TheHodgkin–Huxley model, orconductance-based model, is amathematical model that describes howaction potentials inneurons are initiated and propagated. It is a set ofnonlinear differential equations that approximates theelectrical engineering characteristics ofexcitable cells such as neurons andmuscle cells. It is a continuous-timedynamical system.

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

The typical Hodgkin–Huxley model treats each component of an excitable cell as an electrical element (as shown in the figure). Thelipid bilayer is represented as acapacitance (C_m).Voltage-gated ion channels are represented byelectrical conductances (g_n, wheren is the specific ion channel) that depend on both voltage and time.Leak channels are represented by linear conductances (g_L). Theelectrochemical gradients driving the flow of ions are represented byvoltage sources (E_n) whosevoltages are determined by the ratio of the intra- and extracellular concentrations of the ionic species of interest. Finally,ion pumps are represented bycurrent sources (I_p).^{[clarification needed]} Themembrane potential is denoted byV_m.

Mathematically, the current flowing into the capacitance of the lipid bilayer is written as

${\displaystyle I_c=C_m\frac{\mathrm{d}{V}_m}{\mathrm{d}t}}$

and the current through a given ion channel is the product of that channel's conductance and the driving potential for the specific ion

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

where${\displaystyle V_i}$ is thereversal potential of the specific ion channel.Thus, for a cell with sodium and potassium channels, the total current through the membrane is given by:

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

whereI is the total membrane current per unit area,C_m is the membrane capacitance per unit area,g_K andg_Na are thepotassium andsodium conductances per unit area, respectively,V_K andV_Na are the potassium and sodium reversal potentials, respectively, andg_l andV_l are the leak conductance per unit area and leak reversal potential, respectively. The time dependent elements of this equation areV_m,g_Na, andg_K, where the last two conductances depend explicitly on the membrane voltage (V_m) as well.

In voltage-gated ion channels, the channel conductance is a function of both time and voltage (${\displaystyle g_n(t,V)}$ in the figure), while in leak channels,${\displaystyle g_l}$, it is a constant (${\displaystyle g_L}$ 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.

Using a series ofvoltage clamp experiments and by varying extracellular sodium and potassium concentrations, Hodgkin and Huxley developed a model in which the properties of an excitable cell are described by a set of fourordinary differential equations.^[1] Together with the equation for the total current mentioned above, these are:

${\displaystyle I=C_m\frac{\mathrm{d}{V}_m}{\mathrm{d}t}+{\overline{g}}_{\text{K}}{n}^4(V_m-V_K)+{\overline{g}}_{\text{Na}}{m}^{3}h(V_m-V_{Na})+{\overline{g}}_l(V_m-V_l),}$

${\displaystyle \frac{dn}{dt}={\alpha }_n(V_m)(1-n)-{\beta }_n(V_m)n}$

${\displaystyle \frac{dm}{dt}={\alpha }_m(V_m)(1-m)-{\beta }_m(V_m)m}$

${\displaystyle \frac{dh}{dt}={\alpha }_h(V_m)(1-h)-{\beta }_h(V_m)h}$

whereI is the current per unit area and${\displaystyle {\alpha }_i}$ and${\displaystyle {\beta }_i}$ are rate constants for thei-th ion channel, which depend on voltage but not time.${\displaystyle {\overline{g}}_n}$ is the maximal value of the conductance.n,m, andh are dimensionless probabilities between 0 and 1 that are associated with potassium channelsubunit activation, sodium channel subunit activation, and sodium channel subunit inactivation, respectively. For instance, given that potassium channels in squid giant axon are made up of four subunits which all need to be in the open state for the channel to allow the passage of potassium ions, then needs to be raised to the fourth power. For${\displaystyle p=(n,m,h)}$,${\displaystyle {\alpha }_p}$ and${\displaystyle {\beta }_p}$ take the form

${\displaystyle {\alpha }_p(V_m)=p_{\mathrm{\infty }}(V_m)/{\tau }_p}$

${\displaystyle {\beta }_p(V_m)=(1-p_{\mathrm{\infty }}(V_m))/{\tau }_p.}$

${\displaystyle p_{\mathrm{\infty }}}$ and${\displaystyle (1-p_{\mathrm{\infty }})}$ are the steady state values for activation and inactivation, respectively, and are usually represented byBoltzmann equations as functions of${\displaystyle V_m}$. In the original paper by Hodgkin and Huxley,^[1] the functions${\displaystyle \alpha }$ and${\displaystyle \beta }$ are given by

where${\displaystyle V=V_{rest}-V_m}$ denotes the negative depolarization in mV.

In many current software programs^[2]Hodgkin–Huxley type models generalize${\displaystyle \alpha }$ and${\displaystyle \beta }$ to

${\displaystyle \frac{A_p(V_m-B_p)}{\exp (\frac{V_m-B_p}{C_p})-D_p}}$

In order to characterize voltage-gated channels, the equations can be fitted to voltage clamp data. For a derivation of the Hodgkin–Huxley equations under voltage-clamp, see.^[3] Briefly, when the membrane potential is held at a constant value (i.e., with a voltage clamp), for each value of the membrane potential the nonlinear gating equations reduce to equations of the form:

