This repository includes two Mathematica files (with PDF version): One with details and examples of the mathematical model and another with fits to experimental data. **Abstract:** How can we reconcile the massive fluctuations in neural connections with a stable long-term memory? Two-photon microscopy studies have revealed that large portions of neural connections (spines, synapses) are unexpectedly active, changing unpredictably over time. This appears to invalidate the main assumption underlying the majority of memory models in cognitive neuroscience, which rely on stable connections that retain information over time. Here, we show that such random fluctuations may in fact implement a type of memory consolidation mechanism with a stable very long-term memory that offers novel explanations for several classic memory ‘laws’, namely Jost’s Law (1897: superiority of spaced learning) and Ribot’s Law (1881: loss of recent memories in retrograde amnesia), for which a common neural basis has been postulated but not established, as well as general ‘laws’ of learning and forgetting. We show how these phenomena emerge naturally from massively fluctuating neural connections.