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This site gives basics of how to place electrodes for action potential recording. What I understand is that in order to get better signal, the reference electrode must be placed where there is minimum electrical activity. So, does that mean we can leave it hanging in the air, or attach it to a wooden surface?

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Short answer
In general, a reference electrode should be placed in a well-conducting medium close, or some distance away from the recording electrode.

Background
The idea of a reference electrode is that it picks up any unwanted signals, such as artifacts, that are also present on the recording electrode. Background noise and artifacts can ruin the electrophysiological signal.

As your linked web page explains nicely, a well-placed reference electrode can mitigate noise and artifacts. In the linked page they take the example of recording motoneurons in muscle. If you would place the active electrode in the nerve close to the muscle, and the reference electrode away from the nerve but still close to the muscle, any artifacts arising from, e.g., muscle activity are picked up by the active and reference electrodes simultaneously. The closer the reference electrode is to the recording electrode, the larger the similarity of the artifact will be and the lesser the contamination. This effect can be achieved using a differential amplifier. The subtraction of the reference signal is referred to as common-mode rejection (Kappenman & Luck, 2011).

The terminology encountered is often referred to as monopolar recording and bipolar recording setups (e.g. Beck et al., 2007). Monopolar recordings use a distant reference electrode. This setup comes in handy when the potentials of interest are small and the preparation is devoid of artifacts. Because the distant reference electrode will almost pick up none of the signal, the signal is nicely preserved. However, any local artifacts will not be cancelled out. In case of artifacts and background noise, a bipolar setup is preferred by placing the reference close to the recording electrode. The signal amplitude may drop, but the added benefit of noise cancellation may outweigh that loss. Basically it is a trade-off to optimize the signal-to-noise ratio.

An electrophysiology amplifier records potentials (or currents). This requires a closed system that allows the passage of current through the system (Fig. 1). Large impedances will result in current flow to be minimal and hence the potentials cannot be picked up anymore. As a rule of thumb, in case of EEG recordings and other compound-action potential recordings as in your linked page, low-impedance measurements are best. Therefore, placing the reference on a high-impedance surface as suggested in the question (suspended in air or fastened to an insulator like wood) is the worst thing you can do.

electrodes amplifier
Fig. 1. Diagram of an electrophysiological recording setup. source: InTech. Note that the reference electrode here is called a reference electrode. The terminology on your linked web page is likewise confusing as they call the reference electrode the ground electrode. A ground electrode is, however, different from the reference electrode, as the ground electrode is used to reduce outside influences on your signal, like the 50-Hz mains power or other electrical field emanating from equipment and so forth.

In addition, for the most optimal recordings, impedances should be matched between active and reference electrode (Kappenman & Luck, 2011). When there is a large mismatch between them, common-mode rejection works less effectively and the signal-to-noise ratio (SNR) will decrease. Therefore, placing the active electrode in tissue (low impedance) and the other one in a high-impedance medium will result in the common-mode rejection to fail.

References
- Beck et al., J Neurosci Methods (2007); 166(2): 159-67
- Kappenman & Luck, Psychophysiology (2010); 47(5): 888–904

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