science.sciencemag.org/content/364/6447/eaaw5181/suppl/DC1

Supplementary Material for

Sequential replay of nonspatial task states in the human hippocampus

Nicolas W. Schuck* and Yael Niv*

*Corresponding author. Email: schuck@mpib-berlin.mpg.de (N.W.S.); yael@princeton.edu (Y.N.)

Published 28 June 2019, Science 364, eaaw5181 (2019) DOI: 10.1126/science.aaw5181

This PDF file includes: Materials and Methods Supplementary Text Figs. S1 to S7 Tables S1 and S2 Reference

Materials and MethodsParticipants

Thirty nine participants were selected according to standard fMRI screening criteria (right hand-

edness, 18–35 years of age, normal or corrected-to-normal vision and no contraindication for

fMRI) from the Princeton University community, and were compensated with $20 per hour plus

up to $5 performance-related bonus. Six participants were excluded from analysis due to ei-

ther technical errors (3 participants), violation of performance criteria standards (2 participants

with over 4 times the average error rate in the last two blocks of the experiment) or incomplete

data (1 participant). The final sample consisted of 33 participants (22 female, mean age 23.4

years). All participants provided informed consent and the study was approved by Princeton

University’s Institutional Review Board.

Stimuli

Stimuli consisted of spatially superimposed images of a face and a house (see (7); face images

from http://faces.mpdl.mpg.de/faces described in (53), see Fig. 1 Main Text).

Faces and houses could be classified as either young or old, e.g. a stimulus could show an old

face image blended with a contemporary (i.e., young) house image. Four classes of stimuli were

possible: (1) two old or (2) two young face and house pictures, (3) a young face with an old

house or (4) vice versa.

Task

The task was identical to Schuck et al. (2016) and will be described only briefly. Each trial be-

gan with the display of the mapping of a left and right key to a young and old response (changing

randomly trialwise) below a fixation cross for 1.2s (range: 0.5–3.5s). Then, a compound face-

house stimulus was shown for 3.3s (range: 2.75-5s; Main text Fig. 1) and participants had to

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make an age judgment about one of the two image categories. Participants knew which category

of the stimulus they had to judge by applying the following rules: 1. Before the first trial of

each run, the category to judge was displayed on the screen; 2. Once the age of the relevant

category changed (e.g., from young to old), the judged category changed on the next trial. 3.

No age comparison was necessary on the first trial after a category change, i.e. each category

was judged for at least two trials in a row before a switch. The average trial duration was 4.5s

(range: 3.25-8.5s), all timings were randomly drawn from a truncated exponential distribution

and the response deadline was 2.75s. The category instruction cue at the beginning of a run

was displayed for 4s. Erroneous or time-out responses led to feedback (written above stimulus

for 0.7s) and trial repetition. If an error trial involved an age change (and thus would require a

category switch on the next trial), participants had to repeat the trial before the error as well as

the error trial, giving them the chance to observe the age change. Otherwise, they had to repeat

the trial on which they made the error.

Design

Participants underwent two fMRI sessions. The first session began with the display of written

instructions while participants underwent a functional scan (group 1), or a 5 minute resting-state

scan followed by instructions (group 2). The instructions explained the rules of the task and

contained a training phase in which simple age judgments had to be made on (non-overlapping)

face and house images. The images shown in this period were later used in the task, thus fa-

miliarizing participants with the age judgment aspect of the task as well as the stimuli. The

instructions furthermore involved an annotated walk-through of four trials of the real task (i.e.,

with overlapping images and the requirement to switch attention after an age change). Follow-

ing the instructions, participants performed 4 runs of the task (97 trials per run, 388 total). Each

run lasted about 7-10 minutes and participants were given the chance to rest briefly between

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runs. A 5 minute fieldmap scan was done between runs 2 and 3, resulting in a longer break for

participants. After run 4, participants underwent a resting state scan as well as a structural scan.

