Caravelli F, Traversa F L, Di Ventra M
Invenia Labs, 27 Parkside Place, Parkside, Cambridge CB1 1HQ, United Kingdom.
London Institute for Mathematical Sciences, 35a South Street, London W1K 2XF, United Kingdom.
Phys Rev E. 2017 Feb;95(2-1):022140. doi: 10.1103/PhysRevE.95.022140. Epub 2017 Feb 28.
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 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 results suggest a much richer dynamics of memristive networks than previously considered.
具有忆阻元件（带记忆的电阻器）的网络正被探索用于从非传统计算到大脑模型等各种应用。然而，突出图连通性在记忆动力学中作用的分析结果仍然很少，从而限制了我们对这些重要动力系统的理解。在本文中，我们推导了一个精确的运动矩阵方程，该方程考虑了纯忆阻电路的所有网络约束，并利用它得出关于其弛豫特性的分析结果。我们能够根据底层电路基本回路空间子空间上的正交投影算子来描述记忆演化。这种正交投影明确揭示了忆阻电路空间和时间部分之间的耦合，并简洁地描述了电路拓扑。对于无序图的情况，我们能够使用随机矩阵将幂律弛豫的出现解释为具有广泛尺度范围的指数弛豫时间的叠加。这种幂律也是通用的，即与底层图的拓扑无关，仅取决于回路的密度。相反，对于受交流电压作用的电路，我们能够获得动力学的近似解，并针对特定网络拓扑进行了测试。这些结果表明忆阻网络的动力学比以前所认为的要丰富得多。
