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MBW:Hodgkin-Huxley

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Summary

The Hodgkin-Huxley papers are an informal term for a series of five 1952 papers culminating with A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. The papers are considered a seminal work in the field of biology; they arguably form the foundation of modern neuroscience. Today the final paper – which introduced the Hodgkin-Huxley equation - has over 7000 electronic citations, and the resulting action potential (AP) equation remains vigorously in use as a model for neural impulses. From a scientific standpoint, the Hodgkin-Huxley papers are notable for predicting biological mechanisms not detectable using contemporary techniques. (Further research has been done and simplified models of reaction diffusion have been found. To read more see APPM4390:Traveling Waves in Excitable Media.)

Background

Hodgkin and Huxley both came from educated and upper-class backgrounds, the accomplishments of the Huxley family being particularly notable. They began studying giant squid axons in 1939, but were quickly called into wartime service. They resumed their studies in 1945, and began to build upon the work of several scientists, most notably David Goldman and the team of KS Cole and HJ Curtis [4]. By the late 1940s, it was well understood that the potential difference across a cell membrane was caused by a concentration gradient of ions. Maintaining this voltage for any length of time, however, required a high impedance across the cell wall. Thus, it was clear that some mechanism existed for modifying the conductivity of the membrane, but it was impossible to determine the mechanism through direct experimentation alone.

Voltage Clamp

A notable feature of the Hodgkin-Huxley papers was their use of voltage clamping voltage clamping, at the time an inchoate technology. The first three of the 1952 papers focused on current-voltage measurements taken from the giant axon of the squid. By controlling the membrane voltage, the scientists were able to construct an i-v curve demonstrating that conductance increased rapidly once the membrane was depolarized by 20 mV.

Gated Ion Channels

Hodgkin and Huxley famously predicted the existence of gated ion channels, despite no concrete evidence to support their claim. In previous work, done in conjunction with Bernard Katz, they had speculated that the ions became charge-neutral when combined with a lipid soluble carrier. However, this paradigm predicted that any incoming current be preceded by an outgoing flow of lipid carriers, which was not borne out by experimental data. Thus, they concluded that the ions were being attracted through selective charge-biased pathways, which necessarily had to perform work to overcome the membrane potential.Ion channels

Significance

Even by the standards of Nobel-Prize-winning research, Hodgkin-Huxley's work has proved unusually fruitful. A recent paper by [2] illustrates the ongoing effort to build upon and improve the original 1952 work. It even goes so far as to quote the original paper extensively in the process of justifying the changes being proposed. The authors suggest an alternative model which accounts for entropy conservation and thermal fluctuations, but acknowledge that the resulting calculations sacrifice clarity increased accuracy. Erwin Neher ultimately provided the evidence for selective ion gating, twenty years after they were predicted by Hodgkin and Huxley. Neher's patch clamp allowed isolated current through a ion channel to be recorded and paved the way for the sophisticated study of neuromuscular junctions and the interaction between nerves and other body tissues.

Model Derivation

The first equation introduced describes the total inward current into an axon. Eq1a.jpg

Where the Ionic currents due to sodium, potassium, and other ions, respectively. The picture commonly associated with Equation 1 is seen in Figure 1.

Fig1a.jpg

The Ionic currents are curve-fitted such that they produce the following equations:

Eq3b.jpg


Where Eq4b.jpg are constants. The simplifications made in the Hodgkin-Huxley model is that the Ionic currents can be modeled with accuracy by first order differential equations. These equations govern n,m, and h.


Eq5a.jpg


Where Eq6.jpg are functions of the initial voltage Eq7.jpg alone. In this way, n, m, and h can be solved analytically.


Plugging in the equations for the ionic currents into the first equation yields: Eq9b.jpg


For more information on the stable and oscillatory behavior of this model, see Period Doubling and other Phenomena in the Hodgkin-Huxley equations.


For a propagating wave, we have the relation: Eq10.jpg

Where a is the radius of the fiber and is the specific resistance of the axoplasm (internal medium though which the pulse travels).

We combine this result with the wave equation: Eq11a.jpg

Where theta is the velocity of conduction. With all this effort, we arrive at the finalized equation:

Eq12.jpg

For a look at simple RC circuit mathematics check out Wikipedia RC Circuit.

To see a simplified 2 dimensions model refer to APPM4390:Mathematical neuroscience: from neurons to circuits to systems

Model Categorization

The Hodgkin-Huxley model is an RC circuit which is governed by ordinary differential equations (ODEs). The purpose of the model is to show how the action potential of neurons is activated and propagated through the cells.

More information on neurons and their propogation can be found in APPM4390:Noise in the Nervous System.

Use of Curve Fitting in Model

Fig2.jpg

A large portion of the paper is devoted to curve-fitting experimental data such as the data seen in Figure 2. This was particularly the case with modeling the conductances of the different ions. To fit the entire behavior of the potassium conductance over time, Hodgkin and Huxley decided to curve-fit the long-term behavior with a first order equation. This behavior can be seen in Figure 3 from in the time intervals [2,5]ms. To reconcile this first-order model with the highly nonlinear behavior seen in the short-term time scale, the first order equation was taken to higher powers, for no reason other than to fit the experimental data.

