## Structured Abstract

### Background

The cerebral cortex is divisible into many individual areas, each exhibiting distinct connectivity profiles, architecture, and physiological characteristics. Interactions among cortical areas underlie higher sensory, motor, and cognitive functions. Graph theory provides an important framework for understanding network properties of the interareal weighted and directed connectivity matrix reported in recent studies.

### Advances

We derive an exponential distance rule that predicts many binary and weighted features of the cortical network, including efficiency of information transfer, the high specificity of long-distance compared to short-distance connections, wire length minimization, and the existence of a highly interconnected cortical core. We propose a bow-tie representation of the cortex, which combines these features with hierarchical processing.

### Outlook

The exponential distance rule has important implications for understanding scaling properties of the cortex and developing future large-scale dynamic models of the cortex.

## Abstract

Small-world networks provide an appealing description of cortical architecture owing to their capacity for integration and segregation combined with an economy of connectivity. Previous reports of low-density interareal graphs and apparent small-world properties are challenged by data that reveal high-density cortical graphs in which economy of connections is achieved by weight heterogeneity and distance-weight correlations. These properties define a model that predicts many binary and weighted features of the cortical network including a core-periphery, a typical feature of self-organizing information processing systems. Feedback and feedforward pathways between areas exhibit a dual counterstream organization, and their integration into local circuits constrains cortical computation. Here, we propose a bow-tie representation of interareal architecture derived from the hierarchical laminar weights of pathways between the high-efficiency dense core and periphery.

Because the concepts of localization of function and parcellation into cortical areas are closely intertwined, elucidating the global pattern of areal interactions is central to understanding higher brain functions (*1*–*5*). Cerebral cortex in the macaque monkey is subdivided into a mosaic of ~100 cortical areas, each displaying characteristic features, including cytoarchitecture (*6*). Each area has a characteristic connectivity profile thought to contribute to determining its functional properties (*1*, *7*, *8*). Here, we review how interareal connectivity at the single-cell level (*9*), revealed by quantitative anatomical tract tracing, is relevant to our understanding of large-scale cortical networks and their hierarchical organization (*8*, *10*–*13*).

The circuitry of cerebral cortex is dominated by local (within-area) connections, and interareal connections constitute only about 20% of total cortical connectivity. Hence, the dozens of long-distance projections to areas beyond the immediate neighboring areas account for ~ 5% (*10*). Local networks conform in many ways to a canonical microcircuit that spans all cortical layers (*14*, *15*) and includes recurrent excitation presumed to shape and amplify the sparse input from subcortical and distant cortical sources (*16*).

Felleman and Van Essen (FVE) showed that interareal connectivity obeys hierarchical constraints rooted in the strong anatomical regularities of feedforward and feedback pathways (*17*). In this way, multiple distributed cortical hierarchies form a large-scale model of the cortex (*17*) that reflects the laminar integration of interareal connectivity into local circuits (*18*) and is relevant to sensory (*17*, *19*, *20*), motor (*21*, *22*), and cognitive (*23*–*27*) systems. The structural features of interareal interactions may provide important insights into the observed dynamics of large-scale interareal networks controlling information flow through the cortex (*1*, *28*, *29*).

## Density and Small-World Architectures

Graph theory provides a powerful framework for investigating complex networks such as those found in the brain. Many insights into the functional processes supported by such networks have been gleaned from analysis at the binary level (i.e., connections present or absent; see Glossary for definitions) (*30*). One important class of models that has received much attention is that of small-world (SW) networks, distinguished by high clustering coupled with a short average path length (also called characteristic path length) across the graph (*31*). The relevance of the SW property to understanding the cortex comes from its proposed capacity to optimize essential cortical features, including functional integration and segregation (*32*, *33*).

Several studies based on collations of published anatomical tract tracing data (*34*–*38*) concluded that the cortical interareal network conforms to the SW network model of Watts and Strogatz (*31*). According to this hypothesis, efficient signal propagation through cortical circuits benefits from a modest number of shortcuts connecting different communities across the cortical graph.

The SW hypothesis for interareal connectivity has been challenged by recent studies that used a consistent and optimized methodology to establish a quantitative data-base of macaque interareal connectivity (*8*, *11*). We segmented the cortical sheet into 91 cortical areas and quantitatively analyzed the incoming connections to a subset of 29 areas chosen to represent five major regions of the cortex. By using similar procedures and identical area definitions across brains, this effort set out to overcome the limitations inherent in collated datasets that combine results across many anatomical studies (*39*). These limitations arise from the diversity of procedures used among different anatomical studies, including cross-study differences in parcellation schemes, extent of cortex examined, tracer sensitivity, criteria for accepting the presence or absence of a connection, and the spatial resolution of the analysis.

