found that high-degree neurons (hubs) in NMC are

significantly over-connected compared to the random case (Fig. 3i). We fur-ther confirmed this result by using the Brain Connectivity Toolbox33 to generate more conservative random networks with degree-sequence exactly as prescribed (Supplementary Fig. 6).

normalized vector of motifs’ z-scores. Motifs in large networks tend to display higher z-scores than the motifs in small networks56. Thus, to compare networks with different sizes we employed the significance profile (SP), which is defined by normalizing the z-scores vector of each network to 1

SPz

zi

i

i

=( )∑ 2 1 2 4/ ( )

Thus, it depicts the relative rather than the absolute significance of the triads (Fig. 5b–d).

nmc connectedness. The general connectedness of the NMC was character-ized by identifying connected components of the network. A strongly connected component is a group of nodes in which any node is reachable from any other node through a directed path (a series of nodes and directed edges). Intuitively, a strongly connected component reflects a group of recurrently interlinked neurons that could give rise to an anatomical module with functional specialization.

It was found that the NMC contains one strongly connected giant component of 31,329 ± 5 (99.95% ± 0.01%) neurons. The remaining 17 ± 5 neurons were either presynaptic (9–20) or postsynaptic (1–5) to the giant-component neurons. Thus, the NMC was not fully connected but not very far from it. The tiny fractions of the few ‘outlier’ cells found above led us to further test whether their unique connectivity was an artifact of NMC boundary. By inspecting the 3D coordinates of these cells, we found that they were all positioned on the edges of the NMC. Therefore, the cells did not convey a special biological meaning, and the networks can be regarded as fully connected.

The size of the NMC’s giant component was compared to the size expected from ER random networks. A matching ER random network with nNMC = 31,346 nodes and a sparsity of pNMC = 0.008 is also expected to be fully connected, with a single giant, strongly connected component containing all of the nodes. Thus, we refrained from considering the connectedness of the NMC as a high-order connectivity feature.

Small-world properties. The first property of the small-world analysis is based on the length of the shortest path lij between pairs of nodes in the network. A path length between two nodes in the network is expressed as the number of con-nections along that path. To generalize this property for the entire network, the characteristic path length (l) of a network was used, which is the mean shortest path length averaged over all pairs of neurons

ln n

li j i j

ij=− ≠

∑11

5( )

( ), :

Clearly, this measure is well-defined in connected networks where any node is reachable from any other node. The NMC network contained a single giant component of 31,329 ± 5 neurons that were mutually reachable (99.95% ± 0.01% of all neurons; see above).

The second property of the small-world analysis is captured by the tendency of nodes to cluster together. The overrepresentation of specific dense-triad motifs, as demonstrated above, is one expression of the clustering tendency in the NMC. However, here the focus is on local clustering of individual nodes, which measures the level at which the neighbors of a node are interconnected among themselves. Let the binary (unweighted) adjacency matrix of a directed network be denoted by A; then the local clustering coefficient ci of a node i is defined as

cA A

d d Ai

Tii

i i ii=

+− −

( )

( )( )

( )

3

22 1 26tot tot

where ditot depicts the total degree (in-degree + out-degree) of node i. Essentially,

in directed networks, this definition reflects the ratio between (the number of triangles among a node and its neighbors) and (the number of all possible tri-angles that could have been formed)34. The value of ci ranges from 0 (none of

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the neighbors are connected to each other) to 1 (all neighbors are mutually con-nected). The network-wide clustering coefficient (c) is computed by averaging over all local clustering coefficients

cn

ci

i= ∑1 7( )

Giant components, path lengths and clustering coefficients were computed using the Brain Connectivity Toolbox33 in Matlab.

weighted small-world properties. To quantify the strength of a synaptic chain between two neurons (a path), we relied on the experimental observation that the probability for an extra or missed spike in a cell scales linearly with the total charge arriving at the soma37. In particular, an extra positive charge increases the probability for an extra spike about 3× more than a negative charge of the same magnitude increases the probability for a canceling a spike. Thus, the overall probability for a spike in a neuron can be described as

Pr x s q k q( ) ( ( )) ( )= = + + ⋅1 1 2 8q

where x is a binary random variable indicating a spike event, s denotes the base spontaneous spiking probability, q denotes the amount of arriving charge (posi-tive or negative values), θ(·) is the Heaviside step function and k is the proportion ratio (probability/charge).

Thus, we first note that, for a pair of neurons connected by a synapse, the effect of a spike in the presynaptic neuron on the spike probability of the postsynaptic neuron is expressed by

Pr x x s w k w( | ) ( ) ( )( )post pre= = = + + ⋅1 1 1 2 9q

where xpre and xpost are binary random variables indicating a spike in the pre- and postsynaptic cells, respectively, s denotes the base spontaneous spiking probability of the postsynaptic neuron, w denotes the strength of the synaptic connection, and k is the proportion coefficient. We assume that w is a measure approximat-ing the overall charge flowing through the postsynaptic current on activation of the synapse. By incorporating the excitatory/inhibitory scaling factor into the synaptic weight, w = (1 + 2θ(w))·w, equation (9) can be simplified to

Pr x x s k w( | ) ( )post pre= = = + ⋅1 1 10

We now use equation (10) to quantify the effect between two neurons con-nected through a path consisting of multiple neurons. Let two neurons, i and j, be connected by a path through neurons {h1, h2, …, hm} via a sequence of synaptic weights, { , , .., }w w wih h h h jm1 1 2 . For each neuron hl, let shl denotes its spontaneous spiking probability. Then, given that the first neuron had spiked, we will derive the effect on the probability of the last neuron to spike. Here we use a three-neuron example, but the generalization to any arbitrary case is straightforward. In a three-neuron path, the probability for a spike in the third neuron is given by

