The complex dynamics of memristive circuits: analytical results and universal slow relaxation

TL;DR

This paper provides an analytical framework for understanding the relaxation dynamics of memristive circuits, revealing universal power law behavior and the influence of network topology on memory evolution.

Contribution

It derives an exact matrix equation of motion for memristive circuits and uncovers universal relaxation properties linked to network topology and loop density.

Findings

Power law relaxation emerges as a superposition of exponential times.

Relaxation behavior is independent of specific topology, depending only on loop density.

Approximate solutions for circuits under alternating voltage are validated against specific topologies.

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 a 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 power law relaxation as a superposition of exponential relaxation times with a broad range of scales using random matrices. This power law is also {\it 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 result suggest a much richer dynamics of memristive networks than previously considered.