Injections of retrograde tracers in the 29 areas revealed 36% more connections than previously reported (*8*, *10*). These so-called new-found projections (NFPs) were presumably missed by earlier studies for several reasons, including that they link widely separated areas (long-distance connections) and tend to be sparse, therefore requiring high resolution obtained by optimized sampling frequency for their detection (*40*, *41*). Notably, repeat injections in selected areas and statistical modeling of the variability of projection magnitude demonstrated well-defined weighted connectivity profiles for each area and indicated high consistency for pathways of sufficient strength (*8*, *10*). Inclusion of the NFPs considerably increases the cortical network density (i.e., the number of binary connections that exist relative to the total number of connections possible) (*42*). The density of the full interareal network (FIN), represented by the graph (or matrix) *G** _{91x91}*, remains unknown. However, a dominating set analysis of the currently known interareal network represented as the

*G**subgraph of the FIN predicts that the FIN is itself a densely connected network (*

_{29x91}*8*).

The *G** _{29x29}* interareal subgraph, formed among the injected (target) nodes, is edge-complete (see Glossary), and it has a link density of 66% (i.e., two-thirds of connections that can exist do exist) (

*8*). The

*G**is denser than any subgraph used in previous studies of the cortical network. Figure 1A displays the differences in density and average path length of various published subgraphs that have been used to investigate the large-scale properties of the cortex. FVE analyzed several hundred publications and reported on 32 visual areas and 305 pathways, for a graph density of 32% (*

_{29x29}*17*). In their meta-analysis, many pathways were identified as untested (i.e., the subgraph was edge-incomplete); they predicted a density of 45% if the unknown connections were to be tested. Three subsequent studies added collated data to the FVE data set, thus generating subgraphs of 47 areas

*G**(*

_{47x47}*34*), 71 areas

*G**(*

_{71x71}*43*), and 179 areas

*G**(*

_{179x179}*44*), respectively. The resulting subgraphs were also edge-incomplete, but untested connections were assigned a “nonconnection” status, leading to low density estimates ranging from 5 to 25%. Jouve

*et al*. updated the FVE data set with additional connections reported during the 7-year interval between the two studies, yielding an observed density of 37% (

*45*). They used an inference algorithm based on second-order connection regularities and arrived at a density prediction of 58% for visual cortex.

The aggregate evidence suggesting high density of the FIN of the macaque neocortex led us to explore the implications for putative SW properties of the dense cortical graph. In contrast with *d-*dimensional regular lattices, where the characteristic path length *L* grows as a power-law *L* ∝ *n ^{1/d}* with the number of nodes

*n*, in a SW network path length growth is logarithmic with the number of nodes and hence much slower, i.e., as , where

*k*is the average degree in the network (

*31*,

*46*). Figure 1B examines the effect of density on a hypothetical 1000-node ring lattice network on the interval over which the SW phenomenon of high clustering and short path length exists. The reduction in average path length and increase in clustering of a lattice by random rewiring of connections is density dependent. At densities below about 42%, even limited rewiring substantially decreases average path length while maintaining high clustering, thereby providing shortcuts characteristic of a SW architecture. By contrast, at high densities rewiring barely affects path length, because density alone determines this feature, independently of the more detailed structure of the network. Figure 1C shows the SW coefficient as a function of rewiring probability for regular networks (lattices) with increasing density [defined as in (

*31*)]. These results indicate that the density predicted by FVE (45%) and that reported by Markov

*et al*. (

*8*) would reject the hypothesis of the large-scale interareal cortical network being a Watts-Strogatz–type SW network.

The *G*_{29x29} matrix discussed here has a density of 66% and includes pathways with few labeled neurons. As we have argued elsewhere, weak pathways could fulfill diverse roles (*8*, *10*, *11*, *13*, *39*), and given that injections involve only a fraction of a target area, we do not capture the full complement of neurons for each weak pathway. Thresholding would reduce the range of weights that we report. For example, eliminating pathways with, on average, fewer than 10 neurons per tracer injection would reduce the graph density from 66 to 53%, and the range of connection weights (*8*) from five to four orders of magnitude. At 53%, our conclusions concerning the SW would not change. However, 37% of the pathways that would be eliminated by this thresholding have been reported in earlier publications. Further, these conservative steps would ignore the possibility that much larger injections coupled with higher sampling could potentially reveal larger cell numbers, thereby increasing the range of connection weights, numbers of areas connected, and the graph density with respect to the values reported here.

At graph densities approaching 100%, the point when all pairs of nodes are interconnected, the variability within the graph’s structure approaches zero. Hence, at the high density found in the *G** _{29x29}* matrix, little binary specificity might be expected. However, because the probability of connections drops steeply with distance, the subset of long-distance connections can show binary specificity, as we demonstrate in the next section.

## Binary Specificity in the High-Density Cortical Graph

Figure 2A shows the *G** _{29x29}* matrix with the NFPs indicated in red and organized so that target areas are associated with one of the five regions of the cortex (occipital, temporal, parietal, frontal, and prefrontal; see Fig. 2D). Earlier anatomical studies suggested that cortical areas are preferentially connected to physically nearby areas in the region in which they are located, whereas the additional NFPs mainly interconnect area pairs located in different regions (

*8*). Because regionally related areas tend to have similar connectivity patterns, this raises the question of whether the NFPs are perhaps important because they connect areas having dissimilar connectivity patterns. Using a similarity index (see Fig. 2B), we measured the degree to which two areas receive (or avoid receiving) input from common sources (in-link similarity) (

*11*). This confirmed that within-region similarity was high; between-region similarity was lower and, notably, declined with increasing distance between regions (Fig. 2B). The observation that NFPs tend to interconnect distant areas with low similarity provides one indication of their specificity.

