{"history":[],"article":{"id":"7041c3a5-0d2e-4bb3-9c4e-88518fe4eb01","title":"Hodgkin–Huxley model","slug":"hodgkin-huxley-model","content":":::summary\n:::\n\n# \nThe **Hodgkin–Huxley model**, or **conductance-based model**, is a [mathematical model](https://bliptext.com/articles/mathematical-model) that describes how [action potentials](https://bliptext.com/articles/action-potential) in [neurons](https://bliptext.com/articles/neuron) are initiated and propagated.\nIt is a set of [nonlinear differential equations](https://bliptext.com/articles/nonlinear-system) that approximates the [electrical engineering](https://bliptext.com/articles/electrical-engineering) characteristics of [excitable cells](https://bliptext.com/articles/excitable-cell) such as neurons and [muscle cells](https://bliptext.com/articles/muscle-cell).\nIt is a continuous-time [dynamical system](https://bliptext.com/articles/dynamical-system).\n[Alan Hodgkin](https://bliptext.com/articles/alan-hodgkin) and [Andrew Huxley](https://bliptext.com/articles/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](https://bliptext.com/articles/squid-giant-axon).\nThey received the 1963 [Nobel Prize in Physiology or Medicine](https://bliptext.com/articles/nobel-prize-in-physiology-or-medicine) for this work.\n# [Basic components of Hodgkin–Huxley-type models which represent the biophysical characteristic of cell membranes. The lipid bilayer is represented as a capacitance (C_m). Voltage-gated and leak ion channels are represented by nonlinear (g_n) and linear (g_L) 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 (I_p). | upload.wikimedia.org/wikipedia/commons/9/98/Hodgkin-Huxley.svg]\n\n# Basic components\nThe typical Hodgkin–Huxley model treats each component of an excitable cell as an electrical element (as shown in the figure).\nThe [lipid bilayer](https://bliptext.com/articles/lipid-bilayer) is represented as a [capacitance](https://bliptext.com/articles/capacitance) (C_m).\n[Voltage-gated ion channels](https://bliptext.com/articles/voltage-gated-ion-channel) are represented by [electrical conductances](https://bliptext.com/articles/electrical-conductance) (g_n, where n is the specific ion channel) that depend on both voltage and time.\n[Leak channels](https://bliptext.com/articles/leak-channel) are represented by linear conductances (g_L).\nThe [electrochemical gradients](https://bliptext.com/articles/electrochemical-gradient) driving the flow of ions are represented by [voltage sources](https://bliptext.com/articles/voltage-source) (E_n) whose [voltages](https://bliptext.com/articles/voltage) are determined by the ratio of the intra- and extracellular concentrations of the ionic species of interest.\nFinally, [ion pumps](https://bliptext.com/articles/ion-pump-biology) are represented by [current sources](https://bliptext.com/articles/current-sources) (I_p).\nThe [membrane potential](https://bliptext.com/articles/membrane-potential) is denoted by V_m.\nMathematically, the current flowing through the lipid bilayer is written as\nand the current through a given ion channel is the product of that channel's conductance and the driving potential for the specific ion\nwhere V_i is the [reversal potential](https://bliptext.com/articles/reversal-potential) of the specific ion channel.\nThus, for a cell with sodium and potassium channels, the total current through the membrane is given by:\nwhere I is the total membrane current per unit area, Cm is the membrane capacitance per unit area, gK and gNa are the [potassium](https://bliptext.com/articles/potassium) and [sodium](https://bliptext.com/articles/sodium) conductances per unit area, respectively, VK and VNa are the [potassium](https://bliptext.com/articles/potassium) and [sodium](https://bliptext.com/articles/sodium) reversal potentials, respectively, and gl and Vl are the leak conductance per unit area and leak reversal potential, respectively.\nThe time dependent elements of this equation are Vm, gNa, and gK, where the last two conductances depend explicitly on the membrane voltage (Vm) as well.\n\n# Ionic current characterization\nIn voltage-gated ion channels, the channel conductance is a function of both time and voltage (g_n(t,V) in the figure), while in leak channels, g_l, it is a constant (g_L in the figure).