Lights were turned off during resting-state scans and participants were instructed to stay awake

for the entire duration of the scan (5 minutes, 100 TRs). The second session was identical for

all participants and involved the following scans: resting state, 2 task runs, fieldmap, 2 task

runs, resting state and structural scan. Thus, overall, participants performed 8 task runs and 3

(group 1) or 4 (group 2) resting-state scans. In all other regards, the task design involved the

same characteristics as detailed in Schuck et al. (2016).

Behavioral Analyses

Behavioral analyses were done using mixed effects models implemented in the package lme4

(48) in R (49). The model included fixed effects for Block, Condition, Category and intercept.

Participants were considered a random effect on the intercept and the slopes of all fixed effects.

The reported p-values correspond to Wald chi-square (χ2) tests as implemented in R. Reaction

time (RT) analyses were done on error-free trials only and reflect the median RT within each

factor cell.

fMRI Scanning Protocol

Magnetic-resonance images were acquired using a 3-Tesla Siemens Prisma MRI scanner (Siemens,

Erlangen, Germany) located at the Princeton Neuroscience Institute. A T2*- weighted echo-

planar imaging (EPI) pulse sequence was used for functional imaging (2×2 mm in plane res-

olution, TR = 3000 ms, TE = 27 ms, slice thickness = 2 mm, 53 slices, 96×96 matrix (FOV

= 192 mm), iPAT factor: 3, flip angle = 80◦, A→P phase encoding direction). Slice orienta-

tion was tilted 30◦ backwards relative to the anterior – posterior commissure axis to improve

acquisition of data from the orbitofrontal cortex (46). Field maps for distortion correction were

acquired using the same resolution (TE1 = 3.99ms), and a MPRAGE pulse sequence was used

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to acquire T1-weighted images (voxel size = 0.93 mm). The experiment began 20 seconds after

acquisition of the first volume of each run to avoid partial saturation effects.

fMRI Data Preprocessing

FMRI data preprocessing was done using SPM8 (http://www.fil.ion.ucl.ac.uk/spm)

and involved fieldmap correction, realignment, and co-registration to the segmented structural

images. The task data used to train the classifier were further submitted to a mass-univariate

general linear model that involved run-wise regressors for each state (see below), nuisance

regressors that reflected participant movement (6 regressors) and run-wise intercepts. The re-

sulting voxelwise parameter estimates were z-scored and spatially smoothed (4 mm FWHM).

The resulting activation maps were used as the training set for a support-vector machine (SVM)

with a radial basis function (RBF) kernel using LIBSVM (51). The SVM was trained to predict

the task state from which a particular activation pattern came from. Like the activation maps

used for classifier training, the resting-state data were z-scored and smoothed (4mm FWHM).

Anatomical ROIs were created using SPM’s wfupick toolbox. The hippocampus (HC) was

defined as the left and right hippocampus AAL labels. The orbitofrontal cortex was defined

as in (14). Behavioral analyses and computations within the assumed graphical model of state

space (see below) were done using R (49, 52).

Autonomic nervous system activity during task and rest

In order to assess whether there were any differences in vigilance or arousal between the differ-

ent resting conditions (and the task), we analyzed the data recorded during the experiment from

the Siemens MRI optical pulse sensor and pneumatic respiratory belt. We then used the TAPAS

PhysioIO toolbox (50) to determine participants’ heart rate.

Data was successfully recorded in at least one session in 30 out of 33 participants. The aver-

age heart rate across the entire experiment was 69.7 beats per minute (SD: 10.4). As expected,

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this data showed that participants had generally higher heart rates during the task compared to

the average rest conditions, t29 = 6.2, p < .01, presumably due to heightened vigilance and

arousal (mean difference: 3.8 beats per minute more during the task). Importantly, however,

there was no difference between the pre (PRE + INSTR) and the post rest conditions (all TASK

rest conditions), t22 = 0.7, p = .43 (differences in df ’s arise from physiological recording

failures in some participants/sessions). In addition, there was no difference between the PRE

condition and any of the TASK rest conditions (ps = .23, .96 and .75 for the 1st, 2nd and 3rd

resting state, respectively), between the INSTR and any of the TASK rest conditions (ps = .11,

