Human emotional experiences vary widely in everyday life, yet they are systematically organized within a continuous affective space defined by valence and arousal. Although this circumplex organization has shaped decades of research, its neural implementation remains unclear. Here, we asked whether abstract emotion concepts are encoded using grid-like coordinate principles in the human brain. During fMRI, participants tracked trajectories through affective space conveyed through morphs of facial expressions and sequences of phrases. We found that ventromedial prefrontal cortex (vmPFC) encoded affective space with robust hexadirectional modulation characteristic of grid-like coding. This coordinate representation generalized across stimulus modalities, indicating abstraction beyond perceptual features. The grid-like signal reflected a metric organization of emotion concepts independent of overt behavior, whereas distance-related signals in vmPFC predicted trial-by-trial affective judgements. Together, these findings suggest that emotion knowledge is structured by a grid-like coordinate system in the human brain that can support predictive inference and flexible decision-making.

Fig 1. Experimental paradigm and behavior.

(A) Schematic of a single trial in the two versions of the task (phrase and face). Each trial began with a sequence of two phrases from the same setting, or a morph between two facial expressions of the same identity. After a jittered fixation period, participants had 3 s to indicate whether valence or arousal had increased, decreased, or stayed the same. The probed dimension was split evenly across trials and revealed only on the response screen, so participants could not anticipate which dimension would be queried while viewing the stimuli. The face image shown is an illustrative example similar to the experimental stimulus; the actual stimuli were drawn from the Karolinska Directed Emotional Faces database [54]. (B) Rating screens for one example phrase. After scanning, participants rated each phrase and facial expression on valence and arousal, followed by ratings of seven emotion categories, using continuous scales. (C) Group-average valence-arousal ratings. Each point represents one stimulus and is colored according to the emotion category with the highest rating. (D) Proportion of trials in which in-scanner judgments were consistent with the group-average post-scan ratings across four quantiles of absolute distance in the judged dimension for both valence (blue) and arousal (pink) comparisons. The dotted line indicates chance agreement. Points represent individual participants (n = 18 independent participants), and points from the same participant are connected by gray lines. Bars show group means with 95% confidence intervals. Numerical data are provided in S1 Data.

https://doi.org/10.1371/journal.pbio.3003795.g001

Grid-like coding of affective space in vmPFC

A map-like organization of affect entails that judgments of emotional stimuli are based on a Euclidean space in which emotion concepts are embedded [11]. Indeed, analyses of the post-scan ratings revealed a two-dimensional affective structure broadly similar across participants (S1–S3 Figs; S1 and S2 Tables), stimulus modalities (S4 Fig), and rating types, including direct ratings of valence and arousal (Fig 1C) as well as multidimensional scaling (MDS) of emotion category ratings (S5 Fig). Across these complementary measures, the dominant structure in our data resembled a circumplex, with positive categories (e.g., happy, surprise) clustering toward high valence, negative categories (e.g., fear, sadness, anger, disgust) toward low valence regions, and arousal increasing from calm to highly activating categories like surprise and fear. Because the MDS solution was derived independently of any prespecified valence-arousal axes, its convergence with the geometry recovered from direct valence-arousal ratings (Procrustes disparity = 0.20, permutation p < .0001) provides evidence for an integrated representation of the two affective dimensions.

To examine whether this structure aligned with choice behavior during the fMRI scan, we assessed the relationship between in-scanner judgments of valence and arousal change and the affective structure derived from post-scan ratings. We found that as the distance between stimuli in the judged dimension increased, so did the agreement between individual choices during scanning and the group-average post-scan ratings (logistic regression: β = 0.9137, SE = 0.1212, z = 7.540, 95% CI [0.6770, 1.1599], p < .0001; overall proportion of consistent choices: range = 50.96%–66.35%; median = 56.97%, IQR = 7.21%, chance = 33.33%; balanced consistency across increase, decrease, and no-change trials: median = 57.92%, IQR = 7.05%; Fig 1D). Such a relationship was not observed for distance along the non-judged dimension (β = −0.0560, SE = 0.0443, z = −1.264, p = .2063), nor was there an interaction between distances along the judged and non-judged dimensions (β = −0.0215, SE = 0.0510, z = −0.421, p = .6737). These behavioral results indicate that participants could reliably read out task-relevant changes along the valence and arousal dimensions during scanning in a manner consistent with the group-average post-scan ratings, validating the use of rating-derived coordinates for subsequent tests of grid-like structure in the fMRI data.

