PHYSICAL REVIEW RESEARCH 4, 033089 (2022)

Decoding silence in free recall
Francesco Fumarola ,1,* Zhengqi He,1 Łukasz Kuśmierz,1,2 and Taro Toyoizumi

1,3,†

1

Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, Wako, Saitama 351-0198, Japan2
Department of Neurobiology, Duke University, Durham, North Carolina 27710, USA
3
Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo
113-8656, Japan(Received 12 January 2022; accepted 16 May 2022; published 1 August 2022)
This article is part of the Physical Review Research collection titled Physics of Neuroscience.
In experiments on free recall from lists of items, not all memory retrievals are necessarily reported. Previous
studies investigated unreported retrievals by attempting to induce their externalization. We show that, without any
intervention, their statistics may be directly estimated through a model-free analysis of inter-response times—
the silent intervals between recalls. A delay attributable to unreported recalls emerges in three situations: if
the final item was already recalled (“silent recency effect”); if the item that, within the list, follows the latest
recalled item was already recalled (“silent contiguity effect”); and in sequential recalls within high-performing
trials (“sequential slowdown”). We endeavor to model all these effects through a stochastic process where thediscarding of recalled items without reporting (“bouncing”) occurs either if they are repetitious or, in strategicallyorganized trials, if they are not sequential. Based on our findings, we propose various approaches to further
probing the submerged dynamics of memory retrieval. This article is part of the Physical Review Researchcollection titled Physics of Neuroscience.
DOI: 10.1103/PhysRevResearch.4.033089
I. INTRODUCTION

Human memory recall has come to be increasingly regarded as a multilayered process, driven by the interplay of
free association and higher cognitive operations [1–5]. Various neuroimaging tools are being deployed to study what
has been termed postretrieval mechanisms, a set of mechanisms aimed at monitoring and evaluating retrieved memories[6–10]. Evidence points toward a wide-ranging involvementin the top-down control of memory recall by prefrontal[11,12], frontal [13], and parietal [14] areas. Ventrolateralprefrontal cortex (VLPFC), in particular, has been argued
to implement a postretrieval selection process [10] that isstronger if task-irrelevant representations are dominant; electrodes implanted in VLPFC [5] were observed to increase
their activity a few hundred milliseconds after the onset ofthe recall stage, consistent with a top-down postretrieval feedback. The general breadth and organization of postretrieval
processes, however, remain unknown.
An ideal setting for addressing the question is the experimental paradigm known as free recall [15–19], which hasprovided insight into all facets of human episodic memoryfor over a century [20,21]. Free-recall experiments include a*†francesco.fumarola@riken.jp
taro.toyoizumi@riken.jp

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2643-1564/2022/4(3)/033089(13)033089-1

presentation stage and a recall stage; the former consists inthe presentation of a list of items (often words), and the latter
consists in the recall of those items (usually reported verballyby subjects) with no constraint on the ordering. The aim is
to maximize performance, defined as the number of distinct
items recalled correctly. (For a comprehensive description ofboth experiments and theories, see Ref. [22].)On the basis of existent modeling approaches, results onfree recall that have accumulated in the literature may be
subsumed under two classes: results that have been conventionally modeled without invoking postretrieval mechanisms;
and results that are, more or less implicitly, understood tooriginate in postretrieval feedback.
Within the former group, the most studied effects concern the serial position of recalled items, i.e., their order
within the list presented. The recency and primacy effects [16]are the preferential tendency to recall items from the beginning and the end of the list, respectively. The difference inserial position between two consecutively recalled items iscalled “serial-position lag,” and the lag-recency or contiguity
effect [23] is a bias toward small lags, i.e., the tendency torecall contiguously items that are contiguous within the list
[24,25]. The additional tendency to recall in forward order(“forward asymmetry”) makes lag L = +1 the most frequenttransition, which we will refer to as “sequential” [26]. All
these results have been conventionally modeled without assuming any postretrieval mechanism [22].
In parallel, there is a body of work showing evidence for
postretrieval mechanisms such as, notably, repetition avoidance. Repetition avoidance is indirectly evidenced by various
experimental facts. For example, in serial recall, where items
must be recalled in their order, fewer intrusions (words
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FUMAROLA, HE, KUŚMIERZ, AND TOYOIZUMIPHYSICAL REVIEW RESEARCH 4, 033089 (2022)

reported erroneously but not belonging to the lists) were foundto correlate with better recall of the final item [27]; this was
attributed to the fact that recall of the final item is boosted byrepetition avoidance and becomes harder if fewer items areblocked as repetitious. More detailed descriptions of repetition avoidance were worked out with other models [28,29].
Since it is difficult to examine all possible models, it would bedesirable to develop model-free approaches to extract information on unreported events from ordinary free-recall data.
Another topic that has been the focus of a large amountof attention is retrieval properties (such as recall strategies)
not at the level of individual recalls but observed consistently
within individual trials. This includes chunking, i.e., retrieving
consecutive items in one group, and serializing, i.e., retrieving items in their serial order [1,4,30,31]. Such strategies
are mostly consistent within a trial and considered to be acharacteristic of trials rather than recalls. They were shown
to be effective in boosting performance but are implemented
by a minority of subjects and even by those only in a fractionof trials [30]. In fact, they are often developed and refinedover the length of an experimental session as each subject
learns from his performance in previous trials [32]. Whilesome recall strategies are related to chunking, it has longsince been known that a particularly advantageous strategy
consists in waiving the freedom perk of free recall by serializing [30,31,33]. Sticking to such strategies should involve
discarding any retrieval that does not fit the prescribed recall
sequence.
One approach to unraveling the histories of unreported
recalls consists in directly demanding that subjects externalize
them. This type of interventionism was attempted as early aswith the “unedited recall” experiments of Ref. [34], where
subjects were instructed to express all items that came acrosstheir mind. Such experiments suggest that there are more
items recalled than reported; moreover, they have provided
further support to the idea that termination is predominantly
triggered by repetition [35]. Although such an experimental procedure was shown to yield results coherent with the“edited” (i.e., regular) version of free recall [36] and was consequently proposed as a method to explore age dependence[37], memory capacity [38], and intrusions [29], self-reporting
obviously involves physical bounds on the time resolution
and, potentially, biases created by the additional instruction[39]. Whether or not such data offer an exact representationof thought processes taking place during ordinary free recall
is also a question.
Here, we sidestep both the caveats of the externalization
paradigm and, in our opening foray, the problem of selecting
a model. We focus instead on the model-free analysis of one
specific behavioral observable—the time that elapses between
consecutive recalls, also known as the inter-recall interval or
IRI [40–43].
During the recall stage of a typical free-recall trial, the firstfew recalled items are reported rather quickly; in contrast, towards the end of the trial the IRIs drastically increase [44,45].
Many features of this growth were shown to be reproduced
by a pure-death process, in which items are sampled withreplacement from a pool of listed items [45–47]. Crucially,in that model it is assumed that only items that have not beenreported so far are admissible, and thus, when the other items

