Wiring cost and topological participation of the mouse brain connectome

Results

We studied the topological and spatial properties of the mouse connectome, using data on 461 axonal tract-tracing experiments in wild-type mice, conducted and made publicly available by the Allen Institute for Brain Science (17, 23, 24). We used topographical and developmental mouse-brain ontologies (25) to subdivide the whole mouse brain into 130 bilaterally symmetric cortical and subcortical regions. Each experiment consisted of an anterograde tracer injection into one of the 65 regions on the right side of the brain, followed by whole-brain (intrahemispheric and interhemispheric) mapping of axonal projections. We defined weights of directed interregional axonal projections (edges) as normalized connection densities (NCD), discarded nine nodes on each side of the brain that had no available experiments, and removed spurious connections by using a probabilistic threshold (P < 0.01) based on expert visual review of tract-tracing images (17). We assumed hemispheric symmetry and used the available interhemispheric projections to construct a whole-brain connectome (Fig. 1 A and B, SI Appendix, and Dataset S1).

The resulting anatomical connectivity matrix had 112 nodes, a connection density of 53% (i.e., approximately half of all possible interareal connections existed), and an approximately symmetrical, block-diagonal appearance (Fig. 1D). The interregional NCD weights were highly heterogeneous and followed a log-normal distribution: the strongest edges had NCD four orders of magnitude greater than the weakest edges. The sum of edge weights connecting a given node to the rest of the network (nodal strength) also followed a log-normal distribution. We defined two measures of cost: the axonal distance (in millimeters) of interregional projections, estimated by continuous axonal tract reconstruction from images of tracer propagation, and the axonal wiring cost of interregional projections, computed as the product of axonal distance and the normalized axonal bandwidth. The bandwidth is a marker of axonal cross-sectional area, and axonal wiring cost therefore approximates the normalized volume of a cylindrical axonal projection (Fig. 1 B–F and SI Appendix).

NCD weights of axonal projections w decayed as a function of interregional distance d (17). The form of this weight–distance relationship was significantly better fit by a power law, $w\mathrm{\propto }{d}^{\mathrm{-}\alpha }$, than by an exponential function $w\mathrm{\propto }{e}^{\mathrm{-}\beta d}$ (P < 10⁻⁶, Vuong’s test). The power law for global weight–distance scaling in the connectome had an exponent of 2.05, such that $w\mathrm{\propto }{d}^{\mathrm{-}2.05}$, and clearly differed both from the exponential weight–distance scaling of a 3D spatial lattice (constructed by assigning strongest weights to closest pairs of nodes and, thereby, globally minimizing axonal wiring cost), and from the absent weight–distance scaling of random graphs (constructed by assigning weights to random pairs of nodes) (Fig. 1G). The connectome had a higher probability of high-cost nodes and edges compared with the spatial lattice but a lower probability of high-cost nodes and edges compared with the random graph (Fig. 1 H and I).

We characterized the global organization of the connectome by computing the topological measures of average shortest path length and clustering coefficient, and their normalized ratio known as the small-world index, σ. Simultaneously low path length and high clustering give σ > 1, and define small worldness, a marker of complex network topology. We found σ > 1 across a range of network densities, up to and including the studied 53%, establishing that the dense, weighted, and directed mouse connectome is a small-world network; see SI Appendix for additional discussion.

We detected a stable community structure of the connectome, using a multiscale modular decomposition (Fig. 2, SI Appendix, Table S1, and Movie S1). We found that 48 of the 56 bilaterally symmetric nodes could be reliably classified into four modules. Two of the four modules were further divisible into submodules, reflecting a hierarchically modular organization of functionally specialized brain regions. At the highest level of the hierarchy the four modules comprised: (i) a somatosensory-motor module; (ii) a brainstem-cerebellum module; (iii) an auditory-visual module subtending auditory and visual submodules; and (iv) an olfactory-hippocampal-hypothalamic module subtending olfactory, hippocampal, and hypothalamic submodules. The connectome could also be decomposed into a stable core of high-strength nodes and a periphery of low-strength nodes. The core comprised seven highest-strength nodes (ventral auditory area; anterolateral, anteromedial, lateral, and posterolateral visual areas; temporal and posterior parietal association cortex), all located in the auditory-visual module.