Additional evidence for the NFP-related specificity comes from a dominating set analysis, a graph theoretical method that quantifies the extent to which connections are gathered (“dominated”) by a small set of nodes (*8*, *11*). In the case of directed networks, in-link and out-link domination are specified separately. Because with retrograde tracers all the in-links are revealed for an injected target area (but not the out-links), we focused on in-link domination (*8*, *11*). A subset of nodes is said to be fully in-link dominating if all nodes of the graph each project to at least one edge into this subset. There can be several fully dominating sets of nodes (areas); a minimum fully dominating set (MDS) is the smallest such set. The smaller the MDS, the more concentrated the input, indicating the increased role that this set plays in the network. A more refined measure is given by encoding the percentage of all nodes that project into a group of given target areas (*8*, *11*) (thus 100% corresponds to full domination). The histogram of this fraction over all possible groups of a given size (e.g., all combinations of two or more targets) gives an overall picture of domination; see fig. S3, A and B, in (*11*) for the *G** _{29x91}* network. In this case, the low value of 2 for the MDS along with the high domination percentage of many small groups of target areas indicates that the FIN must also be a dense network [see (

*8*) for more details]. Comparing these histograms with and without the NFPs included shows that the NFPs play a key role in the statistics of the inputs to area groups, as their removal appreciably reduces domination for all sets, and the MDS size jumps from 2 to 5 (

*11*).

Besides suggesting a key role for the NFPs, the dominating set analysis confirms the high density of the cortical graph (*11*). However, this density is not homogeneously distributed. The percentage of areas projecting to a target area is 99% for areas within 10 mm, 85% between 10 and 20 mm, 50 to 60% between 20 and 40 mm, and below 40% at 40 mm or greater. Examination of the cortical areas projecting to a target region shows that a modest number of areas outside that region project to all of the injected areas included in the target region (*11*) (Fig. 2D). Thus each target region has a set of common input areas, which constitute a connectivity signature of the target region (*1*, *7*, *11*). The number of common input areas to a region is substantially larger when NFPs are included (red versus blue in Fig. 2D). Comparisons with randomly permuted networks indicate that the increase in the number of common inputs after inclusion of NFPs significantly exceeds that which would occur from an increase in density by the addition of an equivalent number of random connections (*11*). Further, the specificity of the long-distance connections and hence of the NFP is indicated by the fact that the density of the interregional edge-complete graphs is considerably lower than for those formed by intraregional connections (Fig. 2C). Note, that the specificity of the NFP connectivity is the same as for all long-distance connections, including those already known (*11*).

The impact of the interareal pathways on the physiology of the target areas is constrained by the laminar origins of the parent neurons of the pathway and the cortical layers targeted by their synaptic terminals. These laminar constraints on interareal pathways contribute to determining cortical hierarchies (*17*), which we address in the next section.

## Hierarchical Organization

More than 70% of all the projections to a given locus on the cortical sheet arise from within 1.5 to 2.5 mm, so that cortical connectivity is dominated by short-distance (*10*), local connections that conform to a canonical microcircuit optimized to amplify and shape weaker long-distance cortical inputs (Fig. 3A) (*16*). Therefore, when considering long-distance interareal pathways, it is important to consider not only the strength and the specificity of the connections, but also their pronounced laminar asymmetry determined by the direction of the connection. Hence, feedforward (FF) pathways (mostly directed rostrally) originate principally from supragranular layers and terminate in layer 4 in higher areas (*47*–*49*), whereas feedback (FB) pathways (mostly directed caudally) originate mainly from infragranular layers in higher areas and avoid layer 4 in lower areas (*47*, *49*–*51*). Pairwise comparison of the connections has been used to reveal cortical hierarchies (*17*, *52*). While the FVE model is indeterminate (*53*), it can be partially resolved by using a continuous scale such as hierarchical distance based on the fraction of supragranular layer neurons (SLNs) (Fig. 3B) (*54*–*56*). The SLN index quantifies an order relation between areas as defined by their laminar profiles of connectivity and thereby allows estimation of the hierarchical distance separating them (*57*). Further analysis shows that connections between neighboring areas have the highest weight, defined as the fraction of labeled neurons (FLN) (*10*, *58*), whereas long-distance FF and FB connections have lower weights (Fig. 3, B and C).

