Difference between revisions of "NumHH"

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<math> \frac{dy}{dt} = f(y) </math>
 
<math> \frac{dy}{dt} = f(y) </math>
  
<big> <big> <big> <math> y_k = y_{k-1} + dt \ f(y_{k-1}) </math> </big> </big> </big>
+
<big> <big> <big> <math> y_k = y_{k-1} + \Delta t \ f(y_{k-1}) </math> </big> </big> </big>
  
 
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.
 
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 [[Media:intuit.hoc|code]] for visualizing the numerical solution of differential equations.
 
Click here for [[Media:intuit.hoc|code]] for visualizing the numerical solution of differential equations.

Revision as of 19:37, 23 February 2009

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} + \Delta t \ (-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} + \Delta t \ 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.