  A model of mobile, charged ion channels embedded in a biomembrane is
investigated. The ion channels fluctuate between an opened and a closed state
according to a simple two-state reaction scheme whereas the total number of ion
channels is a conserved quantity. Local transport mechanisms suggest that the
ion channel densities are governed by electrodiffusion-like equations that have
to be supplemented by a cable-type equation describing the dynamics of the
transmembrane voltage. It is shown that the homogeneous distribution of ion
channels may become unstable to either a stationary or an oscillatory
instability. The nonlinear behavior immediately above threshold of an
oscillatory bifurcation occuring at finite wave number is analyzed in terms of
amplitude equations. Due to the conservation law imposed on ion channels
large-scale modes couple to the finite wave number instability and have thus to
be included in the asymptotic analysis near onset of pattern formation. A
modified Ginzburg-Landau equation extended by long-wavelength stationary
excitations is established and it is highlighted how the global conservation
law affects the stability of traveling ion channel density waves.