When participants viewed face morphs and phrase sequences, transitions between emotion concepts could be represented as movement through affective space. If the basis for representing this space is implemented in grid-like population codes, trajectories aligned with a common grid orientation should evoke similar fMRI patterns separated by 60° rotational symmetry. To test this prediction, we estimated putative grid orientation (S3 Table) and quantified hexadirectional modulation using a cross-validated approach in which orientation was estimated and tested on independent data from two pairs of functional runs (see Materials and methods). Examining signals in a region of vmPFC previously identified as exhibiting 6-fold symmetry [4], we distinguished trials with trajectories aligned with the estimated grid axes and computed multivoxel pattern similarity across trial pairs (Fig 2B and 2C).

Consistent with grid-like coding, response patterns in portions of vmPFC (Fig 2A) were more similar for trajectory pairs aligned with the same 60° periodic grid orientation (r = .0801, SD = .0262) than for those offset by 30° (r = .0729, SD = .0262; Δr = 0.0072, 95% CI [0.0031, 0.0115], d = 0.7820, p = .0014; Figs 2D, 2F and S6). This effect was specific to 6-fold periodicity characteristic of grid-like coding (6-fold versus other folds: Δr = 0.0088, 95% CI [0.0046, 0.0133], d = 0.9223, p = .0005; Fig 2E). Simulations showed the magnitude of the pattern similarities observed for the 6-fold effect corresponds to a signal-to-noise ratio of 0.1535 (S7 Fig). We did not observe grid-like responses in the entorhinal cortex (Δr = −0.0002, 95% CI [−0.0027, 0.0022], d = −0.0452, p = .5594; Fig 2D). Given the established evidence for grid cells and grid-like fMRI signals in this region across spatial and abstract domains [3,4,26], this result was unexpected. However, temporal signal-to-noise and mean signal in this region were lower (S8 Fig), which may have limited sensitivity to detect subtle effects.

As a point of comparison, we compared the modulation of response patterns in the vmPFC to those in the hippocampus, which has been found to exhibit a map-like structure [25], but not grid-like coding. Consistent with our account that the hippocampus represents specific concepts embedded within affective space, rather than the low-dimensional grid-like codes as expressed in vmPFC [21], we observed no comparable grid-like coding effect in the hippocampus (Δr = 0.0011, 95% CI [−0.0008, 0.0034], d = 0.2498, p = .1568; Δr_{hippocampus –} Δr_vmPFC = −0.0060, 95% CI [−0.0105, −0.0021], d = −0.6592, p = .0042; Fig 2D).

Having established that a grid-like code of affective space was present in vmPFC, we next conducted a series of tests to assess the sensitivity of this effect to our analytical procedures. Because the number of trial pairs contributing to the aligned and misaligned conditions differed, with an average of 1,514.2 aligned trial pairs (SD = 234.1) and 1,363.8 misaligned pairs (SD = 97.2) per cross-validation fold across participants, we performed a subsampling analysis to match the number of pairs between conditions. Averaged across 10,000 subsampling iterations, the 6-fold modulation remained unchanged (Δr = 0.0072, 95% CI [0.0032, 0.0115], d = 0.7827, p = .0014). Assessing other aspects of the analysis, we observed evidence of 6-fold modulation at multiple local spatial scales (S9 Fig) and localizations of vmPFC from past studies [4,6] (S4 Table). Exploratory searchlight and univariate mapping revealed effects throughout the vmPFC and other regions that have been found to exhibit 6-fold modulation in prior studies, including multiple regions of the default network (S10–S12 Figs and S5 Table).

To account for potential confounds that could increase pattern similarity between aligned trials, including affective features, emotion category, trajectory distances, face identity, phrase setting, and behavioral choice (S13–S15 Figs), we conducted a control analysis incorporating these variables as nuisance regressors. The 6-fold modulation in vmPFC remained robust in this analysis (Δr = 0.0070, 95% CI [0.0033, 0.0109], d = 0.8416, p = .0008; Fig 3B). Because aligned pairs more often shared the same start category, end category, and stimulus identity or setting (S15 Fig), we also repeated the analysis after directly matching aligned and misaligned pairs on these factors. The 6-fold effect was again present in the matched subsample (Δr = 0.0062, 95% CI [0.0022, 0.0106], d = 0.6818, p = 0.0026; Fig 3B).