Participation coefficient p, a measure of connection diversity (0 < p < 1), was generally low for nodes reliably assigned to hierarchical modules, p = 0.50 ± 0.17 (SD), indicating that these nodes tended to have more intramodular than intermodular connections. In contrast, eight nodes in the network (orbital, anterior cingulate, posterior parietal association cortex, caudate nucleus, prethalamus, thalamus, pretectum, and midbrain) were not reliably assigned to hierarchical modules and had diverse connections indicated by significantly higher (P < 0.001, Wilcoxon rank-sum test) participation coefficients, p = 0.77 ± 0.07 (SD). We termed these eight nodes “high-participation” (hi-par) hubs. One hi-par hub, posterior parietal association cortex, was also in the core.

Participation positively correlated with nodal degree and wiring cost (Fig. 3 A and B). Participation also correlated with weight-distance power-law exponents, estimated individually for each node, and there was a significant difference (P = 0.011, Wilcoxon rank-sum test) in the average power-law exponent for hi-par nodes, ${\widehat{\mathrm{\alpha }}}_{\mathrm{hi}\text{-}\mathrm{par}}$ = 1.75 ± 0.72 (SD), compared with the average exponent for low-participation (lo-par) nodes, ${\widehat{\mathrm{\alpha }}}_{\mathrm{lo}\text{-}\mathrm{par}}$ = 2.47 ± 0.63 (SD). In other words, connection weights of hi-par nodes decayed slower as a function of distance (Fig. 3 C–E). We used the global weight–distance scaling relationship $w_{\text{global}}\mathrm{\propto }{d}^{\mathrm{-}2.05}$ (Fig. 3D) to algorithmically bipartition the connectome into low-cost and high-cost subnetworks, such that the low-cost subnetwork predominantly contained weights $w_{\mathrm{low}\text{-}\mathrm{cost}}\mathrm{<}{w}_{\text{global}}$, and the high-cost subnetwork predominantly contained weights $w_{\mathrm{hi}\text{-}\mathrm{cost}}\mathrm{>}{w}_{\text{global}}$ (26). The low-cost subnetwork comprised mainly lo-par nodes, and the high cost subnetwork comprised most of the core and hi-par nodes (Fig. 3F and SI Appendix, Table S1).

We used three deterministic network models to simulate the topological organization of the connectome (Fig. 3 G–I). First, the nonparametric cost-minimized spatial lattice generated a moderately similar community structure to the connectome (normalized mutual information, NMI = 0.66) including modules and a core, but no hi-par hubs (Fig. 3H). Second, a one-parameter model counterpart of global cost minimization, $w\mathrm{\propto }{d}^{\mathrm{-}2.05}$, generated a network broadly similar to the lattice. Third, a topologically adaptive model that assigned weights to edges based on each node’s individual weight-distance power-law exponent (and, thus, assigned strong, long, and costly edges to hi-par nodes) generated hierarchical modules, a core, and seven hi-par hubs (Fig. 3I). The topologically adaptive model also showed a more similar community structure (NMI = 0.69) and substantially higher correlations (compared with the lattice) with many other connectome statistics, including connection weight (r = 0.51 in the lattice model, compare r = 0.64 in the topologically adaptive model), nodal degree (r = 0.47 compare r = 0.82), nodal strength (r = 0.73 compare r = 0.90) and nodal participation (r = 0.42 compare r = 0.64).

We finally explored neurobiological substrates of hi-par nodes by using histological and gene-expression measures quantified from independently acquired datasets. We correlated nodal participation with nodal neuronal density, neuronal and nonneuronal cell counts, and cortical thickness, using previously published data for neocortical regions (27) (SI Appendix, Table S12). Participation negatively correlated with neuronal volume density (r = −0.66, P < 0.001), neuronal surface density (r = −0.60, P = 0.002), and fraction of neurons in the total cell population (r = −0.59, P = 0.003); in contrast, strength positively correlated with neuronal volume density (r = 0.64, P = 0.001) (Fig. 4A and SI Appendix, Fig. S9). These results, as well as the related negative correlation between degree and neuronal volume density (r = −0.48, P = 0.02), are compatible with the negative (albeit nonsignificant) correlation between strength and participation of neocortical regions (r = −0.32, P = 0.14), and with simultaneously high neuronal density and low participation of high-strength core regions (SI Appendix, Figs. S9 and S10). All of the above significant associations of participation and strength remained significant by partial correlation analysis which controlled for topological covariates (SI Appendix, Fig. S11).

