The binomial model in fluctuation analysis of quantal neurotransmitter release.

The mathematics of the binomial model for quantal neurotransmitter release is considered in general terms, to explore what information might be extractable from statistical aspects of data. For an array of N statistically independent release sites, each with a release probability p, the compound binomial always pertains, with <m> = N<p>, p' identical to 1 - var(m)/<m> = <p> (1 + cvp2) and n' identical to <m>/p' = N/(1 + cvp2), where m is the output/stimulus and cvp2 is var(p)/<p>2. Unless n' is invariant with ambient conditions or stimulation paradigms, the simple binomial (cvp = 0) is untenable and n' is neither N nor the number of "active" sites or sites with a quantum available. At each site p = popA, whereas po is the output probability if a site is "eligible" or "filled" despite previous quantal discharge, and pA (eligibility probability) depends at least on the replenishment rate, po, and interstimulus time. Assuming stochastic replenishment, a simple algorithm allows calculation of the full statistical composition of outputs for any hypothetical combinations of po's and refill rates, for any stimulation paradigm and spontaneous release. A rise in n' (reduced cvp) tends to occur whenever po varies widely between sites, with a raised stimulation frequency or factors (tending to increase po's. Unlike <m> and var(m) at equilibrium, output changes early in trains of stimuli, and covariances, potentially provide information about whether changes in <m> reflect change in <po> or in <pA>. Formulae are derived for variance and third moments of postsynaptic responses, which depend on the quantal mix in the signals. A new, easily computed function, the area product, gives noise-unbiased variance of a series of synaptic signals and its peristimulus time distribution, which is modified by the unit channel composition of quantal responses and if the signals reflect mixed responses from synapses with different quantal time course.