The mise en scéne of memristive networks: effective memory, dynamics and learning. Issue 4 (4th July 2018) - Caravelli, Francesco
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. Graphical abstract: We discuss the properties of the dynamics of purely memristive circuits using a recently derived consistent equation for the internal memory of the circuit. In particular, we show that the amount of memory in a memristive circuit is constrained by the conservation laws, and that the dynamics preserves these symmetry by means of a projection on this subspace. We obtain these results both for current and voltage controlled linear memristors. Moreover, we discuss other symmetries of the dynamics under various transformations, and study the weak and strong non-linear regimes. In the strong regime, we derive a constrained conservation law for the internal memory. In particular, we are able to show that for the case of purely passive or active systems, the eigenvalues of the Jacobian are always real, implying that oscillations can emerge only for mixtures. 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. These results provide another direct connection between memristors and learning.