Automatic responses enable us to react quickly and effortlessly, but they often need to be inhibited so that an alternative, voluntary action can take place. To investigate the brain mechanism of controlled behavior, we investigated a biologically-based network model of spiking neurons for inhibitory control. In contrast to a simple race between pro- versus anti-response, our model incorporates a sensorimotor remapping module, and an action-selection module endowed with a Stop process through tonic inhibition. Both are under the modulation of rule-dependent control. We tested the model by applying it to the well known antisaccade task in which one must suppress the urge to look toward a visual target that suddenly appears, and shift the gaze diametrically away from the target instead. We found that the two-stage competition is crucial for reproducing the complex behavior and neuronal activity observed in the antisaccade task across multiple brain regions. Notably, our model demonstrates two types of errors: fast and slow. Fast errors result from failing to inhibit the quick automatic responses and therefore exhibit very short response times. Slow errors, in contrast, are due to incorrect decisions in the remapping process and exhibit long response times comparable to those of correct antisaccade responses. The model thus reveals a circuit mechanism for the empirically observed slow errors and broad distributions of erroneous response times in antisaccade. Our work suggests that selecting between competing automatic and voluntary actions in behavioral control can be understood in terms of near-threshold decision-making, sharing a common recurrent (attractor) neural circuit mechanism with discrimination in perception.