Network analysis of memristive device circuits: dynamics, stability and correlations

TL;DR

This paper develops a comprehensive framework for analyzing memristive circuits, revealing their dynamic behaviors, stability conditions, and correlations, which are crucial for advancing neuromorphic computing and understanding nanoscale systems.

Contribution

It introduces a general circuit analysis method for memristive networks, derives equations of motion, and explores stability and correlation properties using Lyapunov functions and graph theory.

Findings

Memristive circuits can exhibit resonator-like properties.

Lyapunov functions reveal stability conditions and the absence of stable equilibria in some nonlinear cases.

Graph Laplacian relates to memristor dynamics and correlations between devices.

Abstract

Networks with memristive devices are a potential basis for the next generation of computing devices. They are also an important model system for basic science, from modeling nanoscale conductivity to providing insight into the information-processing of neurons. The resistance in a memristive device depends on the history of the applied bias and thus displays a type of memory. The interplay of this memory with the dynamic properties of the network can give rise to new behavior, offering many fascinating theoretical challenges. But methods to analyze general memristive circuits are not well described in the literature. In this paper we develop a general circuit analysis for networks that combine memristive devices alongside resistors, capacitors and inductors and under various types of control. We derive equations of motion for the memory parameters of these circuits and describe the conditions for which a network should display properties characteristic of a resonator system. For the case of a purely memresistive network, we derive Lyapunov functions, which can be used to study the stability of the network dynamics. Surprisingly, analysis of the Lyapunov functions show that these circuits do not always have a stable equilibrium in the case of nonlinear resistance and window functions. The Lyapunov function allows us to study circuit invariances, wherein different circuits give rise to similar equations of motion, which manifest through a gauge freedom and node permutations. Finally, we identify the relation between the graph Laplacian and the operators governing the dynamics of memristor networks operators, and we use these tools to study the correlations between distant memristive devices through the effective resistance.