If neurons fire due to an intracellular influx of positive ions, why are the negative electrodes in tDCS felt more strongly than the others? For example, see "Reducing Procedural Pain and Discomfort Associated with Transcranial Direct Current Stimulation" by McFadden et al.
The underlying mechanism(s) causing the differences between the effectiveness of anodic and cathodic neural stimulation are largely unknown.
As far as I know, the exact difference between the effectiveness of cathodic versus anodic stimulation are largely unknown. For instance, in animals and humans, both anodic and cathodic pulses can evoke neural responses. In human cochlear implant users, the anodic phase is more effective than the cathodic phase. In contrast, in animals (guinea pigs, cats and ferrets) exactly the opposite is observed. The reason for this discrepancy is unclear (Machery & Cazals, 2016).
The authors in your linked article state (McFadden, 2011):
We hypothesize that the mechanism behind this may be associated with a concentrated exiting of current from the scalp at the cathode.
This, honestly, makes little sense as that same current was injected at the anode, barred the electrode sizes were not different.
My best answer: Although the cathode contains negative charge, ions that it influences initially only live in the extracellular space.
Neurons are largely regulated by the voltage difference between inside and outside the cell.
If the cathodal electrode makes the extracellular space more and more negative, it follows that the intracellular space relative to the outside becomes more positive. This leads to increased firing and sensation at the cathodal electrode.
After about 60 seconds of tDCS, the sensation fades. The neurons have grown accustomed to a new default level of positive ions relative to surrounding electrons. The internal space has adopted more electrons.
Now, a follow-up question would be why anodal stimulation is often associated with activation. Of that, I am not sure. Additionally, wouldnt there be firing associated with removal of the anodal electrode if this theory were true?