Rodents explore their environment through coordinated orofacial motor actions, including whisking. Whisking can free-run via an oscillator of inhibitory neurons in the medulla and can be paced by breathing. Yet, the mechanics of the whisking oscillator and its interaction with breathing remain to be understood. We formulate and solve a hierarchical model of the whisking circuit. The first whisk within a breathing cycle is generated by inhalation, which resets a vibrissa oscillator circuit, while subsequent whisks are derived from the oscillator circuit. Our model posits, consistent with experiment, that there are two subpopulations of oscillator neurons. Stronger connections between the subpopulations support rhythmicity, while connections within each subpopulation induce variable spike timing that enhances the dynamic range of rhythm generation. Calculated cycle-to-cycle changes in whisking are consistent with experiment. Our model provides a computational framework to support longstanding observations of concurrent autonomous and driven rhythmic motor actions that comprise behaviors.