The synapses were the "logic junctions" in the nervous system. According to the newly-regeneration Fifth Doctor, when synapses were weak, they picked up all sorts of jumbled signals, much like radio receivers. His needed time to heal in the Zero Room after his recent regeneration.
According to Nyssa, who studied bioelectronics, his adrenaline normalising helped to bridge the synapses. (TV: Castrovalva [+]Christopher H. Bidmead, Doctor Who season 19 (BBC1, 1982).) The Tenth Doctor was similarly incapacitated after his regeneration, and only regained consciousness due to tea fumes in the TARDIS control room: "superheated infusion of free radicals and tannin, just the thing for healing the synapses." (TV: The Christmas Invasion [+]Russell T Davies, Doctor Who Christmas special (BBC One, 2005).)
Miniature clones of the Fourth Doctor and Leela, injected into the Doctor's brain, witnessed a passing thought—an electrochemical reaction in the synapses—that a leg wanted to move. (TV: The Invisible Enemy [+]Bob Baker and Dave Martin, Doctor Who season 15 (BBC1, 1977).)
The Slitheen intended to use a MITRE headset to harvest telekinetic energy from Luke Smith—every spark from every synapse—but he overloaded the system, and used his amplified telekinesis to escape. (TV: The Lost Boy [+]Phil Ford, The Sarah Jane Adventures series 1 (CBBC, 2007).)
The Tenth Doctor's regeneration energy remained dormant in Donna Noble's head until Davros zapped her with electricity, and her synapses got "that little extra spark, kicking them to life". (TV: Journey's End [+]Russell T Davies, Doctor Who series 4 (BBC One, 2008).)
Cybernetics[]
In alternate 2007, advancements in synapse research allowed John Lumic to bond cyberkinetic impulses onto a metal exoskeleton, creating the Cyberman. (TV: Rise of the Cybermen [+]Tom MacRae, adapted from Spare Parts (Marc Platt), Doctor Who series 2 (BBC One, 2006).)
The android Bigon's reasoning chip contained more circuits than the Doctor's brain had synapses. (TV: Four to Doomsday [+]Terence Dudley, Doctor Who season 19 (BBC1, 1982).)