Under the viewscreen on many Federation starships, there are a series of lights that constantly run through a chase or 'marching ants' pattern. What's the reason for these lights? Are they status indicators or some part of the viewscreen projection system?
As to their purpose in TNG, the simple truth seems to be that they're utterly useless and exist solely to create a mental linkage between TOS and TNG for fans
"While creating these designs with Herman Zimmerman, we heard a tremendous amount of angst among the fans that we were 'replacing their [original] show,'" Probert says. "So I included clues that hopefully would calm their concerns." Those clues include chasing lights below the viewscreen, and a ship's diagram and dedication plaque similar to those from TOS. "Instead of being at the back of the bridge, they're on either side, near the front," he adds. "Just above the food replicators — which were never used."
Almost everyone looks towards the main viewscreen, so it's almost certainly all about Red Alert. A whole bunch of other unnecessary/unused lights on the bridge also exist that only turn on during emergencies:
(Actually, the one above the door may usually be on, but without a color tint. I just quickly circled some Red Alert lights from TNG 1x21, The Arsenal of Freedom)
I have no doubt those running lights do the same. There could be some important extra information conveyed through them (perhaps it's a battery meter for antimatter reserves?), but I don't recall the lights ever actually being mentioned in the series.
Well, it was my belief that in the original series (TOS), the chase lights have 2 modes. The first one being 2 lights next to each other - alternating left and right, and the more common line left to right. In Star Trek IV it switches to the format we see today. In TNG, DS9 and VOYAGER they are the equivalent to the CPU lights on a modern PC. Note that in Star Trek IV and later that only have one mode: 2 lights moving outwards towards the middle. Note that Voyager has 2 sets of chase lights: one under the view screen and one above the view screen.