A predictive network model of cerebral cortical connectivity based on a distance rule

Abstract

Recent advances in neuroscience have engendered interest in large-scale brain networks. Using a consistent database of cortico-cortical connectivity, generated from hemisphere-wide, retrograde tracing experiments in the macaque, we analyzed interareal weights and distances to reveal an important organizational principle of brain connectivity. Using appropriate graph theoretical measures, we show that although very dense (66%), the interareal network has strong structural specificity. Connection weights exhibit a heavy-tailed lognormal distribution spanning five orders of magnitude and conform to a distance rule reflecting exponential decay with interareal separation. A single-parameter random graph model based on this rule predicts numerous features of the cortical network: (1) the existence of a network core and the distribution of cliques, (2) global and local binary properties, (3) global and local weight-based communication efficiencies modeled as network conductance, and (4) overall wire-length minimization. These findings underscore the importance of distance and weight-based heterogeneity in cortical architecture and processing.

Figure 1. Distribution of the Pooled Fraction of Labeled Neurons for the 29 Target Areas

(A) FLN values span five orders of magnitude and follow a lognormal distribution in this density plot. Log₁₀(FLN) values were binned (bin size 0.5), the height of each bin (ordinate) corresponds to the fraction of projections with log₁₀(FLN) falling within that bin divided by the bin width. Blue, Gaussian fit with mean at μ_Gauss = − 3.17 (location parameter) and SD of σ_Gauss = 1.42 (scale parameter), both in units of log₁₀(FLN). (B) Right tail (large FLN values) of the distribution exhibits a slow, power-law decay as shown by the double logarithmic plot. Blue line in (B), right tail of the lognormal in (A). This is also a density plot, as in (A). In this case, the binning was done directly on the FLN values with a bin width of 0.025 FLN. With this choice for bin width, the right tail of the distribution is formed by those high FLN values that fall outside of one sigma (σ_Gauss = 1.42) in (A).

Figure 2. Projection Length Distributions in the Macaque Cortex

(A) FLN values (log f_ij) for all 1,615 projections as a function of projection length (d_ij) estimated through the white matter. Red circles, averages within 5 mm distance bins; the red line is an exponential fit to all the black points giving a decay rate of λ_FLN = 0.150 mm⁻¹. (B) Histogram of interareal projection length for all labeled neurons (n = 6,494,974). Blue line, exponential fit with decay rate λ_d = 0.188 mm⁻¹, also shown in (A). (C) Distribution of interareal distances in G_29×91 matrix, a purely geometrical property, is best approximated by a Gaussian (mean 〈d〉 = μ = 26.57 mm; SD σ = 10:11 mm).

Figure 3. Distance Rules-Based Network Models of the Cortex

(A) The only model parameter λ here is set by setting the number of unidirectional links M₁ to that in the data (Markov et al., 2012). (B) Motif fractions in the EDR and CDR models and data. Statistics were carried out on 1,000 random graph realizations; error bars show the SD. (C) Logarithm of motif ratio counts between model and data. (D) The SD of the deviations in (C) as function of λ, optimal agreement (minimum σ_Δ) is at λ_Δ = 0.180 mm⁻¹.

Figure 4. Prediction of Degree Distributions and Eigenvalue Spectra

(A and B) In- and out-degree distributions for unidirectional connections observed in data (red and blue bars, respectively) and the degree distribution for bidirectional connections (black bars). Curves: predictions of the two random graph models (CDR, EDR). EDR (A) better describes the experimental data than does the CDR (B), by capturing both the central tendency and the spread of the degrees. (C) Absolute G_29×29 graph spectra values for EDR and CDR models. Statistics were carried out on 200 random graph realizations; error bars show SD. Abscissa, i index of eigenvalue; ordinate, absolute value of the eigenvalue |θ_i|.

Figure 5. Core-Periphery Structure

(A) The 17 areas participating in the 13 10-clicks. (B) White arrows, missing links between the 17 nodes of the core. (C) Distribution of clicks, showing the average number of clicks of a given size in the model graphs (EDR, CDR), the degree-preserving randomly rewired graph (RAND), and the actual graph G_29×29 (black). The averages were obtained from 10³ realizations of the model graphs. The inset is a magnification of the corresponding region, showing that the EDR is the only model close to the value in the data. (D) FLN distributions within the core-periphery structure of G_29×29. See also Figure S1.

Figure 6. 2D Cortical Surface Map Localizing Core and Periphery

Areal members of the core are shown in red and of the periphery in yellow. See bottom right for areal names (see Markov et al., 2012 for abbreviations).

Figure 7. Global and Local Communication Efficiency

(A and B) Effects of graph density via sequentially deleting weak (blue, green) and strong (black, red) links. Data comparison with (A) EDR model, dashed lines and (B) CDR model, dashed lines. (C–E) Weight-based layout and high-capacity backbone. Kamada-Kawaii, force-based graph-drawing algorithm reveals optimal layout. In this algorithm, links are springs with strengths proportional to the link weight. (C) Full density (all 536 links), all weights taken as unity (binary graph). (D) As in (C), with link weights given by their FLN values. Note the strong clustering by functional lobes. (E) Blue links, 130 strongest connections (0.16 density) left after weak link (thin gray) removal (indicated by the black arrow in A). The high-density core is encircled by the orange curve in (D). See also Figure S2. (See Supplemental Information for definitions.)

Figure 8. Wire Minimization

(A) Histograms of wire lengths (Λ) computed from 2 × 10⁵ random node position permutations of the G_29×29 for binary (left) and FLN-weighted (right) matrices. The wire lengths in the data shown as solid vertical lines. In each case, the wire length of the data is smaller than all random permutations. (B) Histograms of wire lengths obtained from all 29 × 28/2 = 406 transpositions of pairs of areas from the G_29×29 graph for the binary (left) and weighted (right) matrices. Solid vertical lines are as in (A). In the binary case, 27 transpositions lead to marginally shorter wire lengths, while for the weighted case there are only 3. (C) For each of the 50 EDR and 50 CDR networks generated, we performed 2,000 random node permutations. Box plots of the ratios of the wire lengths for each of the 50 networks and their permutations show that EDR generates networks with 30% shorter wire lengths than a random permutation of the same network, while the CDR, on average, generates networks with no differences in wire length from their random permutations. The line in each box indicates median value. The error bars, the extreme data points that are no more than 1.5 times the interquartile range (approximately box height); the data points, values that are outside this range. See also Figures S3.