To test the robustness of the hexadirectional effect in vmPFC to alternative estimates of affective space, we repeated the analysis using trajectory angles derived from multidimensional scaling of emotion category ratings (Fig 3D). The grid-like effect in vmPFC was replicated using this alternative geometry (Δr = 0.0051, 95% CI [0.0008, 0.0096], d = 0.5359, p = .0203; 6-fold versus other folds: Δr = 0.0049, 95% CI [0.0006, 0.0094], d = 0.5091, p = .0245), and remained evident after controlling for potential confounds (Δr = 0.0046, 95% CI [0.0004, 0.0090], d = 0.4935, p = .0288) and matched subsampling (Δr = 0.0058, 95% CI [−0.0003, 0.0124], d = 0.4239, p = .0476; Fig 3E). Notably, hexadirectional modulation was present in vmPFC when affective space was defined directly from self-reported valence and arousal and when it was estimated using data-driven multidimensional scaling, indicating that grid-like representation in vmPFC is robust to independent methods of defining affective space.

Generalization of grid-like codes across facial and linguistic stimuli

Dimensional theories of affect propose that a common cognitive structure underlies judgements about the meaning of words, facial expressions, and subjective experience more generally [11]. If vmPFC implements a grid-like basis for affective space, its orientation should generalize across stimuli that differ in perceptual features but have similar meaning and underlying geometry. Estimating grid orientation from behavioral ratings of phrase stimuli and testing on faces revealed robust 6-fold modulation in vmPFC (Phrase →Face: Δr = 0.0117, 95% CI [0.0013, 0.0224], d = 0.5061, p = .0234). The reciprocal analysis produced comparable results (Face →Phrase: Δr = 0.0072, 95% CI [0.0013, 0.0135], d = 0.5495, p = .0196; Fig 3C). When affective space was defined using MDS of category ratings, cross-modal generalization was observed in the phrase-to-face direction (Δr = 0.0123, 95% CI [0.0038, 0.0218], d = 0.6242, p = .0096), but not from faces to phrases (Δr = 0.0002, 95% CI [−0.0057, 0.0063], d = 0.0187, p = .4693; Fig 3F). These results suggest that hexadirectional coding in vmPFC may reflect a modality-invariant representation of affective space, although cross-modal generalization was not consistently observed across alternative definitions of the space.

Grid-like coding of affective space in vmPFC is distinct from decision-related signals

The ventromedial prefrontal cortex is widely implicated in encoding decision-related variables [27], such as the subjective value of an action [28–30] emotional value [31], and decision confidence [32,33]. To determine whether the observed grid-like signals in vmPFC are linked to decision processes used to evaluate stimuli, we examined how grid-like signals relate to the consistency of judgments made during and after scanning. If the hexagonal modulation we observed resulted from a well-established map of emotion concepts, then its expression should be independent of behavioral choice. Alternatively, if the hexagonal modulation resulted from signals more closely tied to task performance (e.g., a mapping of task sets or action values), then hexagonal modulation should be more closely related to behavioral performance. Comparisons of hexadirectional modulation on trials in which judgements were consistent with consensus ratings and those that were inconsistent revealed minimal differences (Δr_consistent − Δr_inconsistent = 0.0049, 95% CI [−0.0031, 0.0126], d = 0.2880, p = .1220; BF10 = 0.4477; Fig 4B). This finding suggests that the strength of the grid-like signal we observed was not closely tied to choice variables in the task.

Given the strong relationship between affective distance and behavioral consistency (Fig 1D), we reasoned that vmPFC responses might encode affective distance separately from grid-like codes. We therefore fit multivariate encoding models predicting vmPFC responses using Euclidean distance in a two-dimensional affective space, which might correlate with decision confidence, the value of a correct decision, or the valence of stimuli (Fig 4A). Consistent with past work identifying value and decision-related variables in vmPFC [27,28,31], leave-one-participant-out cross-validation revealed encoding models using two-dimensional affective distance predicted vmPFC activity (r = 0.0127, 95% CI [0.0050, 0.0195], d = 0.7994, 38.09% of the noise ceiling, p = .0021) and outperformed less complex models based on valence or arousal alone (Δr = 0.0097, 95% CI [0.0054, 0.0141], d = 1.0296, p = .0001; Fig 4D). This finding suggests that vmPFC activity carries information about affective distance that might be relevant for decision-making.

