Reduced Hodgkin-Huxley Equations

Consider the problem of the passage of a single nervous impulse along a fiber it is described by the Hodgkin-Huxley equations. Our aim is not so much to obtain exact relationships between the speed and various other parameters (Huxley has already done this ), as to find out the most essential features of these equations. On the qualitative level an explanation of the nervous impulse looks very simple. 

Considering that h varies slower than m, we may assume in the excitable region h = const. ( 0.6). Similarly, the term p — Na) can also be approximately considered constant in this region, since (p is small compared to PNa-Thus it is the factor which plays the most important role in determining the sodium current. As regards the variable m proper, we will assume for it a relaxation equation of the type of Eq. (8), wherein t is constant and mooi p) is a stepped function  

Introduce the characteristic sodium current ii = gNa ( Na P ) and choose the following measurement units time C(pjh speed coor- 

Using the above measurement units we can rewrite Eq. (9) in the following form  

The condition of Eq. (40) allows Eq. (38) to be solved, whereupon the conditions of Eq. (39) will enable solution of the potential equation [Eq. (37)]. It would seem that V would still remain unknown, but the condition for determining it will be found in the process of solving. Introduce the notation 

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