.65 and .20, respectively), or between the PRE and the INSTR conditions (p = .15). In addi-

tion, there was no relation between heart rate during the TASK rest condition and sequenceness

effects during the TASK rest condition, r = −0.05, p = .79.

fMRI Classification Analysis

The support vector machines were trained on 8 maps of parameter estimates (“betas”) for each

of the 16 states (one map for each state and run) restricted to the anatomical mask of the hip-

pocampus (back-transformed into each participant’s individual brain space) or the orbitofrontal

cortex. Classification accuracy was assessed with a leave-one-run-out cross-validation scheme

in which data from 7 runs were used for training and the held-out run was used for testing

(Main Text; Fig. 2). The resting-state analysis used a classifier trained on all available task

data (8 runs). This classifier was applied to each volume of the resting-state data and the most

highly classified state was considered as the output of the classifier for that volume. The result-

ing sequence of predictions was the main focus of our analyses (see below). We obtained the

distance to the hyperplane by dividing the decision value by the norm of the weight vector w,

as specified in the libSVM webpage (http://www.csie.ntu.edu.tw/ cjlin/libsvm/faq.htmlf4151).

For each volume, we then calculated the average of the distances of all pairwise comparisons of

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the predicted class against all other classes, to obtain a proxy of how certain the classifier is in

its prediction. Student t tests pertaining to decoding results were one-tailed, given the a priori

expectation of larger-than-chance decoding in the hippocampus.

Sequenceness Analysis

The main question of the sequenceness analysis was whether the state transitions decoded from

resting state scans, T , were related to the distance between states experienced during the task,

D. To this end, we analyzed the neural state transitions T with logistic mixed-effects models,

using the lme4 (48) package in R (49). Because the slow hemodynamic response function leads

to encoding of sequential structure in activity patterns (i.e., there is high similarity between

temporally adjacent patterns), a classifier trained on sequential task data can be biased to de-

code states in a similar sequence to the training data, even if the test data are random (i.e., the

‘sequenceness’ identified in the test data comes from the training data, not the test data). We

therefore applied the trained classifier to matched fMRI noise (see below) and used the resulting

spurious ‘state transitions,’ T [ε], as a covariate that would account of the spurious base rate of

transitions that is due to the classifier rather than the data. Applying these models to control con-

ditions consistently yielded non-significant effects of sequenceness, showing that this analysis

appropriately controls for the above mentioned spurious structure that is observable for instance

in the significant correlations between D and T in the noise data (Main Text; Fig. 3F). Specifi-

cally, our model included the following fixed effects: (1) the distance between states, D, which

was the regressor of interest, and as regressors of no interest (2) the transition probabilities ob-

tained in the above mentioned noise simulations, T [ε], (3) an orthogonal quadratic polynomial

of T [ε] that was included in order to account for as much noise-related variance as possible, and

(4) an intercept. Models of change in sequenceness across PRE, INSTR and TASK conditions

(Main Text; Fig. 4) additionally involved interaction terms of condition with the distanceD and

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condition with the noise transitions T [ε]. Participant identity was included as a random factor

to account for between-subject variability. To capture state-related variability (state frequency

effects affect the distribution of state transitions), state identity sj of a transition from state i to

state j was used as an additional random effect nested within subject. Participant and state were

random grouping factors for all fixed effects with exception of the quadratic expansion of T [ε],

where including these random factors caused problems in fitting the logistic regression models.

