Bi-generation was a variant of regeneration which the Doctor believed to be a myth. However, the Fourteenth Doctor experienced a bi-generation after being gravely wounded by a Galvanic Beam. This resulted in him continuing to exist as an incarnation distinct from the newly created Fifteenth Doctor. The process was assisted by Donna Noble and Melanie Bush who each grabbed and pulled the Doctor's arms to separate the two incarnations who both remembered their whole lifetime prior to the bigeneration.
The Fifteenth Doctor later caused the TARDIS to undergo a bigeneration of its own using the fact that he had won a prize for defeating the Toymaker so that both Doctors had their own TARDISes. (TV: The Giggle [+]Loading...["The Giggle (TV story)"])