Theory, modeling and design of memristor-based min-max circuits

Abstract

Neuromorphic systems have recently emerged as promising candidates for future computing paradigms. Min-Max circuits are indispensable building blocks in Artificial Neural Networks and Fuzzy systems. For instance, the inference engine in Fuzzy controllers that constitutes the decision-making unit in such systems is a Min-Max circuit. Conventionally, transistor-based architectures were adopted in the design of Min-Max circuits. Several designs have been reported that primarily focus on reducing the area consumption (some were voltage mode and others were current mode). However, the miniaturized features of the memristor and the peculiar characteristics it exhibits have driven researchers to use it in state-of-the-art Min-Max circuits. This work addresses the theory, design and modeling of memristor-based Min-Max circuits. Basics of memristor-based Min-Max circuits are addressed through an elaborate explanation of 2-input Min-Max circuits. First, the working principle is explained based on Ohm’s and Kirchhoff’s Laws. Then, the theory is generalized to an arbitrary number ‘N’ of inputs (N-ary memristor-based Min-Max circuits) via a formal mathematical proof. An important feature of the memristor is the existence of a threshold below which no change in the state variable (no switching in the case of Min-Max circuits) occurs. Although some existing models overlook the threshold behavior of memristors, most experimental data does confirm the existence of a threshold and, accordingly, it is essential to incorporate its effect in Min-Max circuits. Furthermore, failure to abide by the threshold restrictions results in a circuit malfunction not just a parametric failure (i.e. increased power consumption, increased delay…. etc.) which further necessitates a careful and thorough modeling of the effect of the threshold on the circuit’s behavior. Modeling of the threshold will be approached in two ways. First, an analytical approach is adopted to derive a closed form expression for the effect of the threshold on the circuit. Then, an algorithm is developed (implemented in MATLAB) that emulates the circuit operation. The algorithm runs exhaustive simulations on memristor states and input voltage vectors for different circuit sizes (number of inputs) to verify the derived model. The implications of the derived model are twofold: (1) it provides a closed formula for designers who wish to design memristor-based Min-Max circuits (2) it demonstrates a clear trade-off between the size and the resolution of the circuit.