User:Differintegral/sandbox

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

Voltage-gated ion channels
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. Together with the equation for the total current mentioned above, these are:


 * $$I = C_m\frac{{\mathrm d} V_m}{{\mathrm d} t} + \bar{g}_Kn^4(V_m - V_K) + \bar{g}_{Na}m^3h(V_m - V_{Na}) + \bar{g}_l(V_m - V_l),$$


 * $$\frac{dn}{dt} = \alpha_n(V_m)(1 - n) - \beta_n(V_m) n$$


 * $$\frac{dm}{dt} = \alpha_m(V_m)(1 - m) - \beta_m(V_m) m$$


 * $$\frac{dh}{dt} = \alpha_h(V_m)(1 - h) - \beta_h(V_m) h$$

where I is the current per unit area, and $$\alpha_i $$ and $$\beta_i $$ are rate constants for the i-th ion channel, which depend on voltage but not time. $$\bar{g}_n$$ 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, the functions $$\alpha$$ and $$\beta$$ are given by
 * $$ \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, Hodgkin-Huxley type models generalize $$ \alpha $$ and $$ \beta $$ to


 * $$ \frac{A_p(V_m-B_p)}{\exp\big(\frac{V_m-B_p}{C_p}\big)-D_p} $$

with appropriate constants.

where the specific values of $$A_p,B_p,C_p, D_p, a_p, $$ and $$ b_p $$ depend on the ionic gate. In the original paper by Hodgkin and Huxley, these values are


 * $$\alpha_p = p_\infty/\tau_p$$


 * $$ \beta_p = (1 - p_\infty)/\tau_p$$.

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

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. 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:


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


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


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

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


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


 * $$I_K(t)=\bar{g}_K n(V_m)^4(V_m-E_K).$$

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


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

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, a modified Gauss–Newton algorithm, is often used to fit these equations to voltage-clamp data.

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.