Asymptotic behavior of memristive circuits

TL;DR

This paper investigates the asymptotic behavior of memristive circuits, showing they can be used for computation through Lyapunov functions and connecting their dynamics to spin glass physics, with implications for circuit topology.

Contribution

It introduces a polynomial Lyapunov function for DC-controlled memristors, links memristive circuits to combinatorial optimization and spin glass models, and estimates stationary points based on topology.

Findings

Existence of a polynomial Lyapunov function for memristive circuits.

Approximation of matrix element distribution with Gaussian in random circuits.

Scaling law for the number of stationary points based on circuit topology.

Abstract

The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence that also purely memristive circuits can be employed for computational purposes. In the present paper we show that a polynomial Lyapunov function in the memory parameters exists for the case of DC controlled memristors. Such Lyapunov function can be asymptotically approximated with binary variables, and mapped to quadratic combinatorial optimization problems. This also shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we show numerically that the distribution of the matrix elements of the projectors can be roughly approximated with a Gaussian distribution, and that it scales with the inverse square root of the number of elements. This provides an approximated but direct connection with the physics of disordered system and, in particular, of mean field spin glasses. Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit. We estimate the number of stationary points of the approximate Lyapunov function and provide a scaling formula as an upper bound in terms of the circuit topology only.