Formally, the model followed the general assumption that the number of transitions Y is

drawn from a binomial distribution of n draws and probability T :

Yijk ∼ B(nk, Tijk)

where nk corresponds to the number of measurements for subject k, and i and j index the

outgoing and incoming states of a given transition. The logit transformed probabilities T (shown

in Main Text Fig. 2D; logit is the canonical link function for binomial models) were then

modeled in a mixed effects regression model with the above mentioned fixed and random effects

structure:

logit(Tijk) = β0 +Dijβ1 + T [ε]ijβ2 + T [ε]2ijβ3 +

γ0k +Dijγ1k + T [ε]ijγ2k +

ζ0kj +Dijζ1kj + T [ε]ijζ2kj + εijk

In the text, we describe the fixed effect of D, β1 in the models, as ‘sequenceness,’ and the

fixed effect of T [ε], β2, as ‘randomness’ (Main Text; Fig.s 4B,C). The subject-specific random

effects of D, γ1k, were used as individual sequenceness indicators in the correlations (Main

Text; Fig.s 4F,G). The state- and subject-specific random effects are indicated by ζ . Correlations

between random effects were estimated. Model comparisons were conducted using likelihood-

ratio tests by comparing models that included the noise transitions T [ε] with versus without the

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fixed effect regressor of distance (sequenceness), or without the condition interaction terms to

the full models that included these terms. The random effects structure was kept constant across

these comparisons.

T-tests pertaining to sequenceness results (number of steps, etc.) are one-tailed, given our a

priori expectation of larger sequenceness in the hippocampus compared to the various controls.

Alternative Task Transition Matrices

As mentioned in the main text, for reasons of model comparability we used the 1-step transitions

of the task as a basis to test alternative replay models. The 1-step transition matrix simply

reflects from which state one could proceed to which other states in one trial. The alternative

task transition matrices were based on the assumption that the hippocampus has access to only

partial state information, and hence correspond to transition matrices defined over subsets of

states.

For instance, to compute the transition matrix of the “stimulus model” we defined SstimFy as

the subset of states in which Fy was the stimulus:

SstimFo = {(Fy)Fy, (Fo)Fy, (Hy)Fy, (Ho)Fy}.

SstimFo , Sstim

Hy , SstimHo were the corresponding subsets of states in which Fo, Hy and Ho were the

stimuli, respectively. The 1-step distance matrix was then computed such that every transition

between two states si and sj in the complete task state diagram was converted into transitions

from all four states aliased with si to all four states that are part of the same subset as sj , that

is Sstimj . All resulting transitions are summed, and normalized so that all exiting transitions

from a state would sum to 1. The new transition between SstimFy and Sstim

Fo would therefore count

within it (Fy)Fy → (Fy)Fo, (Ho)Fy → (Fy)Fo and (Hy)Fy → (Fy)Fo, whereas the

new transition between SstimFy and Sstim

Ho would only count within it (Fo)Fy → (Fy)Ho. After

normalization, these would be 3/8 and 1/8, respectively.

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For the other alternative models, we defined subsets of states that have the same current

attended category, and subsets of states that have the same current and previous attended cat-

egories, and then computed the transition matrices as described above. The 1-step transition

matrices of these alternative models are shown in Figures 5A-C in the Main Text. The reverse

replay transition matrix was simply the transpose of the full task 1-step transition matrix.

Synthetic fMRI Data and Noise Simulations

In order to estimate to what extend training the classifiers on sequential data influenced the se-

quenceness of their predictions, we simulated, for each participant, individually spatio-temporally

matched fMRI noise, and applied the classifiers to these data. For each participant and resting

state session, we first extracted fMRI data from the hippocampus and the orbitofrontal cortex

and calculated the voxel wise mean of that participants’ real resting state data as a baseline for

each TR. We then added temporal noise to this baseline as described below, ensuring that spatial

properties driven by the anatomy and tissue partial volume are reflected in the noise data used

for the simulations (i.e., the average noise data looks like each participants’ real data reflecting

grey/white matter differences etc.).

As in the classification analyses, we applied linear detrending to each voxel. We then esti-

mated the average standard deviation of the voxels within these regions, as well as the average

autocorrelation using an AR(1) model in R. Next, we used the neuRosim toolbox in R (52) to

simulate fMRI noise with the same standard deviation and temporal autocorrelation as the real

data. Finally, we used AFNI’s 3dFWHMx and 3dBlurToFWHM functions to first estimate the

spatial smoothness of the real data, and then smooth the simulated noise until it had the same

effective smoothness.

