The neural circuit was a section of the Human brain related to ESP ability.
In 2265, when the starship USS Enterprise encountered the galactic barrier, nine crewmen whose esper rating had been rated as above average were killed due to damage to their neural circuits. (TOS: "Where No Man Has Gone Before")
In 2268, Leonard McCoy claimed that the Kelvans' paralysis field was a sufficient hold over the Enterprise crew, as "they [could] jam all [their] neural circuits" with it. (TOS: "By Any Other Name")