  We have carried out the transient nonlinear transport measurements on the
layered cobalt oxide Ca$_3$Co$_{4}$O$_9$, in which a spin density wave (SDW)
transition is proposed at $T_{\rm SDW} \simeq 30$ K. We find that, below
$T_{\rm SDW}$, the electrical conductivity systematically varies with both the
applied current and the time, indicating a close relationship between the
observed nonlinear conduction and the SDW order in this material. The time
dependence of the conductivity is well analyzed by considering the dynamics of
SDW which involves a low-field deformation and a sliding motion above a
threshold field. We also measure the transport properties of the isovalent
Sr-substituted systems to examine an impurity effect on the nonlinear response,
and discuss the obtained threshold fields in terms of thermal fluctuations of
the SDW order parameter.
