Hodgkin-Huxley model

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The Hodgkin-Huxley Model is a set of non-linear ordinary differential equations, named after Alan Lloyd Hodgkin and Andrew Huxley, that approximates the electrical characteristics of excitable cells such as neurons and cardiac myocytes. The original Hodgkin-Huxley model described the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon [1]. The model has played a seminal role in biophysics and neuronal modeling, but in 2006 was determined not to give an appropriate description for excitation transfer in cerebral cortex neurons of higher vertebrates [2] as it is based on superseded assumptions of synapse function and were probably in vitro artifacts. For synapses in other contexts such as that from which the model was originally described, the Hodgkin-Huxley model is still considered valid pending further research.

The components of a typical Hodgkin-Huxley model are shown in the figure. Each component of an excitable cell has a biophysical analogue. The lipid bilayer is represented as a capacitance (Cm). Voltage-gated ion channels are represented by nonlinear electrical conductance (gn, where n is the specific ion channel). Leak channels are represented by linear conductances (gL). The electrochemical gradients driving the flow of ions are represented by batteries (Ei 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 time derivative of the potential across the membrane ($\dot{V}_m$) is proportional to the sum of the currents in the circuit. This is represented as follows:

$\dot{V}_{m}= -\frac{1}{C_m} (\sum\limits ^{}_{i} I_{i} ),$

where Ii denotes the individual ionic currents of the model.

A reduced version of the Hodgkin-Huxley model was proposed by Richard FitzHugh and is now known as the FitzHugh-Nagumo model.

Ionic Current Characterization

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

${I}_{i}(V_m,t)= {g_i} (V_m - E_i).$

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 channels (gn(t,V)) are expressed as:

${g}_{n}(V_m,t) = \bar{g}_n \phi^{\alpha} \chi^{\beta}$

$\dot{\phi}(V_m,t) = \frac{1}{\tau_{\phi}} (\phi_{\infty} - \phi)$

$\dot{\chi}(V_m,t) = \frac{1}{\tau_{\chi}} (\chi_{\infty} - \chi)$,

where $\phi$ 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 $\tau_{\phi}$ and $\tau_{\chi}$ are the time constants for activation and inactivation, respectively. $\phi_{\infty}$ 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 [3]. 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:

$\phi(t) = \phi_{0} - [ (\phi_{0}-\phi_{\infty})(1 - e^{-t/\tau_{\phi}})]$

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

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

$I_{n}(t)=\bar{g}_{n} \phi^{\alpha} \chi^{\beta} (V_{m}-E_{n})$.

The Levenberg-Marquardt algorithm [4][5], 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 $g_i$ 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 [6].

References

1. ^ Hodgkin, A., and Huxley, A. (1952): A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117:500–544. PMID 12991237
2. ^ Naundorf, B. et al. (2006): Unique features of action potential initiation in cortical neurons. Nature 440 (7087): 1060-1063. DOI:10.1038/nature04610 HTML abstract Supplementary information
3. ^ Marquardt, D. (1963): An algorithm for the least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11 (2):431–441.
4. ^ Levenberg, K. (1944): A method for the solution of certain non-linear problems in least-squares. Q. Appl. Math. 2 (2):164–168.
5. ^ Johnston, D., and Wu, S. (1997): Foundations of Cellular Neurophysiology, chapter 8. MIT Press, Cambridge, MA. ISBN 0-262-10053-3
6. ^ Hille, B. (2001): Ionic Channels of Excitable Membranes (3rd ed.). Sinauer Associates, Inc., Sunderland, MA. ISBN 0-87893-321-2de:Hodgkin-Huxley-Modell