NumHH

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

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

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

Euler's equation is the simplest way to solve a differential equation numerically. However it is often not the preferred method: often you need to take much smaller time steps with Euler than with some other methods, so it takes longer to get as good a solution. Still if you are doing something complicated, like solving an equation with noise, or Bayesian filtering (to compute likelihood), an argument can be made that a simpler method is desirable -- at least as a first step.

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