If representations of affective distance encoded in vmPFC were used to guide judgments made during the experiment, the predictions of encoding models should be systematically related to behavioral agreement across participants. That is, stimuli that varied more along dimensions of valence and arousal should be more consistently evaluated by participants and produce larger encoding model responses. To test this prediction, we computed trial-wise pattern expression scores by projecting the multivoxel activity onto weights from the distance encoding model. Greater expression of affective distance in vmPFC was associated with more consistent behavioral ratings (β = 0.0788, SE = 0.0330, z = 2.388, 95% CI [0.0156, 0.1430], p = .0169, mixed-effect logistic regression; Figs 4E and S16). This relationship remained after controlling for stimulus modality (β = 0.1048, SE = 0.0532, z = 1.969, 95% CI [0.0029, 0.2089], p = .0489) and did not differ between face and phrase trials (β = −0.0523, SE = 0.0644, z = −0.812, 95% CI [−0.1786, 0.0754], p = .4169), indicating that the association between distance-related activity and behavior did not depend on stimulus type. Moreover, hexadirectional modulation did not vary with trajectory distance (alignment × distance interaction: β = −0.0006, 95% CI [−0.0018, 0.0005], d = −0.2384, p = .8327; Fig 4C), indicating that the distance-related signal associated with behavior is not a simple consequence of stronger grid-like patterning. Together, these findings indicate vmPFC maintains a grid-like representation of affective space and separately encodes distance-related signals more closely related to behavioral judgments.

Participants

We recruited 20 participants from the Atlanta, GA area. Participants were healthy adults fluent in English, with normal or corrected-to-normal vision, and no contraindications for MRI. The study was approved by the Institutional Research Board at Emory University (Protocol Number: 2025P010826), and all participants provided written informed consent prior to participation. All procedures were conducted according to the principles of the Declaration of Helsinki. Data from 18 participants (13 females; age: mean = 39.56, SD = 13.19, range = 21–60 years) were analyzed. One participant withdrew due to claustrophobia, and fMRI data from one participant were excluded due to excessive head motion (maximum framewise displacement >10 mm). Another participant did not perform the post-scan rating task and was therefore excluded from analyses using individual ratings. For the primary one-tailed paired sample comparisons, a sample size of n = 18 at α = 0.05 has 70% power to detect a true effect size of d = 0.53, 80% power for d = 0.61, and 90% power for d = 0.72.

Experimental paradigm

Behavioral data analysis

MRI data acquisition

Gradient-echo echo-planar imaging (EPI) data were acquired on a 3T Siemens MAGNETOM Prisma scanner with 32-channel parallel imaging located at the Facility for Education and Research in Neuroscience at Emory University. T2*-weighted functional data were acquired using a multiband multi-echo EPI sequence with the following parameters: repetition time (TR) = 1,250 ms; echo times (TEs) = 15 ms/ 37.16 ms/ 59.3 ms; flip angle = 55°; number of slices = 56 interleaved; voxel size = 2.5 mm isotropic; field of view = 210 mm; multiband acceleration factor = 4; bandwidth = 2,381 Hz/pixel. The slice angle was set to 10° relative to the anterior-posterior commissure line to reduce signal loss in basal frontal regions [62,63]. A pair of fieldmap scans with reversed phase-encoding directions (posterior–anterior and anterior-posterior) was acquired to enable correction of susceptibility-induced distortions in the functional images. Each fieldmap acquisition used the same voxel size, slice coverage and field of view as the functional scans, with TR = 7,220 ms, TE = 73 ms, and flip angle = 90°.

To facilitate normalization of EPIs to standard space, structural images were acquired using T1 MPRAGE: TR = 1,900 ms; TE = 2.26 ms; flip angle = 9°; number of slices = 192; voxel size = 1 mm isotropic; field of view = 256 mm × 256 mm; GRAPPA acceleration factor = 2; bandwidth = 199 Hz/pixel.

MRI preprocessing

Results included in this manuscript come from preprocessing performed using fMRIPrep 25.0.0 [64,65] (RRID:SCR_016216), which is based on Nipype 1.9.2 [66,67] (RRID:SCR_002502).

