In 1952, Hodgkin and Huxley wrote a series of five papers that described the experiments they conducted that were aimed at determining the laws that govern the movement of ions in a nerve cell during an action potential. The first paper examined the function of the neuron membrane under normal conditions and outlined the basic experimental method pervasive in each of their subsequent studies. The second paper examined the effects of changes in sodium concentration on the action potential as well as the resolution of the ionic current into sodium and potassium currents. The third paper examined the effect of sudden potential changes on the action potential (including the effect of sudden potential changes on the ionic conductance). The fourth paper outlined how the inactivation process reduces sodium permeability. The final paper put together all of the information from the previous papers and turned them into a mathematical models.

A.L Hodgkin and A.F. Huxley developed a mathematical model to explain the behavior of nerve cells in a squid giant axon in 1952. Their model, which was developed well before the advent of electron microscopes or computer simulations, was able to give scientists a basic understanding of how nerve cells work without having a detailed understanding of how the membrane of a nerve cell looked. To create their mathematical model, Hodgkin and Huxley looked at squid giant axons. They used squid giant axons because squids had axons large enough to manipulate and use their specially built glass electrodes on.(Click here for more information on materials and methods) From their experimentation with a squid axon, they were able to create a circuit model that seemed to match how the squid axon carried an action potential.

Current flowing through the membrane can be carried via the charging and discharging of a capacitor or via ions flowing through variable resistances in parallel with the capacitor. Each of the resistances corresponds to charge being carried by different components. In the nerve cell these components are sodium and potassium ions and a small leakage current that is associated with the movement of other ions, including calcium. Each current (INa, IK, and IL) can be determined by a driving force which is represented by a voltage difference and a permeability coefficient, which is represented by a conductance in the circuit diagram. Conductance is the inverse of resistance. These equations can easily be derived using Ohm’s law (V=IR)

gNa and gK are both functions of time and membrane potential. ENa, EK, EL, Cmand gL are all constants that are determined via experimentation.(Click here for more information on currents)

The influences of membrane potential on permeability were discovered to perform as follows. Under depolarization conditions, there is a transient increase in sodium conductance and a slower but more sustained increase in potassium conductance. These changes can be reversed during repolarization. The nature of these permeability changes was not fully understood when Hodgkin and Huxley did their work. They did not know what the cellular membrane looked like on the micro scale. They did not know about the existence of ion channels and ion pumps in the membrane. Based off of their finding, however, they were able to conclude that changes in permeability were dependant on membrane potential and not membrane current. Molecules aligning or moving with the electric field cause a change in permeability. Originally they supposed that sodium ions crossed the membrane via lipid carrier molecules that were negatively charged. What they observed however, proved that this was not the case. Rather, they supposed that sodium movement depends on the distribution of charged particles which do not act as carriers in the usual sense but rather allow sodium to pass through the membrane when they occupy particular sites on the membrane. This turned out to be the case. These charged particles are ion channels. In the case of sodium permeability, the carrier molecules (as they are referred to by Hodgkin and Huxley) are inactivated when there is a high potential difference. Potassium permeability is similar to sodium permeability but there are some key differences. The activating carrier molecules have an affinity for potassium, not sodium. They move more slowly and they are not blocked or inactivated. (Click here for more information on conductances)

To build their mathematical model that describes how the membrane current works during the voltage clamp experiment, they used the basic circuit equation

where I is the total membrane current density (inward current positive), Ii is the ionic current density (inward current positive), V is the displacement of membrane potential (depolarization is negative), Cm is the membrane capacitance, t is time. They chose to model the capacity current and ionic current in parallel because they found that the ionic current when the derivative was set to zero and the capacity current when the ionic current is set to zero were similar. We can enrich this equation further by realizing that

where INa is the sodium current, IK is the potassium current and IL is the leakage current. We can further expand on this model by adding the following relationships:

Where ER is the resting potential. When examining the graph of the potassium current versus the potassium potential difference, you can see that in the beginning, it’s or third order equation will describe it. But at the end, during the end, it seems to be more first order. In order to explain this in the conductance formula, we let

where is a constant, and n is a dimensionless variable that varies from 0 to 1. It is the proportion of ion channels that are open. To further understand where n comes from we can derive the equation

where alpha is the rate of closing of the channels and beta is the rate of opening. Together, they give us the total rate of change in the channels during an action potential. The sodium conductance is described by the equation

whereis a constant and m is the proportion of activating carrier molecules (ion channels) and h is the proportion of inactivation carrier molecules (ion channels). M and h can be further described by

where alpha and beta are again rate constants that are similar to the rate constants for the potassium conductance. (Click here for more information on inactivation)

Our Previous Work

Hodgkin

Huxley

Graph used to determine the values for the potassium conductance rate constants alpha and beta

Graph used to determine the values for the sodium activation conductance rate constants (m), alpha and beta

Graph used to detmine values for the sodium inactivation conductance rate constants (h), alpha and beta