Minimal models of the inspiratory and sigh breathing rhythms of the preBötzinger complex

The preBötzinger complex of the lower brainstem generates two breathing-related rhythms: one for inspiration on a timescale of seconds, and another that produces larger amplitude sighs on the order of minutes. We hypothesize that these two disparate rhythms emerge in tandem wherein recurrent excitation gives rise to the inspiratory rhythm while a calcium oscillator generates sighs; distinct neuronal populations are not required. We present several mathematical models that instantiate our working hypothesis including: (1) an activity (firing rate) model and (2) a minimal spiking network model. Both modeling frameworks corroborate the single-population rhythmogenic hypothesis.