  We study the effective dynamics of black hole horizons in Einstein-Maxwell
theory in a large number of spacetime dimensions $D$. We demonstrate that
horizon dynamics may be recast as a well posed initial value problem for the
motion of a codimension one non gravitational membrane moving in flat space.
The dynamical degrees of freedom of this membrane are its shape, charge density
and a divergence free velocity field. We determine the equations that govern
membrane dynamics at leading order in the large $D$ expansion. Our derivation
of the membrane equations assumes that the solution preserves an SO$(D-p-2)$
isometry with $p$ held fixed as $D$ is taken to infinity. However we are able
to cast our final membrane equations into a completely geometric form that
makes no reference to this symmetry algebra.
