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AbstractBreathing behaviour involves the generation of normal breaths (eupnoea) on a timescale of seconds and sigh breaths on the order of minutes. Both rhythms emerge in tandem from a single brainstem site, but whether and how a single cell population can generate two disparate rhythms remains unclear. We posit that recurrent synaptic excitation in concert with synaptic depression and cellular refractoriness gives rise to the eupnoea rhythm, whereas an intracellular calcium oscillation that is slower by orders of magnitude gives rise to the sigh rhythm. A mathematical model capturing these dynamics simultaneously generates eupnoea and sigh rhythms with disparate frequencies, which can be separately regulated by physiological parameters. We experimentally validated key model predictions regarding intracellular calcium signalling. All vertebrate brains feature a network oscillator that drives the breathing pump for regular respiration. However, in air‐breathing mammals with compliant lungs susceptible to collapse, the breathing rhythmogenic network may have refashioned ubiquitous intracellular signalling systems to produce a second slower rhythm (for sighs) that prevents atelectasis without impeding eupnoea.image Key pointsA simplified activity‐based model of the preBötC generates inspiratory and sigh rhythms from a single neuron population.Inspiration is attributable to a canonical excitatory network oscillator mechanism.Sigh emerges from intracellular calcium signalling.The model predicts that perturbations of calcium uptake and release across the endoplasmic reticulum counterintuitively accelerate and decelerate sigh rhythmicity, respectively, which was experimentally validated.Vertebrate evolution may have adapted existing intracellular signalling mechanisms to produce slow oscillations needed to optimize pulmonary function in mammals.