\nThe current generated by ion pumps is dependent on the ionic species specific to that pump.\nThe following sections will describe these formulations in more detail.\n\n## Voltage-gated ion channels\nUsing a series of [voltage clamp](https://bliptext.com/articles/voltage-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 four [ordinary differential equations](https://bliptext.com/articles/ordinary-differential-equation).\nTogether with the equation for the total current mentioned above, these are:\nwhere I is the current per unit area and \\alpha_i and \\beta_i are rate constants for the i-th ion channel, which depend on voltage but not time.\n\\bar{g}_n is the maximal value of the conductance.\nn, m, and h are dimensionless probabilities between 0 and 1 that are associated with potassium channel [subunit](https://bliptext.com/articles/protein-subunit) activation, sodium channel [subunit](https://bliptext.com/articles/protein-subunit) activation, and sodium channel [subunit](https://bliptext.com/articles/protein-subunit) inactivation, respectively.\nFor 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, the n needs to be raised to the fourth power.\nFor, \\alpha_p and \\beta_p take the form\np_\\infty and are the steady state values for activation and inactivation, respectively, and are usually represented by [Boltzmann equations](https://bliptext.com/articles/boltzmann-equation) as functions of V_m.\nIn the original paper by Hodgkin and Huxley, the functions \\alpha and \\beta are given by where denotes the negative depolarization in mV.\nIn many current software programs\nHodgkin–Huxley type models generalize \\alpha and \\beta to\nIn order to characterize voltage-gated channels, the equations can be fitted to voltage clamp data.\nFor a derivation of the Hodgkin–Huxley equations under voltage-clamp, see.\nBriefly, 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:\nThus, for every value of membrane potential V_{m} the sodium and potassium currents can be described by\nIn order to arrive at the complete solution for a propagated action potential, one must write the current term I on the left-hand side of the first differential equation in terms of V, so that the equation becomes an equation for voltage alone.\nThe relation between I and V can be derived from [cable theory](https://bliptext.com/articles/cable-theory) and is given by\nwhere a is the radius of the [axon](https://bliptext.com/articles/axon), R is the [specific resistance](https://bliptext.com/articles/resistivity) of the [axoplasm](https://bliptext.com/articles/axoplasm), and x is the position along the nerve fiber.\nSubstitution of this expression for I transforms the original set of equations into a set of [partial differential equations](https://bliptext.com/articles/partial-differential-equation), because the voltage becomes a function of both x and t.\nThe [Levenberg–Marquardt algorithm](https://bliptext.com/articles/levenberg-marquardt-algorithm) is often used to fit these equations to voltage-clamp data.\nWhile the original experiments involved only sodium and potassium channels, the Hodgkin–Huxley model can also be extended to account for other species of [ion channels](https://bliptext.com/articles/ion-channel).\n\n## Leak channels\nLeak 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_{leak} is a constant.\nThus, the leak current due to passive leak ion channels in the Hodgkin-Huxley formalism is.\n\n## Pumps and exchangers\nThe membrane potential depends upon the maintenance of ionic concentration gradients across it.\nThe maintenance of these concentration gradients requires active transport of ionic species.\nThe [sodium-potassium](https://bliptext.com/articles/sodium-potassium-pump) and [sodium-calcium exchangers](https://bliptext.com/articles/sodium-calcium-exchanger) are the best known of these.\nSome of the basic properties of the Na/Ca exchanger have already been well-established: the [stoichiometry](https://bliptext.com/articles/stoichiometry) of exchange is 3 Na⁺: 1 Ca²⁺ and the exchanger is electrogenic and voltage-sensitive.\nThe Na/K exchanger has also been described in detail, with a 3 Na⁺: 2 K⁺ stoichiometry.