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Memristor-Based Cellular Nonlinear/Neural Network 1202 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 6, JUNE 2015 Memristor-Based Cellular Nonlinear/Neural Network: Design, Analysis, and Applications Shukai Duan, Member, IEEE, Xiaofang Hu, Student Member, IEEE, Zhekang Dong, Student Member, IEEE, Lidan Wang, Member, IEEE, and Pinaki Mazumder, Fellow, IEEE Abstract— Cellular nonlinear/neural network (CNN) has been regularity of cellular automata and local computation of recognized as a powerful massively parallel architecture capable neural networks can be melded together to accelerate of solving complex engineering problems by performing trillions numerous real-world computational tasks. Many artificial, of analog operations per second. The memristor was theoret- ically predicted in the late seventies, but it garnered nascent physical, chemical, as well as biological systems have been research interest due to the recent much-acclaimed discovery represented using the CNN model [3]. Its continuous-time of nanocrossbar memories by engineers at the Hewlett-Packard analog operation allows real-time signal processing at high Laboratory. The memristor is expected to be co-integrated with precision, while its local interaction between the constituent nanoscale CMOS technology to revolutionize conventional von processing elements (PEs) obviate the need of global buses Neumann as well as neuromorphic computing. In this paper, a compact CNN model based on memristors is presented along and long interconnects, leading to several digital and analog with its performance analysis and applications. In the new CNN implementations of CNN architectures in the form of very design, the memristor bridge circuit acts as the synaptic circuit large scale integrated (VLSI) chips. In [4]–[9], several element and substitutes the complex multiplication circuit used powerful applications of CNNs were reported including in traditional CNN architectures. In addition, the negative differ- pattern and image analysis such as vertical line detection, noise ential resistance and nonlinear current–voltage characteristics of the memristor have been leveraged to replace the linear resistor reduction, edge detection, feature detection, and character in conventional CNNs. The proposed CNN design has several recognition. merits, for example, high density, nonvolatility, and programma- However, to improve the resolution in static and dynamic bility of synaptic weights. The proposed memristor-based CNN image analysis, the size of PEs and the connectivity between design operations for implementing several image processing the PEs describing the control (feedforward) and feedback functions are illustrated through simulation and contrasted with conventional CNNs. Monte-Carlo simulation has been used to templates that are used as programming artifacts in the CNN demonstrate the behavior of the proposed CNN due to the model of computation [5], [6], must be significantly reduced. variations in memristor synaptic weights. The current version of CNN arrays in a CMOS VLSI chip Index Terms— Cellular neural/nonlinear network (CNN), fault is typically limited to 16-K PEs. Emerging technologies such tolerance, image processing, memristor, stability. as resonant tunneling diode [7] and quantum dots [8] have been attempted recently to implement PEs more compactly to I. INTRODUCTION improve the resolution of CNN computation. However, lack ELLULAR nonlinear/neural network (CNN) was of programmability associated with fixed connecting elements Cproposed in [1] and [2] by demonstrating how the between the PEs as well as wide variability of tunneling Manuscript received November 10, 2013; accepted June 21, 2014. Date of currents in RTDs and quantum dots are the major limitations publication July 21, 2014; date of current version May 15, 2015. This work of these emerging technologies. was supported in part by the Program for New Century Excellent Talents in The recent advent of memristors [10], [11] has opened University under Grant [2013]47, in part by the National Natural Science Foundation of China under Grant 61372139, Grant 61101233, and Grant the possibility of significantly enhancing the resolution of 60972155, in part by the Spring Sunshine Plan, in part by the Research on-chip CNN model of computation. The memristor was Project, Ministry of Education of China, under Grant z2011148, in part by introduced [12] as a fundamental circuit element and recently the Technology Foundation for Selected Overseas Chinese Scholars, in part by the Ministry of Personnel in China under Grant 2012-186, in part by nanofabrication technology has shown its superior device the University Excellent Talents