For each existing resting-state run, matched noise data with the same number of TRs and

voxels were generated. Figure S1 shows the temporal and spatial smoothness of the real and

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simulated data separately for each run. In all cases, the properties of the simulated data did not

differ from the real data, paired t-tests, all ps > .05.

Finally, we applied each participant’s classifier to the matched noise data. The classifier was

identical to the classifier that was used in estimating the sequences of states from the real data.

The resulting sequence of predicted states reflects the bias of the classifier to make sequential

predictions because of pattern overlap in the training set, even when applied to noise, as well as

any tendency of the classifier to decode certain states more often than others. We therefore used

the sequence of states from this analysis to construct the nuisance covariate for the mixed effects

models, i.e. the noise ‘transition matrix’, T [ε], and to perform the appropriate comparisons

in the correlation analysis. The average noise transition matrix is shown in Fig. S2. These

comparisons between sequenceness in real data versus simulated noise in the correlation and

mixed effect analyses indicated that the noise sequenceness T [ε] indeed explained a significant

amount of sequential variability of the decoded states (see Main Text Figs. 3F,G, 4B, D), and

thus served as a powerful control. Together with the tests done on permuted data (Main Text;

Fig. 3B-D, 3F,G), the comparisons across brain regions (Main Text; Fig. 4E) and the within-

participant comparisons between the PRE, INSTR and TASK conditions (3B-D, H and 4A-D),

our approach represents a stringent control of potential biases.

Transition exclusion analysis

We assessed the effect of temporal contingencies in the data used to train the classifier on

the sequenceness measure in the resting state scans by excluding certain transitions from the

data used for classification training. Specifically, for all states that had more than one incoming

transition (blue states in Figure S3), we excluded from the classifier training those trials in which

these states were preceded by one of the preceding states (see Fig. S3: trials corresponding to

black arrows were included, trials corresponding to red arrows were not). In two separate

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sets of classifiers, we swapped which of the transitions were excluded as shown in Fig S3.

For instance, the state (Hy)Ho was preceded in 25% of the trials by the state (Fo)Hy, and

in 25% by (Fy)Hy (and in 50% by (Hy)Hy). We therefore trained two separate classifiers,

one that excluded the (Fo)Hy → (Hy)Ho transitions, and one that excluded the (Fy)Hy →

(Hy)Ho transitions. The sequential decoding analysis was then performed on the resting scans

with both classifiers, and we compared how often the excluded versus the matched included

transitions were observed during rest. We took the non-left out transitions ending in the same

state as the matched comparison, such that the number of states, the possible more-than-one step

transitions between them, and the involved terminal state were equated. If pattern similarity

in the classifier training set determines the frequency of decoded transitions, then excluded

transitions should be decoded less often in the resting scans as compared to included transitions.

If, in contrast, sequential replay has a significant contribution to the observed transitions during

rest, the excluded transitions should be observed equally often, or at least more often than

transitions that correspond to a larger distance in the the state space.

Replay simulations

In order to assess the sensitivity of our analysis to effects of fast sequences of neural events on

fMRI data, we simulated ‘neural’ replay events at different speeds and applied our analysis to

the simulated data. Each simulation involved 240 ‘units’, which generated neural activity on the

order of milliseconds. Each unit drove the activity of one voxel, the observational unit in fMRI

(Fig. S4A). Activity in the units was a mixture of signal and noise. The noise component was

drawn from a multivariate Gaussian distribution with a diagonal covariance band that reflected

spatial autocorrelations between neighboring units. The signal reflected sequential activation of

patterns which represented one of 16 possible states. Specifically, the replay was simulated as a

fast sequential activation of the involved states. Each state was activated for 2 ms, and involved