Interest in the hierarchical organization of the cortex is fueled by evidence that FF and FB processes are distinct physiologically. A useful, albeit oversimplified characterization is that FF connections are “driving” and FB connections are “modulatory” (*59*–*63*). Further, FF and FB pathways engage different glutamate receptor subtypes (*64*). If the two polarities indeed have relatively distinct roles, one might predict that any given neuron would contribute to only one type of pathway, rather than supplying axons to both. This has been tested by injecting two distinguishable retrograde tracers in higher and lower cortical areas and examining the intermediate areas for double-labeled neurons (i.e., neurons with a bifurcating axon directed at both pathways). The results showed that the FF and FB neurons in the intermediate areas were virtually all single-labeled, which is striking given the commonality of axonal bifurcation (*57*, *65*). Not only do FF and FB constitute distinct populations, but they also form two segregated streams, consistent with earlier observations (*47*, *66*, *67*). The supra- and the infragranular layers each have a counterstream organization, most pronounced in the supragranular layers (Fig. 3D) (*12*, *68*). Further, the supragranular counterstream showed a point-to-point (i.e., topographical) precise connectivity in both FF and FB directions, whereas the infragranular counterstream has a more diffuse topography showing high divergence and convergence in both directions (*12*). Hence, contrary to previous assertions, topographic precision distinguishes supragranular layer from infragranular connections and not FB versus FF pathways (*12*).

The integration of interareal connections into the canonical microcircuit of a target area presumably determines how top-down and bottom-up streams affect processing within the target area (*18*). The concept of hierarchical processing has influenced theories of cortical computation, predictive coding, and emerging concepts of inference in cortical function (*18*, *69*, *70*). Recent progress in elucidating interareal communication includes the demonstration of gamma-band phase coherence in the supragranular and beta-band coherence in the infragranular layers (*71*, *72*). These differences in coherence reflect differences in interareal synchronization (*73*–*76*), which are thought to facilitate effective communication (*74*, *77*). Recently beta-gamma asymmetries have been shown to correlate with the aforementioned SLN fraction, suggesting a match between anatomically and functionally defined hierarchies (*18*). The SLN analysis revealed infrequent, but potentially important, departures from the reciprocity of FF and FB projections. For instance, instead of FF being invariably reciprocated by FB, a few FF pathways are reciprocated by a FF, thereby forming a strong loop (*54*, *78*, *79*), which is illustrated in Fig. 3E, where the FF projection of V4 to the frontal eye field (area 8L) is reciprocated by a FF and not a FB connection (*12*, *54*). Given the evidence for different physiological roles of FF and FB pathways, such anti-hierarchical pathways may allow circulation of information up and down the cortical hierarchy entirely via FF pathways. The functional importance of these rare so-called strong loops (*79*) remains to be determined.

## Rich-Club and Bow-Tie Structure

Previous studies based on sparse cortical networks have suggested an important structural heterogeneity in the cortex, where hub areas are statistically more interconnected than expected, forming a so-called rich club (*80*–*82*) or central core (*44*). Using methods adapted to high-density graphs (*83*), we have shown that the *G** _{29x29}* subgraph harbors 13 cliques (complete subgraphs in which all possible connections are present) of size 10. This results in a very high, 92% density core formed of 17 nodes in the edge-complete

*G**(*

_{29x29}*13*), a periphery of 12 nodes with a density of 49%, and a 54% density of connectivity between the periphery and core. Because additional injections cannot change the structure of the

*G**subgraph, this implies that the FIN also has a dense core (to which the core of*

_{29x29}

*G**must belong, being 92% dense), possibly larger than that of*

_{29x29}

*G**. The core-periphery distinction is also supported by the average strength of connections, as the fraction of labeled neurons (FLN, see Glossary) between the core and periphery is on average weaker than within either the core or periphery (*

_{29x29}*13*).

Next, we present a method (see captions of Table 1 and Fig. 4) that uses both the FLN weights and the SLN fractions to reveal correlations among core and periphery links (Fig. 4A), thereby providing further information on the large-scale organization of the cortical network. In Fig. 4B, peripheral nodes are split into two groups (Table 1). In the left fan, the preponderant pathways are FF going from the periphery to the core reciprocated by FB from the core to the periphery. In the right fan, the preponderant pathways are FB going from the periphery to the core reciprocated by FF pathways from the core to the periphery (Fig. 4B). The preponderant pathways being FF from left to right and FB in the inverse direction also holds for the small number of direct links between the two fans (see Fig. 4B). The core includes areas in the frontal, prefrontal, and parietal regions [see Fig. 2D for region locations and (*8*) for area members of each region]. The predominant FF inputs from the left wing to the core areas may serve as feeders for the central core processing (consistent with the inclusion of primary sensory and motor areas in this left wing; see Fig. 4B), whereas the FB pathway predominance from the right wing could correspond to a monitoring or coordinating role for the members of this group.

## A Cortical Distance Rule as Cost-of-Wiring Principle: The EDR Model

The above observations suggest a general picture of the interareal network: It is a dense network, but with high binary specificity ensured by long-distance connections and characterized by the existence of a core-periphery structure organized into a bow tie via FF/FB pathways. The specificity of connections increases with projection distance and with decreasing connection weights. Is there a fundamental, biophysical principle–based model that can capture most of these properties with a minimal number of fitting parameters? The answer is yes, as discussed below.

