Retrieving the structure of probabilistic sequences from EEG data during the goalkeeper game

TL;DR

The study investigates whether EEG signals contain fingerprints of the penalty taker’s probabilistic context-tree structure during the Goalkeeper Game. It applies the Context Algorithm to 300 ms EEG segments preceding responses to retrieve context trees and measures their Balding distance $d$ to the penalty taker’s tree, relating this distance to performance via Linear Mixed Effects modeling. Across multiple electrodes, the Balding distance typically decreases as learning progresses, and significant inverse correlations between $d$ and success rates emerge (notably at electrodes like Fp2, F4, P3, Oz), supporting the MDL-based view that neural activity encodes the past-dependency structure of temporal sequences. These findings highlight a neural–behavioral link in statistical learning of temporal structures and demonstrate a framework for decoding sequence structure from EEG in a naturalistic task.

Abstract

This work draws on the conjecture that fingerprints of stochastic event sequences can be retrieved from electroencephalographic data (EEG) recorded during a behavioral task. To test this, we used the Goalkeeper Game (game.numec.prp.usp.br). Acting as a goalkeeper, the participant predicted each kick in a probabilistic sequence while EEG activity was recorded. At each trial, driven by a context tree, the kicker chose one of three options: left, center, or right. The goalkeeper then predicted the next kick by pressing a button. Tree estimation was performed by applying the Context Algorithm to EEG segments locked to the button press (-300 to 0 ms). We calculated the distance between the penalty taker's tree and the trees retrieved per participant and electrode. This metric was then correlated with the goalkeeper's success rates. We observed a clear reduction in the overall distance distribution over time for a subset of electrodes, indicating that EEG dependencies become more congruent with the penalty taker's tree as the goalkeeper learns the sequence. This distance is inversely proportional to the goalkeepers' success rates, indicating a clear relationship between performance and the neural signatures associated with the sequence structure.

Figures (7)

• Figure 1: The Goalkeeper Game task. A) Key/finger/choice mapping and trial configuration: on the left, we show the keyboard design and the fingers' correspondence with the game option: right index finger (0), right middle finger (1), and right ring finger (2). On the right, the duration of each trial section is presented along with the game screen samples. The first section is termed the readiness period, during which the goalkeeper and penalty taker on the screen remain at rest. The second is termed the response time section, marked by the appearance of three arrows at the bottom of the screen informing the goalkeeper to inform his/her prediction. No maximum duration for this period was set. This section ended with the button press, which triggers the feedback animation ($2300ms$). B) Context tree model generating the penalty taker's choices and examples of penalty taker's and goalkeeper's sequences. The bottom trace represents the continuous recording of the EEG signal. The arrows indicate the transition probability following the context tree model; the transition that did not occur is presented in gray.
• Figure 2: Context Algorithm and distance calculation. A) The cartoon on the left shows the spatial distribution of the 32 electrodes following the 10-20 International System from which the EEG signals were recorded. On the right, we show an example of an EEG recorded from a single electrode of one participant. The gray bars on the bottom indicate the readiness period, response time, and feedback sections. The colored windows indicate the 300 ms of the EEG signal of each trial used for the context tree estimation. The color code indicates the context of each trial following the tree representation (top left corner). B) The pruning procedure was performed by (1) selecting not unvisited terminal subtrees of the complete tree. In our example, the branch corresponding to symbol 1. (C) testing the null hypothesis of equality of distributions for the sample of EEG segments associated to the leaves of that subtree, and (D) pruning the subtree if the null hypothesis was not rejected for all pairs of leaves in the subtree or keeping the subtree if the null hypothesis was rejected for at least one pair of leaves. E) The goalkeeper’s estimated context tree was then compared with the penalty taker’s tree using the distance as in Balding2009. The scale shows the possible distances and the corresponding trees in a simplified representation. The red lines indicate branches that should have been pruned during the context tree estimation, and the blue lines indicate branches that should have been spared.
• Figure 3: Processing Pipeline A) Sliding Windows, context trees were estimated for 300 trial windows with 50 trials of displacement. Then, the distances between the penalty taker’s tree and each goalkeeper’s estimated tree were calculated per window and electrode. Top numbers indicate the window’s index. The colorbar shows the distance-color mapping. Mean and standard deviation of the success rate are presented at the bottom of each topo-plot. The time evolution suggests an inverse relationship between the distances and the success rates. B) Linear Mixed Effects Model, success rate as a function of Balding’s distance. The model was used to evaluate the predictability of the success rate based on Balding’s distance. Each graph shows a single electrode highlighted in green. Each blue dot represents a median-based centroid of the distances paired with the median-based success rate centroid for one window (* $p<0.05$ and ** $p<0.01$). The absolute values of the correlations (r) are presented for each graph. The inverse relationship for specific electrodes indicates that the corresponding brain regions are enrolled in the structural learning of the penalty taker's sequence. C) Topological view of relevant electrodes Electrodes that present a significant correlation in the relationship between success rate and Balding's distance.
• Figure : Figure S1: ERP estimation method for signals with different lengths. A) Signals of a single electrode/goalkeeper at the time interval between the end of the feedback and the next button press (right alignment). The picture illustrates the different signal lengths given the variable response times across trials. B) To estimate the event-related potential (ERP) per goalkeeper, the $75\%$ percentile of the distribution of segment lengths was chosen as the ERPs final length. C) EEG signals with a length greater than the threshold were trimmed on the left, while signals with a length smaller than the threshold were flagged. D) The final ERP was calculated by using the traditional point-by-point average, but disregarding the flagged intervals of the smaller signals. As a consequence, the final ERP average has less variability closer to the button press
• Figure : Figure S2: Event-Related Potentials (ERPs) at the time interval between the end of the feedback and the next button press. Electrodes are identified by their respective labels in the 10-20 system. For each electrode, the left vertical axis shows the ERP amplitude per goalkeeper (gray). The right vertical axis shows the amplitude for the grand-average ERP (blue). Significant intervals ($p < 0.05$) of the grand-average ERPs are indicated by the blue shaded bars. Given the variability of the response times, the percentile labels 0.75, 0.50, and 0.25 indicate the percentage of participants with available signal (from the right to the left). For most electrodes, a significant negative deflection is verified in the time interval ranging between -400 and 0 ms.

Theorems & Definitions (1)

• Definition 2.1: context tree