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14 units. Ten of these units were unique to this state, and 4 units were non-unique, that is they

were also activated in response to at least one other state. Neural activity corresponding to the

activation of a sequence of states could then be simulated by activating the state patterns in the

given order (Fig. S4B). The strength of activation caused by the patterns relative to the back-

ground noise was treated as a parameter that was systematically varied across simulations, and

unit-wise deviations of this average activation strength were drawn from a Gaussian distribution

with a standard deviation of 0.5. The strength of the average activation of units during the rep-

resentation of a state reflects the signal-to-noise-ratio (SNR) in the simulated data and affects

decoding accuracy (see below). Activity in units was converted to activity in voxels through

convolution with the canonical double-gamma hemodynamic response function:

h(t) =t5e−t

Γ(6)− t15e−t

6Γ(16)

where t references time and Γ represents the gamma function. The resulting temporo-spatial

activation pattern (Fig. S4C) was then used to train and test the classifier as described below.

With this general setup, we generated a ‘task’ data set and a ‘rest’ data set. In the task data,

the sequences of states were activated in sequential order (all combinations, where face always

comes before house and young before old, i.e. FyFy, FyFo, FyHy, FyHo, FoFy, ...) with the

same temporal spacing as in the task. The task data set contained 8 runs in each of which every

state was activated once. Note that, unlike in the real data, the order of state activation in the

simulated task data was unrelated to the order of replay during rest. This ensures that the effects

found cannot be explained by a bias for sequentiality in the classifier due to the training data

set.

In addition to the task data set, we generated 300 TRs of rest data. During rest, noise was

drawn from the same multivariate Gaussian used for the task data simulation (units exhibited

the same covariance structure). Replay events were assumed to occur spontaneously every 30

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seconds during rest (12), with inter-replay-intervals jittered according to an exponential distri-

bution (intervals were defined as the period from end of the previous sequence to the beginning

of the next sequence). Each replay sequence was between 2 and 10 states long, following an

exponential distribution (3). Replayed states followed the same sequence as the one used in our

main task (but different from the one used to activate states in the simulated task data). The tem-

poral delay between state activations (i.e., the replay speed) was treated as a second parameter,

whose effect on the results will be investigated along with the effect of the SNR.

The resulting unit time-courses were convolved with the HRF as described above, and sub-

sampled to provide one ‘volume’ of brain activity consisting of voxels sampled every 2.9 sec-

onds as in our real data. The subsampled voxel time-courses were then z scored. As in the

analysis presented in the main manuscript, we then trained a classifier on all task data and ap-

plied it to every volume of the resting state data. Finally, we analyzed the resulting sequence

of classifier predictions as in the real data via correlations between pattern transition frequency

and state distance of the underlying replayed sequences.

Supplemental Text

Supplemental ResultsBehavior

Figure S5 shows reaction times (RTs) and error rates separately for each state and session. As

reported in the text, error rates and RTs decreased over time, and were also affected by the state

type.

Mixed effect models: Alternative distance metric

To test for the generality of our results, we repeated the model comparisons reported in the

main text using a different distance metric for task states. Recall that in the main analysis

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we computed distance between each pair of states as the minimum number of steps needed to

traverse from one state to the other, in the state space. Here, instead, we simulated a random

walk within the task’s state space, and obtained the average number of steps it took to tranverse

between two states. This metric was taken as the distance metric D instead of the minimum

number of steps. With this new D we were able to replicate all the results from the mixed

effects analysis reported in the main text. In particular, we again found a significant interaction

between distance and resting state scan timepoint (TASK vs PRE) in modeling the empirical

transition matrix T (significantly better fit when the distance by timepoint interaction term was

included, χ23 = 11.19, p = 0.01), suggesting that there was a significant difference in the

relationship between T and state distance when comparing the resting scan before session 1

with the scan after session 1. Below is a table listing the fixed effects, AICs and whether a

likelihood ratio model comparision test was significant for all models reported in the main text.

Mixed effect models with OFC data

We also repeated the mixed effects models reported in the main text using data from the or-

bitofrontal cortex instead of the hippocampus. The table below lists all results. Importantly,

we did not find any evidence in favor of sequenceness increases from PRE to TASK, or from

INSTR to TASK.

