Spontaneous spatial symmetry breaking in excitatory neuronal networks and its effects in sparsely connected networks.

We explore the dynamics of the preBötzinger complex, the mammalian central pattern generator with N ∼ 10³ neurons, which produces a collective metronomic signal that times the inspiration. Our analysis is based on a simple firing-rate model of excitatory neurons with dendritic adaptation (the Feldman Del Negro model [Nat. Rev. Neurosci. 7, 232 (2006), Phys. Rev. E 82, 051911 (2010)]) interacting on a fixed, directed Erdos–Rényi graph. In the all-to-all coupled variant of the model, there is a type of spontaneous symmetry breaking in which some fraction of the neurons become stuck in a high firing-rate state, while others become quiescent. This separation into firing and non-firing clusters persists into more sparsely connected networks, but is now determined by k-cores in the directed graphs. It produces a number of features of the dynamical phase diagram that violate the predictions of mean-field analysis. In particular, we observe in the simulated networks that stable oscillations do not persist in the large-N limit, in contradiction to the predictions of mean-field theory. Moreover, we observe that the oscillations in these sparse networks are remarkably robust in response to killing neurons, surviving until ∼ 20% of the network remains. This is consistent with experiment.