  We derive the spectral index of $f_{NL}$ and its running from isocurvature
single field and investigate the curvaton models with a negative spectral index
of $f_{NL}$ in detail. In particular, a numerical study of the axion-type
curvaton model is illustrated, and we find that the spectral index of $f_{NL}$
is negative and its absolute value is maximized around $\sigma_*=\pi f/2$ for
the potential $V(\sigma)=m^2f^2(1-\cos{\sigma\over f})$. The spectral index of
$f_{NL}$ can be ${\cal O}(-0.1)$ for the axion-type curvaton model. A
convincing detection of a positive $n_{f_{NL}}$ will rule out the axion-type
curvaton model. In addition, we also give a general discussion about the
detectable parameter space for the curvaton model with a polynomial potential.
