Hodgkin and Huxley (HH 1952d): Nerve action potential for squid giant axon. Quantitative model of time and voltage-dependent transmembrane currents for Na+, K+, and a leak current, Ileak. Centerpiece for Nobel prize.

## Description

The Hodgkin-Huxley model of membrane current is concerned with the flow of electric currents through the surface membrane of a nerve fiber. The total current is given by the first equation (see Equations section, below) and consists of the capacity current, , and an ionic current, split into three component currents carried by sodium ions (I_{Na}), potassium ions (I_{K}) and other ions (I_{l}). The ionic currents are written in terms of conductances, *g*_{Na}, *g*_{K}, and . The conductances from sodium and potassium ions are time-dependent and are related to the terms in the first equation, and , respectively. The equations for *m*, *h*, and *n* are given by , and . The total current, *I*, is specified and the four differential equations are solved for the conductances and the potential, *V*. *V* is related to E in the diagram by , where *E*_{r} is the absolute value of the resting potential.

## Equations

The five governing equations are

The equations for are given in the reference on page 519 of the 1952d paper as

Note that are undefined for and , respectively, but are continuous functions when the limiting values are taken for those values.

The initial condition for *n* is given in the resting state, defined by , in the model, , as . Initial conditions for *m* and *h* are similarly defined. Depolarization means a negative voltage. The temperature dependence of the model is embodied in the parameter ,

which triples the rate constants for every increase of 10°C.

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

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Hodgkin AL and Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. 500-544, 1952d. Cole KS and Moore JW. Potassium ion current in the squid giant axon: dynamic characteristic. Biophys J 1: 1-14, 1960. Hille B. Ionic Channels of Excitable Membranes. Sunderland, MA: Sinauer, 2001. Jack JJB, Noble D, and Tsien RW. Electric Current Flow in Excitable Cells. Oxford, England: Clarendon Press, 1975.

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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

**Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants:** NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.