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The pattern of nerve action potentials produced by unit permeability changes (quantal inputs) occurring at random is considered analytically and by computer simulation methods. The important parameters of a quantal input are size and duration. Varying both the mean and the probability density function of these parameters has calculable effects on the distribution of interspike intervals. Particular attention is paid to the relation between the mean rate of excitatory inputs and the mean frequency of nerve action potentials (input-output curve) and the relation between the coefficient of variation for the interval distribution and the mean interval (variability curve). In the absence of action potentials one can determine the parameters of the voltage distribution including the autocorrelation function and the power spectrum. These parameters can sometimes be used to approximate the variability of interspike intervals as a function of the threshold voltage. Different neuronal models are considered including one containing the Hodgkin-Huxley membrane equations. The negative feedback inherent in the Hodgkin-Huxley equations tends to produce a small negative serial correlation between successive intervals. The results are discussed in relation to the interpretation of experimental results.