You'll need an instrumentation amplifier to measure the difference in potential between two electrodes. The REF pin of the IA can be tied to 2.5V so that you can handle both positive and negative inputs.
50Hz or 60Hz mains interference may be a problem, unless you run everything from batteries and go to the middle of a field to take readings.. You will probably need to follow the IA with a notch filter to remove this, and then an op amp to provide further amplification before the signal is large enough for an Arduino to digitize.
However, you should consider the safety aspects. What would happen if the Arduino was being powered from USB, and the computer developed an earth fault? Medical EEG equipment has a high degree of isolation between the electrode amplifier module and the rest of the equipment, so as to be safe in the event of an equipment fault.
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