A clue to the importance of weight-distance relations for understanding the properties of the cortical network comes from the observation that the FLN weights are highly heterogeneous, following a log-normal distribution varying over five orders of magnitude (*8*, *10*) (Fig. 5A). The log-normal distribution may directly reflect the interplay between metabolic costs associated with projection lengths and a geometrical or spatial property of areal locations (*13*). Axonal projections out to a distance *d* through the white matter come at an energy (metabolic) cost, irrespective of the areas involved. This is suggested by the exponential decay of the number of labeled neurons as a function of projection distance *d*: *p*(*d*) = *c*exp(–λ*d*), corresponding to an exponential distance rule (EDR) (Fig. 5B). The spatial decay constant λ = 0.188 mm^{–1} expresses the growth rate of the metabolic cost with distance. We therefore expect that the FLN between two areas separated by a distance *d* is determined to a first approximation by this cost, independently of areal identity. A relevant spatial property is expressed by the inset in Fig. 5B, showing that the fraction of area pairs separated by a distance *d* is well approximated by a truncated Gaussian. Hence, combining the EDR with the Gaussian distribution of interareal distances, one finds that the distribution of area pairs with a given FLN indeed obeys a log-normal distribution (*13*).

These observations suggest that the EDR, which is a global distance rule, acts as a principle for resource allocation via the probability factor determined by the space constant λ. To what extent is the cortical network determined by this principle? To address this question, we defined two random network models [see (*13*) for details], one based on the EDR and the other on a constant distance rule (CDR, where projection length bears uniform cost). We then compared the statistical properties of these two sets of networks to those of the observed *G** _{29x29}* subgraph. Only the EDR model captures the binary statistical graph properties of

*G**on all scales (from local to global), including motif distributions (*

_{29x29}*84*) and graph spectral properties. Most notably, the EDR but not the CDR generates graphs with a pronounced core-periphery structure, closely tracking the distribution of cliques in the

*G**(Fig. 5C). This figure also demonstrates that the CDR model fails to produce cliques of size 8 and larger. This finding suggests that the existence of the dense core in the cortex reflects the cost-of-wiring principle expressed in the EDR.*

_{29x29}## Efficiency of Information Transfer

A simple measure of bandwidth for information transfer in complex networks can be defined via the average conductance between all source-target pairs in the network (*85*), called global efficiency, or (*E*_{g}). Conductance here is interpreted as in physics, by the inverse resistance of the directed path of minimal total path resistance through the network from the source node to the target node (see Box 1 for definitions of the quantities in terms of FLNs). The path resistance can be interpreted as the negative logarithm of the probability that activity in the source node will generate activity in the target node. Here, we equate bandwidth with axon number as reflected by FLN. Thus, a sequence of edges having large FLN values (or “high bandwidth” edges) directed from source to target would form a path of low resistance (high conductance), providing a high-bandwidth pathway for information transmission. To obtain a graded measure of global efficiency within the structure of *G** _{29x29}* and to understand the role of weak projections, we computed

*E*

_{g}on the remaining network after the sequential removal of the weakest link (smallest FLN) and plotted it as function of the network density. Figure 5D shows that the global efficiency of the network does not change before 76% of the weakest links are removed, indicating the existence of a high global efficiency (high bandwidth) backbone formed by short-range paths (see Fig. 5, E and F). Indeed, the average length of the remaining edges at 24% remaining density is 16 mm compared to the 27-mm average length of the removed edges. This suggests that the cortical network is organized in such a manner as to be independent of the activity along the weak projections for high-bandwidth information transfer. We speculate that these long-range pathways, which we have shown to have a high binary specificity, may contribute to interareal synchronization between cooperating areas.

### Box 1: Efficiency measures for information transfer in networks.

In the simplest approximation, we may interpret the FLN weight *f _{ij}* of the projection from source area

*i*to target area

*j*as the probability

*p*that activity in

_{i→j}*i*will induce activity in

*j*. Assuming that these are independent events, the probability of activity in area

*k*induced along the path

*i→j→k*will be given by the product of the FLNs:

*p*or ln(

_{i→j→k}= f_{ij}·f_{jk}*p*) = ln(

_{i→j→k}*p*) + ln(

_{i→j}*p*). Then, the positive quantity

_{j→k}*w*= –ln(

_{ij}*f*) can be interpreted as a measure of resistance for information transfer along the

_{ij}*i→j*link, as small FLN (weak links) induce a large resistance and vice versa. Additionally, based on our assumption above, the link resistance is additive along network paths. We define the resistance

*r*from an arbitrary node

_{xy}*x*to an arbitrary node

*y*as the smallest sum of link resistances among all directed paths ω

*from*

_{xy}*x*to

*y*, that is, . The highest transmission probability path will be the one, , that achieves this minimum (see figure). The global efficiency measure

*E*

_{g}is defined as the average conductance (inverse resistance) between all the possible

*N(N – 1)*node pairs, where

*N*is the number of nodes in the network as shown in (A).

A local efficiency measure *E*_{l} is defined in (B): We remove a node *i* with all its links and then we compute the average of the resistances between all pairs of its neighbors as measured through the rest of the network, and finally, we average these quantities over all nodes *i*. Thus, *E*_{l} quantifies the degree to which the satellites of a typical city can communicate via paths that avoid the city (B).

**Schematics for efficiency measures.** (**A**) Global efficiency and (**B**) local efficiency. The dashed curves are schematic paths through the rest of the network (not shown). Here *{i}* denotes the set of network neighbors of node *i*.