Definition of regions of interest

Masks of the hippocampus and entorhinal cortex were obtained from the Julich-Brain Cytoarchitectonic Atlas [82]. The vmPFC mask was obtained by drawing a sphere with radius of 5 mm around the peak voxel of the hexadirectional effect reported by Constantinescu and colleagues [4] (MNI peak voxel coordinates: [6, 44, − 10]; 69 voxels). Additional vmPFC peak coordinates reported in prior studies ([6, 46, − 10] [6], [2, 28, − 20] [4], and [−8, 42, 0] [6]) were also examined using identical 5-mm-radius spherical regions of interest. For searchlight visualization within vmPFC, a larger anatomical mask including Brodmann areas 10, 11, 14, 24, 25, and 32 was defined based on a multimodal cortical parcellation of Glasser and colleagues [83]. We also conducted exploratory analyses in additional regions previously implicated in grid-like coding; the coordinates and corresponding effects are reported in S6 Table.

General linear models

First-level general linear models were applied to preprocessed fMRI data using SPM 12 software (Wellcome Trust Centre for Neuroimaging, UK). For each participant, a separate model was estimated that included: (1) one regressor per trajectory of facial expressions or phrases, modeled as a 5 s boxcar function, (2) a regressor for button presses, modeled using a stick function, and (3) a 3-s boxcar regressor for the response period. All regressors were convolved with the canonical hemodynamic response function provided in SPM. To account for head motion, 24 nuisance regressors derived from spatial realignment (i.e., three translations, three rotations, and their first- and second-order temporal derivatives) were included, along with constant terms modeling the mean signal for each imaging session.

Multivariate hexadirectional modulation analysis

To test whether voxel-wise activity patterns reflected 6-fold symmetry within each region of interest or searchlight, we performed a multivariate hexadirectional modulation analysis using a cross-validated procedure following Bao and colleagues [6]. The multivariate approach was used as our primary analysis because it preserves fine-grained information distributed across voxels within a region of interest [84], which can be missed by conventional univariate analysis that average across voxels. For each participant, the data were split into two halves of runs, one for estimating grid orientation (training set) and the other for testing modulation effects (test set). In the training set, we conducted a univariate analysis regressing cos(6θ) and sin(6θ) against single-trial βs estimated for the trajectories. The resulting regression coefficients (β_cos and β_sin) were averaged across voxels within each region. The participant-specific grid orientation φ was then computed as

In the test half, trials were classified as aligned if their trajectory angle θ was within ± 7.5° of φ (mod 60°). We computed the Pearson correlation between all pairs of response patterns for each region. Correlations were averaged across pairs of trials that were both aligned (aligned-aligned pairs, angular difference within 0° ± 15° mod 60°) and mispatched pairs (aligned-misaligned pairs with an angular difference within 30° ± 15° mod 60°). Comparisons where both trials were misaligned were excluded to avoid low-signal, high-noise pairs [6].

Trajectory angles (θs) used in the analysis were obtained either from ratings of valence and arousal or coordinates in a two-dimensional MDS solution derived from pairwise cosine distances of emotion category ratings. To obtain stable MDS-derived angles, we repeated the MDS procedure with 10 random initialization seeds. One solution was selected as the reference, and all remaining solutions were aligned to it using Procrustes analysis. Trajectory angles were then computed separately for the reference and the aligned solutions, and the resulting angles were averaged across the 10 MDS solutions to yield estimates of θ.

To quantify the aligned-misaligned differences in pattern similarity while controlling for potential confounds, we fit a linear regression model in which pairwise pattern similarity was predicted by alignment (1 = aligned, 0 = misaligned) together with 11 nuisance regressors. These regressors captured absolute differences between the two trajectories in starting valence and arousal, ending valence and arousal, distances along the valence and arousal dimensions, and Euclidean distance in the two-dimensional space, as well as whether the pair shared the same starting emotion category, ending emotion category, behavioral choice, and stimulus context (face identity or phrase setting). Including these variables allowed us to test whether hexadirectional modulation remained after accounting for affective features, emotion category, task structure and behavior.

Because aligned trial pairs more often shared the same starting category, ending category, and face identity or phrase setting (S15 Fig), we performed a complementary matched-subsampling analysis. Within each participant and cross-validation fold, aligned and misaligned pairs were grouped according to every combination of shared starting category, shared ending category, behavioral choice, and stimulus context. For each combination, we randomly sampled without replacement an equal number of pairs from the two conditions, determined by the smaller available count. This procedure was repeated 5,000 times, producing samples of the aligned and misaligned conditions with identical numbers of pairs for each combination of the four binary variables.

