How long does the trace of a memory last in the brain? With long-term plasticity one refers to the phenomen by which synapses are modified by neural activity and these modifications last for long times, a day perhaps of the order of days. This phenomenon is often described in contrast to short-term plasticity, where the response of a synapse has a non-linear component that depend on the history of the presynaptic activity in a short time in the past of the order of hundreds of ms.

Plasticity is in fact a much more complicated subject and involves processes at different levels, at the synapse, neuron, network and whole brain (e.g., homeostasis) level. Given our current knowledge of memory systems in the brain, how long are we able to trace even small bits of the memory of a stimulus we were exposed in the past?

Obviously one can answer the question at the cognitive level, i.e., we can rephrase the question and ask instead: What's the oldest memory I can recall? It is well known that our behavior as adults is a reflection of even the oldest memories of our lives. However, I would argue that this is an inderect effect, namely those events that happened in our childhood had an influence on how and what we experience say in the adolescence, and then what we experience as adults is a reflection of what we experienced in adolescence. One known phenomenon that seems to "elongate" memories in the future is the one of replay, i.e., the fact that our brain spontaneously recalls memories of the past like in dreaming.

In computational neuroscience, we can trace bits of patterns stored in a neural network with plastic synapses by probing the network, e.g., by presenting a noisy version of a training pattern and testing if the network is able to recall that pattern, something like an associative memory paradigm. A statistical tool used to test the scaling properties of synapse models is the signal-to-noise ratio analysis. Using this analysis, one is able to address the question of a memory lifetime by assessing whether the synapses maintained some information of a pattern in the past (the signal) with statistical arguments (i.e., with respect to the "noise" of all the other memories interfering with the old one).

Is anybody aware of any experiment where people tried to trace single bits of memories at a single synapse level? In models, the weight of single synapses is maintained for ever unless other memories interfere with that. In reality, it would be interesting to see whether single biological synapses can maintain information on long time-scales (days? months? years?), even considering protein turn-over, or if its the network dynamics that plays a crucial role in maintaining those memories or refreshing them.

Note that answers that go beyond the synapse level are also welcome, that's why the main title doesn't specify to much details.
This is a very difficult question that we don't know the answer to yet.  Here are some references.

Impermanence of dendritic spines in live adult CA1 hippocampus
Alessio Attardo, James E. Fitzgerald &amp; Mark J. Schnitzer
http://www.nature.com/nature/journal/v523/n7562/full/nature14467.html


  Strikingly, CA1 spine turnover dynamics differed sharply from those seen previously in the neocortex7, 8, 9. Mathematical modelling revealed that the data best matched kinetic models with a single population of spines with a mean lifetime of approximately 1–2 weeks. This implies ~100% turnover in ~2–3 times this interval, a near full erasure of the synaptic connectivity pattern. 


Long-Term In Vivo Imaging of Dendritic Spines in the Hippocampus Reveals Structural Plasticity
Ligang Gu, Stefanie Kleiber, Lena Schmid, Felix Nebeling, Miriam Chamoun, Julia Steffen, Jens Wagner, and Martin Fuhrmann
http://www.jneurosci.org/content/34/42/13948.short

Transient and Persistent Dendritic Spines in the Neocortex In Vivo
Anthony J.G.D. Holtmaat, Joshua T. Trachtenberg3, Linda Wilbrecht, Gordon M. Shepherd, Xiaoqun Zhang, Graham W. Knott, Karel Svoboda
http://www.cell.com/neuron/abstract/S0896-6273(05)00004-8  


  Dendritic spines were imaged over days to months in the apical tufts of neocortical pyramidal neurons (layers 5 and 2/3) in vivo. A fraction of thin spines appeared and disappeared over a few days, while most thick spines persisted for months. In the somatosensory cortex, from postnatal day (PND) 16 to PND 25 spine retractions exceeded additions, resulting in a net loss of spines. The fraction of persistent spines (lifetime ≥ 8 days) grew gradually during development and into adulthood (PND 16–25, 35%; PND 35–80, 54%; PND 80–120, 66%; PND 175–225, 73%), providing evidence that synaptic circuits continue to stabilize even in the adult brain, long after the closure of known critical periods. In 6-month-old mice, spines turn over more slowly in visual compared to somatosensory cortex, possibly reflecting differences in the capacity for experience-dependent plasticity in these brain regions.


Stably maintained dendritic spines are associated with lifelong memories
Guang Yang1, Feng Pan1 &amp; Wen-Biao Gan1
http://www.nature.com/nature/journal/v462/n7275/full/nature08577.html  


  Importantly, a small fraction of new spines induced by novel experience, together with most spines formed early during development and surviving experience-dependent elimination, are preserved and provide a structural basis for memory retention throughout the entire life of an animal.