Circuit theory is generally a pretty dry subject, in and of itself. For one thing, there hasn't been a new fundamental circuit discovered in the past 150 years. Or rather, there hadn't been until this year, when Hewlett-Packard researchers announced the realization of the memristor. The discovery process is described in this article from the IEEE Spectrum: How We Found the Missing Memristor.
Think of a resistor as a pipe through which water flows. The water is electric charge. The resistor’s obstruction of the flow of charge is comparable to the diameter of the pipe: the narrower the pipe, the greater the resistance. For the history of circuit design, resistors have had a fixed pipe diameter. But a memristor is a pipe that changes diameter with the amount and direction of water that flows through it. If water flows through this pipe in one direction, it expands (becoming less resistive). But send the water in the opposite direction and the pipe shrinks (becoming more resistive). Further, the memristor remembers its diameter when water last went through. Turn off the flow and the diameter of the pipe “freezes” until the water is turned back on.
That freezing property suits memristors brilliantly for computer memory. The ability to indefinitely store resistance values means that a memristor can be used as a nonvolatile memory. That might not sound like very much, but go ahead and pop the battery out of your laptop, right now—no saving, no quitting, nothing. You’d lose your work, of course. But if your laptop were built using a memory based on memristors, when you popped the battery back in, your screen would return to life with everything exactly as you left it: no lengthy reboot, no half-dozen auto-recovered files.
As described in the Wikipedia article on the memristor, the idea actually goes back to UC Berkeley professor and IEEE Fellow Leon Chua's work in 1971, and it built on the work of other scientists at that time as well.
Two characteristics of memristors are especially intriguing to me. First, unlike a transistor, which holds discrete information values, a memristor's state is infinitely variable, which offers the possibility of freeing us from the tyranny of the bit. Back in the late 1970's and early '80's when I was getting starting in my IT career, we used to joke about discovering a third value for the bit as being the next big breakthrough in computing. Memristors go a long step past that point.
OK, its state is not infinitely variable, that was admittedly hyperbole; but a memristor's state is a continuous value, analogous to a floating-point number vis-a-vis an integer. Analog (living) neural circuits are continuous rather than discrete, hence the apparent potential for mimicking the brain. Harkening back to my days (or rather nights) studying expert systems at the Harvard Extension School, I can see memristors providing an elegant solution to the problem of representing uncertainty as a facet of state.
The second interesting characteristic of memristors is the inherent persistence of state without power consumption (disregarding structural enthalpy for the moment). The example of instant-on computers is trivial, as the referenced article suggests. However, when coupled with continuous variability, the persistence of memristors' state may afford the creation of machine learning systems as effective as neural nets without their inherent opacity. Training data sets "nudge" the neurons in a neural net into an appropriate configuration for emulating the behavior of living neural circuitry, but the "black box" nature of neural net state means it is impossible to use formal methods to predict the limits of its effective behavior (an unfortunate characteristic it shares with its biologic counterpart). A memristor-based neural net might be more conducive to the creation of transparently descriptive and ultimately predictive models of system behavior.
The architecture for such a system would likely be very different from the current neural net architectural paradigms. I'm not in a position to speculate on what form such an architecture would take. What I can do is to marvel at the possibilities. I can't wait to see the first fruits of memristor-based computer architectures.