The latest finding in neuroinformatics indicates that human long-term memory (LTM) is retained as a neurophysiologic network with unconsciously built physiological synaptic connections at each node as a gated diode in electronic circuits [Wang & Fariello, 2012; ICIC, http://www.ucalgary.ca/icic/
]. Therefore, there seems no mechanism to intentionally (consciously) remove or forget such a connection once it is established in LTM except a potential precise surgery in the future. However, one may intentionally bypass a piece of (unwanted) memory or loss the pathes to access a certain memory, for instance, because of aging.
With regards to short-term memory (STM), however, we still don’t fully understand the mechanisms of how we unconsciously and selectively forget certain information but memorize the rest by permanently retaining them into LTM. This is the key mechanism of learning. Guess what one can do if it’s consciously controllable one day.