To assess the specificity of the 6-fold modulation, we repeated the same procedures for alternative periodicities of 3-, 4-, 5-, 7-, and 8-fold symmetry, corresponding to angular symmetries spaced every 120° (3-fold), 90° (4-fold), 72° (5-fold), 51.4° (7-fold), and 45° (8-fold). For each periodicity k, φ was estimated using the same procedure as for the 6-fold model but applying the corresponding periodic factor. Trials were first classified as aligned if their trajectory angle θ was within ± (360°/(8k)) of φ (mod 360°/k). For each aligned trial, we computed correlations with all other trials and labeled each pair as aligned-aligned or aligned-misaligned depending on whether the angular difference between trajectories was near 0° or half of the symmetry spacing (360°/(2k)).

Statistical significance of group-level effects (both against 0 and for differences between regions or folds) was assessed using sign-flipping with 10,000 iterations. For each iteration, participant-specific aligned-misaligned contrast values or regression coefficients for alignment were randomly sign-inverted to generate a null distribution of group means. One-tailed p-values were then computed as the proportion of permuted group means that were greater than or equal to the observed group mean. To estimate uncertainty in group-level effects, we performed nonparametric bootstrap resampling (10,000 iterations) across participants, producing a distribution of group means to derive 95% confidence intervals using the percentile method [85]. Effect size was then calculated by dividing the observed group mean by the standard deviation of the bootstrap distribution, multiplied by the square root of the sample size.

Simulation of single-trial multivoxel response patterns

To estimate the amount of variance within and between subjects in the pattern similarities, we conducted a simulation to generate single-trial patterns comprising three components: (1) a stable subject-specific pattern that was constant across trials, (2) a grid-like signal component that varied with trajectory direction, and (3) residual trial-to-trial multivoxel variation reflecting other neural and non-neural processes. Specifically, for subject s, trial t, and voxel v, the observed pattern of activity Y is:

For each participant, the pattern in the stable component B was derived from the empirical mean multivoxel single-trial beta pattern across trials, and the pattern in the grid-like signal component g was derived from the empirical voxelwise hexadirectional modulation amplitude estimated from the quadrature analysis ( where c_v and s_v are the fitted coefficients for cos(6θ) and sin(6θ), respectively). The amplitude of signals related to both spatial patterns were normalized within participants, allowing their overall contribution to the simulated data to be scaled by the parameters b and A. Subject-specific values of b were drawn from a normal distribution , such that μ_b determined the mean amplitude of the stable component across participants and σ_b determined its between-subject variability. The remaining trial-to-trial variation (ϵ) was generated from a multivariate normal distribution with the empirical cross-voxel covariance estimated after removing the fitted grid-like signal from the single-trial beta patterns.

We analyzed the simulated data using the same cross-validated procedure as the empirical data. We jointly optimized μ_b, σ_b, and A to match four empirical values including the mean raw pattern correlation, the aligned-misaligned difference in pattern correlation, the between-subject SD of the raw correlations, and the between-subject SD of the aligned-misaligned difference. For each measure, the objective was the squared difference between the simulated and empirical value. Optimal parameters were estimated using a multiobjective goal-attainment procedure implemented with MATLAB’s fgoalattain.

Stability of grid orientation estimates

To assess the stability of grid orientation estimates across subsets of the data, we have performed a bootstrap analysis to compare orientation estimates across complementary split pairs. For each bootstrap iteration, we divided trials within a split into 10 contiguous temporal blocks and resampled these blocks with replacement to preserve dependence among neighboring trials. Grid orientation was then re-estimated using data from the sampled blocks. This procedure was repeated 5,000 times, producing a bootstrap distribution of the grid orientation estimates for each split and participant. Orientation stability was assessed for complementary split pairs by computing the difference between values from the two bootstrap distributions. Because the orientation estimates were defined modulo 60° for a 6-fold model, differences were wrapped to the range [−30°, 30°].

Univariate hexadirectional modulation analysis

As a complementary test, we also assessed hexadirectional modulation using a univariate approach. For each participant, we specified a first-level general linear model including: (1) a 5-s boxcar regressor for the stimulus period, with cos(6θ) and sin(6θ) entered as parametric modulators, where θ was the trajectory angle through affective space, (2) a stick-function regressor for button presses, and (3) a 3-s boxcar regressor for the response period. All task regressors were convolved with the canonical hemodynamic response function. The model additionally included 24 nuisance regressors for head motion, derived from spatial realignment (i.e., three translations, three rotations, and their first- and second-order temporal derivatives), together with session-specific constants. The cos(6θ) and sin(6θ) coefficients were jointly tested using an F contrast to identify voxels with large linear combination of the two modulators. For group inference, each participant’s F-statistic map was transformed to a Z-map [86], which was then entered into a second-level one-sample t test. Because the first-level F-statistic depends on estimates of variance of the regression coefficients, biased estimation of variances in autocorrelated fMRI data can lead to inflated F- and transformed Z-statistics [4]. We therefore treated this univariate analysis as descriptive and did not use it to localize regions showing grid-like coding effect.

