View source for Hodgkin-Huxley system

A system of four reaction-diffusion equations (cf. [[Reaction-diffusion equation|Reaction-diffusion equation]]) modelling the electrical activity of nerve cells. The equations have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110210/h1102101.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110210/h1102102.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110210/h1102103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110210/h1102104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110210/h1102105.png" /> are non-linear functions, fitted into experimental data and corresponding to a biochemical model, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110210/h1102106.png" /> is time and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110210/h1102107.png" /> is one-dimensional space.

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110210/h1102108.png" />, undamped travelling-wave solutions, the action potentials (cf. [[Action potential|Action potential]]), have been studied using the Conley index. They include single-pulse solutions, trains of finitely many impulses and periodic solutions.

The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110210/h1102109.png" /> corresponds to a special experimental setting called a current clamp. The equations reduce to a four-dimensional [[Autonomous system|autonomous system]] of ordinary differential equations, its homoclinic and periodic solutions, called stationary action potentials, arising through Hopf (or more degenerate) bifurcations (cf. also [[Homoclinic point|Homoclinic point]]; [[Homoclinic bifurcations|Homoclinic bifurcations]]; [[Hopf bifurcation|Hopf bifurcation]]).

Modifications in the equation that retain the form above, with possibly more variables, abound in the biological literature, accounting for variations in the biochemistry of cells. There is also a simplified version that has been much studied by mathematicians, the FitzHugh–Nagumo equations.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.L. Hodgkin,   A. F. Huxley,   "A quantitative description of membrane current and its application to conduction and excitation in nerve"  ''J. Physiology'' , '''117'''  (1952)  pp. 500–544  (Reprint: Bull. Math. Biology 52 (1990), 25–71)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Rinzel,   "Electrical excitability of cells, theory and experiment: review of the Hodgkin–Huxley foundation and an update"  ''Bull. Math. Biology'' , '''52'''  (1990)  pp. 5–23</TD></TR></table>