The human cortex contains more than 16 billion neurons linked by approximately 150,000 kilometers of axons, forming roughly 164 trillion synapses. How such an astronomically complex network organizes itself, neither completely random nor rigidly regular, has been one of neuroscience’s deepest puzzles.
A study published August 20 in Cell proposes a surprisingly simple answer: the brain’s wiring is constrained by the same geometric principles that govern standing waves on a drumhead.
Researchers at Monash University’s Turner Institute for Brain and Mental Health developed a mathematical model called the Geometric Eigenmode Model (GEM) that treats the cortex as a continuous surface in which neural activity propagates as waves. Just as a drumhead has preferred resonant frequencies, spatial patterns of vibration determined by its shape and tension, the cortex has preferred spatial resonances called geometric eigenmodes. These are the natural standing wave patterns of neural activity, each characterized by a specific spatial wavelength and a topographical layout of nodes (points that remain stationary) and antinodes (points of maximal oscillation).
The model’s central hypothesis is that connections between cortical regions are preferentially preserved and strengthened when they link locations that oscillate in-phase, locations near the antinodes of the same eigenmode. Such connections act as communication channels that reduce the energy needed to excite that mode, making the brain’s dynamics more efficient.
“The model recapitulates key aspects of binary and weighted empirical connectome topology and topography in human and non-human species,” the authors write. “The findings point to a fundamental role of geometry in shaping the multiscale architecture of cortical connectomes that has been conserved across 90 million years of evolution.”
How the model works
The GEM is derived from neural field theory, a class of mathematical models that treat large-scale neuronal dynamics as waves traveling through a continuous cortical sheet. The model requires only two free parameters: k, the number of eigenmodes included (the model used 108 out of a possible 4,386, roughly 2.5%), and rs, a length scale parameter that controls how strongly long-wavelength modes contribute relative to shorter ones.
When fitted to a group-average human connectome mapped using diffusion MRI from 339 healthy adults, the GEM achieved remarkable accuracy. It captured the rank ordering of connection strengths with a correlation of ρ = 0.81, and successfully predicted the spatial topography of network hubs, which brain regions are most highly connected, with spatial correlations of ρ = 0.52 for binary and ρ = 0.47 for weighted degree. It achieved a true-positive rate of 75% for predicting the presence or absence of connections.
Crucially, the model outperformed all existing alternatives: a simple exponential distance rule (connections are more likely between nearby regions), a model based purely on the cortex’s geometric eigenmodes without the neural field theory weighting, and random networks. When validated in held-out test data through split-half cross-validation, the GEM’s superior performance held.
Across species and scales
The model’s power extends far beyond the human brain. The researchers tested it against connectomes from chimpanzees, macaques, marmosets, and mice, mapped using either diffusion MRI or invasive viral tract tracing, methods that operate at fundamentally different spatial scales.
In every species, the GEM reproduced the organization of the cortical connectome with high fidelity. Because the last common ancestor of these five species lived approximately 90 million years ago, the results suggest that the influence of geometry on cortical wiring is a universal, deeply conserved feature of mammalian brain organization.
The model also captured features that were not directly optimized in its fitting, including the modular community structure of cortical networks, the tendency of brain regions to organize into densely interconnected subgroups, and the spectral properties of the connectome.
What it means
Previous models of brain wiring have emphasized the exponential distance rule, the observation that connections become less likely as the distance between regions increases. This rule has been proposed as a universal principle of connectome organization, arising from the metabolic cost of long axons. The GEM does not contradict this principle, but it goes further: it explains not just which regions might be connected based on distance, but exactly which pairs of locations at any given distance are most likely to form strong connections.
“Within our model, the contribution of each eigenmode reflects its relative expression in the assumed mean-field neural dynamics,” the authors explain. Low-order (long-wavelength) eigenmodes dominate because they require less energy to excite, an energy dependence analogous to the Boltzmann distribution in thermodynamics.
The researchers note several important caveats. The model currently considers only intra-hemispheric cortico-cortical connections, excluding inter-hemispheric connections and connections with subcortical structures like the thalamus. It assumes a uniform spatial length scale across the cortex, when in reality different neuronal populations have different characteristic propagation scales. And the model captures the steady-state, time-averaged architecture, not the dynamic co-evolution of geometry and connectivity during development.
Nevertheless, the study provides the strongest evidence to date that the complex, seemingly idiosyncratic wiring of the mammalian brain can be understood through a remarkably simple geometric principle.
Sources
- Normand, F., Gajwani, M., Cao, T., et al. “Geometric constraints on the architecture of mammalian cortical connectomes.” Cell 189, 1-21 (2026). DOI: 10.1016/j.cell.2026.05.048. https://doi.org/10.1016/j.cell.2026.05.048
- Code and data: https://github.com/francisnormand/GeometricEigenModeModel and https://osf.io/rz3hw/