# A mean-field model of memristive circuit interaction

## TL;DR

This paper presents an exactly solvable mean-field model of interacting memristors, analyzing its dynamics, fixed points, and asymptotic behavior using a Lyapunov functional analogous to a long-range Ising Hamiltonian.

## Contribution

It introduces a novel mean-field approach to model interacting memristors with an exactly solvable circuit and a Lyapunov functional for analyzing non-equilibrium dynamics.

## Key findings

- Exact predictions for decay to lower resistance state
- Reasonable predictions for decay to higher resistance state
- Lyapunov functional resembles a long-range Ising Hamiltonian

## Abstract

We construct an exactly solvable circuit of interacting memristors and study its dynamics and fixed points. This simple circuit model interpolates between decoupled circuits of isolated memristors, and memristors in series, for which exact fixed points can be obtained. We introduce a Lyapunov functional that is found to be minimized along the non-equilibrium dynamics and which resembles a long-range Ising Hamiltonian with non-linear self-interactions. We use the Lyapunov functional as an Hamiltonian to calculate, in the mean field theory approximation, the average asymptotic behavior of the circuit given a random initialization, yielding exact predictions for the case of decay to the lower resistance state, and reasonable predictions for the case of a decay to the higher resistance state.