Average reference for dipole fitting

Here is a question that I get asked occasionally. I have slightly edited the question and my answer to it, which both were posted to the EEGLAB email discussion list.

Question: I’m using EEGLAB-DIPFIT to localize independent components using the spherical head model. Apparently the software requires the data to use the average reference. Why is this?

Answer: In principle you could use an arbitrary reference in your source reconstruction. The practical reason to use an average reference over the sampled electrodes in source estimation is that this prevents the solution to be biassed due to forward modelling errors at the reference electrode. Let me give a partially intuitive, partially mathematical explanation.

Assume that you would use left mastoid as reference. That would mean that the measured value “V” at each electrode “x” is V_x, so the list of all measured values in the N channels is
V_C3-V_M1
V_Cz-V_M1
V_C4-V_M1

V_M1-V_M1 (this is zero)
V_M2-V_M1

Those values can be modeled using the source model and the volume condution model. Now, lets assume a spherical volume conduction model. That is especially inaccurate for low electrodes, and the bony structure of the mastoid is definitely not modelled appropriately in a spherical model. So for the model potential “P” we would have the value at each of the N electrode also referenced to the model mastoid
electrode:
P_C3-P_M1
P_Cz-P_M1
P_C4-P_M1

P_M1-P_M1 (this is zero)
P_M2-P_M1

The source estimation algorithm tries to minimize the quadratic error between model potential distribution and the measurement, so the error term to be minimized is
Total_Error
= sum of quadratic error over all channels
= [(V_C3-V_M1)-(P_C3-P_M1)]^2 + ….
= [(V_C3-P_C3)-(V_M1-P_M1)]^2 + …. (here the terms are re-ordered)

So for each channel the error term consists of a part that corresponds to the potential at the electrode of interest, plus a part that corresponds to the reference electrode. The error term corresponding to the reference electrode is identical over all channels (i.e. repeats in each channel), hence for each channel you are adding some error term for the reference electrode. Therefore, the minimum error (“minimum norm”) solution will be one that especially tries to minimize the model error at the reference electrode (since that is included N times). In the case of a mastoid reference we know that there is a large volume conductor model error at M1, hence the source solution would mainly try to minimize that error term. The result would be that the source solution would be biassed, because it tries to reduce the (systematic) error at the reference.

The solution is to use an average reference (average over all measured electrodes). That implicitely assumes that the model error over all electrodes is on average zero, hence the minimum norm solution is not biassed towards a specific reference electrode.

PS the maths in my explanation above are rather sloppy, but the argument still holds for a more elaborate mathematical derivation which would assume the forward model inaccuracies are uncorrelated over electrode sites.