## The spread of potential changes in a cable system

The propagation of the nervous impulse depends not only on the electrical excitability of the nerve membrane, but also on the cable structure of the nerve. We have already seen that the passive electrical properties of a patch of membrane can be represented as a capacitance Cm in parallel with a resistance Rm, so that the circuit diagram of a length of axon is the network shown in Fig. 6.1, where Ro is the longitudinal resistance of the external medium, and R; is the longitudinal resistance of the axoplasm. Such a network is typical of a sheathed electric cable, albeit one with rather poor insulation, because Rm is not nearly as large compared with R, as it would be if the conducting core were a metal. If a constant current is passed transversely across the membrane so as to set up a potential difference V between inside and outside at one point, then the voltage elsewhere will fall off with the distance x in the manner indicated in the lower part of Fig. 6.1. The law governing this passive electrotonic spread of potential is

where the space constant A is given by

A similar argument applies in the case of a brief pulse of current, except that the value of C then has to be taken into account in addition to that of R .

However, it is not necessary to enter here into the detailed mathematics of the

Fig. 6.1. Electrical model of the passive (electrotonic) properties of a length of axon. The graph below shows the steady-state distribution of transmembrane potential when points A and B are connected to a constant current source.

passive spread of potential in a cable system, and it will suffice to note that theory and experiment are in good agreement.

Suppose now that an action potential of amplitude V has been set up over a short length of axon, and that the threshold potential change necessary to stimulate the resting membrane is a certain fraction, say one-fifth of Vo. Because of the cable structure of the nerve, current will flow in local circuits on either side of the active region as indicated in Fig. 6.2, and the depolarization will spread passively. At a critical distance in front of the active region, which with the assumption made above would be about 1.5 times the space constant, the amount of depolarization will just exceed threshold. This part of the axon will then become active in its turn, and the active region will move forwards. Provided that the axon is uniform in diameter and in the properties of its membrane, both the amplitude and the conduction velocity of the action potential will be constant, and it will behave in an all-or-none fashion. Since the amplitude of the action potential is always much greater than the threshold for stimulation, the conduction mechanism embodies a large safety factor, and the spike can be cut down a long way by changes in the conditions

Fig. 6.2. The local circuit currents that flow during a propagated action potential; b, The local circuit currents set up by a battery inserted in the core-conductor model.

that adversely affect the size of the membrane potential before conduction actually fails. It will be appreciated that although there is outward current flowing through the membrane both ahead of the active region and behind it, propagation can only take place from left to right in the diagram of Fig. 6.2, because the region to the rear is in a refractory state. In the living animal, action potentials normally originate at one end of a nerve, and are conducted unidirectionally away from that end. In an experimental situation where shocks are applied at the middle of an intact stretch of nerve, the membrane can of course be excited on each side of the stimulating electrode, setting up spikes travelling in both directions.

It should be clear from this description that conduction will be speeded up by an increase in the space constant for the passive spread of potential, because the resting membrane will be triggered further ahead of the advancing impulse. This is one of the reasons why large axons conduct impulses faster than small ones, for it follows from eqn (6.2) that A is proportional to the square root of fibre diameter. Another factor that greatly affects A is myelinization of the nerve, and we must next discuss this in more detail.

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