Replay simulations

Using simulations we tested whether, in principle, rapid replay events can be reliably decoded

from BOLD data, given their low temporal resolution and correlated noise. To this end, we sim-

ulated replay events in resting data (see Materials and Methods) and attempted to decode them

with classifiers trained on simulated task data. Our simulations showed that indeed classifier

predictions during rest could be related to replay events. In a simulation in which decodability

of states was high (SNR: 15) and replay speed was slow (temporal delays between replayed

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states: 2560 ms), a simulated replay event could be decoded quite reliably in the volumes fol-

lowing the event (Fig. S6A). More interestingly, even when decoding success was realistically

low (only ∼ 12%, as in our experiment) and replay events were very fast (40 ms between sim-

ulated pattern activations), the classifier was able to decode the states belonging to the replayed

sequences in the TRs following the replay event (Fig. S6B, S6C). This indicates that the super-

position of state-related patterns caused by rapid replay events does not hamper the classifiers’

ability to detect the presence of states.

Notably, the decoding accuracy for replay events was similar to the decoding of individual,

well-separated states. This effect has important implications for the feasibility of measuring

replay events in fMRI, where classification accuracy for our 16-state task is low. Let’s call

the decoding accuracy of 11.6% in the hippocampus, which reflects task conditions in which

a volume is characterized by a single true state, a. In the volumes following a replay event,

several states impact the BOLD signal. That each of these states is decodable with a similar

accuracy a (see Fig. S6B) increases our ability to observe sequential events in two consecutive

volumes. In particular, the chance that the states decoded in two consecutive TRs following a

replay of, say, 5 steps, are in the correct order (and therefore result in a low distance measure

that we use in several of our analyses in the main text) is the sum of the probability that the first

TR is the first state of the replayed sequence, times the probability that the prediction on the

next TR is any of the states 2-4, plus the remaining possibilities if the first decoded states are

later in the sequence:

P (ordered pair) ∼ P (C1 = s1)P (C2 = {s2, s3, s4, s5})

+ P (C1 = s2)P (C2 = {s3, s4, s5})

+ P (C1 = s3)P (C2 = {s4, s5})

+ P (C1 = s4)P (C2 = s5) (1)

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where P (C1 = s1) reflects the probability that the decoded class C in the first volume is the

first state of the sequence s1, and P (C2 = {s2, s3, s4, s5}) reflects the probability that any of the

states 2-5 is decoded on the second volume. Because in the volumes following a replay event

all involved states independently have roughly the same chance to be classified as in the regular

classification analysis, we can substitute a for each probability of a single state, and n ·a for the

probability to classify any of n states, giving

P (ordered pair) ∼ 4a2 + 3a2 + 2a2 + a2. (2)

For a given replayed sequence of length l, the probability of an ordered pair can therefore be

written generally as

P (ordered pair) ∼l−1∑i

(l − i)a2 = a2l2 − l

2(3)

Although this relationship only represents a back-of-the-envelope calculation, it illustrates that

the probability to observe an ordered pair is significantly higher than a2, the intuitive estimate of

observing two correctly classified states in a row, assuming the decoding accuracy on task data.

For the example of a replayed sequence of l = 5 steps, and a decoding accuracy of a = 0.116,

as in our case, the probability of observing an ordered pair is approximately 0.1162(25−52

) =

0.13456, i.e. about 13.5%.

Indeed, the superposition of multiple patterns triggered by the replay lasts for several vol-

umes. Even very short replay events therefore allow the recovery of a substantial proportion of

states involved in the replay event. Interestingly, our simulations show that overall decoding ac-

curacy in each volume is equally high under fast compared to slower replay events (Fig. S7A).

Even though under low SNR conditions the order of predicted states following a fast replay

event does not necessarily reflect the order of replayed states, the classifier will decode states

which come from the same sequence. This leads to a lower average number of steps between

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decoded states following a replay event than expected by chance (Fig. S7B), offering a plausi-

ble explanation for the findings reported in the main text (Fig. 3C,D). In other words: because

the replayed states are close by in state space, even a random ordering of these same states will

be more sequential than expected when states are drawn at random from the entire state space.

