Current-Controlled Negative Differential Resistance due to Joule Heating   in TiO2
  We show that Joule heating causes current-controlled negative differential resistance (CC-NDR) in TiO2 by constructing an analytical model of the voltage-current V(I) characteristic based on polaronic transport for Ohm's Law and Newton's Law of Cooling, and fitting this model to experimental data. This threshold switching is the 'soft breakdown' observed during electroforming of TiO2 and other transition-metal-oxide based memristors, as well as a precursor to 'ON' or 'SET' switching of unipolar memristors from their high to their low resistance states. The shape of the V(I) curve is a sensitive indicator of the nature of the polaronic conduction. 
A. S. Alexandrov, A. M. Bratkovsky, B. Bridle, S. E. Savel'ev, D. B.   Strukov and R. Stanley Williams
cond-mat.mes-hall quant-ph