A more local notion of information transfer can be similarly defined (*86*), via the average conductance between all the pairs of nodes that are neighbors of a given node *i*, after the removal of node *i*, then averaging this quantity over all nodes *i* (see Box 1), called local efficiency, *E*_{l}. One can think of *E*_{l} as a measure of accessibility between the satellites of a town via routes that avoid the town. Figure 5D shows this quantity as a function of density (following the same weakest-link removal procedure as above). Intriguingly, local efficiency increases sharply with weak-link removal (or inactivity), compared to the global efficiency. The weak-link removal increases locality because the weak links are long-range. This tends to prune the interregional shortcuts and physiologically would be predicted to decrease interactions between diverse functional modalities, leading to a more localized structure of the remaining network. It eliminates those neighbors of a node to which it connects via weak links, thus disconnecting regions and modalities but preserving dense within-region connections. Accordingly, the paths linking the remaining neighbors of the node (avoiding that node) are all strong, having small path resistance (large conductance), and hence the measure of local efficiency becomes large. The redundancy of local strong paths [the complete triangle is the most abundant three-motif, formed by strong links as shown in figure 3B of (*13*)] guarantees that the number of edges on a path between two nodes is small, which further decreases the path resistance between the nodes and hence further increases the local efficiency.

These changes in local efficiency indicate flexibility of network modularity and long-distance functional interactions—for example, in response to changes in cognitive load (*13*, *87*). This suggests that in the cortical network, local information processing is voluminous, and because of the redundancy of local high-conductance paths, it is much more efficient than global information processing, whose efficiency acts at a constant base line (Box 1). Notably, the organization of the cortical network around a high-efficiency backbone with constant global efficiency and optimal local efficiency behavior is also captured well by the EDR (Fig. 5D). These efficiencies are not binary graph measures, but are based on weights (FLNs). By contrast, the CDR fails completely, especially for the local efficiency measure (Fig. 5D).

## Optimal Placement

The ability of the weight-distance relations of the *G** _{29x29}* to predict numerous features of the cortical network underlines the importance of the embedded nature of real-world networks where nodes are located in three-dimensional (3D) Euclidean space and are linked by weighted and directed edges (

*88*). An important aspect of the embedded cortical network is the spatial layout of cortical areas. Numerous studies have presented evidence that the spatial layout is optimized to minimize total wire length (

*89*–

*91*). A recent study used collated data for 95 areas of the macaque and claimed that optimized component rearrangements based on a simulated annealing algorithm could substantially reduce total wire (

*92*). However, the database used by that study was edge-incomplete and the network density derived from it had a low density. In addition, weights were only classified as being on one of four levels. We adapted evolutionary optimization algorithms and applied them to our database to search for areal placements that minimize total wire length; we found alternative organizations that shorten the binary network by 5%. For the weighted network, the maximum reduction was less than 1% and involved a small number of switches between adjacent areas (Fig. 6) (

*13*). Figure 6A shows a 3D representation of the

*G**network with areas color coded to indicate regional identity. Figure 6B is an example of shuffled areal positions leading to increased wire length and loss of adjacency of areas originally from the same region. An optimization procedure applied to the randomized network leads to regional clustering similar to that observed in the cortex (Fig. 6C). Further, random networks based on the EDR had significantly shorter wire lengths than CDR-generated networks. These results therefore confirm that wire minimization is a constitutive organizational principle of the cortical network (*

_{29x29}*89*–

*91*,

*93*–

*99*) and suggest that this design constraint is at least partially implemented by the operation of the EDR generating the weight distributions of interareal projections (

*13*).

## Concluding Remarks

### Box 2: Glossary of technical terms.

1. **Area:** A region of the cortex with specific cytoarchitecture and associated with a function. **Target area:** An area that received a retrograde tracer injection. **Source area:** An area containing labeled neurons projecting to a target area.

2. **Average path length:** Average value of shortest path lengths between all node pairs in the graph. Length here is measured in hop-counts along directed edges.

3. **Binary specificity:** The degree to which a network or a graph differs in its binary graph theoretical properties from a random graph.

4. **Bisection bandwidth:** The minimum of the number of connections between two, equal-size partitions of the nodes of a graph, taken over all such partitions.

5. **Clustering:** The average of the fraction of connected neighbors of a node (fraction of triangles).

6. **Counterstream:** Refers to the organization principle by which there are streams ascending (supragranular layer) and descending (infragranular layer) the cortical hierarchy (*135*). Recently, this has been extended to include a dual counterstream organization where an ascending and descending stream is identified in each of the two compartments (*12*).

7. **Dominating set:** A set of nodes in a graph such that all nodes of the graph have at least one edge with one of their end-nodes in this set.

8. **Edge:** A link or connection between two nodes directed from one to the other, here interareal pathway. There can be at most two directed links, oppositely oriented between any two nodes.

9. **Edge-complete subgraph:** A subgraph that has exactly the same connections between its nodes as the connections between the same nodes in the larger graph that this subgraph is part of.

10. **FLN:** Fraction of labeled neurons: For a given injection (target area *i*) and source area *j*, the FLN is the ratio *f _{ij}* between the number of labeled neurons in area

*j*and the total number of extrinsic (not in

*i*) labeled neurons for that injection. We use FLN as a measure of weight (

*10*,

*58*).

