Convergence of Neural Networks with a Class of Real Memristors with Rectifying Characteristics

The paper considers a neural network with a class of real extended memristors obtained via the parallel connection of an ideal memristor and a nonlinear resistor. The resistor has the same rectifying characteristic for the current as that used in relevant models in the literature to account for diode-like effects at the interface between the memristor metal and insulating material. The paper proves some fundamental results on the trajectory convergence of this class of real memristor neural networks under the assumption that the interconnection matrix satisfies some symmetry conditions. First of all, the paper shows that, while in the case of neural networks with ideal memristors, it is possible to explicitly find functions of the state variables that are invariants of motions, the same functions can be used as Lyapunov functions that decrease along the trajectories in the case of real memristors with rectifying characteristics. This fundamental property is then used to study convergence by means of a reduction-of-order technique in combination with a Lyapunov approach. The theoretical predictions are verified via numerical simulations, and the convergence results are illustrated via the applications of real memristor neural networks to the solution of some image processing tasks in real time

Memristor Circuits for Simulating Neuron Spiking and Burst Phenomena

Since the introduction of memristors, it has been widely recognized that they can be successfully employed as synapses in neuromorphic circuits. This paper focuses on showing that memristor circuits can be also used for mimicking some features of the dynamics exhibited by neurons in response to an external stimulus. The proposed approach relies on exploiting multistability of memristor circuits, i.e., the coexistence of infinitely many attractors, and employing a suitable pulse-programmed input for switching among the different attractors. Specifically, it is first shown that a circuit composed of a resistor, an inductor, a capacitor and an ideal charge-controlled memristor displays infinitely many stable equilibrium points and limit cycles, each one pertaining to a planar invariant manifold. Moreover, each limit cycle is approximated via a first-order periodic approximation analytically obtained via the Describing Function (DF) method, a well-known technique in the Harmonic Balance (HB) context. Then, it is shown that the memristor charge is capable to mimic some simplified models of the neuron response when an external independent pulse-programmed current source is introduced in the circuit. The memristor charge behavior is generated via the concatenation of convergent and oscillatory behaviors which are obtained by switching between equilibrium points and limit cycles via a properly designed pulse timing of the current source. The design procedure takes also into account some relationships between the pulse features and the circuit parameters which are derived exploiting the analytic approximation of the limit cycles obtained via the DF method

Complete stability of feedback CNNs with dynamic memristors and second-order cells

The paper considers a feedback cellular neural network (CNN) obtained by interconnecting elementary cells with an ideal capacitor and an ideal flux-controlled memristor. It is supposed that during the analogue computation of the CNN the memristors behave as dynamic elements, so that each dynamic memristor (DM)-CNN cell is described by a second-order differential system in the state variables given by the capacitor voltage and the memristor flux. The proposed networks are called DM-CNNs, that is CNNs using a dynamic (D) memristor (M). After giving a foundation to the DM-CNN model, the paper establishes a fundamental result on complete stability, that is convergence of solutions toward equilibrium points, when the DM-CNN has symmetric interconnections. Because of the presence of dynamic memristors, a DM-CNN displays peculiar and basically different dynamic properties with respect to standard CNNs. First of all a DM-CNN computes during the time evolution of the memristor fluxes, instead of the capacitor voltages as for a standard CNN. Furthermore, when a steady state is reached, the memristors keep in memory the result of the computation, that is the limiting values of the fluxes, while all memristor currents and voltages, as well as all currents, voltages, and power in the DM-CNN vanish. Instead, for standard CNNs, currents, voltages, and power do not drop off when a steady state is reached