In NIC1 this is row 128, in NIC2 it is column 128 and in NIC3 column 129. Its appearance seems to be related to the degree pedestal is present. The adjacent rows (or columns) are sometimes affected.
The affected column/row contains the first 256 pixels read out in each quadrant. Since the shading function is very steep and highly nonlinear during this part of the readout, it is the part of the array most sensitive to changes in the detector environment - i.e. the same thing that the pedestal effect is supposed to be. The pixels are not any less sensitive, they have just had an incorrect bias subtracted from them. The result is a row or column of pixels that is either under or over corrected for the shading by the dark reference file.
It is possible to fit a function to the affected column and add it back in as a delta bias, but the data in the affected pixels tends to be rather noisy as well. You may be better off just treating the affected pixels as "bad" and using spatial dithers to recover that information (just as you would do for other bad pixels).