Supporting Foundations of Chongqing under properties such as nonvolatility, binary as well as multiple Grant 2011-65, in part by the University Key Teacher Supporting Founda- memory states, and nanometer geometries that can be shrunk tions of Chongqing under Grant 2011-65, and in part by the Fundamental Research Funds for the Central Universities under Grant XDJK2014A009 and to the ultimate physical dimensions [13]–[16]. These versatile Grant XDJK2013B011. features of memristors have been exploited in showing their S. Duan, Z. Dong, and L. Wang are with the College of Electronics and applications in nonvolatile memory [17], [18], artificial neural Information Engineering, Southwest University, Chongqing 400715, China (e-mail: [email protected]; [email protected]; [email protected]). networks [19]–[23], composite circuits [24], [25], and so on. X. Hu is with the Department of Department of Mechanical and Because its conductance can change in response to the applied Biomedical Engineering, City University of Hong Kong, Hong Kong (e-mail: voltage or current like a biological synapse, the memristor has [email protected]). P. Mazumder is with the Department of Electrical Engineering and been demonstrated an artificial synapse for biological signal Computer Science, University of Michigan, Ann Arbor, MI 48109 USA processing. Recently, Kim et al. [21] and [22] presented a (e-mail: [email protected]). compact memristor bridge synapse in which both weighting Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. and weight programming can be performed at different time Digital Object Identifier 10.1109/TNNLS.2014.2334701 slots. 2162-237X © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. DUAN et al.: M-CNN: DESIGN, ANALYSIS, AND APPLICATIONS 1203 between the two platinum electrodes. The memristance mag- nitude is determined by the electron tunnel barrier, in series with a resistor. In this case, the state variable x is the width of the Simmons tunnel barrier and its dynamics are represented by [13] dx(t) dt⎧ ⎪ − | | ⎪ i − x aOFF − i − x , > ⎨cOFF sinh i exp exp w b w i 0 = OFF c c ⎪ − | | ⎩⎪ i x aON i x cON sinh exp − exp − − − , i <0 iON wc b wc Fig. 1. Physical model of Simmons tunnel barrier memristor [13]. (3) where cOFF, cON, iOFF, iON, aOFF, aON, wc,andb are the fitting parameters, in which the parameters c and c influence In this paper, a novel type of memristor-based CNN OFF ON the change in magnitude of x, the parameters iOFF and ion (M-CNN) is addressed. Specifically, the memristor bridge reflect the current thresholds, and aOFF and aon are the upper circuit is employed to realize the interactions between the and the lower bounds of x, respectively. Equation (3) is also neighboring cells. Different from the memristor circuit in [21] interpreted as the velocity of the oxygen vacancy drift in and [22], the memristor used here is a more practical model TiO2−x material. About this accurate model, two problems based on the experimental data. Moreover, since the memristor have been discussed [15]: 1) there is no explicit relationship exhibits nonlinear current–voltage (I–V ) characteristic with between the current and the voltage of the memristive device locally negative differential resistance, the memristor is also and 2) it is too complicated to be used in numerical and mathe- considered to replace the original linear resistor in a traditional matical analysis. Afterward, an alternative model with simpler CNN cell. Thus, an M-CNN, equipped with nonvolatile and expressions, which can reflect the same physical behavior, is programmable synapse circuits, is more versatile and compact proposed [15]. In this simplified model, the derivation of the and saves the traditional complex output function realization state variable x is given by circuits. ⎧ α ⎪ OFF In Section II, the basics of the memristor are briefly ⎪ i(t) − · ( ), < < ⎪kOFF i 1 fOFF x 0 iOFF i introduced. Then, the topology and dynamics of the proposed ( ) ⎨ OFF dx t = M-CNN are described in Section III. To guarantee the feasibil- 0, iON < i < iOFF (4) dt ⎪ α ity of implementation scheme, the stability and fault tolerance ⎪ ON ⎪ i(t) ⎩kON − 1 · fON(x), i < iON < 0 analysis are investigated in Section IV. Section V presents the iON numerical simulations of the uncoupled and coupled M-CNNs in image processing, including image inversion, horizontal line where kOFF is a positive constant and kON is a negative constant, detection (HLD), edge extraction, noise removal, as well as aOFF and aON are fitting parameters, and iOFF and iON are the a variation analysis based on Monte-Carlo methods. Finally, current thresholds. Correspondingly, there are two window ( ) ( ) conclusion
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