13-3. Verify Eq. 13.6. 13.1 The Nervous System (a) Axon membrane Current inside the axon (b) Ro Current outside the axon Rm www HE Ст www R₁ 179 FIGURE 13.6 (a) Currents flowing through a small section of the axon. (b) Electrical circuit representing a small section of the axon. wi. n a pr. FL Th a 13.1.4 Axon as an Electric Cable In the analysis of the electrical properties of the axon, we will use some of the techniques of electrical engineering. This treatment is more complex than the methods used in the other sections of the text. The added complexity, however, is necessary for the quantitative understanding of the nervous system. Although the axon is often compared to an electrical cable, there are pro- found differences between the two. Still, it is possible to gain some insight into the functioning of the axon by analyzing it as an insulated electric cable sub- merged in a conducting fluid. In such an analysis, we must take into account the resistance of the fluids both inside and outside the axon and the electrical properties of the axon membrane. Because the membrane is a leaky insulator, it is characterized by both capacitance and resistance. Thus, we need four electri- cal parameters to specify the cable properties of the axon. The capacitance and the resistance of the axon are distributed continu- ously along the length of the cable. It is, therefore, not possible to represent the whole axon (or any other cable) with only four circuit components. We must consider the axon as a series of very small electrical-circuit sections joined together. When a potential difference is set up between the inside and the out- side of the axon, four currents can be identified: the current outside the axon, the current inside the axon, the current through the resistive component of the membrane, and the current through the capacitive component of the membrane (see Fig. 13.6). The electrical circuit representing a small axon section of length Ax is shown in Fig. 13.6. In this small section, the resistances of the outside and of the inside fluids are R, and Ri, respectively. The membrane capacitance and resistance are shown as Cm and Rm. The whole axon is just a long series of these subunits joined together. This is shown in Fig. 13.7. Sample values of the circuit parameters for both a myelinated and a nonmyelinated axon of radius 5 × 10-6 m are listed in Table 13.1. (These values were obtained from [13-4].) Note that the values in Table 13.1 are quoted for a 1-m length of the axon. The unit mho for the conductivity of the axon membrane is defined in Appendix B. An examination of the axon performance shows immediately that the circuit in Fig. 13.7 does not explain some of the most striking features of the axon. An electrical signal along such a circuit propagates at nearly the speed of light (3 x 108 m/sec), whereas a pulse along an axon travels at a speed that is at " Table 13.1 Properties of Sample Axons Property Axon radius Resistance per unit length of fluid Nonmyelinated axon Myelinated 5 x 10-6 m axon 5 x 10-m 6.37 x 109 2/m 6.37 x 10º 2/m both inside and outside axon (r) Conductivity per unit length of axon membrane (gm) Capacitance per unit length of axon (c) 1.25 x 104mho/m 3 x 10-7 mho/m 3 x 10-7 F/m 8 x 10-10 F/m most about 100 m/sec. Furthermore, as we will show, the circuit in Fig. 13.7 dissipates an electrical signal very quickly; yet we know that action potentials propagate along the axon without any attenuation. Therefore, we must conclude that an electrical signal along the axon does not propagate by a simple passive process.