\n\n# Mathematical properties\nThe Hodgkin–Huxley model can be thought of as a [differential equation](https://bliptext.com/articles/differential-equation) system with four [state variables](https://bliptext.com/articles/state-variable),, and h(t), that change with respect to time t. The system is difficult to study because it is a [nonlinear system](https://bliptext.com/articles/nonlinear), cannot be solved analytically, and therefore has no [closed-form](https://bliptext.com/articles/closed-form-expression) solution.\nHowever, there are many numerical methods available to analyze the system.\nCertain properties and general behaviors, such as [limit cycles](https://bliptext.com/articles/limit-cycle), can be proven to exist.\n# [A simulation of the Hodgkin–Huxley model in phase space, in terms of voltage v(t) and potassium gating variable n(t). The closed curve is known as a limit cycle. | upload.wikimedia.org/wikipedia/commons/9/9e/Hodgkin///Huxley///Limit///Cycle.png]\n\n## Center manifold\nBecause there are four state variables, visualizing the path in [phase space](https://bliptext.com/articles/phase-space) can be difficult.\nUsually two variables are chosen, voltage V_m(t) and the potassium gating variable n(t), allowing one to visualize the [limit cycle](https://bliptext.com/articles/limit-cycle).\nHowever, one must be careful because this is an ad-hoc method of visualizing the 4-dimensional system.\nThis does not prove the existence of the limit cycle.\nA better [projection](https://bliptext.com/articles/projection-mathematics) can be constructed from a careful analysis of the [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant) of the system, evaluated at the [equilibrium point](https://bliptext.com/articles/equilibrium-point).\nSpecifically, the [eigenvalues](https://bliptext.com/articles/eigenvalues) of the Jacobian are indicative of the [center manifold](https://bliptext.com/articles/center-manifold)'s existence.\nLikewise, the [eigenvectors](https://bliptext.com/articles/eigenvalues-and-eigenvectors) of the Jacobian reveal the center manifold's [orientation](https://bliptext.com/articles/orientation-geometry).\nThe Hodgkin–Huxley model has two negative eigenvalues and two complex eigenvalues with slightly positive real parts.\nThe eigenvectors associated with the two negative eigenvalues will reduce to zero as time t increases.\nThe remaining two complex eigenvectors define the center manifold.\nIn other words, the 4-dimensional system collapses onto a 2-dimensional plane.\nAny solution starting off the center manifold will decay towards the center manifold.\nFurthermore, the limit cycle is contained on the center manifold.\n\n## Bifurcations\nIf the injected current I were used as a [bifurcation parameter](https://bliptext.com/articles/bifurcation-theory), then the Hodgkin–Huxley model undergoes a [Hopf bifurcation](https://bliptext.com/articles/hopf-bifurcation).\nAs with most neuronal models, increasing the injected current will increase the firing rate of the neuron.\nOne consequence of the Hopf bifurcation is that there is a minimum firing rate.\nThis means that either the neuron is not firing at all (corresponding to zero frequency), or firing at the minimum firing rate.\nBecause of the [all-or-none principle](https://bliptext.com/articles/all-or-none-law), there is no smooth increase in [action potential](https://bliptext.com/articles/action-potential) amplitude, but rather there is a sudden \"jump\" in amplitude.\nThe resulting transition is known as a [canard](https://bliptext.com/articles/undefined).\n\n# Improvements and alternative models\nThe Hodgkin–Huxley model is regarded as one of the great achievements of 20th-century biophysics.\nNevertheless, modern Hodgkin–Huxley-type models have been extended in several important ways:\nSeveral simplified neuronal models have also been developed (such as the [FitzHugh–Nagumo model](https://bliptext.com/articles/fitzhugh-nagumo-model)), facilitating efficient large-scale simulation of groups of neurons, as well as mathematical insight into dynamics of action potential generation.