  The ionization state of the gas plays a key role in the MHD of protoplanetary
disks. However, the ionization state can depend on the gas dynamics, because
electric fields induced by MHD turbulence can heat up plasmas and thereby
affect the ionization balance. To study this nonlinear feedback, we construct
an ionization model that includes plasma heating by electric fields and impact
ionization by heated electrons, as well as charging of dust grains. We show
that when plasma sticking onto grains is the dominant recombination process,
the electron abundance in the gas decreases with increasing electric field
strength. This is a natural consequence of electron-grain collisions whose
frequency increases with electron's random velocity. The decreasing electron
abundance may lead to a self-regulation of MHD turbulence. In some cases, not
only the electron abundance but also the electric current decreases with
increasing field strength in a certain field range. The resulting N-shaped
current--field relation violates the fundamental assumption of the
non-relativistic MHD that the electric field is uniquely determined by the
current density. At even higher field strengths, impact ionization causes an
abrupt increase of the electric current as expected by previous studies. We
find that this discharge current is multi-valued (i.e., the current--field
relation is S-shaped) under some circumstances, and that the intermediate
branch is unstable. The N/S-shaped current--field relations may yield
hysteresis in the evolution of MHD turbulence in some parts of protoplanetary
disks.
