The relationship between local field potentials (LFPs) and the electromagnetic fields that give rise to them

Recently there has been a call (Reimann et al., 2013) for a re-evaluation of the genesis of local field potentials (LFPs), a measurement deeply correlated with normal and pathological excitable cell tissue operation (Einevoll et al., 2013; Friston et al., 2014). The lack of a full scientific account of LFP origins additionally means that brain augmentation hardware, a primary tool for which is the manipulation of LFPs, is in effect pulling unmarked levers. How can we knowledgeably control LFPs when LFP origin itself is a mystery? Here we investigate how the task of revisiting LFP origins might best be approached. LFPs originate in the two deeply interconnected fundamental physical fields of the brain: the vector electric field [E(r,t), V/m] and the vector magnetic field [B(r,t), V-s/m2]. Each of these can be Helmholtz-decomposed into the gradient of a scalar potential [say Φ(r,t)] and the curl of a vector potential [say A(r,t)] (Groot and Suttorp, 1972; Landau et al., 1984; Malmivuo and Plonsey, 1995; Jackson, 1999). This means in practice that there are three “potential fields” operating in the brain1. At present it is technologically impossible to directly measure the vector electric field or magnetic field at the resolution of tissue fine structure. Therefore neuroscientists rely on a technically straightforward measurement of voltage (call it LFP(r,t)) that imperfectly accesses the “potential fields” and within which E and B are only indirectly represented. Empirical work over many decades has converged on transmembrane ionic current as the ultimate origin of the LFP (Buzsaki et al., 2012; Destexhe and Bedard, 2013). This means we must address the finest details of the formidably complex tissue ultra-structure typified by Figure ​Figure1A1A (Nicholson and Sykova, 1998; Briggman and Denk, 2006; Kinney et al., 2013)2. This is because the ionic currents originate in the membrane micro-environment indicated by the generic sources d1·sd4 in Figure ​Figure1A.1A. Fundamental field theory tells us that E and B actually mediate LFP expression. This requires us to look at how membrane-related sources first cause E and B and through them, the LFP. We must treat transmembrane currents and their supporting systems of charge as electromagnetic (EM) field sources. Figure 1 EM field origins in nervous tissue ultra-structure. (A) Electron micrograph colored to reveal neuron/glia ultra-structure with (B) the resting state source charge density characteristic centered on the huge transmembrane electric field (106–10 . LFP(r,t) measurement arose as a lab technique nearly 70 years ago (Brooks and Eccles, 1947) and still involves insertion of electrodes that are huge compared to the cyto-architectural scale of the tissue. These electrodes inevitably disrupt the structure around their insertion routes and the eventual measurement points, homogenizing the tissue to some extent and causing an inflammatory response that adds to the disruption. Thus a localized artificial medium is created around each electrode tip, which forms the actual context of the LFP(r,t) measurement. The measurement reveals a spatial average (dependent on the electrode tip geometry) and a temporal average (dependent on sample rate and filters in the measurement equipment) voltage differential relative to a reference electrode elsewhere in the tissue. LFP(r,t) cannot be automatically claimed to access the scalar electric potential Φ(r,t) in the natural tissue. Even if contributions from tissue damage can be ignored, we are not directly measuring Φ. Rather, we are measuring some spatiotemporal average of Φ, the nature of which is not obvious and gets little attention in the literature. This LFP ⇔ Φ mapping needs to be revisited as part of a campaign of elucidating LFP origins. Another important factor affecting the ability to infer EM fields from voltage measurements is that there are an infinity of different E and B fields that can give rise to the same Φ (and therefore the same LFP). This degeneracy of Φ owes its mathematical origin to what is called, in classical electromagnetism, electromagnetic gauge (Jackson, 1999). E and B are not uniquely revealed by Φ. Scalar electric potential Φ is like a height measurement. The lack of specificity that scalar potential has as a reflection of the electric field generating it is analogous to the degeneracy that height has to the terrain. If I have a height of 20 m, am I on my balcony or up a tree? Thus LFPs cannot be properly interpreted or understood without a good theoretical foundation for the origins of E and B based on real tissue ultra-structure knowledge. The LFP is a one-way lens. E and B can “see” Φ but Φ cannot “see” E and B. A practical example of the degeneracy of Φ is in the use of lumped-element circuit models of neurons. These models accurately replicate voltages and currents even though the field (E and B) system of the model is totally unlike that of real tissue. This technique confers a degree of useful predictive utility, but loses contact with the actual underlying tissue physics. The degeneracy in potentials is the reason we can abstract-away E and B physics and is central to the success of circuit theory (Plonsey and Collin, 1961, p. 326). However, degeneracy in electric potentials means that the EM field system implicit in a tissue's circuit-element model cannot be claimed to be the EM field system of the tissue.