The complex dynamics of memristive circuits: analytical results and universal slow relaxation

Exact equations of motion provide an analytical description of the evolution and relaxation properties of complex memristive circuits.

Physical Review E 95, 22140 (2017)

F. Caravelli, F. Traversa, M. Ventra

Our derived equation may serve as the basis
for the analysis of the relaxation properties of circuits
with memory.

Networks with memristive elements (resistors with memory) are being explored for a variety of applications ranging from unconventional computing to models of the brain. However, analytical results that highlight the role of the graph connectivity on the memory dynamics are still a few, thus limiting our understanding of these important dynamical systems. In this paper, we derive an exact matrix equation of motion that takes into account all the network constraints of a purely memristive circuit, and we employ it to derive analytical results regarding its relaxation properties. We are able to describe the memory evolution in terms of orthogonal projection operators onto the subspace of fundamental loop space of the underlying circuit. This orthogonal projection explicitly reveals the coupling between the spatial and temporal sectors of the memristive circuits and compactly describes the circuit topology. For the case of disordered graphs, we are able to explain the emergence of a power law relaxation as a superposition of exponential relaxation times with a broad range of scales using random matrices. This power law is also universal, namely independent of the topology of the underlying graph but dependent only on the density of loops. In the case of circuits subject to alternating voltage instead, we are able to obtain an approximate solution of the dynamics, which is tested against a specific network topology. These result suggest a much richer dynamics of memristive networks than previously considered.

More in Remembering to learn

  • EPL

    Memristive networks

    A simple solvable model of memristive networks suggests a correspondence between the asymptotic states of memristors and the Ising model.

  • EPL

    From memory to scale-free

    A local model of preferential attachment with short-term memory generates scale-free networks, which can be readily computed by memristors.

  • European Physical Journal B

    Solvable memristive circuits

    Exact solutions for the dynamics of interacting memristors predict whether they relax to higher or lower resistance states given random initialisations.

  • International Journal of Parallel, Emergent and Distributed Systems

    Memristive networks and learning

    Memristive networks preserve memory and have the ability to learn according to analysis of the network’s internal memory dynamics.