Outstanding Results

As the Hodgkin-Huxley paper came out only 13 years after the Cole & Curtis paper, they were the first to pioneer modeling the membrane potential of an axon. In this case, the most appropriate “toy example” would be to illustrate the results the model from the Hodgkin-Huxley paper generated. An encouraging figure supporting the model as an appropriate means of describing current flow across an axon is Figure 22 in [1], illustrated as Figure 3 below. Fig3a.jpg

We can see the agreement between these two plots that there is an action potential threshold which is required to be overcome before initiating the electrical pulse as verified by experimentation. We additionally see that for the numerical simulation and the experimental results certain electrical noise below the action potential threshold results in no electrical pulse at all. (For more information on electrical noise see APPM4390:Noise in the Nervous System.) Intuitively, with the amount of electrical noise in the body, there should be some threshold below which noise is “filtered out”, and never results in the electrical pulse to be sent. Note that although these plots have different quantitative values, with the starting temperature set to two different values, we seek the qualitative behaviors associated with the data.

Other Results

However, with all of this curve-fitting, the numerical simulation does fit remarkably well with the experimental results. The electrical properties modeled by this paper included:

  • Form, duration, and amplitude of the voltage spike
  • Conduction velocity
  • Impedance changes during the spike
  • Refractory period
  • Ionic exchanges
  • Subthreshold responses
  • Oscillations

The largest accomplishment of the Hodgkin Huxley paper was that it accurately predicted the presence of ionic channels actively propagating the electrical stimuli without knowing that such channels existed.

Application

Since the Hodgkin-Huxley model is now over 50 years old some people are starting to question its validity in modeling today. C. Meunier and I. Segev[5] discuss the continued usefulness of the HH model in modern neural sciences, particularly the limitations of the original model. As and example the HH model is limited to the two voltage-dependent currents modeled in the squid giant axon and is not suitable for dendrites that have multiple current channels. The authors then present a counter point that the HH model is used as the framework for the expanded model for more current channels. Another point that the authors make is that current computing power has become so great that an over arching model is no longer necessary when one can model the entire system and integrate it numerically.They say that "realistic models, which take into account the morphological and nonlinear electrical properties of neurons at the spatial scale of 10 μm, are now feasible," indicating that the HH model is no longer needed and one can let the computer do the job. But then the majority of the solutions done from this integration are checked against the HH model to determine if the integration is correct.

Even though there are limitations to the Hodgkin-Huxley model it is still a very powerful tool in use today, 50 years after its creation. The model provides a starting framework for more complex problems that may need to be expanded for a proper model. It also can provide a sanity check for computational solutions of large neural networks to ensure the integrations were done correctly.

Project categorization

Mathematics

A system of 4 ODEs is obtained after combing the governing equation of the ionic currents with the wave equation, and 3 more equations describing the 3 gating variables m,n and h.

Model type

The system of ODEs are actually modeling the membrane's potential (the different ionic currents can then be obtained from the membrane's potential) obtained from the state of the different gating variables n,m and h. Those variables are first obtained by solving a system of three first order ODEs. But the important output parameter is the membrane potential since it commands various cell's functions.

Biological system

The biological system studied is the giant squid axon, but can of course be used to describe human neurones (or even other kind of cells such as cardiac cells).

Additional paper discussion

The Hodgkin-Huxley model of a giant squid axon can of course be used to described a human neurone's membrane potential, but also various other types of cell like the cardiomyocytes or cardiac cells. In "Modeling cardiac mechano-electrical feedback using reaction-diffusion-mechanics systems" by R.H. Keldermanna, M.P. Nashb and A.V. Panfilova (Physica D, 238, 11-12), the Hodgkin-Huxley model of membrane potential is modified with the inclusion of the effects of stretch on the different properties of the cell's membrane such as its capacitance or diffusion coefficients. There are very few forces exerted on neurones in the brain whereas the cardiomyocytes are constantly being stretched by their surrounding as the heart beats. They make of their mechano-sensitivity one of the main feed-back mechanisms that allow them to efficiently share the load applied by the blood pressure on the myocardial walls. In this article, the Hodkin-Huxley model is also modified to account for the stretch activated channels (SACs), from which stems the mechano-sensitivity of the cardiac cells by introducing a dependency of the gating variable on the strain felt by the cell's membrane. The SACs will open with a increase of the membrane's strain, resulting in a change of the membrane potential that commands the contraction of the cell, which in return will affect the membrane's strain in a feedback mechanism.

Citations

1. A. Hodgdon and A Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 116, 449-556.

2. T. Heimburg and A Jackson, "On the action potential as a propagating density pulse and the role of anesthetics." Biophys. Rev. Letters 2 (2007) 57-78.

3. (2009, Mar.). Erik Cheever: Superheroes [Online]. Available: http://www.swarthmore/edu/NatSci/echeeve1/Ref/HH/History.htm

4. KS Cole and HJ Curtis, "Electric impedance of the squid giant axon during activity." J. Gen. Phys, 22, 649-670. Online via PubMed Central

5. C. Meunier, I. Segev, Playing the Devil's advocate: is the Hodgkin-Huxley model useful?, Trends in Neurosciences, Volume 25, Issue 11, 1 November 2002, Pages 558-563 Online via ScienceDirect

External Links

For a neat Java application that not only shows simulations deriving from the model, but also various parameters plotted against each other in phase space, try going to http://thevirtualheart.org/HHindex.html.

All five of the Hodkin-Huxley papers can be found on PubMed Central. The links are given below.