Difference of Gaussian filter but using Gaussian CDFs not PDFs
You need to ask yourself why do we use the difference of Gaussians from the first place? The reason is because the difference will give us a measurement for the change in value around the point we apply it to as a function of the variances of the Gaussians. If we have a big change the difference between the Gaussians it means that we have some frequencies that were low passed by the Gaussain with the bigger variance. If the change is low it means that there i no information lost. So we can think of it as a band-pass filter.

A simplified reason why it acts like that is because of the shape of the Gaussian PDF as can be seen in the following figure.



If we would use the CDF of the Gaussains and not the PDF it will give us a positive difference on one side and a negative difference on the other side.