(a)

(b)

FIG. 1. Two features of free recall. (a) Paucity of observed output
repetitions as compared with chance level (for details of chance level
estimation, see Fig. S2). (b) Fraction of sequential recalls in a trial
as a function of trial performance (i.e., number of recalled items).
The mean curve is shown, while the standard error of the mean is too
small to plot. The red arrow highlights the presence of an inflection
point, where the slope starts to increase.

are drawn, they are simply discarded. Because towards the endof a trial there are many previously recalled (hence nonadmissible) items, many samples have to be drawn on average,
and thus IRIs increase. Although the pure-death model is very
simplistic, its structure suggests that a form of postretrieval
editing takes place during the recall phase and that only someof the sampled (retrieved) items are reported.
We will argue that a careful but straightforward analysis ofboth the history of reported recalls and the recorded IRIs canallow us to uncover hidden retrievals that are suppressed by
the participants. We will then buttress our model-free analysisof data by means of a toy model chosen for its simplicity
and capability to explain simultaneously different types of
unreported recalls.
II. TWO FREE-RECALL PHENOMENA

The data we considered, collected by the Computational
Memory Laboratory at the University of Pennsylvania, concern experiments performed with lists of N = 16 words (seeAppendix A). The data set displayed standard serial-position
effects (with recency and primacy emerging as in Fig. S1 of
the Supplemental Material [48]).
We begin by pointing out two general facts in the data.
(a) Although repetitions are not expressly forbidden andare duly recorded by the experimenter, the number ofrepetition-free trials is remarkably large. This statement canbe made precise by adopting as chance level a basic Markov
model of transitions among reported recalls [49] equippedwith a sink state for termination (calculation of the chancelevel is fully detailed in the caption of Fig. S2). We thus
find that the number of repetition-free trials is about twice
the chance level [Fig. 1(a)]. This fact agrees with previous
inquiries (see Introduction) and indicates that subjects havea spontaneous tendency toward not reporting twice the same
recall (see Fig. S2 for a more detailed comparison of therepetition statistics to chance level in the data).
(b) We define “trial sequentiality” as the fraction of sequential recalls in the trial. Trial sequentiality is a peculiar functionof trial performance; as shown in Fig. 1(b), not only does

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PHYSICAL REVIEW RESEARCH 4, 033089 (2022)

is known that a bias toward sequentiality boosts performance[33] and, as will be seen below, it is observed selectively inhigh-performance trials.
In the rest of this paper, we report on our testing of thebouncing hypothesis from two complementary fronts—datamining and mathematical modeling. On the side of data mining, we exploit the large size of the data set to tease out
information not explicitly reported by individual subjects,
with the aim of uncovering what has not been verbally outputted. On the side of modeling, we articulate the above
intuitions as a streamlined theory that represents free recall
without bouncing as a Markov process and bouncing as a
non-Markovian add-on, and compare predictions of the theory
with observed features of the data.

(A4)

ash zoo sun ear toy

zoo ash toy (zoo) (toy) sun

(A1)

(A2)

zoo ash toy sun
(A5)
(A3)

(b)

(B4)

ash zoo sun ear toy

zoo sun (toy) (sun) ear toy

(B1)

(B2)

zoo sun ear toy
(B3)

(B5)

FIG. 2. Schematic depiction of bouncing. Intervention over thefree-association process by “bouncing” simultaneously explains
paucity of observed repetitions and higher sequentiality of highperformance trials. Here, a five-item list is presented in two separate
trials leading to an instance of repetition bouncing (a) or one ofstrategic bouncing (b). During the presentation stage of free recall[(a1) and (b1)] a subject is presented visually or orally with a listof L items (here, L = 5 words). Memories of exposure to each of
the items are stored in such a way that mental transitions amongmemories have unequal probabilities represented in (a2) and (b2) by
the shades of gray of the arrows. During the recall stage (a3), thesubject instantiates a mental trajectory through those memories [(a3)and (b3)]. Nothing prevents the sequence of retrieved memories fromcontaining an unlimited number of repetitions [as, for example, in(a3) and (a4)], but the output is devoid of repetitions (a5), posing the
problem of whether some retrievals are discarded. In addition, a tendency toward sequentiality in high-performing trials is also observed(b5), posing the problem of whether nonsequential transitions havebeen actively avoided [as when stepping back from “toy” to “sun” in(b3) and (b4)].

higher trial sequentiality correspond to higher performancebut also this increasing curve appears to accelerate at an inflection point midperformance.
If one reaches for the simplest explanations by postretrieval
processes, those two seemingly unrelated facts appear to sharesome key features.
(i) In both phenomena, the extra mechanism at work withfree association can be described as the avoidance of options in principle provided by free association—repetitious
retrievals [fact (a)] and nonsequential retrievals [fact (b)].
(ii) While the avoidance has different motives in the twocases and could be implemented in different brain areas, it
may be assumed to operate in a de facto similar way, witheach free association move being allowed or disallowed by a
dedicated module depending on whether the retrieval meets a
specific criterion (novelty or sequentiality).
These empirical observations lead to our working hypothesis, illustrated in Fig. 2. To denote the active avoidance of
free-association recalls, we use hereinafter the word “bouncing.” We distinguish two forms of bouncing: firstly, theavoidance of recalls that would yield repetition (cf. Introduction), which we call “repetition bouncing”, and, secondly, theavoidance of recalls that would not obey a sequential ordering.
We refer to this latter type as “strategic” bouncing because it