11. ** G_{nxm} (sub)graph or matrix:** For every one of the

*n*targets (injected areas), it specifies which of the

*m*sources project into that target (0 if no projection, 1 if there is a projection). Here,

*G*denotes the full graph of interareal connections,

_{91x91}

*G**represents the currently known projections from all areas into the injected 29 areas, and*

_{29x91}

*G**denotes the subgraph formed by the connections among the target areas only. The latter is edge-complete, i.e., the status of connectivity is fully known within this set of nodes.*

_{29x29}12. **Nodes:** Discrete entities represented as points or vertices in graph theory for the purpose of studying the patterns of interactions among them (represented as links or edges). In this case, a node represents a cortical area.

13. **SLN:** The fraction of supragranular labeled neurons is defined for each source area projecting to an injected target area. SLN corresponds to the number of retrogradely labeled neurons located in the supragranular layer divided by the total number of neurons (in infra- and supragranular layers). SLN distinguishes FF and FB pathways and can be used to calculate hierarchical distance (*12*, *54*, *55*).

In summary, the interareal network achieves economy of connectivity and communication efficiency by means of a distribution of weights, spatial organization and a core-periphery structure in the form of a bow tie with a dense core. Interareal connections integrate across the local circuits via dual counter-streams located in the supra- and infragranular layers, which have distinct physiological properties. Further explorations of the impact of the cortical core on the local circuit should improve our understanding of their dynamic interaction.

The broadband connections involving large number of cells that form the global efficiency backbone and that are expected to shape receptive fields in target areas are short range and strongly cluster areas of a given modality. The considerably more numerous long-distance connections are sparse and exhibit high binary specificity; we speculate that these connections serve to promote cortical communication by controlling oscillatory coherence (*77*) possibly via contraction dynamics (*100*). During primate evolution the increase in numbers of areas and their spatially restricted broadband connections raises the possibility that increases in brain size would lead to increased isolation of clustered communities (*101*, *102*). We hypothesize that weak long-distance connections appeared during recent evolution and may facilitate cooperation between distant areas. Assuming that a high-density connectivity is a characteristic feature of the interareal cortical graph, long-distance weak links may allow preservation of the connectedness of the cortical network following cortical expansion. Further, long-distance connections allow accommodation of novel information processing, thereby building over evolutionary time upon the existing repertoire of dynamic functions. There are a number of advantages to be gained (see below) by developing a high efficiency central core that we along with others have observed. Long-distance connections particularly for the higher association areas play a special role in forming the cortical core. The high incidence of long-distance projections involving prefrontal areas (*11*) suggests a particular importance of this cortical region in core-periphery integration, indicating the cortical core as an important component of higher order interareal architectures (*103*, *104*).

Self-organization in cortical development is well established (*105*). The findings summarized above suggest interesting parallels with other, highly functional, self-organizing information-processing networks. The high density, yet highly specific nature, of *G*_{29x29} suggests a heterogeneous, expandable, and cost-efficient information-processing network subject to evolutionary constraints. In particular, we find (i) high connectivity, (ii) high global accessibility, and (iii) large path diversity. Additional evidence supports several further constraints: (iv) a high bisection bandwidth (Fig. 4D); (v) resilience to connectivity failures (global conductance measurements in Fig. 5D show that during sequential removal of the weakest links, a substantial decline in the network’s global efficiency is not reached until a density of 24%); and (vi) incremental expandability that allows addition of new areas to the network during evolution, without a substantial wiring overhaul to maintain or improve performance, also supported by optimal placement studies. The global efficiency and the optimal layout of the *G _{29x29}* are supportive of (vi)

*.*As seen in Fig. 5D, global efficiency stays constant with weak-link removal. Weak links are long-range and might be more prevalent in large brains. Conversely, adding long-range links (during cortical expansion) seems not to affect the global efficiency, much less worsen it (as would be expected in a complete overhaul). In terms of optimal layout, the total wiring in

*G*[see (

_{29x29}*13*)] is minimal as a consequence of the EDR. It remains minimal following the addition of new areas with strong links to their neighbors and progressively weaker links to distant areas. This corresponds to an optimal spatial arrangement, consistent with incremental expandability. In an interesting parallel, a recent study on efficient, massive data-processing networks shows that for their purposes, random structures with a pronounced core-periphery organization allow incremental growth without sacrificing their high bisection bandwidth and throughput capability (

*106*).

Although primate neocortex shares a number of common features with human-made, large-scale information-processing networks, there are also crucial differences, and so the analogies should not be overextended. One important difference is that in current technological information networks, information (including a destination address) is encoded within the units (packets) that are sent along the edges of the network (packet-switching networks), whereas this does not appear to be the case for the brain. Our results show that spatial location does influence the wiring properties of the brain. Further, in the brain, the location and timing of activity (in the context of other firing activities) represent the message (neural coding), thus linking network activity to the structural properties of the cortical network. In the Internet or World Wide Web, however, it does not matter where the routers are physically in space, because the packets are routed on the basis of network addresses encoded into the packet. This key difference suggests that spatio-temporal network models should hold more promise for brain science than generic, purely graph theoretical models such as the SW model.

