Thresholds for detecting sounds in quiet decrease with increasing sound duration in every species studied. The neural mechanisms underlying this trade-off, often referred to as temporal integration, are not fully understood. Here, we probe the human auditory system with a large set of tone stimuli differing in duration, shape of the temporal amplitude envelope, duration of silent gaps between bursts, and frequency. Duration was varied by varying the plateau duration of plateau-burst (PB) stimuli, the duration of the onsets and offsets of onset-offset (OO) stimuli, and the number of identical bursts of multiple-burst (MB) stimuli. Absolute thresholds for a large number of ears (>230) were measured using a 3-interval-3-alternative forced choice (3I-3AFC) procedure. Thresholds decreased with increasing sound duration in a manner that depended on the temporal envelope. Most commonly, thresholds for MB stimuli were highest followed by thresholds for OO and PB stimuli of corresponding durations. Differences in the thresholds for MB and OO stimuli and in the thresholds for MB and PB stimuli, however, varied widely across ears, were negative in some ears, and were tightly correlated. We show that the variation and correlation of MB-OO and MB-PB threshold differences are linked to threshold microstructure, which affects the relative detectability of the sidebands of the MB stimuli and affects estimates of the bandwidth of auditory filters. We also found that thresholds for MB stimuli increased with increasing duration of the silent gaps between bursts. We propose a new model and show that it accurately accounts for our results and does so considerably better than a leaky-integrator-of-intensity model and a probabilistic model proposed by others. Our model is based on the assumption that sensory events are generated by a Poisson point process with a low rate in the absence of stimulation and higher, time-varying rates in the presence of stimulation. A subject in a 3I-3AFC task is assumed to choose the interval in which the greatest number of events occurred or randomly chooses among intervals which are tied for the greatest number of events. The subject is further assumed to count events over the duration of an evaluation interval that has the same timing and duration as the expected stimulus. The increase in the rate of the events caused by stimulation is proportional to the time-varying amplitude envelope of the bandpass-filtered signal raised to an exponent. We find the exponent to be about 3, consistent with our previous studies. This challenges models that are based on the assumption of the integration of a neural response that is directly proportional to the stimulus amplitude or proportional to its square (i.e., proportional to the stimulus intensity or power).