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

TL;DR

This paper analyzes the dynamics of memristive circuits, revealing how conservation laws and symmetries constrain memory states, and demonstrates that their internal memory evolution can be viewed as a constrained gradient descent related to learning.

Contribution

It introduces a comprehensive framework for understanding memristive circuit dynamics, including conservation laws, symmetries, and a connection to gradient descent learning algorithms.

Findings

Memory states are constrained by circuit conservation laws.

Dynamics preserve symmetries through projection on physical subspace.

Internal memory dynamics can be interpreted as constrained gradient descent.

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 symmetries by means of a projection on the physical subspace. Moreover, we discuss other symmetries of the dynamics under various transformations of the internal memory, and study the linearized and strongly non-linear regimes of the dynamics. In the strongly non-linear regime, we derive a conservation law for the internal memory variables. We also provide a condition on the reality of the eigenvalues of Lyapunov matrices describing the linearized dynamics close to a fixed point. We show that the eigenvalues ca be imaginary only for mixtures of passive and active components. Our last result concerns the weak non-linear regime. We show 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.