Fluctuation analysis of nonideal shot noise. Application to the neuromuscular junction

Procedures are described for analyzing shot noise and determining the waveform, w(t), mean amplitude, (h), and mean rate of occurrence, (r), of the shots under a variety of nonideal conditions that include: (a) slow, spurious changes in the mean, (b) nonstationary shot rates, (c) nonuniform distribution of shot amplitudes, and (d) nonlinear summation of the shots. The procedures are based upon Rice's (1944. Bell Telephone System Journal. 23: 282-332) extension of Campbell's theorem to the second (variance), lambda 2, third (skew), lambda 3, and fourth, lambda 4, semi-invariants (cumulants) of the noise. It is shown that the spectra of lambda 2 and lambda 3 of nonstationary shot noise contain a set of components that are proportional to (r) and arise from w(t), and a set of components that are independent of (r) and arise from the temporal variations in r(t). Since the latter components are additive and are limited by the bandwidth of r(t), they can be removed by appropriate filters; then (r) and (h) can be determined from the lambda 2 and lambda 3 of the filtered noise. We also show that a factor related to the ratio (lambda 3)2/(lambda 2)(lambda 4) monitors the spread in the distribution of shot amplitudes and can be used to correct the estimates of (r) and (h) for the effects of that spread, if the shape of the distribution is known and if r(t) is stationary. The accuracy of the measurements of lambda 4 is assessed and corrections for the effects of nonlinear summation of lambda 2, lambda 3, and lambda 4 are derived. The procedures give valid results when they are used to analyze shot noise produced by the (linear) summation of simulated miniature endplate potentials, which are generated either at nonstationary rates or with a distribution of amplitudes.