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T22:41:21.030Z","updated_at":"2025-01-25T22:41:21.030Z","search_vector":"'hodgkin':1 'huxley':2 'model':3","search_content":" \n \n\n# \nThe **Hodgkin–Huxley model**, or **conductance-based model**, is a [mathematical model](https://bliptext.com/articles/mathematical-model) that describes how [action potentials](https://bliptext.com/articles/action-potential) in [neurons](https://bliptext.com/articles/neuron) are initiated and propagated.\nIt is a set of [nonlinear differential equations](https://bliptext.com/articles/nonlinear-system) that approximates the [electrical engineering](https://bliptext.com/articles/electrical-engineering) characteristics of [excitable cells](https://bliptext.com/articles/excitable-cell) such as neurons and [muscle cells](https://bliptext.com/articles/muscle-cell).\nIt is a continuous-time [dynamical system](https://bliptext.com/articles/dynamical-system).\n[Alan Hodgkin](https://bliptext.com/articles/alan-hodgkin) and [Andrew Huxley](https://bliptext.com/articles/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](https://bliptext.com/articles/squid-giant-axon).\nThey received the 1963 [Nobel Prize in Physiology or Medicine](https://bliptext.com/articles/nobel-prize-in-physiology-or-medicine) for this work.\n# [Basic components of Hodgkin–Huxley-type models which represent the biophysical characteristic of cell membranes. The lipid bilayer is represented as a capacitance (C m ). Voltage-gated and leak ion channels are represented by nonlinear (g n ) and linear (g L ) 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 (I p ). | upload.wikimedia.org/wikipedia/commons/9/98/Hodgkin-Huxley.svg]\n\n# Basic components\nThe typical Hodgkin–Huxley model treats each component of an excitable cell as an electrical element (as shown in the figure).\nThe [lipid bilayer](https://bliptext.com/articles/lipid-bilayer) is represented as a [capacitance](https://bliptext.com/articles/capacitance) (C m ).\n[Voltage-gated ion channels](https://bliptext.com/articles/voltage-gated-ion-channel) are represented by [electrical conductances](https://bliptext.com/articles/electrical-conductance) (g n , where n is the specific ion channel) that depend on both voltage and time.\n[Leak channels](https://bliptext.com/articles/leak-channel) are represented by linear conductances (g L ).\nThe [electrochemical gradients](https://bliptext.com/articles/electrochemical-gradient) driving the flow of ions are represented by [voltage sources](https://bliptext.com/articles/voltage-source) (E n ) whose [voltages](https://bliptext.com/articles/voltage) are determined by the ratio of the intra- and extracellular concentrations of the ionic species of interest.\nFinally, [ion pumps](https://bliptext.com/articles/ion-pump-biology) are represented by [current sources](https://bliptext.com/articles/current-sources) (I p ).\nThe [membrane potential](https://bliptext.com/articles/membrane-potential) is denoted by V m .\nMathematically, the current flowing through the lipid bilayer is written as\nand the current through a given ion channel is the product of that channel's conductance and the driving potential for the specific ion\nwhere V_i is the [reversal potential](https://bliptext.com/articles/reversal-potential) of the specific ion channel.\nThus, for a cell with sodium and potassium channels, the total current through the membrane is given by:\nwhere I is the total membrane current per unit area, Cm is the membrane capacitance per unit area, gK and gNa are the [potassium](https://bliptext.com/articles/potassium) and [sodium](https://bliptext.com/articles/sodium) conductances per unit area, respectively, VK and VNa are the [potassium](https://bliptext.com/articles/potassium) and [sodium](https://bliptext.com/articles/sodium) reversal potentials, respectively, and gl and Vl are the leak conductance per unit area and leak reversal potential, respectively.\nThe time dependent elements of this equation are Vm, gNa, and gK, where the last two conductances depend explicitly on the membrane voltage (Vm) as well.\n\n# Ionic current characterization\nIn voltage-gated ion channels, the channel conductance is a function of both time and voltage (g_n(t,V) in the figure), while in leak channels, g_l, it is a constant (g_L in the figure).