This section attempts to explain the Hodgkin-Huxley experiments from a biological perspective. Hodgkin and Huxley's work with the giant squid axon was the first to use mathematical models to represent biological systems. Due to Hodgkin and Huxley's findings, we are able to understand how an action potential propagates along a nerve and the functions of their associated ion channels.

• The Resting Potential
• The Model Cell
• The Constand Field Equation
• The Resting Membrane Potential
• The Action Potential
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The descriptions of the resting potential and action potential have been interpreted using Nicholls and colleagues' fouth edition textbook, From Neuron to Brain. (Nicholls, John, A. Martin, B. Wallace, and P. Fuchs. From Neuron to Brain. Forth Edition. Sinauer Associates, Inc. MA 2001.)

The Resting Potential

At rest the inside of a neuron is more negatively charged relative to the outside of the neuron. Though the intracellular concentration is high for potassium and low for both chloride and sodium, the resting membrane potential opposes potassium and chloride ions from diffusing down their concentration gradients. A change in extracellular chloride potential will eventually lead to a change in intracellular chloride potential; thus, inducing changes in the relative volume of the cell and changes in chloride, potassium, sodium, and internal anion concentrations. However, a change in extracellular chloride potential will not result in a change in the chloride equilibrium potential or membrane potential at steady state. Conversely, a change in extracellular potassium potential will lead to a change in the relative volume of the cell and alter the membrane potential. In addition, a change in extracellular potassium potential will result in changes in chloride, sodium, and internal anion concentrations.

Sodium and potassium ions constantly leak through the membrane. Yet, the sodium-potassium exchange pump maintains the leakage concentration. Activated by ATP produced by metabolism, the sodium-potassium exchange pump pumps three sodium ions into the cell for every two potassium ions pumped out of the cell. Activation of ion channels changes the permeability of the cell membrane to both potassium and sodium. These changes generate electrical signals changing the amount of charge on the cell membrane; thus, changing the membrane potential.

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The Model Cell

In order to understand how the Nernst equation is used to predict ion potentials, Nicholls et al present the model cell. In their model cell the cell membrane is only permeable to potassium and chloride and impermeable to sodium and an internal anion. In order to remain stable, three requirements must be met:

1) Intracellular and extracellular solutions must be electrically neutral.
2) The cell must be in osmotic balance.
3) There can be no net movement of any particular ion into or out of the cell.

Ionic equilibrium is maintained since the cell membrane acts as a capacitor. As positively charged potassium ions diffuse out of the cell, positive charges accumulate on the outer surface while negative charges accumulate on the inner surface. This difference in electric potential continues until the efflux of potassium ions has stopped, or no net potassium ion movement occurs across the membrane. This is the potassium equilibrium potential denoted EK.

Where [K]o is the external potassium concentration and [K]i is the internal potassium concentration. The chloride equilibrium potential, denoted EK, is given by

since the ionic charge, z, is negative one.

Experiments performed on isolated sections of squid axon in salt water have shown EK values of about -0.093V, ECl values of about -0.055V, and membrane potential, Vm, ranging from -0.065V to -0.070V. Potentials are negative with respect to extracellular fluid. The potassium concentration ratio of intracellular potassium to extracellular potassium is 40:1.

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The Constant Field Equation

According to Kirchhoff’s voltage laws, current is dependent on voltage and resistance, or voltage and conductance.

Thus, the inward sodium current is defined by

where gNa is the sodium membrane conductance which is dependent on the average number of open sodium channels at resting membrane potential.

If chloride is in equilibrium, then there is no net movement of chloride ions across the membrane, or

and

Substituting and rearranging,

if chloride is in equilibrium, and

if chloride is not in equilibrium.

The membrane potential can also be expressed in terms of ionic concentrations inside the cell and outside the cell and ion membrane permeability, illustrated by the Goldman, Hodgkin, Katz (GHK) equation

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Resting Membrane Potential

At the resting membrane potential, the cell must be stable, or each ionic current must be zero. Sodium-potassium leakage currents are held constant by sodium-potassium ATPase, increasing metabolic energy needed to maintain steady state. The ratio of sodium ions to potassium ions that ATPase produces is given by

.

The ratio, r, is negative since sodium and potassium ions are pumped in opposite directions. This transport system is electrogenic since each cycle produces a net outward charge of positive one. A positive charge accumulates on the outside of the cell membrane, while a negative charge accumulates on the inside of the cell membrane. The effect of the electrogenic ATPase sodium-potassium exchange can be compared to a non-electrogenic transport system by setting the ratio, r, to 1.

The resting membrane potential is described by

if chloride is in equilibrium. Note that the value of the resting membrane potential is closer to the value of the potassium potential. Thus, a greater driving force is needed for the influx of sodium ions across the membrane.

Assuming that all other permeating ions are in steady state, the GHK equation for the resting membrane potential becomes

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The Action Potential

The action potential can be described as a resting potential activated by a sharp rising phase (depolarization) followed by a rapid falling phase extending below the original resting potential (hyperpolarization). Repolarization is depicted by a gradual return to the beginning resting potential.

In 1939 Hodgkin and Huxley showed that an overshoot occurred at the peak of the action potential. With a positive interior membrane potential, sodium ions would continue to influx, even past zero, until equilibrium is reached. Thus, an overshoot at the peak of the action potential suggests the significance of sodium ions in creating the action potential.

Further work by Hodgkin and Katz in 1949 included reducing the external sodium concentration of the giant squid axon experiment. The reduction in external sodium concentration caused a decrease in the overshoot at the peak of action potential. Follow-up work has shown that the increase in sodium permeability is attributed to the opening of many voltage-activated sodium channels (depolarization).

The rapid falling phase of the action potential can be attributed to another increase in ion permeability caused by the opening of many voltage-activated potassium channels and the efflux of potassium ions through the membrane. The period of which the potassium channels last several milliseconds allowing more potassium ions to efflux through the membrane past the original resting potential (hyperpolarization).

In short depolarization is described by a sudden increase in sodium permeability due to the opening of a large number of voltage-activated sodium channels causing a rapid influx of sodium ions. Positive charge builds up on the interior membrane until the membrane potential reaches ENa at which point the sodium channels close. Repolarization follows with a sudden increase in potassium permeability due to the opening of a large number of voltage-activated potassium channels causing a rapid efflux of potassium ions. The interior membrane continues to lose positive charge until the membrane potential reaches EK at which point the potassium channels close. Normal sodium and potassium exchange continues as the membrane potential returns to resting potential.

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