Dynamical phase separation on rhythmogenic neuronal networks

We explore the dynamics of the preB\"{o}tzinger complex, the mammalian central pattern generator with $N \sim 10^3$ 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 2010 :051911]) interacting on a fixed, directed Erd\H{o}s-R\'{e}nyi network. In the all-to-all coupled variant of the model, there is 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, and is partially determined by $k$-cores in the directed graphs. The model has 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 only $\approx 20 \%$ of the network remains. This robustness is consistent with experiment.