Hodgkin-Huxley model
Encyclopedia
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
(a type of scientific model) that describes how action potential
Action potential
In physiology, an action potential is a short-lasting event in which the electrical membrane potential of a cell rapidly rises and falls, following a consistent trajectory. Action potentials occur in several types of animal cells, called excitable cells, which include neurons, muscle cells, and...
s in neuron
Neuron
A neuron is an electrically excitable cell that processes and transmits information by electrical and chemical signaling. Chemical signaling occurs via synapses, specialized connections with other cells. Neurons connect to each other to form networks. Neurons are the core components of the nervous...
s are initiated and propagated.
It is a set of nonlinear
Nonlinearity
In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...
ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s that approximates the electrical characteristics of excitable cells such as neurons and cardiac myocytes
Cardiac muscle
Cardiac muscle is a type of involuntary striated muscle found in the walls and histologic foundation of the heart, specifically the myocardium. Cardiac muscle is one of three major types of muscle, the others being skeletal and smooth muscle...
.
Alan Lloyd Hodgkin
Alan Lloyd Hodgkin
Sir Alan Lloyd Hodgkin, OM, KBE, PRS was a British physiologist and biophysicist, who shared the 1963 Nobel Prize in Physiology or Medicine with Andrew Huxley and John Eccles....
and Andrew Huxley
Andrew Huxley
Sir Andrew Fielding Huxley, OM, FRS is an English physiologist and biophysicist, who won the 1963 Nobel Prize in Physiology or Medicine for his experimental and mathematical work with Sir Alan Lloyd Hodgkin on the basis of nerve action potentials, the electrical impulses that enable the activity...
described the model in 1952 to explain the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon
Squid giant axon
The squid giant axon is the very large axon that controls part of the water jet propulsion system in squid. It was discovered by English zoologist and neurophysiologist John Zachary Young in 1936...
. They received the 1963 Nobel Prize in Physiology or Medicine
Nobel Prize in Physiology or Medicine
The Nobel Prize in Physiology or Medicine administered by the Nobel Foundation, is awarded once a year for outstanding discoveries in the field of life science and medicine. It is one of five Nobel Prizes established in 1895 by Swedish chemist Alfred Nobel, the inventor of dynamite, in his will...
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 bilayerLipid bilayer
The lipid bilayer is a thin membrane made of two layers of lipid molecules. These membranes are flat sheets that form a continuous barrier around cells. The cell membrane of almost all living organisms and many viruses are made of a lipid bilayer, as are the membranes surrounding the cell nucleus...
is represented as a capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
(Cm). Voltage-gated ion channel
Voltage-gated ion channel
Voltage-gated ion channels are a class of transmembrane ion channels that are activated by changes in electrical potential difference near the channel; these types of ion channels are especially critical in neurons, but are common in many types of cells....
s 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 gradient
Electrochemical gradient
An electrochemical gradient is a spatial variation of both electrical potential and chemical concentration across a membrane; that is, a combination of the membrane potential and the pH gradient...
s driving the flow of ions are represented by batteries (En and EL), the values of which are determined from the Nernst potential
Reversal potential
In a biological membrane, the reversal potential of an ion is the membrane potential at which there is no net flow of that particular ion from one side of the membrane to the other...
of the ionic species of interest. Finally, ion pumps
Ion pump (biology)
In biology, an ion transporter, also called an ion pump, is a transmembrane protein that moves ions across a plasma membrane against their concentration gradient, in contrast to ion channels, where ions go through passive transport...
are represented by current sources (Ip).
The time derivative of the potential across the membrane (
) is proportional to the sum of the currents in the circuit. This is represented as follows:
where Ii denotes the individual ionic currents of the model.
Ionic current characterization
The current flowing through the ion channels is mathematically represented by the following equation:
where
is the reversal potentialReversal potential
In a biological membrane, the reversal potential of an ion is the membrane potential at which there is no net flow of that particular ion from one side of the membrane to the other...
of the i-th ion channel.
In voltage-gated ion channels, the channel conductance gi is a function of both time and voltage (gn(t, V) 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 channelsVoltage-gated ion channel
Voltage-gated ion channels are a class of transmembrane ion channels that are activated by changes in electrical potential difference near the channel; these types of ion channels are especially critical in neurons, but are common in many types of cells....
(gn(t, V)) are expressed as:
where
and 
are gating variables for activation and inactivation, respectively, representing the fraction of the maximum conductance available at any given time and voltage. 
is the maximal value of the conductance. 
and 
are constants and 
and 
are the time constants for activation and inactivation, respectively. 
and 
are the steady state values for activation and inactivation, respectively, and are usually represented by Boltzmann equationsBoltzmann equation
The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in rarefied gas...
as functions of
.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. 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:
Thus, for every value of membrane potential,
, the following equation can be fit to the current curve:
The Levenberg–Marquardt algorithm, a modified Gauss–Newton algorithm, is often used to fit these equations to voltage-clamp data.
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
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.Improvements and alternative models
The Hodgkin–Huxley model is widely 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 Fitzhugh-Nagumo model), facilitating efficient large-scale simulation of groups of neurons, as well as mathematical insight into dynamics of action potential generation.
See also
- Action potentialAction potentialIn physiology, an action potential is a short-lasting event in which the electrical membrane potential of a cell rapidly rises and falls, following a consistent trajectory. Action potentials occur in several types of animal cells, called excitable cells, which include neurons, muscle cells, and...
- Biological neural networkBiological neural networkIn neuroscience, a biological neural network describes a population of physically interconnected neurons or a group of disparate neurons whose inputs or signalling targets define a recognizable circuit. Communication between neurons often involves an electrochemical process...
- Biological neuron modelBiological neuron modelA biological neuron model is a mathematical description of the properties of nerve cells, or neurons, that is designed to accurately describe and predict biological processes...
- Fitzhugh-Nagumo model
- MemristorMemristorMemristor is a passive two-terminal electrical component envisioned by Leon Chua as a fundamental non-linear circuit element relating charge and magnetic flux linkage...
- Soliton modelSoliton modelThe Soliton model in neuroscience is a recently developed model that attempts to explain how signals are conducted within neurons. It proposes that the signals travel along the cell's membrane in the form of certain kinds of sound pulses known as solitons...
- Goldman equationGoldman equationThe Goldman–Hodgkin–Katz voltage equation, more commonly known as the Goldman equation is used in cell membrane physiology to determine the equilibrium potential across a cell's membrane taking into account all of the ions that are permeant through that membrane.The discoverers of this are David E...
External links
- 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 a Description in BioModels DatabaseBioModels DatabaseBioModels Database is a free and open-source database for storing, exchanging and retrieving published quantitative models of biological interest...
- Direct link to Hodgkin-Huxley paper #1 via PubMedCentral
- Direct link to Hodgkin-Huxley paper #2 via PubMedCentral
- Direct link to Hodgkin-Huxley paper #3 via PubMedCentral
- Direct link to Hodgkin-Huxley paper #4 via PubMedCentral
- Direct link to Hodgkin-Huxley paper #5 via PubMedCentral
- Neural Impulses: The Action Potential In Action by Garrett Neske, The Wolfram Demonstrations Project