  We present a theory of Mott memristors whose working principle is the
non-linear carrier avalanche multiplication in Mott insulators subject to
strong electric fields. The internal state of the memristor, which determines
its resistance, is encoded in the density of doublon and hole excitations in
the Mott insulator. In the current-voltage characteristic, insulating and
conducting states are separated by a negative-differential-resistance region,
leading to hysteretic behavior. Under oscillating voltage, the response of a
voltage-controlled, non-polar memristive system is obtained, with retarded
current and pinched hysteresis loop. As a first step towards neuromorphic
applications, we demonstrate self-sustained spiking oscillations in a circuit
with a parallel capacitor. Being based on electronic excitations only, this
memristor is up to several orders of magnitude faster than previous proposals
relying on Joule heating or ionic drift.
