The positronic brain

The positronic brain is an extremely sophisticated computation device capable of artificial sentience, created by Dr. Noonien Soong. This device consists of an artificial neural network, designed to imitate the humanoid brain. The construction of a positronic brain is extremely complex, and Dr. Soong is the only scientist to have done so successfully – on at least seven occasions: three unspecified prototypes, B-4, Lore, Data and Juliana Soong (although the latter three were the only truly stable units).
In early 2366, Data allowed nanites to enter his neural network in order to communicate verbally with the crew of the USS Enterprise-D. (TNG: "Evolution")
Later that year, Soong's creation, Data, attempted to emulate his father's work, and created the android Lal. However, Lal's positronic net suffered a cascade failure, and ceased functioning. (TNG: "The Offspring")
On Omicron Theta, Dr. Soong experimented with scanning the synaptic patterns of the colonists' temporal lobes and programmed them into Data's neural nets. (TNG: "Silicon Avatar")
In early 2368, Data's positronic links were severed, cutting off his brain from the rest of his body. (TNG: "The Game")
A plasma shock was enough to overload Data's neural net. (TNG: "Power Play")

Medical Uses

Dr. Julian Bashir implanted Bareil Antos with a positronic matrix to keep him alive after part of his brain stopped functioning. Bareil awoke but said that the sensations he was experiencing did not seem real and seemed like a distant memory. Later, when other parts of Bareil's brain stopped functioning, Bashir chose to let Bareil die rather than replace the rest of his brain with positronic implants. (DS9: "Life Support")
"It is not mentioned whether or not this technology is the same technology used in Data's brain.

Mirror Universe

During the USS Rosenante mission Into the Darkness, Go I it was discovered that Ensign Robert Smith of that universe had several cybernetic components installed in his body. These included a positronic net, cybernetic arms and legs as well as a high capacity data dump storage device installed in his chest. As a result of this work, performed by the Mirror Universe version of Doctor Soong, Smith was able to directly interface with the computers of the ISS Rosenante. The installation of the technology that was installed in Smith was performed as a result of a raid on the [Bynars|Bynar]] Homeworld of Bynaus. the raid was undertaken by the Cybernetic Division of the Imperial Daystrom Institute and the attack was conducted by the institute's tactical division using the ISS Soong as a base of operations for the assault. Later the Bynars would be subjegated and would join the IUP.