A neuron in the brain is like a person handing out business cards at a crowded conference. At first, making acquaintances is easy, but as the address book fills up, each new contact requires more effort: there's little room left. The authors based their model on this 'crowding' rule: the more incoming connections a neuron already has, the harder it is to form another one.

The one-parameter model yielded an exact solution: the average number of connections grows only logarithmically with network size, and the spread remains bounded. This aligns with the fact that the brain allows neither overload nor underload of neurons. When new connections are added in order of spatial proximity, a broad distribution of connection lengths emerges, close to a power law — without any explicit rules favoring long or short connections.

The authors then showed that the statistics of connections determine the boundaries of stable states in the dynamics of threshold networks. Local clustering, in turn, influences the appearance of long-lived unstable states near these boundaries. Thus, simple local mechanics of connection growth explain the macroscopic organization and dynamics of neural circuits.