Heavy-tailed neuronal connectivity arises from Hebbian self-organization

In networks of neurons, the connections are heavy-tailed, with a small number of neurons connected much more strongly than the vast majority of pairs. Yet it remains unclear whether, and how, such heavy-tailed connectivity emerges from simple underlying mechanisms. Here we propose a minimal model of synaptic self-organization: connections are pruned at random, and the synaptic strength rearranges under a mixture of Hebbian and random dynamics. Under these generic rules, networks evolve to produce scale-free distributions of connectivity strength, with a power-law exponent γ = 1 + 1/p that depends only on the probability p of Hebbian (rather than random) growth. Extending our model to include correlations in neuronal activity, we find that clustering (another ubiquitous feature of neuronal networks) also emerges naturally. We confirm these predictions in the connectomes of several animals, suggesting that heavy-tailed and clustered connectivity may arise from general principles of self-organization, rather than the particulars of individual species or systems.