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Generative models of the human connectome
Richard F. Betzel, Andrea Avena-Koenigsberger, Joaquín Goñi, Ye He, Marcel A. de Reus, Alessandra Griffa, Petra E. Vértes, Bratislav Mišić, Jean-Philippe Thiran, Patric Hagmann, Martijn van den Heuvel, Xi-Nian Zuo, Edward T. Bullmore, Olaf Sporns
TL;DR
The paper examines generative rules shaping connectome organization by expanding and quantitatively comparing generative models. It finds that the best-fitting model combines homophilic attraction with geometric constraints, reproduces clustering and edge-length distributions, and changes progressively with age.
Problem
Connectome organization is important for shaping brain-network features, but the generative rules underlying it remain incompletely characterized.
Method
The study expands generative-model approaches and quantitatively compares different sets of models incorporating geometric and topological factors.
Results
The best-fitting model combines homophilic attraction, measured by matching index, with geometric constraints and accurately reproduces connectome clustering and edge-length distributions.
Takeaways & Limitations
The model comparison indicates that connectome organization reflects combined geometric and homophilic factors, whose parameters progressively change with age.
Takeaways & Limitations
The discussion identifies exploring alternative energy functions and additional model classes as important directions for future work.
Abstract
from arXiv · showhide
The human connectome represents a network map of the brain's wiring diagram and the pattern into which its connections are organized is thought to play an important role in cognitive function. The generative rules that shape the topology of the human connectome remain incompletely understood. Earlier work in model organisms has suggested that wiring rules based on geometric relationships (distance) can account for many but likely not all topological features. Here we systematically explore a family of generative models of the human connectome that yield synthetic networks designed according to different wiring rules combining geometric and a broad range of topological factors. We find that a combination of geometric constraints with a homophilic attachment mechanism can create synthetic networks that closely match many topological characteristics of individual human connectomes, including features that were not included in the optimization of the generative model itself. We use these models to investigate a lifespan dataset and show that, with age, the model parameters undergo progressive changes, suggesting a rebalancing of the generative factors underlying the connectome across the lifespan.
1. Introduction
The paper asks which generative wiring rules produce human connectomes that balance wiring cost with topological organization. It compares geometric and non-geometric mechanisms, finding that geometry combined with homophily best reproduces observed networks and changes across the lifespan.
- Motivation: Connectome topology is thought to shape task-evoked and spontaneous brain activity.
- Motivation: Because connection cost increases with length, human connectomes contain disproportionately many short-range connections.
- Problem: Wiring minimization alone cannot explain all topological features, including costly hubs and rich clubs.
- Approach: The study fits generative models to individual human connectomes to identify rules producing synthetic networks with observed properties.
- Approach: The models combine geometric distance preferences with topological rules including preferential attachment, assortativity, and homophilic attraction.
- Findings: The best-fitting model penalized long connections while increasing connections between regions with similar connectivity profiles, and its fit worsened with age.
2. Methods
The authors generate synthetic connectomes probabilistically from geometric and topological wiring rules, then optimize model parameters against observed network distributions. They evaluate fitness using a maximum-discrepancy energy and search parameter space with an iterative Monte Carlo procedure.
- Data: The dataset includes 40 healthy participants whose cortex was parcellated into 219 regions, retaining 108 right-hemisphere regions.
- Data: Connectomes were constructed at approximately 10% density using a threshold of 27 streamlines for anatomical connections.
- Generative algorithm: For each unconnected node pair, connection probability combines Euclidean distance E(u, v)^η with a non-geometric relationship K(u, v)^γ.
- Generative algorithm: The exponent η favors short connections when η < 0, whereas γ controls the importance of the non-geometric rule K(u, v).
- Model family: The analysis tests alternative non-geometric rules, including preferential attachment, degree assortativity, and homophily, while restricting models to two components.
- Fitness: Fitness is the maximum Kolmogorov-Smirnov discrepancy across degree, clustering, betweenness, and edge-length distributions.
