Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo.

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I've just set up an account, and have uploaded two papers by Alan Hodgkin and Andrew Huxley, in which they describe their classic experiments on the giant axon of the squid. With these experiments, which were performed in the early 1950s, Hodgkin and Huxley elucidated the mechanism of the action potential, and were subsequently awarded the Nobel Prize for Physiology or Medicine.

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Their equation is paradigmatic of Mathematical Biology the way that Maxwell's equations (in Heaviside's version) are paradigmatic of Mathematical Physics. But I have caveats.

Hodgkin-Huxley requires eyeballing graphs of voltage-clamp measurements and determining what part of the graph corresponds to what part of the model. A conference that I went to a couple of years had a paper on an artififical intelligence program that would do this grad student work, essentially lining up the epicycles. I don't accept that nerves and brains work according to the paradigm.

The equations also have too many parameters. Simpler models have gained popularity. To excerpt and adapt from "FitzHugh-Nagumo model"

[JVP: which I've slightly adapted in format and ASCII presentation]

The FitzHugh-Nagumo Model (named after Richard FitzHugh, 1922-2007) describes a prototype of an excitable system, i.e., a neuron.

If the external stimulus I_ext exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables v and w relax back to their rest values.

This behaviour is typical for spike generations (= short elevation of membrane voltage v) in a neuron after stimulation by an external input current.

The equations for this dynamical system read

[JVP: notationally vdot = dv/dt]

[JVP: notationally wdot = dw/dt]

vdot= v - v^3 - w + I_ext

tau wdot = v - a - bw.

The dynamics of this system can be nicely described by zapping between the left and right branch of the cubic nullcline.

The FitzHugh-Nagumo model is a simplified version of the Hodgkin-Huxley model which models in a detailed manner activation and deactivation dynamics of a spiking neuron. In the original papers of FitzHugh this model was called Bonhoeffer-van der Pol oscillator, since it contains the van der Pol oscillator as a special case for

a = b = 0.

See also

* Biological neuron models

* Hodgkin-Huxley model

* Computational neuroscience

References

* FitzHugh R. (1955) Mathematical models of threshold phenomena in

the nerve membrane. Bull. Math. Biophysics, 17:257--278

* FitzHugh R. (1961) Impulses and physiological states in

theoretical models of nerve membrane. Biophysical J. 1:445-466

* FitzHugh R. (1969) Mathematical models of excitation and

propagation in nerve. Chapter 1 (pp. 1-85 in H.P. Schwan, ed.

Biological Engineering, McGraw-Hill Book Co., N.Y.)

* Nagumo J., Arimoto S., and Yoshizawa S. (1962) An active pulse

transmission line simulating nerve axon. Proc IRE. 50:2061-2070.

Professor Tim Poston emailed me this critique of the above comment.

Too many for what?

FitzHugh-Nagumo is popular among dynamicists, because it achieves in the plane much of what HH does in 4D, and thus makes the math (both exposition and theorem-proving) far simpler.

The HH variables are far more biological, and its broad picture of ion flows has stood the test of time -- but of course there is a far more detailed picture now. Modelling that, mathematically, needs more parameters, not fewer.

"Too many", like "too few", is relative to one's current purpose.

You can be too thin if you belong to a culture that fixes bride-price on a per-lb basis on the wedding day, or too rich if you need bodyguards just to go for a walk.