Hodgkin and Huxley formulated the ion currents through the three voltage-gated ion channels Na, K, and L as

$$\sum_{k}I_{k}=g_{\rm Na}\,m^{3}h\,(u-E_{\rm Na})+g_{\rm K}\,n^{4}\,(u-E_{\rm K% })+g_{L}\,(u-E_{L}).$$

From Gerstner et al, Neuronal Dynamics, 2.2 Hodgkin-Huxley Model

What they probably didn't know - but is known today - is, that the sodium channel possesses four voltage-sensors (S4), opposed to one ball (IFM) which may close the channel and is described by $h$:

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From Wikipedia, Ball and chain inactivation

Do I interpret Hodgkin and Huxley correctly, in that they found experimentally that three of the four voltage sensors must be in the up (= open) state for the pore to be open? Does this hold "exactly", or should the exponent in $m^3$ not be taken too "literally"? For example, the four voltage-sensors might be slightly different and have different deviations (when in their resp. up states), and only their "average" deviation counts, which is somehow reflected in the exponent $3$? The question then arises, why the exponent is an integer number. Couldn't it be any real number then?



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