The lateral inhibition of Mach bands plays a part here, but other brain processing also influences those dark regions you "see" at the intersections. The brightness we observe at the intersections depends on how our eyes are scanning the pattern and even somewhat on suggestions we are given about what to look for.
A good reference on these "induced brightness" illusions is Visual and Auditory Perception by Gerald M. Murch, Bobbs-Merril, 1973. (Gerald Murch was in PSU's Psychology Department several years ago.)use "Back" to "AARGHH!!"
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