The mise en scéne of memristive networks: effective memory, dynamics and learning

Abstract

We discuss the properties of the dynamics of purely memristive circuits using a recently derived consistent equation for the internal memory variables of the involved memristors. In particular, we show that the number of independent memory states in a memristive circuit is constrained by the circuit conservation laws, and that the dynamics preserves these symmetry by means of a projection on the physical subspace. Moreover, we discuss other symmetries of the dynamics under various transformations of the involved variables, and study the weak and strong non-linear regimes of the dynamics. In the strong regime, we derive a conservation law for the internal memory variable. We also provide a condition on the reality of the eigenvalues of Lyapunov matrices. The Lyapunov matrix describes the dynamics close to a fixed point, for which show that the eigenvalues can be imaginary only for mixtures of passive and active components. Our last result concerns the weak non-linear regime, showing that the internal memory dynamics can be interpreted as a constrained gradient descent, and provide the functional being minimized. This latter result provides another direct connection between memristors and learning.