Synaptic crowding as an exactly solvable wiring rule: degree statistics, emergent small-world wiring, and threshold-network basins

Neurons form synapses under limited fan-in (dendritic crowding) and predominantly local encounters. We introduce a minimal crowding rule for directed wiring: when assembling the in-neighborhood of a target, the first edge is accepted and subsequent candidates are accepted with probability exp(−αr), where r is the number of previously accepted incoming edges. This defines an exactly solvable directed-graph ensemble. We derive an exact finite-N recursion and a generating-function iteration for the in-degree distribution Pα(k), implying ⟨k⟩∼α−1 log N and, in the sparse regime, Var(k) ≈(2α)−1. For synchronous threshold dynamics we obtain a heterogeneous mean-field map and a finite-size absorbing Markov approximation that predict basin (hitting) probabilities and show that basin boundaries depend on the full shape of Pα(k), not only ⟨k⟩. In spatial embeddings, proposing candidates in order of distance leaves Pα(k) unchanged but yields broad wiring lengths P(d) ∝1/d, emergent small-world structure, and a Kleinberg-type kernel P(i→j) ∝r−D without imposing a distance law. A shortcut-rewiring interpolation at fixed Pα(k) separates degree-driven from clustering-driven dynamical effects, providing a minimal and analytically controlled baseline for connectomic wiring and attractor stability.