Mammals show a wide range of brain sizes, reflecting adaptation to diverse habitats. Comparing interareal cortical networks across brains of different sizes and mammalian orders provides robust information on evolutionarily preserved features and species-specific processing modalities. However, these networks are spatially embedded, directed, and weighted, making comparisons challenging. Using tract tracing data from macaque and mouse, we show the existence of a general organizational principle based on an exponential distance rule (EDR) and cortical geometry, enabling network comparisons within the same model framework. These comparisons reveal the existence of network invariants between mouse and macaque, exemplified in graph motif profiles and connection similarity indices, but also significant differences, such as fractionally smaller and much weaker long-distance connections in the macaque than in mouse. The latter lends credence to the prediction that long-distance cortico-cortical connections could be very weak in the much-expanded human cortex, implying an increased susceptibility to disconnection syndromes such as Alzheimer disease and schizophrenia. Finally, our data from tracer experiments involving only gray matter connections in the primary visual areas of both species show that an EDR holds at local scales as well (within 1.5 mm), supporting the hypothesis that it is a universally valid property across all scales and, possibly, across the mammalian class.

It was recently shown that the network of connections between different areas of the macaque cortex has strong structural specificity in terms of the strength of connections as a function of the distance between areas. This has led to a model of cortex connectivity that predicts many observed architectural features, including the existence of a strong core-periphery organization. When viewed across species, increases in brain size are accompanied by a relative decrease in connectivity, and thus an important question is whether there are architectural commonalities in the cortical networks within the mammalian branch. Here, based on tract tracing data from the folded macaque brain and the smooth mouse brain, we introduce a common model framework that allows network comparisons between species. We show that despite important differences in size, the cortices of both species share several network invariants, suggesting that the mammalian cortex exhibits universal architectural principals. This framework also captures differences between the two brains, including the fact that, unlike the macaque, the mouse core includes primary areas and that there is a relative decrease in the frequency of long-distance connections in the large macaque cortex compared to mouse. This approach allows network architectural extrapolations to the human cortex.

Funding: SzH was supported by the "Programme Avenir Lyon Saint-Etienne" of the Université de Lyon (ANR-11-IDEX-0007), within the Program "Investissements d'Avenir" operated by the French National Research Agency (ANR) http://palse.universite-lyon.fr/ , and Marie Curie Program European Union’s Seventh Framework (FP7/2007-2013) No. PCOFUND-GA-2013-609102, PRESTIGE coordinated by Campus France http://www.campusfrance.org/en/prestige . MER was supported by the UNESCO- L’Oreal National Fellowship "For Women in Science" http://www.fwis.fr/ , by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 668863 https://ec.europa.eu/programmes/horizon2020/ , and in part by the GSCE-30260-2015 "Grant for Supporting Excellent Research" of the Babeş-Bolyai University http://ubbcluj.ro/ . DCVE and LM were supported by grant National Institutes of Health (NIH) R01-MH-60974 https://grants.nih.gov . AB was supported by National Institutes of Health (NIH) R01 EY016184 https://grants.nih.gov and the McDonnell Center for Systems Neuroscience (to AB and RG) http://centerserv.wustl.edu/ . ZT was supported, in part, by grant FA9550-12-1-0405 jointly from the US Air Force Office of Scientific Research (AFOSR) and Defense Advanced Research Projects Agency (DARPA) http://www.darpa.mil/ , and by grant No. HDTRA-1-09-1-0039 from Defense Threat Reduction Agency (DTRA) http://www.dtra.mil/ . HK was supported by ANR-11-BSV4-501 (CORE-NETS), ANR-14-CE13-0033 (ARCHI-CORE), ANR-15-CE32-0016 (CORNET) http://www.anr.fr and LabEx CORTEX (ANR-11-LABX-0042) http://www.labex-cortex.com/ of Université de Lyon, within the program "Investissements d’Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.

