The standard kind of Joule thief as (approximately) described in a recent Wired article uses a transformer to establish negative feedback in a bipolar transistor. The key parameters in the system are the inductance of the transform coils (I use 1µH which is easy to achieve), the base resistor (this limits the maximum current in the LED and sets the frequency).
Here is a correct circuit that simulates the operation. I have used four ordinary diodes as an approximation of the operation of a real LED.
Here is what the output voltage across the diodes along with the diode current looks like:
It takes about 2µ seconds to get the oscillator going, but from that point the oscillator puts out pulse of about 3 volts (limited by the diodes) at about 5MHz. If we increase the inductance to 10µH, the frequency drops by about a factor of three. In practice, we may need to include parasitic capacitances in the inductors and from the base to ground to make the simulation more stable (and more reflective of reality).