We used the Allen Institute data on gene expression in adult wild-type mice to estimate the normalized regional expression of 3,380 genes (all genes which were assayed more than once and passed quality control) (23). We explored the association between gene-expression profiles and nodal participation by using partial least squares, a versatile technique that combines dimensionality reduction with regression and is well suited for high-dimensional gene-expression data (28). We found that 48% of the variance (r = 0.69, P < 0.001) in participation was predicted by a linearly weighted combination of gene expression profiles, such that hi-par hubs overexpressed positively weighted genes (and underexpressed negatively weighted genes) (Fig. 4B). Gene ontology analysis of the top 25% most positively weighted genes demonstrated significant enrichment in three biological processes: cognition, learning and memory, and single organism behavior (P < 0.001), and many of the most positively weighted genes are known to be linked to neuronal phenotypes (Fig. 4C and SI Appendix, Tables S3 and S5). In contrast, the bottom 25% most negatively weighted genes were not significantly enriched for specific biological processes, although many of these genes have been individually associated with neurodevelopmental phenotypes (SI Appendix, Table S4).

Discussion

We have reported an extensive topological and spatial analysis of the tract tracing-based mouse brain connectome. The results showed, in support of our original hypotheses, that the connectome had a complex topology that could not be explained entirely by global minimization of wiring cost. More specifically, high-participation hubs of the connectome mediated many connections between modules, had high axonal wiring costs, slow decay of weight with distance, low neuronal density, high expression of genes enriched for cognitive and behavioral functions, and were not present in cost-minimized network models.

Many aspects of the complex topology revealed by this analysis concur with earlier studies of brain networks. Small-world, hierarchically modular, and core-periphery (alternatively known as rich club) organizations have all been previously reported in graph theoretical analyses of macroscale MRI-based brain networks of humans, and the microscale neuronal network of C. elegans (29–31). The hierarchical modular organization is also consistent with prior knowledge of functional localization in the mouse brain. The topological and spatial segregation of visual, auditory, somatosensory, and olfactory regions is not surprising. The auditory-visual module reflects the shared role of auditory and visual cortical areas in monitoring the external environment. Olfactory links to the hippocampus and hypothalamus represent forebrain systems vital to survival for many species in early vertebrate evolution. The integration between mammalian motor and somatosensory cortical systems is strong, and both are linked to the brainstem and cerebellar motor command-and-control systems (25). Moreover, these algorithmically discovered modules accurately reproduce known patterns of developmental gene expression. For example, during early cortical development, the Fgf8 gene is selectively expressed in rostral (motor, somatosensory, visceral sensory) areas, whereas Wnt and BMP genes are selectively expressed in caudal (visual and auditory) areas (32). These and other patterns of developmental gene expression are reflected in the segregated modular affiliation of rostral and caudal areas in the adult connectome.

In contrast to the “classical” modules, the hi-par hubs represent a less immediately familiar grouping. Most hi-par hubs correspond to the cortical (anterior cingulate, posterior parietal association, and orbital regions) and subcortical (striatal, diencephalic, and brainstem regions) components of the medial orbital network of the mouse (33). Hi-par hubs also seem to be closely related to the default-mode network (DMN) observed in functional neuroimaging studies of other species (18, 34). For instance, a recent study of the rat brain (34) identified six cortical regions of the DMN (orbital, anterior cingulate, temporal association, posterior parietal association, retrosplenial, and retrohippocampal). In our analysis, three of these regions (posterior parietal association, orbital, and anterior cingulate cortex) were hi-par hubs, two (posterior parietal and auditory association cortex) were in the core, and one (retrohippocampal region) had a high-participation coefficient, but was reliably assigned to a hierarchical module. We additionally found hi-par hubs in subcortical regions (caudate, diencephalon, and midbrain), which are known to be strongly connected to cortical DMN regions (35) but which have not been consistently reported in previous, cortico-centric, functional MRI studies.

