Asymptotic Behavior of Memristive Circuits

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.

Global minimization via classical tunneling assisted by collective force field formation

“…The technical condition lim N → ∞ N 〈 a ij 〉 A = c ij applies in the case of memristive dynamics if we consider the fact that Ω is a projector operator. As shown in ( 41 ), typically, because of the condition Ω 2 = Ω. As such, if we identify A = χΩ X , because we impose 0 < x i < 1 dynamically, and χ < 1, then we have that , which satisfies the technical condition.…”

Memristors for the Curious Outsiders

“…For arbitrary memristor components, the generalization of Equation 36is not known. In the approximation R off = pR on , with p of order one, the equation above can be recast in the form of a (constrained) gradient descent [107], which is reminiscent of the fact that the dynamics of a purely memristive circuit has an approximate Lyapunov function [108,109]. In the simplified setting of purely memristive circuit without any other components, it can be shown that these circuits execute Quadratically Unconstrained Binary Optimization [110].…”

“…Ideas along these lines can be pushed further in order to explore memristors as complex adaptive systems [122] able to self-organize with the guidance of the circuit topology and the control of external voltages. Using Equation (36), it is possible to obtain approximate solutions, for instance, of the combinatorial Markowitz problem [109]. Hybrid CMOS/Memristive circuits can, in principle, tackle harder problems via a combination of external control and self-organization [123].…”

Evaluating Ising Processing Units with Integer Programming

“…The RANF-k problem is a simple variant of RAN-k where the values of h are also selected uniformly at random from (11). As we will later see, RAN-1 and RANF-1, where h, J ∈ {−1, 1}, are an interesting subclass of this problem.…”