In addition to the discovery of the EDR in the macaque, the consistency and completeness of this tract tracing data [ 32 ] has led to a deeper insight into the interareal network properties of the macaque cortex [ 30 , 33 ]: it revealed a much denser (ρ = 0.66) interareal cortical graph than previously reported (network density is defined as , where N is the number of areas and M is the number of connected ordered area pairs, see glossary). High density graphs have low specificity at the binary level (areas connected or not), so that what distinguishes one area from another is the particular combination of areas it is connected to, combined with the weights of the connections, i.e., their connectivity profile or fingerprint [ 33 – 36 ]. Because the range of weights spans many orders of magnitude (five in the macaque), the specificity of individual connectivity profiles is actually very high [ 5 , 30 , 37 ].

Note that the EDR is purely a property of the distribution of the physical lengths of individual axons, without regard to any network topological structure. The EDR states that there are many fewer long-range axons than short ones and quantifies this: the number of axons of length d that we find in the cortex is proportional to e −λd . In general, to experimentally establish the EDR, we do not need to work with brain areas as nodes of a network; we only need to be able to count neurons and measure the corresponding axon lengths. In this sense, the EDR is a more basic and general property than the description of cortical connectivity as a network at some coarse-grained (e.g., mesoscale) level. Once the level of description is defined (e.g., areal), the network properties are, however, consequences of the distribution of the axonal lengths connecting the vertices. Since connectomes are embedded in physical space, the EDR property effectively constrains the topological structures that connectomes can form across different levels, ranging from the single neuron to the areal level [ 31 ].

Recent retrograde tract tracing data in macaque [ 30 ] provides supporting evidence precisely of such a wiring constraint, in the form of an exponential decay of the wiring probability p(d) with projection distance d: p(d)~e −λd , with a decay length (~1/λ) that is short relative to hemispheric dimensions (in the macaque λ ≅ 0.19 mm −1 , corresponding to a decay length of ). A simple way to think of the decay length is that every increase by in projection length leads to a decrease in the number of projections by a factor of (i.e., 37%). Note that using the base of the natural logarithm is convenient, as in this case is equal to the average projection length, providing a simple, intuitive interpretation. We refer to this decay property of connection density with distance as the Exponential Distance Rule (EDR). Retrograde labeling using fluorescent tracers (see Materials and Methods section) is an accurate labeling method that reveals all incoming connections j→i to an injected (target) area i by labeling the cell bodies of the neurons in source area j whose axons make connections in area i. Importantly, there is no transneuronal labeling, so the retrograde labeling method used yields only one-step incoming connections to the injected nodes of the network.

Expansion of the cerebral cortex is accompanied by an increase in the proportion of white matter relative to brain size [ 7 – 10 ]. However, this increase is not rapid enough to maintain a constant neuronal connection density (defined as the fraction of neuron-to-neuron connections compared to all possible ones). Thus, an increase in brain size is expected to result in a reduction in the long-distance connectedness of cortical areas [ 11 – 14 ]. The reduction of the fraction of connections with cortical expansion and the minimization of the metabolic costs are important design features of the cortex [ 4 , 15 – 26 ]. One can hypothesize that this wire minimization constitutes a critical constraint for the optimal placement of areas in the cortex, serving to increase communication efficiency in larger brains [ 11 , 27 – 29 ], and is supported by recent evidence suggesting reduction of long-distance connectivity with increases in brain size [ 28 ].

Here we show that the cortical networks in the macaque and the mouse in fact do exhibit a common organizational principle despite their very different evolutionary trajectories and large differences in brain size. Supplemented by partial tract tracing data in the microcebus (the mouse lemur) we suggest that this principle and the associated network model is a universal determinant of the interareal network across mammals, allowing tentative predictions for the human brain.

Published connectivity maps using consistent interareal tract tracing studies, first in the macaque [ 4 ] and more recently in the mouse [ 5 , 6 ], allow consideration of the network as a directed, spatially embedded and weighted graph (weights representing neuronal connection densities projecting between areas). The absence of full homology between the nodes (areas) and edges (projections) of the networks of the two species makes it difficult to determine commonalities and similarities between them. However, if generic, global organizational principles exist (constraining the adaptation and growth of cortical connections in similar ways), then we expect to see similarities at the statistical level between the network features in the two species.

