A mathematical model of the medullary respiratory oscillator, composed of two mutually inhibiting populations (inspiratory and expiratory) of computer-simulated neurons, is presented. Each population consists of randomly interconnected subpopulations of excitatory and inhibitory neurons, is presented. Each population consists of randomly interconnected subpopulations of excitatory and inhibitory neurons. Neuronal coupling is such that either the inspiratory or expiratory population alone is capable of cyclic activity. Weak inhibitory connections between inspiratory and expiratory populations provide satisfactory reciprocating activity independent of the natural frequency of either population alone. Initiation and persistence of rhythmic activity is dependent on a diffused noncyclic excitatory input. Vagal discharge, simulated by phasic inhibition of inspiratory neurons, results in increased respiratory frequency with decreased inspiratory activity. In the absence of simulated vagal discharge, uniform facilitation of synaptic connections increases averaged activities of inspiratory and expiratory populations, with minor effect on frequency. In the presence of simulated vagal discharge, facilitation of synaptic connections increases both frequency and amplitude. The simulated effects of synaptic facilitation, with and without vagal discharge, mimic the physiological response to CO2 in the intact and vagotimized animal.