III. MODEL-FREE ANALYSIS OF RECALL DELAYS

If a mechanism exists by which the free-association process is corrected, unwinded, or rebooted, given that thisprevents the outputting of some retrieved items, it will notleave direct traces within the record of recalled items. Yet,it may produce temporal delays in recall transitions where theprocess is activated. In Fig. 2(a), for example, the word “sun”
will be recalled more slowly because first the word “zoo” wasretrieved and discarded. A natural approach is thus to hunt for
traces of those delays in the recorded IRIs. To do so, the firstobligatory step is an a priori analysis of the main potential
confounders.
Following convention [22], we will call the ordered position of an item within the sequence of recalls reported during
the recall stage the “output position.” It can be a positive
number if counted from the beginning of the recall process ora negative number if counted from the end (number of extant
recalls to the end of the trial). For example, in the trial of
Fig. 2(a), the word “ash” has output position = +2 or = −3.
If trials with different performance are included, there is no
fixed correspondence between positive and negative countingof outputs. It was shown in Ref. [45] that the distributionof IRIs is characterized by negative output position. This isindeed apparent in the data set under consideration, wheredependence on the negative output position alone explains
about 20% of the variance (Fig. S3). Note that the time interval from the onset of the recall stage to the first recall at
output position = +1 is not an IRI but will be included forconvenience among the IRIs and this does not qualitatively
change our results.
Upon subgrouping all recall events by their negative output
position, we compare events occurring before and after theitem in a given serial position has been recalled [Fig. 3(a)]. After recall of an item, its repetitious retrieval becomes possible.
If such a retrieval comes to mind, it would be associated witha positive time delay. In other words, systematic bouncing
against the item in a given serial position will be expected tomake the postrecall IRI larger than the prerecall IRI for every
output position (see Appendix B for details of the analyses).
This turns out to be true for the last item in the list regardlessof the output position [Figs. 3(a) and 3(b)]. If the final item has
already been recalled, all else being equal, a delay is found tooccur in the recall process.

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(a)

(a)

(b)

(b)

FIG. 3. Empirical evidence for a higher-order “silent recency
effect” in free recall. (a) Difference between mean IRIs when subsampling on the basis of whether the item with a given serial positionon the list has already been recalled or not. A histogram was made of
the IRIs excluding recalls of the item with the given serial position
also when it is not yet recalled. The serial position concerned isshown on the x axis, and plot markers encode the significance as
defined by a Mann-Whitney U -test P value lower than 10−3 (star,significant; circle, nonsignificant). (b) Histogram (hist.) of the IRIs
for recalls of items other than the final one, plotted separately foreach output position and for the two conditions where the final item
has already been recalled or not. The scale of the IRIs is logarithmic.
The P value from the Mann-Whitney U test is encoded as follows:
circle, P > P∗ = 10−3 ; one star, P∗ = 10−3 > P > 10−4 ; two stars,
10−4 > P. A previously occurring recall of the final item is statistically associated with a delay in the recall of other items. Subjects areseen to rush for the final output because of the imposed time-out.

The fact that the final item appears to be a preferential
bouncing target is a nontrivial manifestation of the recency
effect (cf. Introduction). The delay associated with potential
repetition of the final item can thus be thought of as a “silent
recency effect”; the quantitative strength of memory depends
on serial position, and given that the attraction of the mostrecent memory from the list is the strongest, it stays such evenafter that memory has been recalled. Therefore subsequent
recalls are slowed down through reversions to that item. The
absence of a corresponding silent primacy, on the other hand,is an unexpected finding. It may be related to the observationthat recency typically appears earlier than primacy in therecall process [Fig. S1(b)] as the early recall of an item makes
it more likely to be bounced afterwards.
A similar analysis can be performed by comparing theconditions in which the retrieval transition with a given serialposition lag would pass the repetition screening or not. The
mean IRI in these two conditions can again be compared,
to check for a delay associated with the possible discard ofsuch retrievals (technical aspects in Appendix B). A consideration that can guide expectations on the outcome is that, as
mentioned in the Introduction, the most likely lag to occur is

FIG. 4. Empirical evidence for a “silent contiguity effect.”(a) Difference between mean IRIs when subsampling on the basisof whether the item situated at a certain fixed lag from the last recall
has already been recalled or not. The lag position concerned is shownon the x axis, and plot markers encode significance as defined bya Mann-Whitney U-test P value lower than 10−3 (star, significant;circle, nonsignificant). (b) Histogram of the IRIs for recalls of items
other than the sequential one (i.e., the one that was presented rightafter the latest recalled item) shown separately for each output position and for the two conditions where the sequential item has alreadybeen recalled or not. The scale of the IRIs is logarithmic. The P
value from the Mann-Whitney U test is encoded as follows: circle,
P > P∗ = 10−3 ; one star, P∗ > P > 10−4 ; two stars, 10−4 > P. A
previously occurring recall of the sequential item is statistically associated with a delay in the recall of other items.

L = +1 (sequential recalls). If this type of transition remains
the most probable one also when it can cause bouncing (i.e.,
when it would yield an item already recalled), we expect it tobe associated with the largest delay.
The conjectures are confirmed by inspection of the data
(Fig. 4). Thus we find that another silent serial-position effectis the predominance of delays associated with the potential
discarding of sequential retrievals (“silent contiguity effect”).
IV. A MINIMAL MODEL

To further test the bouncing hypothesis as applied to repetition avoidance, it is necessary to rely on forward models that
describe it in a predictive fashion.
Models of free recall have covered a spectrum ranging
from dual-store memory search models [3,50] to powerfultheories of temporal context dependence [51]. It is not our
ambition to provide a wide-ranging model of top-down control or in any way revise what is known from existing modelsof recall (see Introduction); because our focus is entirely on
the concept of bouncing, we rely on the most drastic simplification compatible with the general features of free recall.

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The three assumptions we make to simplify the problem
are the following [see Fig. 2(a)]:(i) In the absence of any bouncing, the reported recall
process would be identical to the retrieval process, and bothwould be a Markov chain in the space of the list items.
(ii) In the presence of bouncing, the retrieval process can
hit an undesired memory item and bounce back to the previously recalled item, and this step does not affect the sequenceof reported items.
(iii) Every repetition bounce entails a probability q of terminating the process, and we will refer to the parameter q as
impatience.
Thus the only parameters are the Markov transition matrix
and the impatience q (one additional parameter will be introduced later when we study strategic bouncing).
We have explored alternative assumptions for both thetermination mechanism (static sink states, abrupt terminationthresholds) and the non-Markovian add-on (one example isanalyzed in detail in Appendix E), and they have proven lesspredictive than the model we are presenting.

