Why the Same Dose of Botulinum Toxin A Is Less Predictable in Small Muscles: A Discrete Threshold Model

Pharmacodynamic models of botulinum toxin type A map a mean concentration onto a smooth dose-response curve, yet the outcome is binary and arises from a finite population of near-threshold motor units. We analyse a discrete model in which N units are silenced once local SNAP-25 cleavage exceeds a unit-specific threshold, and a functional block of neuromuscular transmission follows once the silenced fraction exceeds a collective threshold. Whether that collective threshold is a fraction of the population or a fixed count changes the location of the dose-response curve and not its steepness: under a fraction, potency is invariant to unit number at fixed dose-concentration gain, and the transition contracts in absolute dose where fixed-count architectures require it to widen. The width contracts as N-1/2, an exponent that is prior and not claimed here. One consequence needs no measurement: a quantal terminal response caps the silenced fraction below unity, so any fixed count fails above a target size it determines, whereas the toxin block targets differ by orders of magnitude. A second reinterprets existing data, flat cohort dose-response curves being what a near-step individual response predicts once convolved with between-subject dispersion. Outcome is accordingly less repeatable in the smallest targets. No parameter is fitted.