The decay isn't as effective as you might first think, and Erik's comment is probably correct for most cases. It's conceivable that one could carefully tune the cap value and mux rate to work together. Adding a moving display to the formula complicates it considerably, but may be possible. The protion of all cases, in which it will work satisfactorily, is reduced considerably, however. It's all in the timing.

There's another thing about retinal persistence (fatigue) that works against you in this case, too. At first, with maximal current flow, the LED is at peak brightness. As the cap charges, the current flowing into it is reduced. That reduces the brightness as one approaches the "fully charged" state. Ideally, the LED will still be conducting slightly when the next cycle begins, so as to eliminate the turn-on delay of the LED. Unfortunately, the peak charge rate is at the wrong end of the curve, so the brightness decrease coincides with the maximal recovery of the retina. There may be a way to reverse this effect, but I haven't skulled it out.

There's no single-bit memory concept involved, BTW, in what I proposed. It's just a slow, and ~logarithmically decreasing current sink. You discharge it (or not, if it wasn't a '1') quickly through a series resistor to the NPN and to GND, and charge the cap through its series resistor via the cathode of the LED. That resistor is connected at the LED's cathode, of course.

RE