IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 9, SEPTEMBER 2001 1027 Evaluation of the Cable Model for Electrical Stimulation of Unmyelinated Nerve Fibers Veit Schnabel and Johannes J. Struijk* Abstract—The cable model, used to calculate the membrane po- tential of an unmyelinated nerve fiber due to electrical stimula- tion, is reexamined under passive steady-state conditions. The va- lidity of two of the assumptions of the cable model are evaluated, namely that the membrane potential be a function of the axial co- ordinate only and that the extracellular potential due to the pres- ence of the nerve fiber be negligible. The membrane potential cal- culated from the passive steady-state cable model is compared with the membrane potential obtained from an analytical three-dimen- sional (3-D) volume conductor model of a nerve fiber. It is shown that for very small electrode-fiber distances (of only a few fiber radii), both assumptions are violated and the two models give quite different results. Over a wide range of the electrode-fiber distance (about 0.1 mm to 1 cm), both assumptions are fulfilled and the two models give approximately the same results. For very large dis- tances (more than 10 cm, independent of fiber diameter) only the second assumption is satisfied, but a modification of the activating function of the cable model allows to calculate the membrane po- tential in agreement with the 3-D model. Index Terms—Activating function, Bessel functions, cable model, electric stimulation, peripheral nerve, volume conductor. I. INTRODUCTION S INCE McNeal [1] presented a mathematical model of elec- tric nerve stimulation using electrodes that are not in direct contact with the nerve fiber, the excitation site and threshold due to electric field stimulation have been calculated for many con- figurations [2]–[4]. In this type of model, the membrane prop- erties of the nerve fiber [5]–[10] are included into a one-dimen- sional cable model where the forcing function is determined as a function of the extracellular potential, called the activating func- tion. The extracellular potential is calculated assuming that the presence of the nerve fiber does not influence the distribution of the electric field. Although it was predicted that this is not a valid assumption for very small fiber-electrode distances [1], this assumption has not been analyzed quantitatively. The application of the cable model requires further that the intra- and extracellular potentials be rotationally symmetric along the fiber’s axis. This assumption is clearly violated when Manuscript received October 13, 2000; revised June 2, 2001. This work was supported by the Danish Technical Research Council under Grant 9700845. As- terisk indicates corresponding author. V. Schnabel is with the Center for Sensory-Motor Interaction, Aalborg Uni- versity, DK-9220 Aalborg, Denmark. *J. J. Struijk is with the Center for Sensory-Motor Interaction, Aalborg University, Fredrik Bajers Vej 7 D3, DK-9220 Aalborg, Denmark (e-mail: jjs@smi.auc.dk). Publisher Item Identifier S 0018-9294(01)07446-8. there is a considerably large component of the electric field transverse to the nerve fiber, e.g., for bipolar stimulation if the axis of the electrodes is perpendicular to the fiber and both electrodes have the same distance to the fiber. However, there exists only a limited number of investigations of this problem [3], [11]. In this paper, these two assumptions are evaluated quantita- tively. A passive steady-state cable model of an unmyelinated nerve fiber is compared with a three-dimensional (3-D) volume conductor model of a nerve fiber, from very small to very large distances of the point current source to the nerve fiber. This means that the capacitive and nonlinear properties of the nerve fiber are neglected in this paper. A range of fiber-electrode dis- tances is given for which the cable model is valid. It is shown that for large distances, the use of a modified activating function, originally proposed for magnetic nerve stimulation [12], is more appropriate. This modified activating function also takes into ac- count the maximal steady-state response of the nerve fiber due to the part of the electric field transverse to the fiber. II. METHODS A. Three-Dimensional Volume Conductor Model The volume conductor model was based on an analytic model for calculating the electric scalar potential induced in a cylin- drical conductive medium, presented by Altman and Plonsey [13]. This model was extended to a volume conductor divided into an arbitrary number of infinitely long concentric cylinders with finite radii . The geometry of the model is given in Fig. 1. The region enclosed by the cylinder with radius has an isotropic conductivity ( ). The region surrounding the outermost cylinder has a conductivity . The point source imposing the current A (cathode) was placed in the region at . Following the procedure for the isotropic monodomain in [13], the electric scalar potential was calculated for all regions separately and was split up into the primary po- tential of a point source in an infinite medium and the sec- ondary potential due to the inhomogeneities at the cylin- drical interfaces of the contiguous regions (1) In the source free regions, the potentials have to obey Laplace’s equation ( , ) (2) 0018–9294/01$10.00 © 2001 IEEE