Employing Vector Field Techniques on the Analysis of Memristor Cellular Nonlinear Networks Cell Dynamics

TL;DR

This work addresses the limitations of Dynamic Route Map (DRM) and its 2nd-order extension (DRM2) for analyzing 2nd-order Memristor Cellular Nonlinear Networks (M-CNNs) by introducing a vector-field–augmented phase portrait for the M-CNN cell state pair $$(V_C,N_d)$$. The cell dynamics are governed by the 2D system with $\dot{V_C} = \frac{I_{ext}+I_{A_{0,0}}}{C} - \frac{V_C}{RC} - \frac{V_C}{R_M(V_C,N_d)}\dot{N_d}$ and $\dot{N_d} = -\frac{I_{ion}(V_C,N_d)}{z_{V_O} e A l_d}$, where $R_M$ is determined by the JART VCM memristor model. The authors propose embedding the vector field into the phase portrait by normalizing vector magnitudes to an ellipse and encoding magnitude via color, with the angle $\theta = \tan^{-1}\left(\frac{\dot{N_d}}{\dot{V_C}}\right)$ and nullclines for $\dot{V_C}=0$ and $\dot{N_d}=0$ to locate equilibria, enabling direct visualization of bistable/monostable dynamics and transitions. This approach enhances M-CNN design for edge computing hardware by revealing subtle dynamical variations that DRM/DRM2 cannot readily capture.

Abstract

This paper introduces an innovative graphical analysis tool for investigating the dynamics of Memristor Cellular Nonlinear Networks (M-CNNs) featuring 2nd-order processing elements, known as M-CNN cells. In the era of specialized hardware catering to the demands of intelligent autonomous systems, the integration of memristors within Cellular Nonlinear Networks (CNNs) has emerged as a promising paradigm due to their exceptional characteristics. However, the standard Dynamic Route Map (DRM) analysis, applicable to 1st-order systems, fails to address the intricacies of 2nd-order M-CNN cell dynamics, as well the 2nd-order DRM (DRM2) exhibits limitations on the graphical illustration of local dynamical properties of the M-CNN cells, e.g. state derivative's magnitude. To address this limitation, we propose a novel integration of M-CNN cell vector field into the cell's phase portrait, enhancing the analysis efficacy and enabling efficient M-CNN cell design. A comprehensive exploration of M-CNN cell dynamics is presented, showcasing the utility of the proposed graphical tool for various scenarios, including bistable and monostable behavior, and demonstrating its superior ability to reveal subtle variations in cell behavior. Through this work, we offer a refined perspective on the analysis and design of M-CNNs, paving the way for advanced applications in edge computing and specialized hardware.

Figures (4)

• Figure 1: (a) M-CNN Cell's equivalent circuit.
• Figure 2: Illustration of M-CNN Cell's SDRs for $N_d=N_{d,min}$ and $N_d=N_{d,max}$ with $R$ selected as (a) $R=\qty{1}{\kilo\Omega}<R_{M,min}<R_{M,max}$ and, (b) $R_{M,min}<R=\qty{3}{\kilo\Omega}<R_{M,max}$.
• Figure 3: Capacitor voltage derivative for (a) $R=\qty{1}{\kilo\Omega}<R_{M,min}<R_{M,max}$, and (b) $R_{M,min}<R=\qty{3}{\kilo\Omega}<R_{M,max}$, and (c) memristor state derivative within M-CNN cell's phase plane.
• Figure 4: Enhanced graphical illustration of M-CNN cell's dynamics including the cell's vector field. (a) Fully bistable M-CNN cell design. (b)-(c) Hybrid mono- and bistable M-CNN cell design for different $C$ values. In all plots, the $V_C$ and $N_d$ nullclines are illustrated in blue dashed and gray dotted lines, respectively, while the solid lines are cell trajectories starting from the circle marker at $\{V_{C,0}, N_{d,0}\}$ and end at the cross marker at $\{V_{C,t}, N_{d,t}\}$ after a simulation with duration $t=\qty{1}{\m\s}$.