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The mammalian breathing rhythm putatively originates from Dbx1-derived interneurons in the preBötzinger complex (preBötC) of the ventral medulla. Cumulative deletion of ∼15% of Dbx1 preBötC neurons in an in vitro breathing model stops rhythmic bursts of respiratory-related motor output. Here we assemble in silico models of preBötC networks using random graphs for structure, and ordinary differential equations for dynamics, to examine the mechanisms responsible for the loss of spontaneous respiratory rhythm and motor output measured experimentally in vitro. Model networks subjected to cellular ablations similarly discontinue functionality. However, our analyses indicate that model preBötC networks remain topologically intact even after rhythm cessation, suggesting that dynamics coupled with structural properties of the underlying network are responsible for rhythm cessation. Simulations show that cumulative cellular ablations diminish the number of neurons that can be recruited to spike per unit time. When the recruitment rate drops below 1 neuron/ms the network stops spontaneous rhythmic activity. Neurons that play pre-eminent roles in rhythmogenesis include those that commence spiking during the quiescent phase between respiratory bursts and those with a high number of incoming synapses, which both play key roles in recruitment, i.e., recurrent excitation leading to network bursts. Selectively ablating neurons with many incoming synapses impairs recurrent excitation and stops spontaneous rhythmic activity and motor output with lower ablation tallies compared with random deletions. This study provides a theoretical framework for the operating mechanism of mammalian central pattern generator networks and their susceptibility to loss-of-function in the case of disease or neurodegeneration.