Complex dynamics of memristive circuits: Analytical results and universal slow relaxation
Abstract
Networks with memristive elements (resistors with memory) are being explored for a variety of applications ranging from unconventional computing to models of the brain. However, analytical results that highlight the role of the graph connectivity on the memory dynamics are still few, thus limiting our understanding of these important dynamical systems. In this paper, we derive an exact matrix equation of motion that takes into account all the network constraints of a purely memristive circuit, and we employ it to derive analytical results regarding its relaxation properties. We are able to describe the memory evolution in terms of orthogonal projection operators onto the subspace of fundamental loop space of the underlying circuit. This orthogonal projection explicitly reveals the coupling between the spatial and temporal sectors of the memristive circuits and compactly describes the circuit topology. For the case of disordered graphs, we are able to explain the emergence of a powerlaw relaxation as a superposition of exponential relaxation times with a broad range of scales using random matrices. This power law is also universal, namely independent of the topology of the underlying graph but dependent only on the density of loops. In the case of circuits subject to alternating voltage instead, we are able to obtain an approximate solution of the dynamics, which is tested against a specific network topology. These results suggest a much richer dynamics of memristive networks than previously considered.
 Publication:

Physical Review E
 Pub Date:
 February 2017
 DOI:
 10.1103/PhysRevE.95.022140
 arXiv:
 arXiv:1608.08651
 Bibcode:
 2017PhRvE..95b2140C
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Nonlinear Sciences  Adaptation and SelfOrganizing Systems
 EPrint:
 12 pages double column (article + supplementary material)