The non-Markovian contribution from bouncing against a
recalled item is, in this framework, of a peculiarly simple type;
namely, the transition probability to a given item becomeszero after that item has been recalled and stays zero up to
the end of the recall process. This non-Markovianity can be
described as a progressive masking of the transition matrix.
Let π̂ be the naked transition matrix (i.e., the zero-diagonal
Markov transition matrix of the bounce-free model) with matrix elements π (y|x) determining the transition probability
from item x to
item y and, for any set S of serial positions,let π (S|x) ≡ y∈S π (y|x).
Further naming N the length of the presented lists, T thenumber of trials in the sample, mα the performance of trial
α measured by the number of distinct reported items, and xkα
the serial position of the kth recall in the trial, and using thenotation where xi:α j is the set of serial positions recalled in between output positions i and j, both included, the normalized
log likelihood of the bouncing model is

mα
T
T m
T 
α −1
 1
 1

 α
 α

 α
1
α
α
L[π̂, q] = ln q +ln π x1:mα −1 |xmα +ln π xn+1 |xn −ln 1 − (1 − q)π x1:n−1
|xnα
T α=1
T α=1 n=1
T α=1 n=1

(for the derivation, see Appendix C). Fitting the data set
with this likelihood (Appendix D) yields an impatience parameter q ≈ 0.1 and a transition matrix that encapsulates all
known serial-position effects including some degree of primacy [Fig. S4(a)]. The number of bounces underlying every
recall event can then be calculated by simulating the modelwith these parameters [for sample trials, see Fig. S4(b)].
To test the model, we assume for simplicity that the IRI
is proportional to the bouncing count. We first compare themean of relevant variables characterizing a recall event. The
(a)

(c)

dependence on negative output position, retrieved serial position, and the recall lag are all qualitatively recovered (Fig. 5).
We then test for the silent serial-position effects along thesame lines followed for real data (Figs. 3 and 4), with the only
difference that instead of measuring a mean temporal delay
between two conditions, we now measure a difference in themean number of bounces occurring within a recall event. We
report that all main phenomenological features of silent serialposition effects, discussed above, are found in our pseudodata,
both as concerns silent recency [Figs. 6(a) and 6(b)] and silent
contiguity [Figs. 6(c) and 6(d)].

(b)

V. MODEL-FREE ANALYSIS OF SEQUENTIALSLOWDOWN(d)FIG. 5. Comparison between mean IRIs and the mean number of
unreported retrievals of the model. The result of averaging is shownfor fixed values of the recall event’s output position (a), the serial
position of the recalled item (b), and the serial-position lag from thepreceding recall (c). In (d), a scatterplot of the values shown in (a)–(c) is presented.

We turn then to the distinctive shape of the sequentialityperformance curve [Fig. 1(b)]. We hypothesized that it
might emerge if a partial discarding of nonsequential retrievals allows the implementation, in some trials, of a moremarkedly sequential recall strategy associated with higher
performances—yielding the upturn of the curve for highperforming trials. [This interpretation would also justify the
related fact that the output position of the final item becomes
bimodally distributed for high-performing trials (Fig. S5),
where the peak around the first output could be related to therecency effect and the peak around the final output could be related to the sequential strategy.] If that were the case, it should
also leave traces in the statistics of the IRIs. Namely, we would
expect that, at least among high performers, recalls performed
by strategizing would be slower than those performed without
strategizing.
To differentiate between strategic and nonstrategic trials,
we can use the fact that the former have by construction a

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(a)

(c)

(b)

(d)

FIG. 6. Silent serial-position effects emerging from the bouncing model. (a) Selective variations in the mean bouncing counts estimatedfrom model simulations. The mean number of bounces within a recall event has been averaged in the two subsamples of recalls where the list
item in a given serial position is or is not bounceable, and the difference between the two means is plotted as a function of the serial position.
Values on the x axis refer to the item by the bounceability of which subsamples are discriminated. (b) Bouncing-count histograms displayedside by side refer to recall events where the final item in the list is or is not bounceable. Corresponding mean curves show the mean bouncing
increment after recall of the final item (“silent recency effect”). (c) Selective variations in the mean bouncing count per recall estimated frommodel simulations. The mean number of bounces within a recall event has been averaged in the two subsamples of recall events where agiven lag does or does not lead to a bounceable item, and the difference between the two means is plotted as a function of the given lag.
(d) Bouncing-count histograms displayed side by side refer to recall events where the sequential candidate for recall is or is not bounceable.
Corresponding mean curves show the mean bouncing increment when the sequential item was already recalled (“silent contiguity effect”).

higher trial sequentiality. An obvious approach is to separate
each given sample of trials in the submedian and supramediansubsamples according to its distribution of trial sequentialities, and test whether recalls are faster in the submediansubsample.
We take into account three confounders. First, the dependence of the IRI on the negative output position can again
play the role of a strong confounder. Second, since sequential strategies are used to achieve higher performances (seeIntroduction), the effect we are seeking may only be present
in the more high-performing trials. We must thus proceedby considering separately event sets defined both by a given
output position and by their occurring in trials with given
performance. The third confounder is the difference in IRIs
for sequential and nonsequential recalls within each trial. Indeed, sequential recalls are mostly faster than nonsequential
ones across different performance levels and output positions[Fig. S6(a)].
For every condition defined by trial performance andoutput position, we consider thus the distribution of trial sequentiality among trials with that performance that contains asequential recall in that output position [Fig. 7(a)]. We thensplit the sample by the median value of trial sequentiality
and compare the IRIs of recall events in the two subsamples [Fig. 7(b)]. We only compare sequential recall events
because highly sequential trials contain few nonsequential
recalls. Thresholding out all sample pairs that do not pass asignificance test, we calculate the sign of the delay associatedwith higher trial sequentiality. (Results are qualitatively un-

varied whether we partition using a median computed overthe sequentialities of all trials in a performance group or computed separately over those associated with sequential events
at each given output position.)
We thus find the following [see Fig. 7(c)].
(i) In all cases where our significance requirement is met,
the delay is positive, meaning that sequential recalls that occurred in sequentially biased trials have taken longer. In thefollowing we refer to this phenomenon as sequential slowdown.
(ii) Strikingly, the phenomenon is concentrated in the highperformance regime. This is in fact just where it is to be found
if the sequential slowdown is due to the implementation ofrecall strategies.
We will focus on the hypothesis that the slow sequential recalls are due to a subselection of the retrievals (forthe problem with alternative explanations, see Discussion)or, equivalently, these delays are punctuated by the active
rejection of what nonsequential retrievals are provided by freeassociation, which is what we termed strategic bouncing.
VI. SEQUENTIAL SLOWDOWN IN THE BOUNCING
MODELWith model parameters coming from the abovementioned
fit, we only add into the model a finite probability s (“strategicity”) of bouncing back from a retrieved item if it isnonsequential. We are not necessarily assuming a common
mechanism for strategic bouncing and repetition bouncing

