ENGR 093: Biomedical Directed Reading Spring 2004  


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.
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 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) 
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 
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