Nonlinear synaptic patchiness in planar neuronal networks emerges from dendritic intersection lattices under coarse-grained geometry

Spatially embedded networks often exhibit heterogeneous organization that cannot be inferred from element density or total element number alone. In planar neuronal cultures, biological synapses form spatially clustered patterns whose structural origin remains debated. Here we introduce a minimal geometric framework, motivated by reconstructed dendritic networks in planar cultures, to isolate the contribution of network geometry to synaptic organization. Dendrites are modeled as straight line segments embedded in two dimensions, whose overlap generates a lattice of dendritic intersections. Synapses are distributed along dendrites with a uniform linear density and undergo nonlinear enrichment at intersections, with scaling exponents constrained by experimental measurements. We show that both synapse abundance and synaptic patchiness are governed, to leading order, by an effective geometric parameter associated with the dendritic intersection lattice. Increasing lattice complexity produces a monotonic increase in patchiness, whereas total synapse number depends approximately linearly on the same parameter in absolute terms; normalized measures reveal the superlinear contribution of intersection multiplicities. Beyond mean behavior, the density field becomes increasingly heterogeneous, with a subset of regions exhibiting elevated synapse density. Empirical image-based analysis supports this geometric dependence: area-based patchiness increases with local intersection density over the experimentally supported range, the corresponding derivative shows no evidence of a sharp singularity, and patch-size statistics show that regions with higher local intersection density contain systematically larger and more numerous synaptic patches.