A purely bottom-up approach to deriving global brain function from local circuitry is currently intractable [ 2 ]. In contrast, a meso-scale approach is more feasible, focusing on the network of interactions between the elements of a mosaic of distinct areas representing the loci of function-specific computation (visual, auditory, somatosensory, motor, etc.). As the mammalian brain is shaped by evolution, morphological and areal network level inter-species comparisons will help identify those features that are conserved across species from those that are species-specific. This will lead to a better understanding of network structural properties and provide valuable clues to the evolution of brain function [ 3 ]. However, progress in this direction has been hindered due to the absence of (i) the necessary data to address the physical properties of the network between areas and (ii) adequate theoretical network comparison methods.

Understanding brain networks is arguably one of the major challenges of the 21st century [ 1 ]. The mammalian cortex is an extraordinary computational device, and analysis of its network properties with 10 7 –10 10 neurons and 10 11 –10 15 synaptic connections is still largely unresolved. In the brain, activity of a single neuron encodes relatively little information; instead, that is achieved via population coding, through spatially distributed temporal activity patterns of cell assemblies. This contrasts with packet-switching information technology (IT) networks, which encode information directly into the packets and the network merely ensures routing between any two nodes. Since the spatiotemporal activity of cell populations is strongly determined by their connectivity and physical layout, cortical network structure and its spatial embedding play a significant role in the brain’s processing algorithm, in sharp contrast with IT networks.

Results

We first give a schematic description of EDR-based network models (Fig 1) before developing a formal methodology for comparing EDR model graphs with experimentally obtained graphs, thereby allowing a quantification of the predictive power of the EDR network model for a given brain. This sets the stage for empirical measurements in the mouse brain, which are required for the construction of a mouse EDR model (Fig 2) and to examine how well the mouse EDR graphs fit with selected local and global mouse network properties obtained from empirical data (Fig 3). We next identify the core-periphery organization in the mouse network and show that it is well captured by the mouse EDR model. The following section is dedicated to a comparison of the capacity of the mouse and macaque EDR models in predicting empirically measured motif distributions (Figs 5 and 6) [38]. Analysis of network motif distribution is a recognized method of capturing the functional features of a network. The motifs analyses suggest the existence of common architectural features in the networks of both species; the following section analyses these structural commonalities by investigating the connection similarity index profiles between all node-pairs as a function of their spatial separation. However, in order to be able to perform comparisons involving distances in brains of very different sizes, we first introduce a common spatial template by an appropriate dimensional rescaling of the two brains. This allows us to show that, effectively, there is a common distribution of similarity indices as a function of adimensional separation in both brains (Figs 7 and 8). The finding that similarity changes across the cortex are only relatively consistent in the two species naturally leads us in the following section to consider the differences in cardinal features governing functional layout and to relate these differences to species characteristic properties of the cortex such as size and cortical folding (Fig 9). We conclude with a Discussion (Fig 10) in which we hypothesize that the EDR is a universal property across scales, i.e., valid also locally (through the gray matter), not just globally (through the white matter), and across the mammalian branch. As preliminary evidence supporting this hypothesis, we present results of tracer experiments (Fig 11) involving local connections only within the gray matter in three species—mouse, macaque, and microcebus—and quote results from other experiments in the rat. We conclude with mathematical arguments that further support the universal character of the EDR and speculate on the importance of these findings for understanding the human brain.

EDR-Based Network Model of the Cortex To what extent does the EDR, as a connectivity constraint, determine the properties of the interareal network? To address this issue, one needs (i) a family of EDR-based network models and (ii) a method of comparison between the model-generated networks and the experimental data network. The exponential decay rule ~e−λd in the macaque was obtained from collating all the labeled neurons (over 6.4 million) following tract tracing experiments in different areas and constructing an interareal distance matrix, the latter estimated as the distances between the area barycenters through the white matter (WM), along the shortest paths. Here axonal p(d) should be interpreted as an average property (see Fig 1A), the probability that an axonal bundle projects to a distance d, independently of the specific functional nature of the areas. At this level of description, the strength of the connection between areas, expressed as the fraction of labeled neurons (FLN), depends uniquely on their geometrical separation. Thus, the network is viewed as a spatial, directed, and weighted graph dependent on the matrix D = {d ij } of interareal distances d ij . We emphasize here that the EDR arises from the estimated probability distribution of axon lengths. Although the strength-distance relation is consistent with the EDR, the probability distribution of axons lengths provides a more compelling demonstration of the property and leads naturally to the parametric EDR model described below. The probability density function, q(d), of the distances in the matrix D is typically a unimodal distribution (Fig 1B), which, when combined with the exponential decay p(d), leads to a log-normal distribution of edge weights, confirmed by the empirical FLN data [4,39]. PPT PowerPoint slide

