  We discuss the electrostatic contribution to the elastic moduli of a cell or
artificial membrane placed in an electrolyte and driven by a DC electric field.
The field drives ion currents across the membrane, through specific channels,
pumps or natural pores. In steady state, charges accumulate in the Debye layers
close to the membrane, modifying the membrane elastic moduli. We first study a
model of a membrane of zero thickness, later generalizing this treatment to
allow for a finite thickness and finite dielectric constant. Our results
clarify and extend the results presented in [D. Lacoste, M. Cosentino
Lagomarsino, and J. F. Joanny, Europhys. Lett., {\bf 77}, 18006 (2007)], by
providing a physical explanation for a destabilizing term proportional to
$\kps^3$ in the fluctuation spectrum, which we relate to a nonlinear ($E^2$)
electro-kinetic effect called induced-charge electro-osmosis (ICEO). Recent
studies of ICEO have focused on electrodes and polarizable particles, where an
applied bulk field is perturbed by capacitive charging of the double layer and
drives flow along the field axis toward surface protrusions; in contrast, we
predict "reverse" ICEO flows around driven membranes, due to curvature-induced
tangential fields within a non-equilibrium double layer, which hydrodynamically
enhance protrusions. We also consider the effect of incorporating the dynamics
of a spatially dependent concentration field for the ion channels.
