Equation 5 was constructed from the product of two functions: an exponential (
Equation 6) and an inverted cumulative Gaussian function (
Equation 7):
\begin{equation}{F_1}\left( {SF} \right) = {e^{\lambda _W^{}*{{\log }_2}(SF)}}\end{equation}
\begin{equation}{F_2}\left( {SF} \right) = 1 - \frac{1}{2}*(1 + erf\left( {\frac{{{\mu _{HC}} - {{\log }_2}SF}}{{sqrt(2)*{\sigma _{HC}}}}} \right)\end{equation}
where λ
W, μ
HC, and σ
HC are three free parameters. Note that the absolute scaling of
Equation 6 is unconstrained, because
Wi is only ever used to calculate the weight ratio,
WR =
\(\frac{{{W_1}}}{{{W_2}}}\), for a given pair of SFs.
WRs of three subjects, shown in
Figure 3A–C, were very well fit by
Equation 5:
r2 = 0.998, 0.995, and 0.996 for subjects BMS, EJF, and TH, respectively.
Figure 3D–F plots
n (color-coded small squares; see color bar to the right of
Figure 3F) as a function of
SF1 and
SF2 in a two-component stimulus
3 for three subjects. Once again, each SF pairing contributed two symbols to the plot, with the same value in symmetrical locations around the main diagonal (the value of
n does not depend on the order of SF
1 and SF
2). Pilot experiments, which tested much higher number of SF pairings (
Appendix C), allowed to suggest a model, describing the dependence of
n upon SF
1 and SF
2, which was then tested in 3 subjects. There are two notable features in the model. First,
n is largest close to the identity line (when the SF ratio is close to 1) and falls off smoothly as the SF ratio increases. We described this with a Gaussian of the
log SF ratio:
\begin{equation}n = {A_n}*{e^{ - {\rm{ }}\frac{{{{\left[ {{{\log }_2}\left( {\frac{{S{F_1}}}{{S{F_2}}}} \right)} \right]}^2}}}{{2*{\sigma _n}^2}}}} + 1\end{equation}