Distance encoding

We built multivariate encoding models to test whether different distance measures capture variation in BOLD signal in vmPFC. Partial least squares regression (SIMPLS algorithm [87]) was used to predict multivoxel patterns of single-trial β estimates from distances in a cross-validated way. Encoding models were trained using data from all participants except one and tested on held-out participants (leave-one-participant-out cross-validation). For each training fold, the predictor matrix X (size 3,536 × 1) contained the distance values for each trial, and the outcome matrix Y (size 3,536 × 69) contained the corresponding single-trial β estimates for each voxel in vmPFC, concatenated across the 17 training participants (208 trials per participant × 69 voxels). The trained model was then used to predict responses in the held-out participant from their own distance values (test X, size 208 × 1), yielding predicted responses that were compared against the held-out participant’s observed data (test Y, size 208 × 69). Encoding performance was quantified as the Pearson correlation between the predicted and observed multivoxel single-trial responses in the test participant, computed across trials and then averaged across voxels within the region of interest to yield a single performance value per region and participant. To estimate an upper bound on model performance, the noise ceiling was calculated by correlating each held-out participant’s observed time series of brain activity with the mean time series of all other participants. One noise ceiling value was obtained for each region by averaging across held-out participants and voxels. Group-level inference for encoding model performance was conducted using the same sign-flipping and bootstrap resampling procedures as above (see ‘Multivariate hexadirectional modulation analysis’).

Relationship between distance-related brain activity and choice behavior

To test whether the distance-related brain pattern predicts choice behavior in the scanner, we computed a pattern expression score for each trial of the held-out participant. Each participant’s observed multivoxel activity pattern (Y_test) was projected onto the distance-encoding weights (β_d) obtained from other participants (Y_test ⋅ β_d), producing one value reflecting how strongly the response pattern on each trial expressed the distance-coding axis. These trial-wise pattern expression scores were then entered into a mixed-effects logistic regression model to predict agreement between the judgements made in the scanner and the post-scan ratings (see ‘Behavioral data analysis’ for how consistency measures were obtained). The model included random intercepts and slopes for each participant to account for individual variability. In the control analysis, we tested whether this relationship was driven by lower-level stimulus properties by including stimulus modality (face versus phrase) and its interaction with the pattern expression score as fixed effects in the model.

All analyses were performed using CanlabCore Tools (https://github.com/canlab/CanlabCore) and SPM 12 (https://www.fil.ion.ucl.ac.uk/spm/software/spm12/), as well as custom MATLAB (R2024a), python (3.9.18) and R (4.4.1) code.

S1 Fig. Agreement between individual and group-average post-scan ratings.

(A) First-order agreement. Pearson correlations between individual and group-average ratings for face and phrase stimuli. (B) Second-order agreement. Spearman correlations between individual and group-average dissimilarity matrices derived from valence-arousal ratings (Euclidean distance) and emotion category ratings (cosine distance) for face and phrase stimuli. Black circles and error bars denote the mean and 95% confidence interval. Each point represents one subject (n = 18 independent subjects). All numerical data are provided in S5 Data.

https://doi.org/10.1371/journal.pbio.3003795.s001

(TIF)

S2 Fig. Valence and arousal ratings for each participant.

Each subplot shows data from one participant. Each point represents one stimulus (including both faces and phrases) and is colored according to the emotion category with the highest rating.

https://doi.org/10.1371/journal.pbio.3003795.s002

(TIF)

S3 Fig. Multidimensional scaling (MDS) of emotion category ratings for each participant.

Each subplot shows the two-dimensional MDS solution for data from one participant. Each point represents one stimulus (including both faces and phrases) and is colored according to the emotion category with the highest rating.

https://doi.org/10.1371/journal.pbio.3003795.s003

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S4 Fig. Low-dimensional embeddings of emotions conveyed by face and phrase stimuli.

