  The development of neuromorphic systems based on memristive elements -
resistors with memory - requires a fundamental understanding of their
collective dynamics when organized in networks. Here, we study an
experimentally inspired model of two-dimensional disordered memristive networks
subject to a slowly ramped voltage and show that they undergo a first-order
phase transition in the conductivity for sufficiently high values of memory, as
quantified by the memristive ON/OFF ratio. We investigate the consequences of
this transition for the memristive current-voltage characteristics both through
simulation and theory, and uncover a duality between forward and reverse
switching processes that has also been observed in several experimental systems
of this sort. Our work sheds considerable light on the statistical properties
of memristive networks that are presently studied both for unconventional
computing and as models of neural networks.