## Future Perspectives

While the high density of the binary interareal graph is not consistent with a SW architecture at the computational level of a cortical area, the SW concept may nonetheless be relevant at finer spatial scales. For example, the high clustering and short path lengths indicative of SW architecture are suggested by diffusion magnetic resonance imaging (MRI) analysis at a node scale of a few millimeters (about the size of the voxels) (*107*). Here, the importance of the SW parameters revealed by this approach must be considered with respect to the methodological thresholding imposed by the whole-brain imaging technique. The nonrandom connectivity at the single-cell level raises the issue of SW architecture at this scale (*108*, *109*). The rich variety of cell identities posits a conceptual challenge for the potential of SW network at the single-cell level. Large-scale circuit mapping at the single-cell level is not currently feasible, although innovative approaches might conceivably provide the appropriate data in mice (*110*). At the interareal level, our results indicate that the spatially embedded nature of the cortical network determines many of its properties, such as structural and weighted graph characteristics. Could similar weight-distance relations operate for single neurons, thereby suggesting a similar operational logic over different scales? Like interareal connectivity, local connectivity shows an exponential density decay (*10*), reflecting the decrease in the likelihood of synaptic contact with distance (*111*). Also, log-normal distributions like that of interareal weights have been observed for the distribution of synaptic strengths of single neurons (*109*).

The EDR that we describe shows that the spatial constant of interareal density decay accounts for many of the binary and weighted features of the cortical graph, as well as the important design feature of wire minimization. Given the high scalability of mammalian neocortex, accommodating five orders of magnitude range of brain weight (*112*), we examined the spatial location of cortical areas in the *G** _{29x29}* subgraph and found that when magnitude of connections is taken into account, it shows an optimal placement of area positions indicating minimization of metabolic expenditure on total wire length, also well predicted by the EDR. Given these results it will be instructive to explore the weight-distance relations in different species to determine how the decay parameter λ changes with brain size. This may provide insights into a common rule governing scaling properties of the brain and also allow improved extrapolation of our understanding of the connectivity of the macaque brain obtained using invasive techniques to the larger and less directly accessible human brain.

A core-periphery structure has been observed in other self-organized information-processing architectures, both human-made such as the World Wide Web (*113*) and the Internet (*114*, *115*), and in biological networks, such as in metabolism (*116*, *117*), the immune system (*118*), and cell signaling (*119*–*121*). The resulting bow tie is an evolutionarily favored structure for a wide variety of complex networks (*122*, *123*), expressing the fact that functional or living systems have an input interface, a processing unit, and an output interface. This is because these systems are not in thermodynamic equilibrium and are required to maintain energy and matter flow through the system. While the overall bow-tie structure is common, we have seen that for the brain it emerges through a counterstream organization of the directed links between core and periphery, showing its specific nature when compared to other bow ties in biology or technical networks. However, the full details of this structural organization will only become evident when additional tracer-based anatomical data are incorporated.

Although the interareal connectivity data explored here has revealed interesting features of the cortical network, many additional analyses remain to be done. It will be important to complement the quantification of connectivity by functional and improved molecular characterization of cortical areas and also of the parent neurons (*124*–*126*). A necessary and complementary development will be to use anterograde tracers to examine the laminar integration of interareal connectivity, combining quantification and morphological characterization at the synaptic level (*127*, *128*). Finally, our use of the term “canonical microcircuit” in the sense of a stereotyped circuit constituting a fundamental cortical building block should be tempered by the evidence for large regional differences in cell densities and dendritic arbor sizes (*129*, *130*). While it is generally accepted that the local circuit exhibits cell-specific connectivity across the cortex (thereby conforming to a canonical circuit) and likewise that there are consistent input-output patterns across the different areas (*131*), there is nevertheless only a single quantitative interlaminar connectivity map, namely, for cat area 17 (*132*). The concept of the canonical microcircuit provides a coherent framework for thinking about neocortical function, and evidence of variations of local connectivity point to the need to establish additional quantitative interlaminar maps. Dynamic models built on existing quantitative interlaminar maps (*132*, *133*) give realistic dynamics (*134*). Extending this type of modeling to include interareal relations becomes a reasonable next step.

## References and Notes

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- ↵For criteria for inclusion of injection sites, see Markov
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**Acknowledgements:**We acknowledge C. Lamy and P. Misery for histology and M. A. Gariel for data analysis. This work was supported by FP6-2005 IST-1583 (H.K.); FP7-2007 ICT-216593 (H.K.); ANR-11-BSV4-501 (H.K.); LabEx CORTEX (ANR-11-LABX-0042) (H.K.); PN-II-RU-TE-2011-3-0121, FP7-PEOPLE-2011-IIF-299915 (M.E.-R.); National Institute of Mental Health grant R01 MH60974 (D.C.V.E.); and Notre Dame’s Interdisciplinary Center for Network Science and Applications (iCeNSA) funds and, in part, by grant FA9550-12-1-0405 from the U.S. Air Force Office of Scientific Research and Defense Advanced Research Projects Agency (Z.T.).