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PHYSICAL REVIEW RESEARCH 4, 033089 (2022)
(a)

(b)

(c)

(d)

(c)

(b)

FIG. 7. “Sequential slowdown” in free-recall data. For each performance level, we list all trials that perform a sequential report at
a given output position and study how its IRI changes in highly
sequential trials. The sequentiality value associated with each trial issimply the fraction of sequential recalls in the trial. (a) The histogramof this quantity for output position 8 and trial performance 15 isfurther subdivided for the sake of the analysis into the set of trials
above and below its median. (b) The corresponding distribution of
IRIs in the two subsamples, with vertical lines with green and redcolors indicating the respective mean values. (c) Sign of the difference in the mean values whenever the difference is significant by aMann-Whitney test with threshold P∗ = 10−3 . Sequentially biased
trials are characterized by a slowdown of their sequential recalls,
specifically in high-performing trials. Squares on the grid are coloredas follows: white if data are not available from both subgroups (as is
the case here because output position K does not exist for trials withperformance < K); yellow if the Mann-Whitney P value between thetwo corresponding subsamples is larger than P∗ = 10−3 ; blue if the
P > P∗ and the difference is positive, i.e., the supramedian trials perform significantly slower than sequential recalls; and brown for the
opposite occurrence, which would be submedian trials performing
significantly slower on sequential recalls, but strikingly, this is neverobserved.

(the fact that strategic bouncing happens only in a minorityof trials may point to the opposite) but only that the basic
retrieval rejection step can also be described as a bounce.
A priority needs to be established between the two typesof bounce; we assume that subjects who are actively implementing a sequential strategy check first for sequentiality
and then for repetitiousness and do not associate any impatience increment with strategic bounces. This proves to be
the sensible choice from the requirement that the performance
should increase as a function of strategicity [Fig. 8(a)] despite
repetition bouncing (Fig. S1).
The analysis of sequential slowdown in data suggests that
this increase in performance should come at the price of aslowdown in individual recalls, measured by the number ofbounces. We test this statement on simulations of the modeland find that indeed it holds true for simulated recalls with any
possible value of their negative output position [Fig. 8(b)].
We then move on to replicating the results of data analysis
for the dependence of trial sequentiality on performance thatwas presented at the outset [Fig. 1(b)]. Rather than aiming

FIG. 8. Simulations of the bouncing model with strategizing. We
simulated the model by adding in a given probability of strategic
bouncing (i.e., the probability of rejecting a nonsequential retrieval),
which we termed “strategicity.” (a) Increasing strategicity also increases performance, coherently with the theory. (b) Increasing
strategicity slows up the process, which agrees with the slowdown ofsequential recall in highly sequential trials seen in the data. (c) The
introduction of strategic bouncing leads to the emergence of theinflection point (red arrow). The features of real data that were shownin Fig. 1(b) are recovered here by merging a dominant population
without strategicity, s = 0 (5 × 105 trials corresponding to the blue
curve) with a smaller one having strategicity, s = 0.5 (5 × 103 trials).
(d) Here, for each joint condition of performance and output position,
sequential recalls are sampled above and below the median value oftrial sequentiality. The mean difference in bouncing count proves
to be always positive and increases in the area where sequential
slowdown was detected in the data [compare Fig. 7(d)].

at numerical accuracy, we seek to understand the shape ofthe curve, namely, its inflection in midperformance, followed
by a drastic rise for larger performances. First we note that
repetition bounces alone do not explain this feature [Fig. 8(c)].
We then simulate a mixture of two populations, a dominant
one without strategicity, s = 0 (5 × 105 trials), and a smaller
one with strategicity, s = 0.5 (5 × 103 trials). When plotting
the sequentiality of simulated trials versus their performance,
we find that even such a small percentage of strategic trials is
sufficient to inflect the curve [Fig. 8(c)].
Finally, we agnostically interrogate the mixed-set pseudodata model for the signatures of strategic slowdown weencountered in Fig. 7(c). We subsample recall events by theoutput position and trial performance characterizing them; foreach joint condition, we use the sequentiality values of therelevant trials to subdivide the sample in a supramedian andsubmedian group and compute the bouncing increment, i.e.,the difference between their mean bouncing counts.
This difference proves to be always positive [Fig. 8(d)],signifying a delay in time as found in real data. In Fig. 7(c), we
also showed that the time delay found in real data grows into
significance in the high-performance region. The modeling
result is in agreement with real data because whenever thedata show a statistically significant difference in IRIs, the033089-7

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corresponding subsample of the model-generated pseudodata
features relatively high values of the bouncing increment. This
result supports the notion that sequential slowdown is due tostrategic bouncing [Fig. 8(d)].
VII. DISCUSSION

Free-recall experiments are an essential behavioral probe
in the study of human memory. In this paper, we undertook
the rigorous mining of a large data set to uncover a numberof features: (i) The number of repetitions observed in theseexperiments is vastly smaller than a basic expectation given
by chance Markovian transitions; (ii) recalls are delayed if the
final item has already been recalled (“silent recency effect”);(iii) recalls are delayed if sequential recall would yield a repetition (“silent contiguity effect”); (iv) the overall abundanceof sequential recalls in the trials has a distinctively inflectedshape when regarded as a function of trial performance; and(v) a slowdown in certain recall events is associated with theoverall prevalence of sequential recalls in high-performing
trials (“strategic sequential slowdown”).
In addition to reporting on the above, we proceeded tosuggest a unifying explanation coherent with the current bodyof knowledge on free recall (see Introduction) and basedon the notion that free-association retrievals are sometimes“bounced,” i.e., discarded. In a majority of cases this happens
because they are repetitious (repetition bouncing) and in a minority because they are not sequential (strategic bouncing). To
test the validity of our explanation, we used a minimal modelwith trajectories driven by heterogeneous transition probabilities and reset by bouncing. Bouncing of repetitious transitions
occurs systematically for repetitious retrievals and adds to the“impatience” that, probabilistically, leads to termination ofa recall trial; strategic bouncing of nonsequential transitions
occurs in high-performing trials with a fixed probability. Upon
fitting on the data and using the number of bounces as a proxy
for the inter-recall time intervals (IRIs), the model was shownto qualitatively account for all the above features of free recall.
The minimal model accounts in one breath for repetition
bouncing and strategic bounding, and may account also fora gradual release from repetition avoidance [52–54] by additionally incorporating what is known about its dependence on
cognitive rather than physical time [55]. However, the features
we uncovered by our model-free analysis of data should be
simultaneously recoverable from a variety of models, suchas the stochastic model of Ref. [56] and the generally valid
framework of the context maintenance and retrieval model ofRefs. [2,29]. All of the effects highlighted here can be explained on the basis of quantitative differences in the effective
memory strength of different items; they do not require orrule out qualitative differences, such as the final item possibly
being recalled from working memory, which may be fasterthan associative recall involving other sources.
Our work underscores the importance of unreported retrievals in determining behavioral observables of memory
experiments. It is indeed easy to see that alternative explanations would run into several problems. For example,
one might hypothesize that the strategy at play in certain
high-performing trials consists merely in waiting longer before surrendering even if no new memory is being retrieved.

