Phase Transitions in Networks of Memristive Elements

The memory features of memristive elements (resistors with memory), analogous to those found in biological synapses, have spurred the development of neuromorphic systems based on them (see, e.g., [1]). In turn, this requires a fundamental understanding of the collective dynamics of networks of memristive systems. Here, we study an experimentally-inspired model of disordered memristive networks in the limit of a slowly ramped voltage and show through simulations that these networks undergo a first-order phase transition in the conductivity for sufficiently high values of memory, as quantified by the memristive ON/OFF ratio. We provide also a mean-field theory that reproduces many features of the transition and particularly examine the role of boundary conditions and current- vs. voltage-controlled networks. The dynamics of the mean-field theory suggest a distribution of conductance jumps which may be accessible experimentally. We finally discuss the ability of these networks to support massively-parallel computation. Work supported in part by the Center for Memory and Recording Research at UCSD. [1] Y.V. Pershin and M. Di Ventra, Proc. IEEE, {\bf 100}, 2071 (2012).