Abstract-The structure of myelinated axons is quite similar to that of a high-loss coaxial cable. An electromagnetic analysis of TM waves shows that distributed effects cannot be neglected. The attenuation and phase constants are obtained as a function of frequency. Predicted finite wave delay in the internodal segment approaches measurements. Keywords -Nerves, models, elctromagnetism, waves, velocity, delay, attenuation, myelin
I. INTRODUCTIONElectric stimulation of nervous system can restore motor functions [1]. Electrical properties of nerves are ususally described by means of a cable model [2], [10]. This model is frequently simplified, and each internodal segment is modeled by a lumped impedance. This can be a series resistor [3]- [7], but sometimes parallel capacitive impedances are added [27]. A lumped resistor is unable to model a finite conduction velocity in the internodal segment, whereas the presence of capacitors modelling the myelin sheath can explain delays [27]. Despite this fact, resistor models are more frequently used, and the delay is usually assigned only to nonlinearities in Ranvier nodes [15].The existence of travelling waves in internodal segments could linearly explain, at least partially, the measured conduction speed [14]. The objective of this paper is to determine whether under any circumstances a distributed circuit [28] covering the effects of both the capacitive and resistive effects would be more suitable than a lumped model. To answer this question we shall analyze an internodal segment through which a transverse magnetic (TM) wave is guided [20], [26]. This approach links with the spatially distributed description of electric and magnetic fields found in literature [7], [11]-[13], [18], [19].From an electromagnetic point of view, lumped element models are obtained as a quasi-static approximation [20]. Clark et al. [21] calculate the wave number k as ( )For axoplasm at 1kHz with conductivity 5 mhos/m, the resultant wave number approximates 0.198 rad/m. In 10 cm., the phase shift is less than 0.02 radian or 1.46º, so in [21] conclude the field is quasi-static. This conclussion was later adopted explicitly [22] or implicitly in lumped element models [3]- [7], [27]. We shall show that the waves are not defined by k, but rather by the propagation constant h, and that quasi-static approximations don't apply always, specially for relatively high frequencies.
II. METHODOLOGYFor sake of simplicity, we shall consider an infinitely long segment, surrounded by an infinite extracellular medium. Even though the results will not be the same found in physiological situations, qualitative conclusions might be useful to gain understanding into nerve conduction. The structure is quite similar to a high-loss coaxial cable ( fig. 1). Medium 1 is the axoplasm, medium 2 is the myelin sheath, and medium 3 is the extracellular fluid. [18]. We don't assume this hypothesis [27]. All media will be generic with permitivity ε i , permeability µ 0 , and conductivity σ i , where subscript i refers to mediu...