Meaning of "external demands" in a paper describing the gut-brain axis
From the context, it seems to refer to those demands outside of the human body. i.e. perceptions of the external world. Examples could include a perceived threat, diunal cycles, or stress about that public presentation you're giving later. 

If you are about to be chased by a dangerous predator, a sudden urge to poop would probably impair your chances of survival. Luckily, when our sympathetic nervous system kicks in, our GI motility slows down, so our body can focus on surviving the external threat and live to poop another day.