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) 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)
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)
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)
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)
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)
The android Bigon's reasoning chip contained more circuits than the Doctor's brain had synapses. (TV: Four to Doomsday)