Gaussian psychometric function ($P(\mathrm{yes}\mid x)=\Phi ((\mu (x)-c)/\sigma )$)

Let internal evidence be $R=\mu (x)+\epsilon$ with zero-mean Gaussian noise of standard deviation $\sigma$, and respond yes when $R>c$. Taking a distribution tail gives $1-\Phi ((c-\mu )/\sigma )$, which equals the displayed expression by symmetry. Its derivative is $\varphi ((\mu -c)/\sigma ){\mu }^{\prime }(x)/\sigma$. Thus sensory gain, variability and criterion all influence the observed relation, and a sigmoid in one stimulus coordinate need not remain symmetric in another.