The Department of Thoughts appears to be a building inside Patrick's brain (which appears to be purple). It has a blue sign on top that states the name and blue doors and stairs.
Role in the series
It is shown that there are lots of Patrick's in the Department of Thoughts when Patrick can't read the Chum Bucket slogan, "Chum is Metabolic Fuel." By the time Patrick had his resignation, the Department of Thought was completely burned up and exploded, yet somehow Patrick was able to make a slogan when he was hired by Mr. Krabs.