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By definition, the cathode is defined as the terminal through which current exits a polarized device. But in the context of neuromodulation, such as spinal cord stimulation, deep brain stimulation, the "cathodic" pulse refers to a negative current pulse that elicits action potentials. A negative current is by definition a current flow that enters the electrode.

Can anyone explain where this terminology comes from and help me resolve this apparent contradiction? It's driven me mad for the past years.

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The answer to your (excellent) question is a matter of definition, which has its roots in the 18th century.

A cathode is the electrode where conventional positive current leaves a current source. Conventional current is positive, due to historic reasons. Around 1750, Benjamin Franklin described the flow of current between wool and wax after electrostatically charging the former. He defined the flow of current as being positive, flowing from the surplus side (which he defined as positive) to the current sink (defined as negative). This unfortunate choice is sometimes referred to as the Franklinian approach to electricity, and is still widely used today to describe electronic circuits and referred to as the conventional flow notation (Fig. 1) (Efron, 1960).

conventional flow
Fig. 1. Franklinian approach: the conventional flow of current still in use today for electronic circuits. Source: All about circuits.

The electrical current was only later found to consist of negatively charged particles, namely electrons. The work of Luigi Galvani on electrical stimulation in frog muscles around 1780 and the invention of the electric battery by Alessandro Volta in 1800 eventually led to the discovery of the electron in 1897, attributed to J.J. Thomson and his work on cathode rays shown in Fig. 2. As the figure shows, the true flow of electrical current is the reverse, namely an electron current flowing from from negative to positive. In the meantime, however, the convention of positive current flowing from the positive pole (cathode) to the negative (anode) was already established. Note that this holds for a discharging device. In a charging device, such as the battery, the cathode is negative, and the anode is regarded as positive.

enter image description here
Fig. 2. Cathode ray showing electron flow. Source: TutorVista

Hence, in a stimulating (discharging) device, the cathode is in truth negative, because it is the source of negative current (electrons). Cathodic stimulation therefore means electrical stimulation with a negative current. A cathodic pulse is therefore injection of a negative current pulse.

Reference
- Efron. Chapman & Hall London, Rider Publisher New York (1960)

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    $\begingroup$ Thanks for this @Christiaan . You write that the positive pole is the anode, is this a typo (in all my electronics courses, confirmed by Wikipedia, the cathode is the positive terminal)?. I do however understand what you mean. This leaves me to assume that in biology or neuromodulation, the cathode is defined as the terminal through which the "charge" leaves the device? In this case it would indeed be the negative terminal. So basically, in electronics we still use the conventional definition of current flow, whereas in biology we use a more accurate, charge based definition? $\endgroup$ – Moppentapper Jan 13 '16 at 0:47
  • $\begingroup$ @Moppentapper No worries, and yes, it was a typo - and corrected. And yes, your comments are correct. In Bio the 'real' situation is used, because it is important to know what really happens in a prep; negative current has often opposite effects over anodic stimulation. $\endgroup$ – AliceD Jan 19 '16 at 9:54
  • $\begingroup$ Excellent @Christiaan thanks so much. It's incredible how simple things are made confusing when different conventions are used! $\endgroup$ – Moppentapper Jan 19 '16 at 10:23

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