- Optimization: Parameter optimization uses repeated Monte Carlo sampling, energy evaluation, Voronoi partitioning, and preferential sampling from low-energy regions.
3. Results
Across tested models, combining geometry with topology improved fit to human connectomes, with matching-index homophily performing best. Applying this model to lifespan data revealed age-related parameter changes and poorer fits for older connectomes.
- Model comparison: Twelve dual-factor models outperformed the pure geometric model, with homophily-based models outperforming clustering- and degree-based alternatives.
- Model comparison: The absolute best model used matching-index homophilic attraction, a normalized measure of overlap between two nodes’ neighborhoods.
- MI model: The MI model achieved mean energy E = 0.12 ± 0.02 with η = −0.98 ± 0.37 and γ = 0.42 ± 0.04.
- MI model: The MI model reduced discrepancies across all four energy components relative to the geometric model.
- Cross-validation: For clustering, modularity, path length, and efficiency, synthetic-network scores were always within 5.5% of observed-network scores.
- Lifespan: With age, η decreased in magnitude, indicating a weakening penalty on long-distance connections, while E, KSe, and KSc increased.
- Lifespan: Older connectomes were fit increasingly poorly, primarily because edge-length and clustering-coefficient distributions were mismatched.
4. Discussion
The study compares generative wiring rules for individual human connectomes and finds that geometric constraints combined with homophilic attachment best reproduce many observed network properties. Applying the matching-index model across the lifespan suggests weakening distance penalties and increasing mismatch with age, while methodological limits constrain interpretation.
- Contributions: The study quantitatively compares multiple generative-model classes fitted to individual human connectomes rather than relying on composite or random-network benchmarks.Earlier work often used composite connectivity matrices or compared proposed models only against random generative models.
- Model comparison: The best-fitting model combines geometric constraints with homophilic attachment based on matching index.This adds non-geometric information about pairwise node similarity to distance-based wiring rules.
- Model comparison: The matching-index model reproduced degree, betweenness centrality, clustering, edge-length distributions, path length, efficiency, modularity, and local degree and clustering sequences.Several of these properties were additional graph-theoretic characteristics not included in the optimization energy function.
- Robustness and lifespan: With age, the distance penalty weakened while energy and mismatches in clustering and edge-length distributions increased.The authors interpret this as the matching-index model becoming an increasingly poor model for older participants’ connectomes.
- Model comparison: Pure geometric models performed worst because spatial constraints alone could not reproduce all topological aspects of brain networks.Strong distance penalties also create mismatches in clustering and edge-length distributions.
- Limitations: The lifespan analysis was not intended to model connectome growth and development, and alternative wiring rules, energy functions, parcellations, and network definitions remain open possibilities.Diffusion imaging and tractography can also introduce false-positive and false-negative connections.
5. Appendix
The appendix tests whether the principal generative-model findings persist across alternative datasets, densities, parcellations, and geometric specifications.
- Robustness across datasets: The appendix reproduces the principal findings using alternative datasets.Figures S1-S9 show model energies for the additional datasets, reproducing Figure 2 from the main text.
- Replication datasets: Two replication datasets comprise HCP participants (N = 214) and NKI participants (N = 126).
- Model evaluation: The appendix includes energy distributions for the 100 best-fitting synthetic networks for each model type.
Additional Datasets
Additional datasets and group-average construction extend the analysis across subjects, preprocessing choices, and connectome representations while addressing limitations of conventional composite matrices.
- CHUV variants: CHUV variants maintain approximately 5%, 10%, or 15% network density and include fiber-density thresholding.Fiber density is defined as streamline count divided by the sum of the two regions’ surface areas.
- Rationale: The alternative procedure boosts signal-to-noise by emphasizing connections consistently expressed across subjects.
- Limitations of composite matrices: Conventional group-average matrices overexpress short-range connections because short-range connections are more consistently reconstructed, whereas long-range connections are more error-prone.The conventional method also requires an ad hoc threshold for including connections.