PowerPoint slide PNG larger image

larger image TIFF original image Download: Fig 1. Schematic of EDR-based model of the cortex. (a) The exponential distance rule (EDR) expresses the empirical observation that the probability of axons of length d decay exponentially [4] with a decay rate λ. (b) Interareal distances d ij are measured between the barycenters of the cortical areas i and j along the shortest paths through the white matter, avoiding the sulci and subcortical obstacles. The interareal distances follow a unimodal (Gaussian-like) distribution q(d) (i.e., q(d)Δd gives the fraction of interareal distances with lengths between d and d + Δd), as there are more area pairs separated at medium distances than at long or short distances, an observation valid for both smooth and folded brains (see Fig 7C). (c) The EDR network model (with the algorithm described in the text) generates strong connections (large bandwidth) between physically neighboring areas and exponentially decreasing strengths between areas that are increasingly far apart [4,40]. https://doi.org/10.1371/journal.pbio.1002512.g001 The EDR distribution with the corresponding distance matrix D in a given brain naturally defines a parametric family of random graphs, called EDR random graphs (Fig 1C), parameterized by the decay rate λ. For these model graphs we make the choice p(d) = λe−λd, where now λ is the (only) model parameter. To distinguish the decay rate parameters in these models from the experimentally measured ones, we denote the latter as λ exp , e.g., for macaque . We also employ, as a null model, the constant distance rule (CDR) family of random graphs, where there is no dependence of connection probability on distance, corresponding to the λ→0 limit, i.e., to the choice p(d) = const. The EDR family of random graphs is defined via a simple algorithm [4] in the spirit of the Maximum Entropy Principle, i.e., it is based only on the given information (p(d) and D), while all else is uniformly random. The algorithm proceeds as follows: First, we randomly draw a connection length d from the distribution p(d). Second, we choose uniformly at random an area pair whose separation distance in the matrix D falls in the same distance bin as d, according to some binning criterion (bin sizes used in this study were typically 5 mm for the macaque and 0.4 mm for the mouse) and finally, insert a randomly oriented connection between them. Multiple connections between the same area pair in the same direction generate the weights for the directed edges with a log-normal distribution. These steps are then iterated until the graph density in the model reaches the observed value in the experimental network.

Network Fitting and Comparison We denote the data network obtained from the experiments by G exp (e.g., for the macaque we use , and for the mouse ). Our goal is to compare the properties of the EDR model networks with the properties of G exp . Since the model networks are only based on distance-dependent connection probabilities, one cannot expect perfect agreement (edge-by-edge) with the biological connectivity graph G exp , however, if the distance rule is a strong determinant of the interareal network, the model graphs should be statistically similar to G exp . The comparisons are performed via parameter matching of network properties [4]: for a given network property P, the interareal distance matrix D and parameter λ is used to generate a large ensemble of EDR graphs . By varying λ we determine the value λ P via minimizing the deviation |P(G exp ) − 〈P(λ)〉|, with the average 〈∙〉 taken over at least 103 EDR graph realizations from . Thus the model parameter is determined so that the average of P in the model is as close as possible with the value of P observed in the data network. We then compare the fitted value λ P with λ exp , the decay rate obtained directly from the experiments. If the two are close, then the EDR is a strong determinant for the measure P of the cortical network. Thus, the extent a particular measure in the EDR model and in the data network agree, i.e., |P(G exp ) − 〈P(λ P )〉| with respect to the same comparison with the CDR model, i.e., with |P(G exp ) − 〈P(λ = 0)〉|, expresses the degree to which the EDR influences that particular measure in the cortical network. This analysis is repeated with several local and global network measures. The more measures for which there is an agreement between λ P and λ exp , the stronger the effect of the EDR in shaping the interareal network. This method also has the added advantage of identifying those network properties that are not well described by the EDR, and thus, based on the nature of these measures, providing us with clues for additional network mechanisms. In the macaque, the EDR model predicts very well many local, global and weighted network properties of the interareal network (see [4] for details), and thus it is a strong determinant for the large-scale network organization of the macaque cortex. It also captures its pronounced core-periphery organization (i.e., a densely connected set of areas—core, with feedback and feedforward links to/from a more loosely connected set of peripheral areas), with the core strongly dominated by associative areas [4,40]. The EDR network model of cortical connectivity represents a radical departure from previous, purely topological models of cortical networks, which do not take into account their physical, i.e., weighted and spatially embedded nature, and this has now been well documented in the recent literature [41,42]. The spatial clustering and geometrical positioning of the nodes in the EDR model in the macaque is observed to strongly echo the functional layout of the cortex as revealed by numerous physiological and anatomical studies [36,43].

