The Hodgkin–Huxley model, or conductance-based model, is a mathematical model
that describes how action potentials in neurons are initiated and propagated.
It is a set of nonlinear differential equations that approximates the
electrical characteristics of excitable cells such as neurons and cardiac
myocytes, and hence it is a continuous time model, unlike the Rulkov map for
example.

Alan Lloyd Hodgkin and Andrew Fielding Huxley described the model in 1952 to
explain the ionic mechanisms underlying the initiation and propagation of
action potentials in the squid giant axon. They received the 1963 Nobel Prize
in Physiology or Medicine for this work.

## Mathematical properties

The Hodgkin–Huxley model can be thought of as a differential equation with four
state variables, v(t), m(t), n(t), and h(t), that change with respect to time
t. The system is difficult to study because it is a nonlinear system and cannot
be solved analytically. However, there are many numeric methods available to
analyze the system. Certain properties and general behaviors, such as limit
cycles, can be proven to exist.

## Alternative Models

The Hodgkin–Huxley model is regarded as one of the great achievements of 20th-century biophysics. Nevertheless, modern Hodgkin–Huxley-type models have been extended in several important ways:

* Additional ion channel populations have been incorporated based on experimental data.

* The Hodgkin–Huxley model has been modified to incorporate transition state
  theory and produce thermodynamic Hodgkin–Huxley models.

* Models often incorporate highly complex geometries of dendrites and axons,
  often based on microscopy data.

* Stochastic models of ion-channel behavior, leading to stochastic hybrid
  systems

Several simplified neuronal models have also been developed (such as the
FitzHugh–Nagumo model), facilitating efficient large-scale simulation of groups
of neurons, as well as mathematical insight into dynamics of action potential
generation.


## References

The body of this page is from Wikipedia (see below).

#### Papers

"The dual effect of membrane potential on sodium conductance in the giant axon
of Loligo". *The Journal of Physiology*. **116** (4): 497–506. April 1952.
doi:10.1113/jphysiol.1952.sp004719.

"Currents carried by sodium and potassium ions through the membrane of the
giant axon of Loligo". *The Journal of Physiology*. **116** (4): 449–72. April 1952.
doi:10.1113/jphysiol.1952.sp004717.

"The components of membrane conductance in the giant axon of Loligo". *The
Journal of Physiology*. **116** (4): 473–96. April 1952.
doi:10.1113/jphysiol.1952.sp004718.

"The dual effect of membrane potential on sodium conductance in the giant axon
of Loligo". *The Journal of Physiology*. **116** (4): 497–506. April 1952.
doi:10.1113/jphysiol.1952.sp004719.

"A quantitative description of membrane current and its application to
conduction and excitation in nerve". *The Journal of Physiology*. **117** (4):
500–44. August 1952. doi:10.1113/jphysiol.1952.sp004764.

#### Interactive Models on the Web

* ModelDB: [Squid axon (Hodgkin, Huxley 1952)](https://senselab.med.yale.edu/ModelDB/ShowModel.cshtml?model=5426)
* Wolfram Demonstrations:
  [Interactive Hodgkin-Huxley](http://demonstrations.wolfram.com/HodgkinHuxleyActionPotentialModel/)
  by Shimon Marom and
  [Neural Impulses: The Action Potential in Action](http://www.demonstrations.wolfram.com/NeuralImpulsesTheActionPotentialInAction/)
  by Garrett Neske
* [Hodgkin-Huxley Simulation with Javascript](http://myselph.de/hodgkinHuxley.html)
  by Hubert Eichner, which creates static plots in the browser.
* BioModels database: [](http://www.ebi.ac.uk/biomodels-main/BIOMD0000000020)

#### Other Links

* Wikipedia: [Hodgkin–Huxley model](https://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model)
* [Summary of the Hodgkin-Huxley model](http://ecee.colorado.edu/~ecen4831/HHsumWWW/HHsum.html)
* [Hodgkin-Huxley model in R](http://www.magesblog.com/2012/06/hodgkin-huxley-model-in-r.html)