The operation is simple

The 555 is wired as an astable and the capacitor is charged only through the 4,7Kohm trimmer (notice the diode) and discharged only through the 2.2 Kohm trimmer, making the duty cycle full adjustable. The square wave is then feed in a totem pole made up of a 2N3904 and a 2N3906, which are cheap, and easy to find. The totem pole ensures the gate being charged and discharged very fast (approx 50nS i think). The IRF840 is a cheap (i found it for 4euros) reliable and powerful power mosfet, it has current capability of 8 A continuous and 32A pulse, 800V drain source voltage, protecting internal zener diode.? There is a snubbing network to ensure that voltage spikes are kept low (unless the insulation of the? transformer start to leak) protecting both transistors and 555 IC. 100 ohm is a compromise between decay time and voltage spike.

Comments and specifications:

The 100 ohm snubber must me a 5W resistor, or it will burn at long operations
The led is only for safety purposes
Use a dead man switch (pushbutton) for safety
The power supply must supply at least 2-3 A if you want decent arcs (20000 KV)

Dangers:
The flyback driven in this way can supply a significant current, aldough the heart fibrillation starts at 30mA
I recommend caution to avoid painful arc-burns.
The arc is a hot plasma, never operate the circuit in presence of flammable substances.
Charging high voltage capacitors is a serious life threat, so if you arent unexperienced just draw arcs and no more
This device when rectified generates static voltage that can be a little annoying.... (or fun, i sprayed with corona
a plastic pen from positive terminal and then i was able to attract little pieces of paper)