There has been an attempt to explain duotronics.

The Spaceflight Chronology from 1980, featured log entries from all the historic events in ST history up to TMP.

Of course, this duotronics account was written in 1980, with a 1980's understanding of computer technology.

In any case...

On Starbase 6, Daystrom records his log entry announcing the breakthrough in duotronics. (Incidentally, this happened three years before the first human was transported and 17 years before the Constitution class was launched).

I'll quote from Daystrom's log entry:

*"Previous computers have relied on essentially binary process: Yes/No--On/Off. No matter how sophisticated the microcircuitry, data storage has been built on pathways of Yes/No information bits. But the growth of galactic knowledge has been outstripping the capacity of standard processing; even at the speed of light there is a limit to how much can be done in a linear pathway.*
*I believe I've found the answer to this problem. Instead of a binary, Yes/No information bit, I propose a basic bit that is Yes or No or all the gradations of maybe in between these two dualities. It is Yes and No in a calculated ratio, and the number of possible ratios is theoretically infinite, though processing materials will place a limit to the duotronic capabilities of a single bit.*
*But that limit is just for a single information bit. A pathway of two bits squares the number of possibilities; a three-way pathway cubes them, etc. Now a typical program may involve millions of bits in sequence. With the duotronic bit of d gradations, a million bit program would have possibilities of:*
*(d x 10^6) ^d x 10^6 or (d^d x 10^6 x 10^6d) = 10^36*
*Since 10^36 alone is a literally astronomical number, and a million bit program is modest nowadays, duotronics has the eventual theoretical possibility of processing information concerning every atom in the galaxy.*
*Of course, duotronics would only be a mathematician's dream were there no material available to realize this duotronic potential. But certain of the new generation of space-made, super-density synthetic crystals have internal structures that would be excellent duotronic conductors in the properly designed computer configuration. Such configurations, along with the mathematical theory behind duotronics, can be found in the patent application that follows as an ancillary transmission to the log."*
So there's one explanation.