This would predict a slowdown preferentially happening atlater output positions, where surrendering is most prominent.
However, such an explanation can be ruled out because thesequential slowdown is observed at a wide range of output
positions [Fig. 7(c)].
In experiments where subjects are instructed to recall items
not belonging to a specified category or items not beginningwith certain letters [8], other types of bouncing might takeplace, but their detection can be attempted by the very means
we have introduced.
Measurements performed in free recall are not limited
to behavioral observables. A parallel study of unreported
retrievals could be attempted through the decoding of suitable neurophysiological data. The timing and magnitude of
postretrieval effects such as those already observed in electrocorticography (ECoG) of prefrontal cortex [5] could becompared with the timing and quantity of the bounces inferredby an analysis of behavioral data.
The issue of whether bouncing is consciously perceived by
the participants is more delicate and could be partly addressedin the future through comparisons with externalized free recall[34,36,37] with separate controls for the accuracy and consistency of self-reporting.
Data used in this paper were generously made available atthe University of Pennsylvania and can be downloaded from[57]. The PYTHON/PYTORCH library we developed to process
free-recall data sets is available upon request.
ACKNOWLEDGMENTS

The study was supported by RIKEN Center for
Brain Science, Brain/MINDS from AMED under Grant
No. JP15dm0207001, and JSPS KAKENHI Grant No.
JP18H05432. F.F., Z.H., Ł.K., and T.T. worked as a team;credit is not individualizable. All the authors reviewed themanuscript together. The authors declare no competing interests.
APPENDIX A: ARCHIVAL DATA

We base our analysis on a publicly available data set,collected as part of the Penn Electrophysiology of Encoding
and Retrieval Study at the University of Pennsylvania (fora full account, see Ref. [43]). The data were acquired fromconsenting human subjects in compliance with the Universityof Pennsylvania’s Institutional Review Board protocol.
Participants were given 75 s to attempt to recall aloud anyof the presented items. Because of the limited amount of time,quantities pertaining to the first negative output position areindeed somewhat affected by the final rush to recall.
In our analysis, all trials (27 198 in total) were used. In
the minority of trials containing some repetition [Fig. S2(a)],repetitions were ignored for the main analyses. Intrusions
from outside the presented lists were all removed at the outset.
APPENDIX B: MODEL-FREE ANALYSES

We classified all recalls into the prerecall group or postrecall group with respect to a target item. The target item wascharacterized either by its serial position (Fig. 3) or by its

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serial-position lag with respect to the latest item recalled (inwhich case the serial position of the target item varies fromevent to event; Fig. 4). Recall events are then classed as“postrecall” if they occur when the target item has alreadybeen recalled within the trial, and they are classed as “prerecall” if they occur at a moment of the trial when the target item
has not yet been recalled.
As discussed in the main text, we aim at finding evidence of
whether the target item may be retrieved and unreported during the recall events. Thus we are interested in the difference
between the IRIs in the prerecall and postrecall conditions.
Moreover, we studied whether the above changes in the
IRIs were output position dependent because IRIs are known
to systematically increase with output positions. For this purpose, for each target item, we also divided recalls by their
negative output positions and tested whether the IRI distributions of the prerecall and postrecall groups were significantly
different at each output position. We checked whether the IRIsof the pre- and postrecall groups were statistically different
using the Mann-Whitney U test because the IRI distributions
were highly non-Gaussian.
To make the analysis of IRIs rigorous, we took the following precautions.
(i) We broke down the sampling of recall events by negative rather than positive output position to account for thecharacter of the output position dependence highlighted inFig. S3.
(ii) Events where the target item is reported should in
principle be classed as prerecall, because it was not recalled
yet while the recall process occurred. However, we excluded
these events from the prerecall condition. Note that reported
recalls of the target item could never figure in the postrecall
sample for that target. Therefore including such events in theprerecall sample could unbalance the two comparison of thetwo groups.
(iii) In the case of unreported retrievals with a fixed lag,we count out all conditions where the lag would not yield anitem within the list (due to the finite list length), as that makes
recall unavailable rather than bounceable.
In all the subsample comparisons of the IRIs, we systematically filtered out information by testing for significancethe difference between the IRIs in each pair of subsamples.
Since the distribution of IRIs in any given subsample is highly
non-Gaussian [see, e.g., Fig. 3(a)], rather than using a t test we
adopted a Mann-Whitney U test [58].
For example, Fig. S6(a) should be strictly understood asstating that if we pool trials regardless of their sequentiality,
the statement that sequential recalls are faster withstands asignificance test with few exceptions on the triangular grid
spanned by performance and output position values.
The P-value threshold we chose at the outset was P∗ =−3
10 and has been uniformly applied to all the analyses.

