Namespaces
Variants
Actions

Difference between revisions of "Hodgkin-Huxley system"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
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
 
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>
+
$$\frac{\partial V}{\partial t}=\delta\frac{\partial^2V}{\partial x^2}+I+F(V,y_1,y_2,y_3),$$
  
<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>
+
$$\frac{dy_i}{dt}=\gamma_i(V)y_i+\alpha_i(V),\quad i=1,2,3,$$
  
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.
+
where $F$, $\gamma_i$ and $\alpha_i$ are non-linear functions, fitted into experimental data and corresponding to a biochemical model, $t$ is time and $x$ 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.
+
When $\delta=1$, 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]]).
+
The case $\delta=0$ 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.
 
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.

Latest revision as of 12:01, 26 July 2014

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

$$\frac{\partial V}{\partial t}=\delta\frac{\partial^2V}{\partial x^2}+I+F(V,y_1,y_2,y_3),$$

$$\frac{dy_i}{dt}=\gamma_i(V)y_i+\alpha_i(V),\quad i=1,2,3,$$

where $F$, $\gamma_i$ and $\alpha_i$ are non-linear functions, fitted into experimental data and corresponding to a biochemical model, $t$ is time and $x$ is one-dimensional space.

When $\delta=1$, undamped travelling-wave solutions, the action potentials (cf. Action potential), have been studied using the Conley index. They include single-pulse solutions, trains of finitely many impulses and periodic solutions.

The case $\delta=0$ corresponds to a special experimental setting called a current clamp. The equations reduce to a four-dimensional 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 bifurcations; 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

[a1] 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)
[a2] 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
How to Cite This Entry:
Hodgkin-Huxley system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hodgkin-Huxley_system&oldid=32546
This article was adapted from an original article by I.S. Labouriau (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article