User:Differintegral/sandbox

I should move away from the overly pithy style, because the function doesn't actually follow that form

Using a series of voltage clamp experiments and by varying extracellular sodium and potassium concentrations, Hodgkin and Huxley developed a model in which the properties of an excitable cell are described by a set of four ordinary differential equations.^[1] Together with the equation for the total current mentioned above, these are:

${\displaystyle I=C_{m}{\frac {{\mathrm {d} }V_{m}}{{\mathrm {d} }t}}+{\bar {g}}_{K}n^{4}(V_{m}-V_{K})+{\bar {g}}_{Na}m^{3}h(V_{m}-V_{Na})+{\bar {g}}_{l}(V_{m}-V_{l}),}$

${\displaystyle {\frac {dn}{dt}}=\alpha _{n}(V_{m})(1-n)-\beta _{n}(V_{m})n}$

${\displaystyle {\frac {dm}{dt}}=\alpha _{m}(V_{m})(1-m)-\beta _{m}(V_{m})m}$

${\displaystyle {\frac {dh}{dt}}=\alpha _{h}(V_{m})(1-h)-\beta _{h}(V_{m})h}$

where I is the current per unit area, and ${\displaystyle \alpha _{i}}$ and ${\displaystyle \beta _{i}}$ are rate constants for the i-th ion channel, which depend on voltage but not time. ${\displaystyle {\bar {g}}_{n}}$ is the maximal value of the conductance. n, m, and h are dimensionless quantities between 0 and 1 that are associated with potassium channel activation, sodium channel activation, and sodium channel inactivation, respectively. In the original paper by Hodgkin and Huxley,^[2] the functions ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are given by

${\displaystyle {\begin{array}{lll}\alpha _{n}={\frac {.01(V_{m}-10)}{\exp {\big (}{\frac {V_{m}-10}{10}}{\big )}-1}}&\alpha _{m}={\frac {.1(V_{m}-25)}{\exp {\big (}{\frac {V_{m}-25}{10}}{\big )}-1}}&\alpha _{h}=.07\exp {\bigg (}{\frac {V_{m}}{20}}{\bigg )}\\\beta _{n}=.125\exp {\bigg (}{\frac {V_{m}}{80}}{\bigg )}&\beta _{m}=4\exp {\bigg (}{\frac {V_{m}}{18}}{\bigg )}&\beta _{h}={\frac {1}{\exp {\big (}{\frac {V_{m}-30}{10}}{\big )}+1}}\end{array}}}$

while in many current software programs,^[3] Hodgkin-Huxley type models generalize ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ to

${\displaystyle {\frac {A_{p}(V_{m}-B_{p})}{\exp {\big (}{\frac {V_{m}-B_{p}}{C_{p}}}{\big )}-D_{p}}}}$

with appropriate constants.

where the specific values of ${\displaystyle A_{p},B_{p},C_{p},D_{p},a_{p},}$ and ${\displaystyle b_{p}}$ depend on the ionic gate. In the original paper by Hodgkin and Huxley, these values are

${\displaystyle n}$	${\displaystyle m}$	${\displaystyle h}$	Units
${\displaystyle A_{n}=0.01}$	${\displaystyle A_{m}=0.1}$	${\displaystyle A_{h}=0.01}$	mV,${\displaystyle ^{-1}}$msec${\displaystyle ^{-1}}$
${\displaystyle B_{n}=-10}$	${\displaystyle B_{m}=-25}$	${\displaystyle B_{h}=0.01}$	mV,${\displaystyle ^{-1}}$
${\displaystyle C_{n}=10}$	${\displaystyle C_{m}=10}$	${\displaystyle C_{h}=0.01}$	mV,${\displaystyle ^{-1}}$
${\displaystyle D_{n}=1}$	${\displaystyle D_{m}=1}$	${\displaystyle D_{h}=0.01}$
${\displaystyle a_{n}=.125}$	${\displaystyle a_{m}=4}$	${\displaystyle a_{h}=0.01}$	mV,${\displaystyle ^{-1}}$msec${\displaystyle ^{-1}}$
${\displaystyle b_{n}=80}$	${\displaystyle b_{m}=18}$	${\displaystyle b_{h}=0.01}$	mV,${\displaystyle ^{-1}}$

${\displaystyle \alpha _{p}=p_{\infty }/\tau _{p}}$

${\displaystyle \beta _{p}=(1-p_{\infty })/\tau _{p}}$.

${\displaystyle n_{\infty }}$ and ${\displaystyle m_{\infty }}$, and ${\displaystyle h_{\infty }}$ are the steady state values for activation and inactivation, respectively, and are usually represented by Boltzmann equations as functions of ${\displaystyle V_{m}}$.

In order to characterize voltage-gated channels, the equations are fit to voltage clamp data. For a derivation of the Hodgkin–Huxley equations under voltage-clamp, see.^[4] Briefly, when the membrane potential is held at a constant value (i.e., voltage-clamp), for each value of the membrane potential the nonlinear gating equations reduce to equations of the form:

${\displaystyle m(t)=m_{0}-[(m_{0}-m_{\infty })(1-e^{-t/\tau _{m}})]\,}$

${\displaystyle h(t)=h_{0}-[(h_{0}-h_{\infty })(1-e^{-t/\tau _{h}})]\,}$

${\displaystyle n(t)=n_{0}-[(n_{0}-n_{\infty })(1-e^{-t/\tau _{n}})]\,}$

Thus, for every value of membrane potential ${\displaystyle V_{m}}$ the sodium and potassium currents can be described by

${\displaystyle I_{Na}(t)={\bar {g}}_{Na}m(V_{m})^{3}h(V_{m})(V_{m}-E_{Na}),}$

${\displaystyle I_{K}(t)={\bar {g}}_{K}n(V_{m})^{4}(V_{m}-E_{K}).}$

In order to arrive at the complete solution for a propagated action potential, one must write the current term I on the left-hand side of the first differential equation in terms of V, so that the equation becomes an equation for voltage alone. The relation between I and V can be derived from cable theory and is given by

${\displaystyle I={\frac {a}{2R}}{\frac {\partial ^{2}V}{\partial x^{2}}}}$,

where a is the radius of the axon, R is the specific resistance of the axoplasm, and x is the position along the nerve fiber. Substitution of this expression for I transforms the original set of equations into a set of partial differential equations, because the voltage becomes a function of both x and t.

The Levenberg–Marquardt algorithm,^[5][6] a modified Gauss–Newton algorithm, is often used to fit these equations to voltage-clamp data.^{[citation needed]}

While the original experiments treated only sodium and potassium channels, the Hodgkin Huxley model can also be extended to account for other species of ion channels.