1. Each cell in the nervous system acts independently of all others. CORRECT
2. Each cell in the nervous system goes its own way. INCORRECT IN THIS CONTEXT. However, if you personified the cell (figuratively, poetically) it's possible to use this phrase.
To go [one's] own way refers to a person's preferred behavior. (Although I suppose you could say, "My dog doesn't listen to me; he just goes his own way.")
go one's way
Also, go one's own way.
1. Do what one pleases, especially differing from what others are doing, as in You go your way and I'll go mine, or He always insisted on going his own way. This expression, dating from about 1400, originally alluded to moving on in a particular direction. It can still do so, as in The delivery truck went on its way. By the late 1500s it was also being used figuratively.
2. Proceed according to one's plans or wishes, as in Let's hope things will go my way this time. Applied to both events and people's actions, this thought is often expressed as everything's going one's way or going my way . For example, With her husband in charge, everything's going her way, or I trust you'll be going my way when we vote on this issue.