Let us label the items by their serial position x = 1, . . . , N,
where N is the length of the memorized list. The first recalled item of every trial is picked according to a distribution
p1 (x) = ρinit (x) which plays the role of aninitial condition.
The second item has probability p2 (x) = x1 π (x|x1 )p1 (x1 ),where π (y|x) is the naked transition matrix. This matrix isassumed diagonal-free, so π (x1 |x1 ) = 0, and no repetition canoccur in the second recall.
At the third recall, for every previous history (x1 , x2 ), there
is a finite probability π (x1 |x2 ) of ending up in a repetition. If
this happens, the trajectory goes back to x2 without outputting
either retrieval, which we refer to as a “bounce.” In otherwords, there is a set J of undesired retrievals that contains
in this case only item x1 (J = {x1 }).
If an infinite number of bounces is allowed, the transition
from the retrieval of x2 to the next is governed by a matrix
0 (x|x2 ; J ) defined as
π (y|x)1[y ∈/ J],
0 (y|x; J ) = 
π
(y|x)
y∈J
/

where 1[X ] = 1 if statement X is true, and 0 otherwise.
In other words, the transition matrix is defined with thetarget items in the bounceable set masked away, and every
other element is normalized so the columns sum to unity.
Notice the special case 0 (y|x; ∅) = π (y|x).
This is, however, only correct if any number of bounces isallowed; in practice, it is realistic that the subject will give up
after a certain number of bounces. Every repetition bounce
is thus associated with a fixed surrender probability q wecall “impatience” (although different modelings of impatience
may also be devised).
Call Jn = [x1 , x2 , . . . , xn−1 ] the (unordered) set of all items
recalled up to time n. The probability pn+1 (y) of item y being
the (n + 1)th recall is given by(y|xn ; Jn ) = 0 (y|xn ; Jn )[1 − sq (xn ; Jn )],(C2)

where sq (x; J ) is the probability of surrendering, i.e., terminating the process, during the recall step starting out from recallof item x, given that the bounceable set is J.
At each step the process has three possibilities: recalling,bouncing, or surrendering. Recall happens if a new word is
retrieved before surrendering; surrendering occurs with probability q whenever a repetitious word is retrieved; and bouncing
occurs with probability 1 − q when a repetitious word is retrieved, returning the process to the latest recalled item.
Calling the three corresponding probabilities s, b, and r, weclearly havesq (x, J ) + bq (x, J ) + rq (x, J ) = 1.

(C3)

For a single retrieval step, their elemental values are

APPENDIX C: BOUNCING MODEL

In the bouncing model as set up in the main text, upon
retrieving a repetitious item, the process reverts to the latest
previously retrieved item and makes a new “try” from there.
Here, we formalize the concept mathematically, writing down
the log likelihood for the model.

(C1)sq(1) (x, J ) = π (J|x)q,

(C4)

b(1)
q (x, J ) = π (J|x)(1 − q),(C5)¯rq(1) (x, J ) = π (J|x),(C6)

where J¯ is the allowed set, i.e.,the complement of J, andwe used the notation π (S|x) = y∈S π (y|x) for an arbitrary
subset S of the N list items.

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With a potentially infinite number of steps we must writeα= {x1α , . . . , xNα }
x1:N

∞

n
 (1)
sq (x; J ) =bq (x, J ) s(1) (x, J )q
n=0

=

∞


[π (J|x)(1 − q)]n π (J|x)q.

where T is the number of trials and we used the notation(C7)(note that in this convention the last item is also included).
The one-trial log likelihood appearing in Eq. (C12) is defined asn=0

L[x1:N ; π̂ , q] =Merging Eqs. (C2) and (C7), we arrive at∞
(y|x; J ) = 0 (y|x; J ) 1 −[π (J|x)(1 − q)]n π (J|x)q .
n=0

(C8)
We can now merge in Eq. (C1) to eliminate 0 , obtaining
π (y|x) 1[y ∈/ J],
(y|x; J ) =1 − (1 − q)π (J|x)∀y = 0,(y|0; J ) = 1[y = 0].

(C10)

As for the eventuality of recall termination (transition froma list item to the sink), it corresponds to a probability
(0|x; J ) =π (J|x)q.
1 − (1 − q)π (J|x)(C11)

The normalized log likelihood is a function of the scalarparameter q and of the matrix parameter π̂ and can be writtenasT
1  αL[π̂ , q, λ] =L x1:N ; π̂ , q ,T α=1

L[x1:N ; û, q] = ln q + ln u(x1:mα −1 |xmα ) +

(C12)
m
α −1

ln (xn+1 |xn ; x1:n−1 )

(C13)

n=1

with  computed according to Eqs. (C9)–(C11) for the given
point in the (π̂, q) parameter space.
Calling mα the number of words recalled in the αth trial,
we can merge the above formulas intoL[x1:N ; π̂ , q] =(C9)where we made it explicit that this does not provide the probability of transition to silence, which can be described in termsof an effective sink state ysink ≡ 0.
The only missing ingredient is now the equivalent of theprobabilities (C9) for the case of transitions to the sink state.
Transitions from the sink state have probability 1 of staying
there and are therefore not problematic:

N−1


m
α −1
n=1

+ ln

ln

π (xn+1 |xn )1 − (1 − q)π (x1:n−1 |xn )π (x1:mα −1 |xmα )q,
1 − (1 − q)π (x1:mα −1 |xmα )(C14)from which, separating the arguments of the logarithms,
L[x1:N ; π̂ , q] = ln q + ln π (x1:mα −1 |xmα )
(C15)
m
mα
α −1

+
ln π (xn+1 |xn ) −ln[1 − (1 − q)π (x1:n−1 |xn )].
n=1

n=1

To account for the column-wise normalization of thestochastic matrix , instead of using Lagrange multipliers,
we opt for adding a degree of freedom by writing π (y|x) =u(y|x)/u(all|x) in terms of non-negative auxiliary variablesu(y|x) defined for x = y without any normalization constraint,
and withu(all|x) =N

u(y|x);

(C16)

y=1

the û matrix can be assumed to have a zero diagonal.
In terms of the matrix û, the one-trial log likelihood (C13)becomesln u(xn+1 |xn ) −

n=1

mα


ln[u(all|xn ) − (1 − q)u(x1:n−1 |xn )].

(C17)

n=1

The corresponding q component of the log likelihood’s gradient is

 α
T mα
xnα
u x1:n−1
∂
1
1 

 α
.