- Group-average construction: The alternative group-average procedure matches the typical single-participant edge-length distribution while retaining the most consistently expressed edges at each length.It estimates the number of connections at each length, then selects the most consistent connections among node pairs at comparable distances.
5.5. CHUV Group-average matrix with fiber length - See Figures S10-S11
This analysis tests fiber length as an alternative to Euclidean distance in geometric wiring models, while noting interpolation uncertainty and strong distance–length correspondence.
- Caveat: Fiber-length interpolation necessarily introduces an additional source of uncertainty because observed fiber lengths exist only for completed streamlines.
- Distance–length relationship: Euclidean distance accounted for 66%, 32%, and 79% of fiber-length variance in CHUV, NKI, and HCP, respectively.
- Modeling assumption: Euclidean distance is retained as a reasonable proxy for connection cost despite representing endpoint separation rather than integrated curved-tract length.
- Alternative distance measure: The analysis tests whether replacing Euclidean distance with fiber length changes the generative-model results.An interpolated fiber-length matrix estimates hypothetical tract lengths for node pairs lacking observed connections.
- Fiber-length interpolation: Hypothetical fiber length is estimated from the mean lengths of connections involving geometric neighbors of the two nodes.The method uses geometric neighbors within E(u, v) × τ, with τ = 0.2; regression coefficients estimate lengths when no such connections exist.
5.6. CHUV Group-average matrix with exponential rule - See Figure S12
The appendix tests an exponential geometric wiring rule and a finer cortical parcellation as alternatives to the main analysis specification.
- Exponential geometric rule: The geometric power-law function is replaced with an exponential function and the analyses are rerun.This tests a proposal that an exponential function better captures the relationship between edge length and connection probability.
- Finer parcellation: The finer-resolution CHUV group-average matrix contains n = 223 cortical parcels in the right hemisphere.The main analysis used an intermediate scale with n = 219 parcels overall and n = 108 in the right hemisphere.
- Finer parcellation: The finer-resolution group-average matrix is generated using the same procedure described earlier.
5.8. Graph theory
This section defines graph-theoretic measures used to characterize network topology, including node-level, global, and community-structure measures.
- Node degree counts a node’s total number of connections in an undirected network.
- Network density is the fraction of existing connections out of all possible connections.
- Degree assortativity measures the extent to which nodes with similar degree connect, commonly using a Pearson correlation of edge-end degrees.
- Clustering coefficient measures the density of a node’s neighborhood through the fraction of connected neighbor pairs.
- Characteristic path length averages shortest-path distances between node pairs, whereas global efficiency averages their reciprocals.
- Network diameter is the longest shortest path between any two nodes.
- Modularity evaluates community structure by comparing within-community connections with connections expected under a null model.
7. Funding information
The paper acknowledges financial and institutional support from multiple foundations, universities, agencies, and fellowship programs.
- The authors report support from the National Science Foundation and the Leenaards Foundation.
- Additional support came from VENI of the Netherlands Organisation for Scientific Research and the Brain Center Rudolf Magnus.
- Other listed sponsors include Chinese national research programs and the Swiss National Science Foundation.
- Several authors received support from the Wellcome Trust, Medical Research Council, and Canadian and Indiana research fellowships.
8. Author contributions statement
The statement assigns responsibility for conceiving, conducting, and analyzing the experiments, contributing data, and reviewing the manuscript.
- R.F.B., P.E.V., E.T.B., and O.S. conceived the experiments.
- R.F.B., A.A-K., B.M., J.G., and O.S. conducted and analyzed the experiments.
- Y.H., M.A.R., A.G., J-P.T., P.H., M.H., and X-N.Z. contributed data.
- All authors reviewed the manuscript.
9. Additional information
The authors disclose one author’s external employment and stock ownership and state that no other competing financial interests were declared.
- E.T.B. is employed half-time by the University of Cambridge and half-time by GlaxoSmithKline.
- E.T.B. holds stock in GlaxoSmithKline.
- The authors declare no other competing financial interests.