We investigated the relationship between topological and spatial features of the connectome by defining the wiring cost of an axonal projection as the product of axonal distance and axonal bandwidth, a marker of axonal cross-sectional area. This cost metric approximates the normalized volume of a cylindrical axonal projection; by this definition, expensive tracts are longer and/or have greater bandwidth. This is an innovative cost metric in mammalian connectomics, because most previous studies have defined wiring cost more crudely as the interregional Euclidean distance, which is bound to underestimate true connection distance (because axonal tracts are often curved rather than straight, SI Appendix), and neglects the cost implications of variable bandwidth.

We used this more anatomically accurate measure of wiring cost to show that modules were well approximated by network models that minimized cost, but hi-par hubs could only be generated by a topologically adaptive model that allowed longer distance connections to nodes with high participation. These observations are compatible with previous models of the connectome based on the trade-off between wiring cost and topological integration (36, 37). Our findings suggest that functionally specialized and anatomically localized modules may be selected by minimization of wiring cost, but hi-par hubs require a competitive selection pressure for topological integration. We further hypothesize that topological integration, by reducing the number of synaptic junctions that must be traversed to pass a message between regional nodes, will tend to reduce conduction delay. In Ramón y Cajal’s language, the conservation of time competes with the conservation of material in the selection of brain networks (1).

It is increasingly important and tractable to understand the neurobiological—genetic, molecular and cellular—correlates of such topological integration (38, 39). We made substantial progress in this direction by reporting associations among nodal participation, low neuronal density, and high expression of genes enriched for cognition, learning and memory, and behavior, and associated with specific neuronal phenotypes. The association between participation and low neuronal density in neocortical regions of the mouse brain is broadly convergent with prior reports of associations between topological centrality and low neuronal density in the macaque connectome (40) and between topological centrality and low cytoarchitectonic differentiation (partly indicative of reduced width, density, and granularity of cortical layer IV) in the cat connectome (41). Reduced neuronal density has been linked to more extensively space-occupying dendritic arborizations and higher synaptic spine counts (42–46). Thus, the observed association between hi-par topology and reduced neuronal density may be mechanistically explicable in terms of greater synaptic density of neurons in hi-par hubs. The association between nodal participation and expression of genes for “higher order” functions provides independent and convergent support for the economical hypothesis that the high cost of topological integration is justified by the role of hi-par hubs in supporting valuable cognitive, learning, and behavioral functions (47).

The Allen Institute has provided one of the largest and richest databases available for any mammalian connectome, but, like any other experimental data, these largely automated measurements are noisy. We applied strict quality control criteria to the tract tracing data to select the most homogeneous injections and applied a probabilistic threshold to eliminate weak and spurious weights. These and other steps in data analysis inevitably involved making assumptions and choices. In particular, there is a trade-off between spatial resolution and signal-to-noise ratio. Finer grained parcellations of the mouse brain are possible and interesting but are also more noisy. To complement the main results, obtained from a relatively coarse-grained but denoised parcellation, we have extensively explored a range of reasonable methodological options in analyzing these data, and whenever possible cross-validated our results with independent datasets. We are confident that our key findings are robustly representative of the organizational principles of the mouse connectome.

We have reaffirmed that complex topology is a general characteristic of brain networks and that wiring-cost minimization is an important but insufficient explanation of mouse brain connectome topology. Our results are compatible with the economical hypothesis—dating back to Ramón y Cajal’s laws of conservation—that nervous systems are selected by competitive pressures for both cost minimization (conservation of material) and topological integration (conservation of time) (48). The adaptive value of expensive but integrative hubs is reinforced by the observation that topological participation is associated with overexpression of genes enriched for higher order cognitive, learning, and behavioral functions.

Materials and Methods

See SI Appendix, SI Materials and Methods for a full description of materials and methods. The analyses utilized Mouse Connectivity (connectivity.brain-map.org), Developing Mouse Brain (developingmouse.brain-map.org), and Mouse Brain (mouse.brain-map.org) atlases made freely available by the Allen Institute for Brain Science.

Supplementary Material