\nThe current generated by ion pumps is dependent on the ionic species specific to that pump.\nThe following sections will describe these formulations in more detail.\n\n## Voltage-gated ion channels\nUsing a series of [voltage clamp](https://bliptext.com/articles/voltage-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 four [ordinary differential equations](https://bliptext.com/articles/ordinary-differential-equation).\nTogether with the equation for the total current mentioned above, these are:\nwhere I is the current per unit area and \\alpha_i and \\beta_i are rate constants for the i-th ion channel, which depend on voltage but not time.\n\\bar{g}_n is the maximal value of the conductance.\nn, m, and h are dimensionless probabilities between 0 and 1 that are associated with potassium channel [subunit](https://bliptext.com/articles/protein-subunit) activation, sodium channel [subunit](https://bliptext.com/articles/protein-subunit) activation, and sodium channel [subunit](https://bliptext.com/articles/protein-subunit) inactivation, respectively.\nFor 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, the n needs to be raised to the fourth power.\nFor, \\alpha_p and \\beta_p take the form\np_\\infty and are the steady state values for activation and inactivation, respectively, and are usually represented by [Boltzmann equations](https://bliptext.com/articles/boltzmann-equation) as functions of V_m.\nIn the original paper by Hodgkin and Huxley, the functions \\alpha and \\beta are given by where denotes the negative depolarization in mV.\nIn many current software programs\nHodgkin–Huxley type models generalize \\alpha and \\beta to\nIn order to characterize voltage-gated channels, the equations can be fitted to voltage clamp data.\nFor a derivation of the Hodgkin–Huxley equations under voltage-clamp, see.\nBriefly, 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:\nThus, for every value of membrane potential V_{m} the sodium and potassium currents can be described by\nIn order to arrive at the complete solution for a propagated action potential, one must write the current term I on the left-hand side of the first differential equation in terms of V, so that the equation becomes an equation for voltage alone.\nThe relation between I and V can be derived from [cable theory](https://bliptext.com/articles/cable-theory) and is given by\nwhere a is the radius of the [axon](https://bliptext.com/articles/axon), R is the [specific resistance](https://bliptext.com/articles/resistivity) of the [axoplasm](https://bliptext.com/articles/axoplasm), and x is the position along the nerve fiber.\nSubstitution of this expression for I transforms the original set of equations into a set of [partial differential equations](https://bliptext.com/articles/partial-differential-equation), because the voltage becomes a function of both x and t.\nThe [Levenberg–Marquardt algorithm](https://bliptext.com/articles/levenberg-marquardt-algorithm) is often used to fit these equations to voltage-clamp data.\nWhile the original experiments involved only sodium and potassium channels, the Hodgkin–Huxley model can also be extended to account for other species of [ion channels](https://bliptext.com/articles/ion-channel).\n\n## Leak channels\nLeak 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_{leak} is a constant.\nThus, the leak current due to passive leak ion channels in the Hodgkin-Huxley formalism is.\n\n## Pumps and exchangers\nThe membrane potential depends upon the maintenance of ionic concentration gradients across it.\nThe maintenance of these concentration gradients requires active transport of ionic species.\nThe [sodium-potassium](https://bliptext.com/articles/sodium-potassium-pump) and [sodium-calcium exchangers](https://bliptext.com/articles/sodium-calcium-exchanger) are the best known of these.\nSome of the basic properties of the Na/Ca exchanger have already been well-established: the [stoichiometry](https://bliptext.com/articles/stoichiometry) of exchange is 3 Na + : 1 Ca 2+ and the exchanger is electrogenic and voltage-sensitive.\nThe Na/K exchanger has also been described in detail, with a 3 Na + : 2 K + stoichiometry.