Here we analyze the problem of determining whether experimentally measured spontaneous miniature end-plate currents (MEPCs) indicate that quanta are composed of subunits. The properties of MEPCs at end plates with or without secondary clefts at the neuromuscular junction are investigated, using both stochastic and deterministic models of the action of a quantum of transmitter. It is shown that as the amount of transmitter in a quantum is increased above about 4000 acetylcholine (ACh) molecules there is a linear increase in the size of the MEPC. It is possible to then use amplitude-frequency histograms of such MEPCs to detect a subunit structure, as there is little potentiation effect above 4000 ACh molecules. Autocorrelation and power spectral analyses of such histograms establish that their subunit structure can be detected if the coefficient of variation of the subunit size is less than about 0.12 or, if electrical noise is added, about 0.1. Positive gradients relate the rise time and half-decay times of MEPCs to their amplitude, even in the absence of potentiating effects; these gradients are shallower at motor nerve terminals that possess secondary clefts. The effect of asynchronous release of subunits is also investigated. The criteria determined by this analysis for identifying a subunit composition in the quantum are applied to an amplitude-frequency histogram of MEPCs recorded from a small group of active zones at a visualized amphibian motor-nerve terminal. This did not provide evidence for a subunit structure.