Asymptotic behavior of memristive circuits and combinatorial optimization

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. We show that a Lyapunov function, polynomial in the internal memory parameters, exists for the case of DC controlled memristors. Such Lyapunov function can be asymptotically mapped to quadratic combinatorial optimization problems. This shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we provide numerical evidence that the distribution of the matrix elements of the couplings can be roughly approximated by a Gaussian distribution, and that they scale with the inverse square root of the number of elements. This provides an approximated but direct connection to 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 Lyapunov function, and provide a scaling formula as an upper bound in terms of the circuit topology only. In order to put these ideas into practice, we provide an instance of optimization of the Nikkei 225 dataset in the Markowitz framework, and show that it is competitive compared to exponential annealing.