Geometric developmental principles for the emergence of brain-like weighted and directed neuronal networks

TL;DR

The results reveal that activity-independent geometric constraints during neural development can account for the conserved architectural principles observed across evolutionarily distant species, suggesting universal mechanisms governing neural circuit assembly.

Abstract

Brain networks exhibit remarkable structural properties, including high local clustering, short path lengths, and heavy-tailed weight and degree distributions. While these features are thought to enable efficient information processing with minimal wiring costs, the fundamental principles that generate such complex network architectures across species remain unclear. Here, we analyse single-neuron resolution connectomes across five species (C. Elegans, Platynereis, Drosophila M., zebrafish and mouse) to investigate the fundamental wiring principles underlying brain network formation. We show that distance-dependent connectivity alone produces small-world networks, but fails to generate heavy-tailed distributions. By incorporating weight-preferential attachment, which arises from spatial clustering of synapses along neurites, we reproduce heavy-tailed weight distributions while maintaining small-world topology. Adding degree-preferential attachment, linked to the extent of dendritic and axonal arborization, enables the generation of heavy-tailed degree distributions. Through systematic parameter exploration, we demonstrate that the combination of distance dependence, weight-preferential attachment, and degree-preferential attachment is sufficient to reproduce all characteristic properties of empirical brain networks. Our results reveal that activity-independent geometric constraints during neural development can account for the conserved architectural principles observed across evolutionarily distant species, suggesting universal mechanisms governing neural circuit assembly.

Figures (8)

• Figure 1: Empirical connectomes deviate from random networks.a) Visualization of the empirical connectome from the mouse visual cortexmicrons_2025, neurons were placed at random locations on a lattice. The size of each node is determined by the degree, and the size of the edges by the connection weight. b) Schematic of properties found in all empirical connectomes. c) Pairplot of network properties: local clustering $\mathcal{C}$, average path length $\mathcal{L}$, weight heavy-tailedness $\mathcal{H}_w$, and degree heavy-tailedness $\mathcal{H}_k$. Properties of random networks (M=1840 sample networks of N=100 neurons per network, each sample has a different density $\rho \in [0.01, 0.95]$) and of empirical networks shown.
• Figure 2: Distance dependence leads to high clustering and short path lengths, but not heavy-tailed degrees nor weights.a) Schematic representation of distance-dependent connection probability with an exponential decay kernel. b) Distribution of synapses found at a given distance in the mouse visual cortex connectome. The empirical distribution (filled histogram) is well approximated by the combination of the expected distances within a bound sphere and an exponential decay kernel (bold line). c) Scatter plot of the clustering $\mathcal{C}$ and average path length $\mathcal{L}$ produced by the $D$ model with varying $\lambda$, especially tuned for mouse. d) Distributions of weight (left), outdegree (centre), and indegree (right) for the mouse visual cortex, random Poisson model and $D$ model. For the models, the distributions from the best fit parametrization are shown. Both models fail to match the distributions from the mouse connectome.
• Figure 3: Weight-preferential attachment principle.a) Schematic of expected synapse distribution. Synapses originating from the same pre-synaptic neuron are expected to form with the same neurites, and therefore in close proximity to each other. b) Distribution of distance between synapses for all neurons in the mouse visual cortexmicrons_2025. The distance between synapses with the same pre-synaptic neuron is significantly smaller than the full distribution of distances between all synapses. c) Scatter plot of the clustering $\mathcal{C}$ and average path length $\mathcal{L}$ produced by the $D+W$ model. $N=100$ realizations with the same parameters. d) Distributions of weight (left), outdegree (centre), and indegree (right) for the mouse visual cortex and the best fit $D+W$ model.
• Figure 4: Degree-preferential attachment principle.a) Correlation between the total axonal length and outdegrees in the mouse visual cortex. b) Correlation between the total dendritic length and indegrees in the mouse visual cortex. Insets at the top of panels a and b are example morphologies from the data points indicated with stars. c) Scatter plot of the clustering $\mathcal{C}$ and average path length $\mathcal{L}$ produced by the $D+W+K$ model. $N=100$ realizations with the same parameters. d) Distributions of weight (left), outdegree (centre), and indegree (right) for the mouse visual cortex and the best fit $D+W+K$ model.
• Figure 5: Parameter scans reveal capabilities and limitations of each model.a) Error of best-fit model for each model and empirical connectome. Lower errors indicate a better match between the model and empirical networks. b) Results of parameter scan for the mouse visual cortexmicrons_2025. Each extension of the model increases the range of possible networks, but only $D+W+K$ can produce networks matching all the properties of the empirical network. c) Summary of whether each model can produce each of the four properties. "No*" dennotes that these properties cannot be achieved simultaneously with those marked as "Yes". For example, the $D$ model can generate networks with brain-like $\mathcal{L}$ and $\mathcal{H}_w$, but then $\mathcal{C}$ is incorrect.