  In this study, we analyze the dynamical stability of the D3-D7 model dual to
a holographic conductor with a constant current under an external electric
field. We particularly focus on the stability around the parameter region where
the multivalued relation between the external electric field and the current is
shown due to nonlinear conductivity. The dynamical stability of the states can
be analyzed by considering linear perturbations in the background states and
computing the quasinormal modes. In the multivalued region, we find that the
states in one branch with a low electric current can be dynamically unstable.
The turning point in the $J$-$E$ characteristic coincides with the stability
switching. Further, we also find that the perturbations around the unstable
states can become stable with finite wavenumber. In other words, the
perturbations in the background states become static at a critical wavenumber,
implying the existence of inhomogeneous steady states with current filaments.
