Fundamental Law of Memory Recall

Human memory appears to be fragile and unpredictable. Free recall of random lists of words is a standard paradigm used to probe episodic memory. We proposed an associative search process that can be reduced to a deterministic walk on random graphs defined by the structure of memory representations. The corresponding graph model can be solved analytically, resulting in a novel parameter-free prediction for the average number of memory items recalled ($R$) out of $M$ items in memory: $R=\sqrt{3\pi M/2}$. This prediction was verified with a specially designed experimental protocol combining large-scale crowd-sourced free recall and recognition experiments with randomly assembled lists of words or common facts. Our results show that human memory can be described by universal laws derived from first principles.

Figure 1

Associative search model of free recall. (a) SM (similarity matrix) for a list of 16 items (schematic). For each recalled item, the maximal element in the corresponding row is marked with a black spot, while the second maximal element is marked with a red spot. (b) A graph with 16 nodes illustrates the items in the list. Recall trajectory begins with the first node, and proceeds to an item with the largest similarity to the current one (black arrow) or the second largest one (red arrow) if the item with the largest similarity is the one recalled just before the current one. When the process returns to the 10th item, a second subtrajectory is opened up (shown with thinner arrows) and converges to a cycle after reaching the 12th node for the second time. (c) Comparison between simulations with random symmetric similarity matrix (blue line) and SM defined by random sparse ensembles with sparsity $f=0.01$ (yellow line), $f=0.05$ (magenta line), $f=0.1$ (green line), and $N=100000$ number of neurons. Each point is the mean of 10 000 simulations. Black line corresponds to theoretical $\sqrt{\frac{3}{2}\pi L}$.

Figure 2

Human recall and recognition performance. (a) Average number of words recalled as a function of the number of words presented. Black line: Eq. (3). Yellow line: experimental results for presentation rate $1.5\sec /\mathrm{word}$. Green line: experimental results for presentation rate $1\sec /\mathrm{word}$. The error in $R$ is a standard error of the mean. (b) Estimated average number of encoded words or sentences for lists of different lengths. Black dashed line corresponds to perfect encoding, green line corresponds to presentation rate $1\sec /\mathrm{word}$ and yellow line to presentation rate $1.5\sec /\mathrm{word}$; blue line corresponds to lists of short sentences (see text for details). The error in $M$ is computed with bootstrap procedure [30]. (c) Average number of words (sentences) recalled as a function of the average number of encoded words (sentences). Green line: experimental results for presentation rate $1\sec /\mathrm{word}$. Yellow line: experimental results for presentation rate $1.5\sec /\mathrm{word}$. Blue line: experimental results for short sentences. The error in $R$ is a standard error of the mean, while the error in $M$ is computed with bootstrap procedure (see the Supplemental Material [12] for details).