\n\n# Mathematical properties\nThe Hodgkin–Huxley model can be thought of as a [differential equation](https://bliptext.com/articles/differential-equation) system with four [state variables](https://bliptext.com/articles/state-variable),, and h(t), that change with respect to time t. The system is difficult to study because it is a [nonlinear system](https://bliptext.com/articles/nonlinear), cannot be solved analytically, and therefore has no [closed-form](https://bliptext.com/articles/closed-form-expression) solution.\nHowever, there are many numerical methods available to analyze the system.\nCertain properties and general behaviors, such as [limit cycles](https://bliptext.com/articles/limit-cycle), can be proven to exist.\n# [A simulation of the Hodgkin–Huxley model in phase space, in terms of voltage v(t) and potassium gating variable n(t). The closed curve is known as a limit cycle. | upload.wikimedia.org/wikipedia/commons/9/9e/Hodgkin///Huxley///Limit///Cycle.png]\n\n## Center manifold\nBecause there are four state variables, visualizing the path in [phase space](https://bliptext.com/articles/phase-space) can be difficult.\nUsually two variables are chosen, voltage V_m(t) and the potassium gating variable n(t), allowing one to visualize the [limit cycle](https://bliptext.com/articles/limit-cycle).\nHowever, one must be careful because this is an ad-hoc method of visualizing the 4-dimensional system.\nThis does not prove the existence of the limit cycle.\nA better [projection](https://bliptext.com/articles/projection-mathematics) can be constructed from a careful analysis of the [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant) of the system, evaluated at the [equilibrium point](https://bliptext.com/articles/equilibrium-point).\nSpecifically, the [eigenvalues](https://bliptext.com/articles/eigenvalues) of the Jacobian are indicative of the [center manifold](https://bliptext.com/articles/center-manifold)'s existence.\nLikewise, the [eigenvectors](https://bliptext.com/articles/eigenvalues-and-eigenvectors) of the Jacobian reveal the center manifold's [orientation](https://bliptext.com/articles/orientation-geometry).\nThe Hodgkin–Huxley model has two negative eigenvalues and two complex eigenvalues with slightly positive real parts.\nThe eigenvectors associated with the two negative eigenvalues will reduce to zero as time t increases.\nThe remaining two complex eigenvectors define the center manifold.\nIn other words, the 4-dimensional system collapses onto a 2-dimensional plane.\nAny solution starting off the center manifold will decay towards the center manifold.\nFurthermore, the limit cycle is contained on the center manifold.\n\n## Bifurcations\nIf the injected current I were used as a [bifurcation parameter](https://bliptext.com/articles/bifurcation-theory), then the Hodgkin–Huxley model undergoes a [Hopf bifurcation](https://bliptext.com/articles/hopf-bifurcation).\nAs with most neuronal models, increasing the injected current will increase the firing rate of the neuron.\nOne consequence of the Hopf bifurcation is that there is a minimum firing rate.\nThis means that either the neuron is not firing at all (corresponding to zero frequency), or firing at the minimum firing rate.\nBecause of the [all-or-none principle](https://bliptext.com/articles/all-or-none-law), there is no smooth increase in [action potential](https://bliptext.com/articles/action-potential) amplitude, but rather there is a sudden \"jump\" in amplitude.\nThe resulting transition is known as a [canard](https://bliptext.com/articles/undefined).\n\n# Improvements and alternative models\nThe Hodgkin–Huxley model is regarded as one of the great achievements of 20th-century biophysics.\nNevertheless, modern Hodgkin–Huxley-type models have been extended in several important ways:\nSeveral simplified neuronal models have also been developed (such as the [FitzHugh–Nagumo model](https://bliptext.com/articles/fitzhugh-nagumo-model)), facilitating efficient large-scale simulation of groups of neurons, as well as mathematical insight into dynamics of action potential generation.","revisionCount":0},"hasMore":false,"nextCursor":null}

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