1138 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO.8, AUGUST2002 Synthesis of Nonlinear Multiport Resistors: A PWL Approach Marco Storace, Member, IEEE, Pedro Julián, Member, IEEE, and Mauro Parodi, Member, IEEE Abstract—This paper presents a method for the approximate synthesis of nonlinear multiport resistors. According to some fundamental circuit theory results, the general problem of syn- thesizing a multiport resistor with given constitutive equations corresponds to that of the synthesis of nonlinear controlled sources. Following this idea, in this paper, we focus on the design of nonlinear controlled sources using a piecewise-linear (PWL) approach. The constitutive equations are first approximated by resorting to canonical expressions for continuous PWL functions, and then implemented using a set of elementary building blocks. The proposed method is applied to the synthesis of the nonlinear resistive part of an equivalent circuit of the Hodgkin–Huxley nerve membrane model. Index Terms—Hodgkin–Huxley nerve membrane model, multi- port resistor synthesis, nonlinear circuit theory, piecewise-linear approximation. I. INTRODUCTION T HE FOLLOWING sentence—written by L. O. Chua in a tutorial paper on device modeling [1]—was a source of inspiration for the present work, and is the starting point for the forthcoming results: […] one of the most fundamental problems in device modeling and nonlinear network synthesis is that of re- alizing a prescribed nonlinear n-port resistor using a pre- scribed set of simple building blocks. The results leading to such a statement can be summarized as follows [1]. 1) Any algebraic n-port can be synthesized by using only mutators and one (generally, nonlinear) n-port resistor. 2) Any R-, L-, C-, M-dynamic n-port can be synthesized by using only mutators, linear two-terminal capacitors (or in- ductors) and one (generally, nonlinear) multiport resistor. 3) Any lumped n-port element can be synthesized by using only a finite number of linear two-terminal capacitors (or inductors) and one (generally, nonlinear) multiport re- sistor. In particular, the last result can be regarded as the circuit-theo- retic version of a theorem stated by Wiener [2]. It points out the Manuscript received February 9, 2001; revised April 11, 2002. This work was supported in part by the Ministero dell’Università e della Ricerca Scientifica e Tecnologica, Rome, Italy, and in part by the University of Genoa, Genoa, Italy. M. Storace and M. Parodi are with the Department of Biophysical and Electronic Engineering, University of Genoa, Genova I-16145, Italy (e-mail: storace@dibe.unige.it; parodi@dibe.unige.it). P. Julián is with the Electronics Research Laboratory, Department of Elec- trical Engineering and Computer Science, University of California, Berkeley, CA 94720 USA and also with CONICET, Argentina and on leave from the Uni- versidad Nacional del Sur, Bahia Blanca, Argentina (e-mail: pjulian@ieee.org). Publisher Item Identifier 10.1109/TCSI.2002.801253. basic role that multiport resistors play in the synthesis of any lumped dynamic multiport element. In this paper, we shall propose a method for the approximate circuit synthesis of quite a large class of n-port resistors. The method is based on explicit representations of the PWL approx- imations of the n-ports’ constitutive equations. 1 In general, PWL functions have been used in the literature for analysis of nonlinear systems and nonlinear circuits, mainly due to the fact that such functions often lead to more convenient mathematical formulations. Moreover, for some problems (see, e.g., [5]), the PWL approach is the only way to obtain a solution, given the current state of science. On the contrary, this paper (as well as, for instance, [6, Ch. 2]) exploits the good properties of PWL functions for the synthesis of nonlinear circuits. In the following, we shall consider time-invariant nonlinear n-port resistors, and we shall assume that their constitutive equa- tions can be written in the form where the s are (generally, nonlinear) functions defining the dependence of the port variables ’s on the variables . As pointed out, for instance, in [1], from the well-known Kol- mogorov’s Theorem [7] it follows that—if the compactness hy- pothesis is verified—it is possible to express each function as sum of compositions of functions of one variable. This im- plies that can be derived from the contributions of a finite number of nonlinear two-terminal resistors and linear controlled sources. Unfortunately, no systematic procedure to obtain such decomposition of is currently available. Therefore, the proce- dure proposed by Yamamura [8] for the reduction of a multidi- mensional function is noteworthy, although the covered class of functions is not universal. 2 However, when the decomposition of is known, each composing function can be approximated by a piecewise-linear (PWL) expression, thus laying the basis for an approximate synthesis through PWL two-terminal resis- tors [6], [10]. By contrast, the approach presented in this paper consists in the direct approximation of through a PWL 1 There is another approach based on implicit representations to the PWL syn- thesis of nonlinear multiport resistors (for an overview on the PWL represen- tations, see [3]). Following this approach, for instance, it is possible to define a synthesis procedure resulting in a hybrid linear multiport terminating in ideal diodes [4]. 2 In particular, Yamamura, as well as other researchers who used the same method (see, e.g., [9]), considered the class of functions composed of five bi- nary operations and unary operations. The corresponding de- composition algorithm requires the analytical expression for the function to be approximated. 1057-7122/02$17.00 © 2002 IEEE