It is still unclear whether the respiratory-like rhythm observed in slice preparations containing the pre-Bötzinger complex is of pacemaker or network origin. The rhythm persists in the absence of inhibition, but blocking pacemaker activity did not always result in rhythm abolition. We developed a computational model of the slice to show that respiratory-like rhythm can emerge as a network property without pacemakers or synaptic inhibition. The key currents of our model cell are the low- and high-threshold calcium currents and the calcium-dependent potassium current. Depolarization of a single unit by current steps or by raising the external potassium concentration can induce periodic bursting activity. Gaussian stimulation increased the excitability of the model without evoking oscillatory activity, as indicated by autocorrelation analysis. In response to hyperpolarizing pulses, the model produces prolonged relative refractory periods. At the network level, an increase of external potassium concentration triggers rhythmic activity that can be attributed to cellular periodic bursting, network properties, or both, depending on different parameters. Gaussian stimulation also induces rhythmic activity that depends solely on network properties. In all cases, the calcium-dependent potassium current has a central role in burst termination and interburst duration. However, when periodic inhibition is considered, the activation of this current is responsible for the characteristic amplification ramp of the emerged rhythm. Our results may explain controversial results from studies blocking pacemakers in vitro and show a shift in the role of the calcium-dependent potassium current in the presence of network inhibition.