  Unraveling the interplay of excitation and inhibition within
rhythm-generating networks remains a fundamental issue in neuroscience. We use
a biophysical model to investigate the different roles of local and long-range
inhibition in the respiratory network, a key component of which is the
pre-B\"otzinger complex inspiratory microcircuit. Increasing inhibition within
the microcircuit results in a limited number of out-of-phase neurons before
rhythmicity and synchrony degenerate. Thus, unstructured local inhibition is
destabilizing and cannot support the generation of more than one rhythm. A
two-phase rhythm requires restructuring the network into two microcircuits
coupled by long-range inhibition in the manner of a half-center. In this
context, inhibition leads to greater stability of the two out-of-phase rhythms.
We support our computational results with in vitro recordings from mouse
pre-B\"otzinger complex. Partial excitation block leads to increased rhythmic
variability, but this recovers following blockade of inhibition. Our results
support the idea that local inhibition in the pre-B\"otzinger complex is
present to allow for descending control of synchrony or robustness to adverse
conditions like hypoxia. We conclude that the balance of inhibition and
excitation determines the stability of rhythmogenesis, but with opposite roles
within and between areas. These different inhibitory roles may apply to a
variety of rhythmic behaviors that emerge in widespread pattern generating
circuits of the nervous system.