Pr x s kw Pr x s kw s kw Pr x

s kw s

( ) ( ) ( )( )3 3 23 2 3 23 2 12 1

3 23

1 1 1= = + = = + + =

= + 222

12 23 1 11+ k w w s ( )

Given that the first neuron had spiked, the probability for a spike at last neuron is increased to

Pr x x s kw s k w w( ) ( )3 1 3 23 22

12 231 1 1 12= = = + +|

Thus, the difference in probability, due to this chain, is given by

∆p Pr x x Pr x k w w s k w w13 3 1 32

12 23 12

12 231 1 1 1 13= = = − = = − ∝( ) ( ) ( ) ( )|

Namely, the strongest path between neurons i and j, among all paths with length (m + 1), { , , .., }w w wih h h h jm1 1 2 , is the one that maximizes the absolute value of the product w w wih h h h jm1 1 2 ⋅ ⋅ ⋅ . We define the optimal path strength between

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(8)(8)

(9)(9)

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neurons i and j, sij, to be the strength of the strongest path, among all paths with the minimal distance between them (lij). The shortest weighted distance between neurons i and j, lij

W , is defined as the inverse of sij. By assigning negative weights for inhibitory synapses, lij

W also indicates whether this path is effectively excita-tory (positive) or inhibitory (negative). Thus, the statistic for the mean shortest path length (weighted and signed) is

ln n

li j i j

ijW W=

− ≠∑1

114

( )( )

, :

We note that in a homogenous network where all weights are identical, wij = w and |w| < 1, for any two nodes i and j, the measure log | |W( )lij is proportional to the binary shortest path-length lij between them. This can be easily shown:

log(| |) log( / | |) log(| / | ) log(| / |)l w w l w lijl l

ij ijij ijW = = = ⋅ ∝1 1 1 (( )15

To search for the optimal weighted paths between all pairs of neurons, we used Dijkstra’s algorithm, which was modified to evaluate all shortest paths between two neurons and select the strongest path among them. We applied it to networks whose weighted connectivity matrix is derived from the contacts or conductance matrices in two steps. First, to ensure that all weights were < 1, they were nor-malized by (wmax + 1), where wmax is the largest weight of all NMC matrices. Second, weights corresponding to inhibitory synapses were given a negative sign and were scaled by 1/3. By doing so, we assume that the number of synapses and their strengths (measured here by total peak conductance of each synapse) approximate the EPSC/IPSC of the corresponding synapses.

For the weighted clustering coefficient, we used a recent generalization that takes into account directed and weighted connectivity34. The standard clustering coefficient defined in equation (6), which is based on the binary adjacency matrix A, was extended to the weighted adjacency matrix, W, as follows:

cW W

d d AiW

Tii

i i ii=

+− −

( ( ) )

( )( )

( )

[ / ] [ / ]1 3 1 3 3

22 1 216tot tot

where W[1/3] denotes an element-wise cubic root of W entries, and ditot depicts

the total degree (in-degree + out-degree) of node i (as in the standard clustering- coefficient). This measure is also applicable for signed connectivity. Its values vary in [−1,1] and it reflects the balance between the strengths of negative and positive triangles in the node’s neighborhood.

The new definition for the weighed small-world index (wsi) is depicted in equation (3). The logarithms in the denominator keep the new measure at the same scale of the standard path-length measure, and the sign operator retains the E/I effect (provided that |lW|,|lW(dd)| > 1, as was observed in the present networks).

Weighted path lengths were computed in Python. The code, accompanied by descriptive examples, is available in Supplementary Software. A Supplementary methods checklist is available.

data and code availability. The experimental data, digital reconstructions and analysis code that support the findings of this study are available at the Neocortical Microcircuit Collaboration Portal (NMC Portal; https://bbp.epfl.ch/nmc-portal).

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51. Reimann, M.W., King, J.G., Muller, E.B., Ramaswamy, S. & Markram, H. An algorithm to predict the connectome of neural microcircuits. Front. Comput. Neurosci. 9, 120 (2015).

52. Ramaswamy, S. et al. The neocortical microcircuit collaboration portal: a resource for rat somatosensory cortex. Front. Neural Circuits 9, 44 (2015).

53. Erdos, P. & Rényi, A. On random graphs I. Publicationes Mathematicae (Debrecen) 6, 290–297 (1959).

54. Alstott, J., Bullmore, E. & Plenz, D. Powerlaw: a Python package for analysis of heavy-tailed distributions. PLoS One 9, e85777 (2014).

55. Newman, M.E., Strogatz, S.H. & Watts, D.J. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118 (2001).

56. Milo, R. et al. Superfamilies of evolved and designed networks. Science 303, 1538–1542 (2004).

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NATURE NEUROSCIENCE

CO R R I G E N DA

Corrigendum: Rich cell-type-specific network topology in neocortical microcircuitryEyal Gal, Michael London, Amir Globerson, Srikanth Ramaswamy, Michael W Reimann, Eilif Muller, Henry Markram & Idan SegevNat. Neurosci.; doi:10.1038/nn.4576; corrected online 14 June 2017

In the version of this article initially published online, the number of neurons for layer 6 was given in Figure 1a as 1,2715 instead of 12,715, the single asterisks in the left panels of Figure 4b and Figure 4c should have been double asterisks, and “left” and “right” were reversed in the legend to Figure 4c. The errors have been corrected in the print, PDF and HTML versions of this article.

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