How long does the trace of a memory last in the brain?
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.