Memristive Networks: from Graph Theory to Statistical Physics

Finding the shortest path in a graph has applications to a wide range of optimization problems. However, algorithmic methods scale with the size of the graph in terms of time and energy. We propose a method to solve the shortest path problem using circuits of nanodevices called memristors and validate it on graphs of different sizes and topologies. It is both valid for an experimentally derived memristor model and robust to device variability. The time and energy of the compu...

We study the phase diagram of memristive circuit models in the replica-symmetric case using a novel Lyapunov function for the dynamics of these devices. Effectively, the model we propose is an Ising model with interacting quenched disorder, which we study at the first order in a control parameter. Notwithstanding these limitations, we find a complex phase diagram and a glass-ferromagnetic transition in the parameter space which generalizes earlier mean-field theory results fo...

We present a tutorial on the properties of the new ideal circuit element, a memristor. By definition, a memristor M relates the charge q and the magnetic flux $\phi$ in a circuit, and complements a resistor R, a capacitor C, and an inductor L as an ingredient of ideal electrical circuits. The properties of these three elements and their circuits are a part of the standard curricula. The existence of the memristor as the fourth ideal circuit element was predicted in 1971 based...

A memristor is a two-terminal nanodevice that its properties attract a wide community of researchers from various domains such as physics, chemistry, electronics, computer and neuroscience.The simple structure for manufacturing, small scalability, nonvolatility and potential of using inlow power platforms are outstanding characteristics of this emerging nanodevice. In this report,we review a brief literature of memristor from mathematic model to the physical realization. Wedi...

Biological neuronal networks are characterized by nonlinear interactions and complex connectivity. Given the growing impetus to build neuromorphic computers, understanding physical devices that exhibit structures and functionalities similar to biological neural networks is an important step toward this goal. Self-organizing circuits of nanodevices are at the forefront of the research in neuromorphic computing, as their behavior mimics synaptic plasticity features of biologica...

Many realistic networks are scale-free, with small characteristic path lengths, high clustering, and power law in their degree distribution. They can be obtained by dynamical networks in which a preferential attachment process takes place. However, this mechanism is non-local, in the sense that it requires knowledge of the whole graph in order for the graph to be updated. Instead, if preferential attachment and realistic networks occur in physical systems, these features need...

This article presents a review on the main applications of the fourth fundamental circuit element, named "memristor", which had been proposed for the first time by Leon Chua and has recently been developed by a team at HP Laboratories led by Stanley Williams. In particular, after a brief analysis of memristor theory with a description of the first memristor, manufactured at HP Laboratories, we present its main applications in the circuit design and computer technology, togeth...

It is observed that the inductive and capacitive features of the memristor reflect (and are a quintessence of) such features of any resistor. The very presence of the voltage and current state variables, associated by their electrodynamics sense with electrical and magnetic fields, in the resistive characteristic v = f(i), forces any resister to accumulate some magnetic and electrostatic fields and energies around itself, i.e. L and C elements are always present. From the cir...

An analysis of the literature shows that there are two types of non-memristive models that have been widely used in the modeling of so-called "memristive" neural networks. Here, we demonstrate that such models have nothing in common with the concept of memristive elements: they describe either non-linear resistors or certain bi-state systems, which all are devices without memory. Therefore, the results presented in a significant number of publications are at least questionabl...

We extend the notion of memristive systems to capacitive and inductive elements, namely capacitors and inductors whose properties depend on the state and history of the system. All these elements show pinched hysteretic loops in the two constitutive variables that define them: current-voltage for the memristor, charge-voltage for the memcapacitor, and current-flux for the meminductor. We argue that these devices are common at the nanoscale where the dynamical properties of el...