where R is the gas constant, 8.314 J/mol K; F is the Faraday constant, 9.65 x 104 C/mol; T is the Kelvin temperature; [Na+]o and [Na+]i are the concentrations of sodium ions outside (460 mM) and inside (46 mM) the membrane, respectively, in mol/l for squid axon. ENa < 0 means the electrical force on Na+ is inward. For squid axon at 293 K (20°C), VNa can be calculated:
VNa = (8.314 x 293/9.65 x 104) ln (460/46) = +57.4 mV 1.2-22
As Section 1.4.1 will show, the net potential energy forcing Na+ ions into the axoplasm is defined as ENa = (Vm - VNa) electron volts (eV). ENa = -127.4 meV for squid axon where the resting potential, Vm0 = 70 mV.
The sequence of events required for the initiation (and propagation) of a nerve impulse is as follows: Some combination of presynaptic events (or stimulus trans-duction in the case of sensory neurons) causes the transmembrane potential to depolarize, reaching the spike initiation threshold in a region of active membrane (e.g., the spike generator locus). There, voltage-triggered sodium channels abruptly increase their conductance for sodium ions, allowing enough Na+ ions to rush in to cause the local transmembrane potential to depolarize. This strong inward flow of Na+ ions causes Na+ and K+ ions on the outside of the membrane in front of the initiation region to move toward the region where JNa is highest through the membrane. gNa (xo, t) continues to rise, reaches a peak of about 30 mS/cm2 at about 0.5 ms, then spontaneously deactivates and returns to its resting level of about 5 |S/cm2 in about 1 to 2 ms. At the same time, depolarization-activated potassium ion channels open more slowly, reaching a peak in about 1 ms, allowing K+ ions to flow out of the core axoplasm, helping to restore the axon resting potential. gK has a resting value of about 0.5 mS, and its peak is about 13 mS. As a result of the outward JK, Vm(t) falls below the resting potential as much as 7 mV for about 1.5 to 5 ms, slowly returning to Vm0. From voltage clamp studies, it is known that gK remains high (potassium ion channels remain open) as long as Vm is depolarized; it does not spontaneously deactivate like gNa.
So how does the spike propagate, and what factors control its velocity? Assume that some event has depolarized Vm at x = xo to the threshold voltage 9 > Vm0 and the gNa begins to increase. Because of the initial inrush of Na+, the membrane depolarizes further, causing gNa to increase rapidly. Vm(t) now goes positive in response to the strong, inward JNa . gNa spontaneously, rapidly returns to its resting value. Depolarization-activated K+ channels allow potassium ion to leave the axon in front and in back of the sodium-active area. The inward JNa and the outward JK are accompanied by axial currents inside and outside the axon, as ions respond to electrostatic and diffusion forces caused by the radial ion currents. The membrane capacitance acts like a low-pass filter in limiting the rate of change of Vm. Cm must charge with some net current density, Jc(t) = Cm Vm (t) in order that Vm change.
Because of the axial currents, iL(x, t), Vm reaches the firing threshold in front of the region of peak gNa activation, causing the Na channels here to open. The Na channels in back of the active region now close. Thus, the spike moves forward in a traveling wave. If an axon is artificially depolarized in its middle, the action potential will propagate in both directions. This is an unnatural situation, however. Artificial, antidromic stimulation can be used as an electrophysiological tool to relate axons to cell bodies in dense nervous tissue such as the spinal cord.
The velocity of propagation of the action potential has been found experimentally to be proportional to the square root of the unmyelinated axon diameter. That is, v = ko + k VD . Thus, evolution has selected certain motoneuron axons to be large and to have high conduction velocities whenever survival requires a rapid escape reflex. The prime example of the need for a giant axon can be seen in the squid, whose giant motor axons can be as large as 1 mm diameter. (A 0.2-mm-diameter squid axon conducts at about 20 m/s.) The crayfish also has giant motor axons to effect its quick tail-flip escape maneuver. In vertebrates, one also can see a grading of nerve axon diameters that can be correlated with the survival importance of the information rapidly reaching the target organs.
Figure 1.2-2 illustrates the recording setup for measuring the transmembrane potential, Vm(t), and the surface potential, Vs(t), during a propagated action potential on an unmyelinated axon. Figure 1.2-3 illustrates Vm(xo, t) as a spike is initiated at x = xo, and also the conductances, gNa(xo, t) and gK(xo, t). The approximate distribution of Vm(x, to), gNa (x, to) and gK(x, to) along the x axis at t = to for a propagating spike are illustrated in Figure 1.2-4. Note that the transmembrane voltage reaches the threshold for sodium ion channel opening in front of the spike peak, keeping the spike propagating at a constant velocity, v.
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