Function aligns with geometry in locally connected neuronal networks

The geometry of the brain imposes fundamental constraints on neuronal network organization and dynamics, yet how these constraints give rise to observed patterns of brain activity remains unclear. Here, we investigate how geometric eigenmodes relate to functional connectivity gradients in three-dimensional neural systems using a combination of generative network simulations and cellular-resolution calcium imaging in larval zebrafish. We show that functional connectivity gradients emerging from network activity naturally align with the geometric eigenmodes of the underlying spatial embedding when connectivity is predominantly local. By systematically increasing the prevalence of long-range connections, we reveal a robust geometry-function correspondence that progressively deteriorates as local connectivity is disrupted. We then show that spatial filtering can artificially imprint geometric patterns on functional gradients, highlighting an important methodological confound. To validate our computational results, we conduct volumetric calcium imaging experiments at cellular resolution in the optic tectum of zebrafish larvae, uncovering functional gradients that closely align with geometric eigenmodes. As predicted from simulations, the eigenmode-gradient mapping exhibits a cutoff point that quantitatively reflects the spatial extent of the regions connectivity kernel, inferred from single-neuron morphologies. This geometry-functional alignment disappears at brain-wide scale, where long-range connections are more prevalent. Our findings demonstrate how short-range anatomical connectivity anchors functional connectivity gradients to the brains geometry.