Analytically tractable model of synaptic crowding explains emergent small-world structure and network dynamics

Neural circuits must balance local connectivity constraints against the need for global integration. Here we introduce a minimal wiring rule motivated by synaptic crowding: as a neuron accumulates incoming connections, each additional synapse becomes progressively harder to form. This single parameter model yields exact, closed-form predictions for network connectivity statistics at any system size for synchronous threshold dynamics. We show that mean connectivity grows only logarithmically with network size while variance remains bounded consistent with homeostatic regulation of synaptic density. When combined with spatially local wiring, the crowding rule produces small-world networks with broad, multi-scale connection-length distributions, without requiring an explicit distance-dependent wiring law. We further demonstrate that the induced connectivity statistics largely determine attractor basin boundaries in threshold network dynamics, while local clustering primarily modulates the prevalence of additional dynamical states near these boundaries. The model provides falsifiable predictions linking local developmental constraints to macroscopic network organization and dynamics, offering a minimal baseline for interpreting connectomic data and designing sparse artificial networks.