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Membrane potential (mV)

dVm dt ,

Vm (t)

Time derivative of membrane potential (mV/msec)

Na+ conductance turns off in response to prolonged depolarization. One gating variable, the n gate, describes voltage-dependent activation of the K + conductance GK. An additional conductance (the leak conductance GL) is voltage independent and small. In mathematical terms, the Hodgkin-Huxley equation is written as follows, with symbols defined in Table I:

Cm dt

VNa)

Gn dm dt

In response to a step change from an initial value of membrane potential (often referred to as the holding potential, Vhold) to the clamp potential, Vclamp, each of the Hodgkin-Huxley gating variables (m, h, and n) changes from an initial value to a steady-state value with an exponential time course (Figs. 4b and 4c). The steady-state values (mN, hN, and nN) and exponential time constants (tm, th, and tn) are determined solely by the current value of Vm, which equals Vclamp for voltage-clamp experiments (Figs. 4e and 4f). The initial values of the gating variables are determined by the holding potential. GNa is proportional to m3 x h; Gk is proportional to n4 (Figs. 4b-4d). The powers to which the gating variables m and n are raised were used by Hodgkin and Huxley to induce small delays in activation of the conductance in order to better match experimental data.

C. The Hodgkin-Huxley Model Accounts for the Basic Properties of the Action Potential

Although the Hodgkin-Huxley model is incorrect in some details (see Section III.A), it can account for many of the basic properties of the neuronal action potential. For example, two properties of the m gate account for the phenomenon of the all-or-nothing action potential with a distinct threshold in response to a particular type of stimulus:

1. Because the m gate activates (opens) with depolarization, and its activation leads to further depolarization, this gate is prone to autocatalytic "positive feedback'' that can magnify a small depolarization into a full-blown action potential. The current threshold for firing an action potential is the amount of current required to engage this cycle of positive feedback (see Section III.D for a discussion of the threshold value of membrane potential). In contrast with the m gate, the h and n gates react to oppose depolarization (h because it becomes smaller with depolarization, and n because its activation increases the size of an outward K+ current).

2. The m gate is much faster than the h and n gates (Fig. 4e). The speed of the m gate means that the rapid rising phase of the action potential can occur before the stabilizing influences of the h and n gates can engage to bring Vm back to near resting potential. The speed of inactivation of h (i.e., its decrease with depolarization) and activation of n determine the width of the action potential.

Traces of each of the gating variables during the course of a depolarization-induced action potential are shown in Fig. 5a. Like spike threshold and spike duration, the phenomena of absolute and relative refractory periods can be accounted for by tracking gating variables. The ARP is associated with elevated values of n and, most important, greatly reduced values of h after a spike (Fig. 5a). These factors make it impossible for a second spike to be elicited soon after the first. The RRP lasts as long as it takes for the h and n gates to return to their baseline values (about 15 msec in squid giant axon).

At resting potential, the voltage-gated Na+ conductance is partially inactivated in that the h gate is partly closed and the n gate is partly open (Figs. 4 and 5). Hyperpolarizing the neuron below resting potential increases the value of h, in a process called deinactiva-tion, and decreases the value of n, in a process called deactivation. Deinactivation of the Na+ conductance and deactivation of the K+ conductance can leave the neuron more excitable after hyperpolarization, and thus can account for anodal break excitation (Fig. 5b), also known as rebound spiking. Neurons typically fire only one rebound spike because after that spike the Na+ and K+ conductances return to their baseline states and the cell returnes to its resting level of excitability.

The first major success of the Hodgkin-Huxley model was that this relatively simple model, derived from voltage-clamp experiments, accounted successfully for many aspects of the action potential in the current-clamped and space-clamped axon. Even more impressive, and strikingly demonstrative of the level of understanding this model represents, was its ability to accurately account for the shape and velocity of the propagating action potential. The quantitative arguments involved are complex and will not be discussed here, but Fig. 6 shows qualitatively how action potentials propagate in unmyelinated axons.

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