# Hodgkin–Huxley model

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Basic components of Hodgkin–Huxley-type models. Hodgkin–Huxley type models represent the biophysical characteristic of cell membranes. The lipid bilayer is represented as a capacitance (Cm). Voltage-gated and leak ion channels are represented by nonlinear (gn) and linear (gL) conductances, respectively. The electrochemical gradients driving the flow of ions are represented by batteries (E), and ion pumps and exchangers are represented by current sources (Ip).

The Hodgkin–Huxley model is a mathematical model (a type of scientific model) that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear ordinary differential equations that approximates the electrical characteristics of excitable cells such as neurons and cardiac myocytes.

Alan Lloyd Hodgkin and Andrew Huxley described the model in 1952 to explain the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon.[1] They received the 1963 Nobel Prize in Physiology or Medicine for this work.

## Basic components

The components of a typical Hodgkin–Huxley model are shown in the figure. Each component of an excitable cell has a biophysical analog. The lipid bilayer is represented as a capacitance (Cm). Voltage-gated ion channels are represented by nonlinear electrical conductances (gn, where n is the specific ion channel), meaning that the conductance is voltage and time-dependent. This was later shown to be mediated by voltage-gated cation channel proteins, each of which has an open probability that is voltage-dependent. Leak channels are represented by linear conductances (gL). The electrochemical gradients driving the flow of ions are represented by batteries (En and EL), the values of which are determined from the Nernst potential of the ionic species of interest. Finally, ion pumps are represented by current sources (Ip).

The current flowing through the ion channels is mathematically represented by the following equation:

$I_i = {g_i}(V_m - V_i) \;$

where $V_i$ is the reversal potential of the i-th ion channel and $V_m$ is the membrane potential, as measured with respect to the resting potential.

Thus the total current through the membrane is given by:

$I = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t} + g_K(V_m - V_K) + g_{Na}(V_m - V_{Na}) + g_l(V_m - V_l),$

where I is the total membrane current per unit area, Cm is the membrane capacitance per unit area, gK and gNa are the potassium and sodium conductances per unit area, respectively, VK and VNa are the potassium and sodium reversal potentials, respectively, and gl and Vl are the leak conductance per unit area and leak reversal potential, respectively. The time and voltage dependence of the potassium and sodium conductances are dictated by the differential equations below.

## Ionic current characterization

In voltage-gated ion channels, the channel conductance gi is a function of both time and voltage (gn(tV) in the figure), while in leak channels gi is a constant (gL in the figure). The current generated by ion pumps is dependent on the ionic species specific to that pump. The following sections will describe these formulations in more detail.

### Voltage-gated ion channels

Under the Hodgkin–Huxley formulation, conductances for voltage-gated channels (gn(tV)) are expressed as:

${g}_n(V_m) = \bar{g}_n \varphi^\alpha \chi^\beta\,$
$\dot{\varphi}(V_m) = \frac{1}{\tau_\varphi} (\varphi_\infty - \varphi)$
$\dot{\chi}(V_m) = \frac{1}{\tau_\chi} (\chi_\infty - \chi),$

where $\varphi$ and $\chi$ are gating variables for activation and inactivation, respectively, representing the fraction of the maximum conductance available at any given time and voltage. $\bar{g}_n$ is the maximal value of the conductance. $\alpha$ and $\beta$ are constants and $\tau_\varphi$ and $\tau_{\chi}$ are the time constants for activation and inactivation, respectively. A dot over a variable indicates its time derivative. $\varphi_\infty$ and $\chi_\infty$ are the steady state values for activation and inactivation, respectively, and are usually represented by Boltzmann equations as functions of $V_m$.

In order to characterize voltage-gated channels, the equations will be fit to voltage-clamp data. For a derivation of the Hodgkin–Huxley equations under voltage-clamp see.[2] Briefly, when the membrane potential is held at a constant value (i.e., voltage-clamp), for each value of the membrane potential the nonlinear gating equations reduce to linear differential equations of the form:

$\varphi(t) = \varphi_{0} - [ (\varphi_{0}-\varphi_{\infty})(1 - e^{-t/\tau_\varphi})]\,$
$\chi(t) = \chi_{0} - [ (\chi_{0}-\chi_{\infty})(1 - e^{-t/\tau_\chi})].$

Thus, for every value of membrane potential, $V_{m}$, the following equation can be fit to the current curve:

$I_n(t)=\bar{g}_n \varphi^\alpha \chi^\beta (V_m-E_n).$

The Levenberg–Marquardt algorithm,[3][4] a modified Gauss–Newton algorithm, is often used to fit these equations to voltage-clamp data.[citation needed]

### Leak channels

Leak channels account for the natural permeability of the membrane to ions and take the form of the equation for voltage-gated channels, where the conductance $g_i$ is a constant.

### Pumps and exchangers

The membrane potential depends upon the maintenance of ionic concentration gradients across it. The maintenance of these concentration gradients requires active transport of ionic species. The sodium-potassium and sodium-calcium exchangers are the best known of these. Some of the basic properties of the Na/Ca exchanger have already been well-established: the stoichiometry of exchange is 3 Na+:1 Ca2+ and the exchanger is electrogenic and voltage-sensitive. The Na/K exchanger has also been described in detail.[5]

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

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. Another new model, the Soliton model, explains why an action potential traveling along an axon results in a slight local thickening and outward displacement of the membrane. It also accounts for a slight increase in temperature, followed by a decrease in temperature, during an action potential.

## References

1. ^ Hodgkin, A. L.; Huxley, A. F. (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve". The Journal of physiology 117 (4): 500–544. PMC 1392413. PMID 12991237.
2. ^ Gray, Daniel Johnston; Wu, Samuel Miao-Sin (1997). Foundations of cellular neurophysiology (3rd. ed.). Cambridge, Mass. [u.a.]: MIT Press. ISBN 9780262100533.
3. ^ Marquardt, D. W. (1963). "An Algorithm for Least-Squares Estimation of Nonlinear Parameters". Journal of the Society for Industrial and Applied Mathematics 11 (2): 431–000. doi:10.1137/0111030. edit
4. ^ Levenberg, K (1944). "A method for the solution of certain non-linear problems in least squares". Qu. App. Maths. 2: 164.
5. ^ Hille, Bertil (2001). Ion channels of excitable membranes (3. ed. ed.). Sunderland, Mass.: Sinauer. ISBN 9780878933211.