On the relation between summation and facilitation

Summation refers to the small reduction of threshold contrast as a signal is extended in space or time, or when multiple signals are presented simultaneously. Facilitation refers to the large reduction in the threshold contrast increment when it is superimposed on a near-threshold pedestal contrast. Separate hypotheses relate each of these effects to distinct measures of the log-log steepness of the psychometric function. This note shows that for 2-alternative forced-choice (2afc) proportion correct, the Weibull function P(c)= 1-(1-γ)exp[-(c/α)^β] and the d′ power law d′(c) = (c/c_0.76)^b are practically indistinguishable. Furthermore, their steepness parameters are proportional to each other, β = 1.247 b. This implies a simple relation between 2afc summation and facilitation which can be tested without measuring the psychometric function.