The generation of the action potential


It is now possible to follow, step by step, the events leading to the potential rise and fall during an action potential. Suppose that we apply a small depolarizing current pulse to our axon with the axial wire (which also prevents propagation). Plate 3 shows a series of snapshots of the electrical equivalent circuit of the membrane including the voltage dependent Na and K channels. The branches are indicated with their reversal potentials and conductances: Na (for sodium), K (for potassium) and L (for leakage). The equilibrium potentials are indicated next to the corresponding batteries. The membrane potential is indicated in the bottom of the capacitor branch, and, as usual, the external side (top, in the figure) is referred as ground or zero potential. To the right of each panel there is a plot of the membrane voltage (V) and the total ionic current (Ii). Concomitant with Plates A-F we must look at Fig 1. Fig 1 shows a plot of the membrane action potential (V) as a result of the stimulating current pulse Is and the underlying Na and K currents through the membrane.


In the resting condition, there is no current circulation between inside and outside which means that the sum of currents through all the four branches is equal to zero (Plate A and Fig. 1). We also know that in this resting condition the value of gNa is much smaller than gK and gL, therefore most of the current will be across the K branch and the leakage branch. As the resting potential is more positive than EK, the current in the K branch will be outward and as it is more negative than the value of EL the current will be inward in the leakage branch. The direction of the currents are indicated by the arrows and their intensity by the darkness in Plates A-F.



Figure 29. Ionic currents during the impulse.



To initiate the action potential, we apply a current pulse directed outward by making the inside more positive. This is shown in Plate B as a current generator connected between the outside and inside of the axon when the switch is closed. As we have described above, the first event will be the charging of the membrane capacitance that will occur as most of the current will initially go in the capacitive branch and subsequently through the Na, K and leakage branches in proportion to their conductances which will be preferentially through the K and leakage at short times (plate B). As the capacitance gets charged, the potential becomes more positive and gNa starts increasing (or RNa decreasing) which produces an inward current through the Na branch. Then the pulse of current is terminated and the total membrane current is again zero. However, there is outward K and leakage current, and inward Na current which keeps increasing until it becomes equal to the sum of IL and IK (Fig. 1). The reason why the inward sodium current increases is because the membrane has been depolarized and the voltage-sensitive Na channels have been activated while K channels, which are slower to activate, have not responded yet to the depolarization. This is clearly seen in the trace of Ii shown Fig 1, which crosses the zero line. At that time, if gNa is still increasing, inward Na current will predominate (see Fig 1) and as more Na channels open, more inward current will flow which will depolarize the membrane even more, which will open more Na channels, etc.




In this fashion, the membrane potential will tend to go to ENa, producing the upstroke of the action potential (Plate C).


During this time, Na inactivation is increasing due to the depolarization. At the same time the K conductance, slower in turning on, starts increasing to the point that the total ionic current becomes outward through the K branch and the leakage branch, and eventually they cancel the Na current making the total ionic current equal zero (Plate D).





After this, K outward current predominates and initiates the repolarization phase of the action potential (Plate E and Fig 1). Later, gK predominates pushing the potential towards the value of EK seen as the underswing of the action potential where gK is high but the current is low because V is too close to EK (see Plate F and Fig 1). After some time at this membrane potential, gK will again become small, producing a return of the membrane potential towards its normal resting value (as in Plate A).





The Propagation of the Nerve Impulse


Our analysis of the generation of the action potential has been carried out in an axon with an axial wire. This was done to simplify the situation by preventing the propagation of the changes in membrane potential. In fact, with the axial wire, the voltage of the membrane along the whole stretch of axon changes simultaneously. Now we have to apply this knowledge to the real axon which does not have an axial wire in it. This means that we have to understand the consequences of the geometry of the axon on the current circulation across the membrane and along the axon. To proceed, we will first analyze the properties of a cylindrical axon when we apply very small current pulses. We will first use small pulses to prevent the initiation of the action potential, allowing us to understand the membrane potential changes and current circulation without the complications introduced by the voltage dependent conductances. After we have a clear idea of the current flow and membrane potential changes along the axon will we be in a position to extend the analysis for larger pulses that elicit changes of the Na and K conductances and

from there follow the series of events that produce the propagation of the action potential.


Passive properties of a cylindrical axon (Cable properties of the axon).


The simple RC model of the membrane analyzed at the beginning of these notes is only valid provided there are no potential gradients in the inside or outside solutions. Here we are going to extend our analysis to include the geometry of the axon. It is important to realize that this analysis will be first done with a passive RC circuit, that is, with no voltage dependent conductances. Once we understand this simpler situation, we will incorporate the properties of the channels as we studied them before.



Figure 2. Longitudinal section of an axon showing a few lines of current flow near the electrode impaled at x=0. The other electrode is outside and far away.


Figure 2 shows a portion of an axon immersed in a conductive solution. There is an electrode penetrating the surface, which is supplying current with respect to another electrode located outside, and the current circulation is pictured by a few lines of flow. If the membrane were a perfect insulator, no current would flow towards the external electrode. But due to the finite permeability of the axon, some current will circulate longitudinally in the axon interior, and exit through the surface membrane to be finally collected by the external electrode. It is clear then that due to the current circulation and the resistivity of the axoplasm, there will be voltage drops along the axon and none of the two sides of the membrane will be isopotential. Consequently we cannot use our simple membrane model to analyze the membrane potential along the axon.


The detailed analysis of this problem takes into account the geometry in three dimensions plus the time variable; the resulting equation is a partial differential equation in four independent variables. Considerable simplification can be achieved by assuming that all the lines of flow are either parallel or perpendicular to the axon membrane and that there is no potential gradient radially inside or outside the axon (Fig. 3, upper panel). This seems like an arbitrary set of assumptions but it can be demonstrated that it is a good approximation for most of the practical cases. Under these conditions, our model gets reduced to a one-dimensional case with time as the second variable and it can be represented as a series of our membrane elements connected by internal and external resistances as shown in the lower panel of Fig. 3.



Figure 3: Equivalent circuit of the Axon.



The space constant

The space constant (l) gives an indication of how far the current passively spreads down along the axon. l represents the distance that one must travel from the current injection site to record a voltage that is 0.37 (or 1/e) the value of the voltage at the injection site. As l increases, the spread of the voltage change also increases.



Where  rm=Rm/2Ba, where Rm is the specific membrane resistance in Scm2 and a is the radius of the fiber. Also ri=Ri /Ba2, where Ri is the specific resistance of the axoplasm in Scm.


A long space constant will allow the depolarization of the active patch to reach further than a short space constant. Consequently a long space constant will help in a faster propagation of the action potential. Factors that influence l, such as the internal resistance or external resistance along the axon will have an influence in the conduction velocity. The value of l is also proportional to the square root of the fiber diameter. If all other factors are maintained constant, such as resistivity, specific membrane conductance and capacitance, then we would predict that increasing fiber diameter increases conduction velocity. The giant axon of the squid, which is about 500 Ám in diameter, can propagate the action potential at about 18 m/s, whereas C fibers in mammalians are around 1 Ám in diameter and propagate the action potential at about 1 m/s.


Another strategy for fast propagation is utilized by vertebrates with myelinated axons. In this case, the circulation of current between patches is indeed discrete because there is a very good insulator (the myelin) that ensheats the axon between nodes of Ranvier, where the activation of the Na conductance occurs in the exposed axon. An active node of Ranvier will excite its neighboring node by the same mechanism outlined above, except that because they are so far apart, the actual speed of propagation is increased. For this reason this propagation has been called saltatory conduction.