Asymptotic Behavior of Memristive Circuits

Area of Science:

• * Condensed matter physics
• * Computational neuroscience
• * Electrical engineering

Background:

• * Memristors show promise for memory and computation, especially with CMOS technology.
• * Their functionality is influenced by nonlinear dynamics and circuit topology.
• * Purely memristive circuits can be utilized for computational tasks.

Purpose of the Study:

• * To demonstrate that purely memristive circuits can be used for computation.
• * To establish a connection between memristive circuits and spin glass physics.
• * To estimate the number of stationary points in memristive circuit computations.

Main Methods:

• * Development of a polynomial Lyapunov function for DC-controlled memristors.
• * Asymptotic approximation of the Lyapunov function using binary variables.
• * Numerical analysis of Erdos-Renyi random circuits and projector matrix elements.
• * Application of concepts from disordered systems and mean-field spin glasses.

Main Results:

• * A polynomial Lyapunov function exists for DC-controlled memristors, mappable to quadratic combinatorial optimization problems.
• * This reveals a parallel between memristive circuits and the Hopfield-Little model.
• * Projector matrix elements in random circuits approximate a Gaussian distribution, scaling with the inverse square root of elements.
• * An upper bound for stationary points is derived based on circuit topology.

Conclusions:

• * Purely memristive circuits offer a viable platform for computation.
• * The study establishes a link between memristive circuit behavior and disordered physical systems.
• * Computational complexity in these circuits can be estimated using circuit topology.