This effect also leads to the correlation between D and T that we report in the main paper (Fig.

3F-H), which we find to be detectable at 12% decoding success already for replay speeds of

about 40-60ms (Fig. S7C,D).

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A

1.0

2.5

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DataSimulation

Spat

ial S

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19

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Figure S2. Spurious state transitions estimated in synthetic data. The matrix shows the(non-normalized) number of transitions between states found when the classifier was applied tomatched noise data (see Main Text and SI). These results reflect the bias of the used classifiersto report certain transitions more often than others. Individual estimates per participant andconditions were used in the mixed effects analysis as the nuisance regressor T[ε], see MainText. Values shown in Figure reflect the average across all participants and all POST conditions(see color scale, lighter = more frequent). Note that the colors of states in the Main Text aredifferent. There, green was replaced with brown.

20

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Figure S3. Transition exclusion analysis. In two separate sets of classifiers, we excluded oneof the transitions leading to states with several incoming transitions (red arrows). The resultingclassifiers therefore differed in the training set. Importantly, although each classifier was trainedon data pertaining to all states, the data used to estimate the neural patterns evoked by the statesmarked in blue were based on different subsets of data, compared to the standard classifier. Foreach training set, the frequency of excluded transitions being decoded from the rest scans wascompared to a matched set of included transitions (included transitions to the same blue states).Results were averaged across both training sets.

21

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Figure S5. Effects of sessions and state identity on behavior. Reaction times (A) and errorrates (B) are shown as a function of state identity (see x label) and session. Colors indicate thestate class, but note that the colors of states in the Main Text are different. There, green wasreplaced with brown. Bars represent standard errors of the mean.

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24

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25

Condition Rand. Seq. Cond × Seq AICPRE 23.16 – – 3651.2PRE 22.48 0.84 – 3648.6

TASKpre 16.59 – – 3661.5TASKpre 14.86 1.10 – 3656.9*

PRE & TASKpre 25.44 0.75 – 7238.9PRE & TASKpre 28.66 0.41 0.67 7233.7*

INSTR 38.65 – – 10044INSTR 36.93 0.80 – 10044

TASKinstr 38.59 – – 10166TASKinstr 36.04 0.91 – 10155*

PRE & TASKinstr 50.17 0.86 – 19997PRE & TASKinstr 51.20 0.78 0.10 19998

Table S1. Mixed effects modeling results. The structure of the models is the same as in themain text, see Methods. Here we used the long term average number of steps between statesinstead of the minimum number of steps between two states as a distance metric. Otherwisethe models are identical to the models reported in the main text. The first column indicatesfrom which condition(s) the data came. TASKpre refers to data from the task condition thatwas matched in length to the PRE condition, to equate power. TASKinstr refers to TASK datamatched to the INSTR condition. Columns 2-4 indicate fixed effects estimates derived from themodels. AIC: Aikaike information criterion. *: significant difference in a log likelihood ratiotest comparing the indicated model to the model in the previous line (performed only for nestedmodels).

26

Condition Rand. Seq. Cond × Seq AICPRE 11.51 – – 3583.6PRE 11.92 0.05 – 3584.1

TASKpre 4.17 – – 3431.4TASKpre 3.07 -0.09 – 3431.8

PRE & TASKpre 13.56 -0.01 – 6925.6PRE & TASKpre 15.15 0.06 -0.129 6925.4

INSTR 20.11 – – 9912.0INSTR 19.29 -0.10 – 9905.7*

TASKinstr 15.96 – – 9817.6TASKinstr 14.98 -0.10 – 9811.6*

PRE & TASKinstr 25.37 -0.08 – 19505PRE & TASKinstr 26.27 -0.08 -0.001 19507

Table S2: Mixed effects modeling results. The table has the same structure as above (TableS1). Importantly, here predicted sequences of states derived from data from the orbitofrontalcortex were used.

27

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