Network analysis of memristive device circuits: dynamics, stability and correlations

Networks with memristive devices are a potential basis for the next generation of computing devices. They are also an important model system for basic science, from modeling nanoscale conductivity to providing insight into the information-processing of neurons. The resistance in a memristive device depends on the history of the applied bias and thus displays a type of memory. The interplay of this memory with the dynamic properties of the network can give rise to new behavior, offering many fascinating theoretical challenges. But methods to analyze general memristive circuits are not well described in the literature. In this paper we develop a general circuit analysis for networks that combine memristive devices alongside resistors, capacitors and inductors and under various types of control. We derive equations of motion for the memory parameters of these circuits and describe the conditions for which a network should display properties characteristic of a resonator system. For the case of a purely memresistive network, we derive Lyapunov functions, which can be used to study the stability of the network dynamics. Surprisingly, analysis of the Lyapunov functions show that these circuits do not always have a stable equilibrium in the case of nonlinear resistance and window functions. The Lyapunov function allows us to study circuit invariances, wherein different circuits give rise to similar equations of motion, which manifest through a gauge freedom and node permutations. Finally, we identify the relation between the graph Laplacian and the operators governing the dynamics of memristor networks operators, and we use these tools to study the correlations between distant memristive devices through the effective resistance.