This coalesces a series of notes from a reddit MachineLearning post, for anyone interested in how an individual neuron might be capable of back propagation.
In his Deep Learning in the Brain slides from Montreal Summer School , Blake Aaron Richards describes the missing piece of the backprop-in-the-brain puzzle, “As it is, backprop through time requires time-stamped records of activity patterns and inputs – not an easy ask for a group of real neurons”.
The G quadruplex form of DNA forms a channel capable of holding an array of different cations (complexed in a rotating cage of carbonyl oxygens). I and a coauthor delve into single-ion selectivity at length in a 1994 paper . Difficulties with molecular mechanical methods led us to propose a quantum effect, suggesting conductivity, which could occur in arrays of folded junk DNA, i.e. nanowires in the nucleus. In addition to conductivity, sequences of different cations could encode information, perhaps gated by porphyrin-like molecules, which align beautifully with quadruplex DNA. Ions could be gated in, or folded in in a coded array; similarly reading could be sequential or parallel, given some mechanisms.
Assuming such a means of computation, possibly local ionic gradients around the nucleus conditioned by neural activity at the surface could gate ions or currents into the nuclear computer (can I compete for the biggest size of hand-waves?)
It turns out that conductivity in quadruplex DNA was demonstrated with atomic force microscopy on a mica substrate in 2014: “Currents ranging from tens of picoamperes to more than 100 pA were measured in the G4-DNA over distances ranging from tens of nanometres to more than 100 nm. Our experimental results, combined with theoretical modelling, suggest that transport occurs via a thermally activated long-range hopping between multi-tetrad segments of DNA. These results could re-ignite interest in DNA-based wires and devices, and in the use of such systems in the development of programmable circuits.” “First-principles calculations have suggested that G4-DNA molecules behave as wide-bandgap semiconductors…”  (and commentary ).
On the mechanical side, a switch has been synthesized using the preferred quadruplex cation. 
I hope this note might inspire folk to fill in the hand-waves.