  The preB\"otzinger Complex, the mammalian inspiratory rhythm generator,
encodes inspiratory time as motor pattern. Spike synchronization throughout
this sparsely connected network generates inspiratory bursts albeit with
variable latencies after preinspiratory activity onset in each breathing cycle.
Using preB\"otC rhythmogenic microcircuit minimal models, we examined the
variability in probability and latency to burst, mimicking experiments. Among
various physiologically plausible graphs of 1000 point neurons with
experimentally determined neuronal and synaptic parameters, directed
Erd\H{o}s-R\'enyi graphs best captured the experimentally observed dynamics.
Mechanistically, preB\"otC (de)synchronization and oscillatory dynamics are
regulated by the efferent connectivity of spiking neurons that gates the
amplification of modest preinspiratory activity through input convergence.
Furthermore, to replicate experiments, a lognormal distribution of synaptic
weights was necessary to augment the efficacy of convergent coincident inputs.
These mechanisms enable exceptionally robust yet flexible preB\"otC attractor
dynamics that, we postulate, represent universal temporal-processing and
decision-making computational motifs throughout the brain.