Core-Periphery Structure in the Mouse Cortex A clique (see glossary in S1 Text) is a complete subgraph of a network, i.e., it carries the maximum number of possible edges between its nodes. In dense graphs (thus with many cliques) the size (number of nodes) distribution of the cliques provides insight into the network’s heterogeneity [4]. The largest cliques in dense graphs can be used to define the cortical network core [4,40]. As in macaque [4,40], the clique distribution analysis in the mouse (Fig 4A) reveals a distinct core-periphery structure. The mouse connectome, , includes a dense core of 12 nodes organized into the two largest cliques each of size 11, plus a periphery of 21 nodes. There are a total of M cc = 131 links within the core, M cp = 190 links from the core to the periphery, M pc = 170 from periphery to core, and M pp = 228 links within the periphery. Densities for the mouse are the following: core 99% (versus 92% in macaque), periphery 54% (versus 49%) and the links between the core and periphery, 71% (versus 54%). The likelihood of a core having 12 nodes in a random graph on 33 nodes with the same density ρ = 0.681 as in is vanishingly small: (versus 10−17 in the macaque). To see how well the EDR model reproduces the clique distribution, we define a scalar deviation measure σ cl (λ) between the clique-size distributions in the data and the EDR model as the root mean square (RMS) of the clique-count log-ratios. The best agreement between the two distributions is achieved at (Fig 3C) and the clique distributions in the model and data are rather close at this value (Fig 4A). PPT PowerPoint slide

PowerPoint slide PNG larger image

larger image TIFF original image Download: Fig 4. Clique distribution and core-periphery structure in mouse. (a) Clique distribution compared between empirical data, EDR model (λ = 0.93 mm−1, best fit from Fig 3C), CDR model, and a randomized network with the same degree sequence as the data. (b) top, mouse network core composed of two cliques of size 11, shown as two rows of squares, each square representing an area that is part of the clique. Three primary areas are present (MOp, SSp-ll, SSp-tr); bottom, in-degrees of mouse cortical areas. Dots mark core areas, largely centered on the highest in-degree areas, consistent with the macaque [30]. (c) The white arrow (ECT → VISam) shows the single missing link between the 12 members of the core. (d) Flat map of mouse cortex; gray color represents network core members. https://doi.org/10.1371/journal.pbio.1002512.g004 Anatomically, the cortical core in the mouse shows significant differences with that previously reported in macaque, the most striking being that the mouse core includes portions of primary somatosensory cortex (SSp-ll and SSp-tr) and primary motor cortex (MOp) (Fig 4B). While additional injections may well expand the core membership in macaque, primary areas in the macaque core are extremely unlikely, given the rarity of connections linking primary areas [30]. This contrasts with the mouse where the inter-primary area subgraph has a density of over 80% [5,6]. In agreement with the presence of primary areas in the mouse core, the two-dimensional map of the flattened cortex (Fig 4D) shows that the mouse cortical core might be spatially more widespread across brain regions compared to that of the macaque, where the core appears concentrated in frontal and parietal areas [4]. Note that in both mouse and macaque, the core areas have overall, higher in-degrees than non-core areas (Fig 4B for the mouse). The wider spatial spread of the mouse compared to the macaque core may reflect the relative expansion in primates of higher-level association cortex with respect to the primary areas [3]. These differences in the cortical core of the mouse and macaque need to be considered in light of the proposal that in primates at least, the core is related to cognitive architectures such as the global workspace, thought to be involved in consciousness [40,44].