${\displaystyle m(t)=m_0-[(m_0-m_{\mathrm{\infty }})(1-e^{-t/{\tau }_m})]}$

${\displaystyle h(t)=h_0-[(h_0-h_{\mathrm{\infty }})(1-e^{-t/{\tau }_h})]}$

${\displaystyle n(t)=n_0-[(n_0-n_{\mathrm{\infty }})(1-e^{-t/{\tau }_n})]}$

Thus, for every value of membrane potential${\displaystyle V_m}$ the sodium and potassium currents can be described by

${\displaystyle I_{\mathrm{N}\mathrm{a}}(t)={\overline{g}}_{\mathrm{N}\mathrm{a}}m(V_m{)}^{3}h(V_m)(V_m-E_{\mathrm{N}\mathrm{a}}),}$

${\displaystyle I_{\mathrm{K}}(t)={\overline{g}}_{\mathrm{K}}n(V_m{)}^4(V_m-E_{\mathrm{K}}).}$

In order to arrive at the complete solution for a propagated action potential, one must write the current termI on the left-hand side of the first differential equation in terms ofV, so that the equation becomes an equation for voltage alone. The relation betweenI andV can be derived fromcable theory and is given by

${\displaystyle I=\frac{a}{2R}\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{x}^2},}$

wherea is the radius of theaxon,R is thespecific resistance of theaxoplasm, andx is the position along the nerve fiber. Substitution of this expression forI transforms the original set of equations into a set ofpartial differential equations, because the voltage becomes a function of bothx andt.

TheLevenberg–Marquardt algorithm is often used to fit these equations to voltage-clamp data.^[4]

While the original experiments involved only sodium and potassium channels, the Hodgkin–Huxley model can also be extended to account for other species ofion 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${\displaystyle g_{leak}}$ is a constant. Thus, the leak current due to passive leak ion channels in the Hodgkin-Huxley formalism is${\displaystyle I_l=g_{leak}(V-V_{leak})}$.

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. Thesodium-potassium andsodium-calcium exchangers are the best known of these. Some of the basic properties of the Na/Ca exchanger have already been well-established: thestoichiometry 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, with a 3 Na⁺: 2 K⁺ stoichiometry.^[5][6]

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The Hodgkin–Huxley model can be thought of as adifferential equation system with fourstate variables,${\displaystyle V_m(t),n(t),m(t)}$, and${\displaystyle h(t)}$, that change with respect to time${\displaystyle t}$. The system is difficult to study because it is anonlinear system, cannot be solved analytically, and therefore has noclosed-form solution. However, there are many numerical methods available to analyze the system. Certain properties and general behaviors, such aslimit cycles, can be proven to exist.

Because there are four state variables, visualizing the path inphase space can be difficult. Usually two variables are chosen, voltage${\displaystyle V_m(t)}$ and the potassium gating variable${\displaystyle n(t)}$, allowing one to visualize thelimit cycle. However, one must be careful because this is an ad-hoc method of visualizing the4-dimensional system. This does not prove the existence of the limit cycle.

A betterprojection can be constructed from a careful analysis of theJacobian of the system, evaluated at theequilibrium point. Specifically, theeigenvalues of the Jacobian are indicative of thecenter manifold's existence. Likewise, theeigenvectors of the Jacobian reveal the center manifold'sorientation. The Hodgkin–Huxley model has two negative eigenvalues and two complex eigenvalues with slightly positive real parts. The eigenvectors associated with the two negative eigenvalues will reduce to zero as timet increases. The remaining two complex eigenvectors define the center manifold. In other words, the 4-dimensional system collapses onto a 2-dimensional plane. Any solution starting off the center manifold will decay towards the center manifold. Furthermore, the limit cycle is contained on the center manifold.

If the injected current${\displaystyle I}$ were used as abifurcation parameter, then the Hodgkin–Huxley model undergoes aHopf bifurcation. As with most neuronal models, increasing the injected current will increase the firing rate of the neuron. One consequence of the Hopf bifurcation is that there is a minimum firing rate. This means that either the neuron is not firing at all (corresponding to zero frequency), or firing at the minimum firing rate. Because of theall-or-none principle, there is no smooth increase inaction potential amplitude, but rather there is a sudden "jump" in amplitude. The resulting transition is known as acanard.

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 incorporatetransition state theory and producethermodynamic Hodgkin–Huxley models.^[7]
• Models often incorporate highly complex geometries ofdendrites andaxons, often based on microscopy data.
• Stochastic models of ion-channel behavior, leading to stochastic hybrid systems.^[8]
• ThePoisson–Nernst–Planck (PNP) model is based on amean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential.^[9]

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

Ion-channel parameters derived from Hodgkin–Huxley-type models fitted to single-cell electrophysiological data have been shown to correlate with single-cell transcriptomic profiles across neuronalcell types, underscoring the biological realism of such models.^[10]

• Interactive Java applet of the HH model Parameters of the model can be changed as well as excitation parameters and phase space plottings of all the variables is possible.
• Direct link to Hodgkin–Huxley model and aDescription inBioModels Database
• Neural Impulses: The Action Potential In Action by Garrett Neske,The Wolfram Demonstrations Project
• Interactive Hodgkin–Huxley model by Shimon Marom,The Wolfram Demonstrations Project
• ModelDB A computational neuroscience source code database containing 4 versions (in different simulators) of the original Hodgkin–Huxley model and hundreds of models that apply the Hodgkin–Huxley model to other channels in many electrically excitable cell types.
• Severalarticles about the stochastic version of the model and its link with the original one.

• Nonlinear systems
• Electrophysiology
• Ion channels
• Computational neuroscience

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