NumericalDiffEqs

Equilibrium Potentials

You can find the equilibrium potential for given channel conductances. Set the injected current to zero I = 0. Then fix and absorb the gating variables and maximal conductances $(g_L, g_{Na}, g_{K})$ into single conductance variables $(G_L, G_{Na}, G_{K})$.

For example

$G_{K} = n^4 g_K$

Then the differential equation for the membrane is

$C \frac{dV}{dt} = - G_L (V - E_L) - G_{Na} (V - E_{Na}) - G_K (V - E_K)$

The equilibrium potential is the value of V such that $\frac{dV}{dt}$ is zero. You can plug in $\frac{dV}{dt} = 0$ and solve for $V_{equilibrium}.$

$V_{equilibrium} = \frac{G_L E_L + G_{Na} E_{Na} + G_K E_K}{G_L + G_{Na} + G_K}$

The equilibrium potential is the weighted average of the reversal potentials -- weighted by the corresponding conductances. Note that the weights add to one.

Numerical Solution of Differential Equations

Remember the equation for the cell with only leak channels.

$C \frac{dV}{dt} = I(t) - g_L(V - E_L)$

Let's simplify: suppose there is no injected current and that the reversal potential for the leak channels is $E_L = 0$. Then our equation is

$\frac{dV}{dt} = - \frac{g_L}{C} V$

Using different letters for the variables (because this is done in the software linked below):

$\frac{dy}{dt} = - k y$

Here k is the rate constant, 1/k is the time constant, 1/k is $\frac{C}{g_L} = RC$ in the notation above. A leaky cell is what is called an RC circuit -- a resistor and capacitor together in a circuit. The time constant of an RC circuit is RC. The bigger k, the higher the rate of convergence, and the smaller the time constant 1/k. The time constant is the time it takes the solution to decay to 1/e of its value.

Solution of differential equations happens at discrete times: $y_k$, separated by small time intervals dt.

The simplest way of solving this equation is with Euler's method:

$y_k = y_{k-1} + dt (-k y_{k-1})$

This is a special case of the general formula for Euler's method applied to the (vector) differential equation

$\frac{dy}{dt} = f(y)$

$y_k = y_{k-1} + dt f(y_{k-1})$

Click here for code for visualizing the numerical solution of differential equations.