D [û, q] =L= −∂q
q T α=1 n=1 u all xnα − (1 − q)u x1:n−1
xnα
q

(C18)

As for the u derivatives, we haveu
Dyx[û, q] =

+

1
T


(α,k,l ):xkα =x,xlα =y

1
∂
1 

 α
L=−
∂uyx
T (α,k): u(all|x) − (1 − q)u x1:k−1xxkα =x1(l = k + 1)
(1 − q)1(k > l )1(k = mα )
+
 .
 α
 α
+
u(y|x)xu x1:mα −1 xu(all|x) − (1 − q)u x1:k−1

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DECODING SILENCE IN FREE RECALLPHYSICAL REVIEW RESEARCH 4, 033089 (2022)

As a sanity check for this last formula, one can isolate
the contribution from trials where xNα = x and find that it isexactly zero, as it should be because no transitions to the sinkare observed from the last recalled item in perfect-recall trials(mα = N), and therefore the likelihood for such trials is notaffected by the uyxN parameter for any y.
During the numerical search for an optimum, it is convenient to remove the lower bounds at zero from the elements ofû and the [0,1] bounds from q. We do so by mapping both sets
of variables to the full real axes. For the sake of convenience
we used the transformation
1− 1 , v = log(u),(C20)
s = logqso that the gradient becomes


es
1
q
D [v̂, s] =D exp(v̂),,(1 + es )2
1 + es1vu,
Dyx [v̂, s] = exp[v(y|x)] Dyx exp(v̂),
1 + es
s

(C21)
(C22)

or in terms of the physical variables,s[û, q],Ds = −q(1 − q)Dyx

(C23)

v
u
Dyx
= u(y|x) Dyx[û, q].

(C24)

APPENDIX D: SIMULATIONS AND FITTING

To run simulations of the bouncing model with and without
strategicity, we extract the normalized histogram of the full
data set at output position = 1 and use it as our initial condition. Simulation of 105 trials takes minutes on an ordinary
laptop.
To fit the model, we write down the gradient equations (C23) and (C24) and search for the minimum from arandom initial condition using the limited-memory BroydenFletcher-Goldfarb-Shanno algorithm for bound constraints(L-BFGS-B algorithm) [59]. This converges in < 102 iterations to the matrix shown in Fig. S4(a) and to the optimal
impatience q ≈ 0.1.
Notice that sequentiality partially comes from the Markov
transition probabilities and partially from strategicity. Our focus is not to achieve the best quantitative model but to explain
how the strategicity sequentializes only high-performancetrials, which yields a major difference between the two mechanisms of increasing sequentiality. For simplicity we used alltrials to fit transition probabilities, which in principle may
overestimate nonstrategic sequentiality.
Fitting and simulations based on an alternative “skipping
model” are described in Appendix E.
APPENDIX E: THE SKIPPING MODEL

An alternative model through which one may attempt toreproduce the silent serial-position effects is what we willrefer to as the skipping model, where the Markovian retrieval
process is not interfered with by the higher-order process thatmerely censors the reporting of repetitious retrievals.
A repetitious retrieval is thus not discarded as in the bouncing model, but adopted as the starting point for the nexttransition (which will be described as a “skip” rather than as a“bounce”; see Fig. S7).
As in the bouncing model, an N × N Markov transition
matrix π is defined such that πxy is the probability of retrieving word x after retrieving word y. We assume again that
this probability is independent of the time step throughout
the retrieval process. We also include among the states of theMarkov chain a sink state accounting for termination.
The transition from the nth to the (n + 1)th recall is governed by a “recall propagator” T{x1 ,...,xn } , defined as a matrix
that depends parametrically on the potentially repetitious(hence unreportable) set of serial positions {x1 , . . . , xn }, whichis the set of words recalled up to that moment.
In other words, for any set of serial positions S, we aredefining the matrix TS such that [TS ]xy is the probability ofrecalling x after recalling y, given that the items in the set S
cannot be reported, having already been recalled (though they
can be retrieved). Since recalls do not happen at all retrieval
events, skipping repetitious retrievals, for x ∈ S or y ∈/ S wehave [TS ]xy = 0 [Fig. S8(a)].
Since the data set samples transitions driven by the TS
matrix, the π matrix can be obtained via maximizing the
likelihood of the TS matrix given the data set. Unfortunately,even if infinite skips are allowed in a single recall event andno sink state is included, a closed formula such as Eq. (C1) is
not available for TS . Although a Dyson-like summation canbe performed for this propagator, when computing the log
likelihood on data, implementing it leads to a combinatorial
blowup of the number of possible paths.
Thus, while the retrieval process in the skipping model issimpler than in the bouncing model (because it stays completely Markovian), the fitting is much less straightforward.
On the other hand, once the π matrix is given, the T
matrix can always be calculated iteratively. In order to practically estimate the π matrix, we exploited the fact by making
an extra assumption and putting an upper limit nmax on thepossible number of bounces. We were thus able to build arecurrent neural network (RNN)-style computational graph as
illustrated in Fig. S8(b).
For any set of serial positions S, let us define the matrix DS
such thatδyx δxz .
[DS ]yz =x∈SGiven a certain source item x and a repetitious set S + x =S ∪ x, the T matrix can be calculated as
TS =

nmax


DS̄ π (DS+x π )n ,

(E1)

n=0

where S̄ is the allowed output set and S + x is the unreportable(i.e., repetitious) set; x, which is the source, is included in theunreportable set.
π DS̄ is here a skip-connect indicating the end of a bounce
when an element recalled is in the reportable set. This graph
now also allows transitions from x to itself, a hitch that canbe solved by masking out diagonal elements of the π matrix.
Even without masking, however, the diagonal elements of theπ matrix vanish after fitting the data.

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We implement the computational graph with PYTORCH.
The negative log-likelihood loss function between the calculated source-to-target log probabilities with the current πmatrix and observed target sample is used as the loss function. A certain batch number of unreportable sets and reported
targets is randomly picked from the data set to carry out astochastic-gradient-descent parameter update of the π matrix.
Let us call xmα the serial position of the word recalled in themth recall of trial α. Call b the size of minibatches. Each minibatch selection consists of a set α of b random trial indices αkand of a set m of corresponding random output positions mk .
Here, k = 1, . . . , b, and for each k, 1  αk  NT (NT being
the number of trials) and 1  mk  L (L being the list length).
The loss function for a particular batch is thenL(π ; α, k) = − log T{xαk ,...,xαk } xαk ,xmαk k=1,...,b ,where the average is used instead of a sum to keep the valuesof the loss function numerically low even with many batches.
It takes approximately tens of minutes to obtain convergenceon a conventional laptop.
The resulting matrix parameter π , shown in Fig. S8(c),can be used to simulate the model starting from the empirically observed initial condition. Both the silent recency effectand the silent contiguity effect are successfully reproduced(Fig. S9).
However, an additional “silent primacy effect” (where past
recall of the first item in the list engenders a delay in subsequent recalls) emerges nearly as conspicuously as silent
recency (Fig. 6). This is compatible with the existence of acorresponding primacy effect in reported recall frequencies[16] but is not compatible with the data (Fig. 3).

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