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

Abstract

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 {\it 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.

arXiv comment12 pages double column (article + supplementary material); new results

References in corpus (18)

• Emergent complex neural dynamics
• Circuit elements with memory: memristors, memcapacitors and meminductors
• Memory and information processing in neuromorphic systems
• Memory effects in complex materials and nanoscale systems
• Nanobatteries in redox-based resistive switches require extension of memristor theory
• Memristive model of amoeba's learning
• Phase-transition driven memristive system
• Memcomputing: a computing paradigm to store and process information on the same physical platform
• Solving mazes with memristors: a massively-parallel approach
• Universal Memcomputing Machines
• Polynomial-time solution of prime factorization and NP-hard problems with digital memcomputing machines
• Memcomputing NP-complete problems in polynomial time using polynomial resources and collective states
• How memory generates heterogeneous dynamics in temporal networks
• Self-organization and solution of shortest-path optimization problems with memristive networks
• Memory models of adaptive behaviour
• Scale-free networks as an epiphenomenon of memory
• Complex dynamics and scale invariance of one-dimensional memristive networks
• Phase Transitions and Duality in Adiabatic Memristive Networks

Cited by in corpus (15)

• Memcomputing: Leveraging memory and physics to compute efficiently
• Taming a non-convex landscape with dynamical long-range order: memcomputing Ising benchmarks
• Mean field theory of self-organizing memristive connectomes
• Locality of interactions for planar memristive circuits
• Artificial Spin Ice Phase-Change Memory Resistors
• Asymptotic behavior of memristive circuits
• Ergodicity, lack thereof, and the performance of reservoir computing with memristive networks
• Memristive Networks: from Graph Theory to Statistical Physics
• The mise en scene of memristive networks: effective memory, dynamics and learning
• The Computational Capacity of LRC, Memristive and Hybrid Reservoirs
• A mean-field model of memristive circuit interaction
• A scalable method to find the shortest path in a graph with circuits of memristors
• Projective Embedding of Dynamical Systems: uniform mean field equations
• Fully analog memristive circuits for optimization tasks: a comparison
• Switching synchronization in 1-D memristive networks: An exact solution