Evaluation of the cable model for electrical stimulation of unmyelinated nerve fibers.

The cable model, used to calculate the membrane potential of an unmyelinated nerve fiber due to electrical stimulation, is reexamined under passive steady-state conditions. The validity of two of the assumptions of the cable model are evaluated, namely that the membrane potential be a function of the axial coordinate only and that the extracellular potential due to the presence of the nerve fiber be negligible. The membrane potential calculated from the passive steady-state cable model is compared with the membrane potential obtained from an analytical three-dimensional (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 distances (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 potential in agreement with the 3-D model.

Calculation of electric fields in a multiple cylindrical volume conductor induced by magnetic coils.

A method is presented for calculating the electric field, that is induced in a cylindrical volume conductor by an alternating electrical current through a magnetic coil of arbitrary shape and position. The volume conductor is modeled as a set of concentric, infinitely long, homogeneous cylinders embedded in an outer space that extends to infinity. An analytic expression of the primary electric field induced by the magnetic coil, assuming quasi-static conditions, is combined with the analytic solution of the induced electric scalar potential due to the inhomogeneities of the volume conductor at the cylindrical interfaces. The latter is obtained by the method of separation of variables based on expansion with modified Bessel functions. Numerical results are presented for the case of two cylinders representing a nerve bundle with perineurium. An active cable model of a myelinated nerve fiber is included, and the effect of the nerve fiber's undulation is shown.

Magnetic and electrical stimulation of undulating nerve fibres: a simulation study.

Mathematical models of myelinated nerve fibres are highly stylized abstractions of real nerve fibres. For example, nerve fibres are usually assumed to be perfectly straight. Such idealizations can cause discrepancies between theoretical predictions and experimental results. One well-known discrepancy is that the currently used models predict (contradictory to experimental findings) that an activation of nerve fibres is not possible with a pure transverse electric field. This situation occurs when a magnetic coil is placed symmetrically above a straight nerve fibre for magnetic nerve stimulation, or when an anode and a cathode are placed equidistantly on a line perpendicular to the fibre in the case of electrical stimulation. It is shown that this discrepancy does not occur if the physiological undulation of peripheral nerve fibres is included in the models. Even for small undulation amplitudes (e.g. 0.02 mm), it is possible to activate the fibre in these positions. For physiological undulations, as found in the literature, and favourable (off-centre) positions, the typical reduction of the thresholds is in a range between one and five, compared with perfectly straight fibres.