(A) Group-average valence-arousal ratings of face and phrase stimuli. Faces and phrases exhibited highly similar structures (Gromov–Wasserstein distance = 0.19, permutation p < .0001). (B) Multidimensional scaling solutions derived from group-average category ratings for face and phrase stimuli, aligned to the valence-arousal space in A using Procrustes analysis. Faces and phrases exhibited highly similar structures in category ratings (Gromov–Wasserstein distance = 0.01, permutation p < .0001). Each point represents one stimulus and is colored according to the emotion category with the highest rating.

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S5 Fig. Multidimensional scaling (MDS) of emotion category ratings.

(A) Scree plot of Kruskal stress for MDS solutions on cosine distances between group-average ratings of emotion categories. Note the elbow at two dimensions (Kruskal stress = 0.16). (B) Two-dimensional MDS solution aligned to the valence-arousal space (see Fig 1C in the main text) using Procrustes analysis (disparity = 0.20, permutation p < .0001). Each point represents one stimulus and is colored according to the emotion category with the highest rating.

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S6 Fig. Power analysis for the observed grid-like coding effect in ventromedial prefrontal cortex.

Power curve was estimated using a repeated random subsampling analysis. For each target sample size, we randomly sampled whole participants without replacement 10,000 times, performed the group-level one-sample test for each subset, and calculated power as the proportion of subsets that exhibited an effect at a < 0.05. This analysis revealed that the observed grid-like coding effect in the independently defined ventromedial prefrontal cortex were detected with a power exceeding 99.8% by a sample size as small as 12 participants.

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S7 Fig. Simulation of the effects of stable signal and grid-like signal strength on pattern correlations.

The panels show (A, D) the between-subject standard deviation (SD) of the aligned correlations, (B, E) SD of the aligned-misaligned difference, and (C, F) mean aligned-misaligned difference as a function of the between-subject SD in the strength of the stable component of the simulated single-trial beta patterns. Colored lines indicate different grid-like signal amplitudes in A–C, and different mean amplitudes of the stable component in D–F. The μ_b was held at its fitted value in A–C, and A was held at its fitted value in D–F. Solid lines and points show means across 500 replicates of the simulation, shaded regions show the 2.5th–97.5th percentiles across simulations, and dashed lines indicate the corresponding empirical values. Optimization of parameters to match the empirical pattern similarity and variability measures yielded μ_b = 0.44, σ_b = 0.067, and A = 0.25, which produced a mean aligned-misaligned difference of 0.0072 (95% SI [0.0029, 0.0118]; empirical Δr = 0.0072), an SD of the aligned correlations of 0.0294 (95% SI [0.0207, 0.0393]; empirical SD = 0.0262), and an SD of the aligned-misaligned difference of 0.0110 (95% SI [0.0073, 0.0152]; empirical SD = 0.009). At these parameter values, the grid-signal-to-noise ratio was 0.1535 on average across voxels and participants.

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S8 Fig. Whole-brain maps of signal quality.

(A) Group temporal signal-to-noise ratio (tSNR) map. For each participant and run, voxelwise tSNR maps were created as the mean preprocessed BOLD signal divided by its standard deviation over time. The parametric map shows tSNR values averaged across runs and participants (n = 18 independent participants). (B) Mean BOLD signal across participants. For each participant, voxelwise signal intensity maps were created by averaging preprocessed BOLD timeseries across all runs. The parametric map shows the group mean of signal intensity divided by the standard deviation across participants (n = 18 independent participants). Note that the ventral surface of the brain (including entorhinal cortex outlined in yellow) exhibits lower tSNR and reduced mean signal intensity.

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S9 Fig. Grid-like coding effect across sphere sizes in ventromedial prefrontal cortex.

Each point represents the pattern similarity difference between aligned-aligned and aligned-misaligned trial pairs for one subject (n = 18 independent subjects). Boxplots show the median and interquartile range; whiskers extend to 1.5× the interquartile range. Black circles and error bars denote the mean and 95% confidence interval. Across radii ranging from 3 mm (17 voxels) to 12 mm (901 voxels), the direction of effect was consistent, although its magnitude varied. We observed a “small” effect at a radius of 3 mm (d = 0.3826), “medium” effects at radii of 4 mm (35 voxels; d = 0.7687) and 5 mm (69 voxels; d = 0.7820), and decreasing effect sizes as the searchlight radius increases to 12 mm (d = 0.4116; Δr_5mm − Δr_12mm = 0.0042, 95% CI [0.0004, 0.0082], Δd = 0.4973, p = .0255). Numerical data are provided in S6 Data.

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S10 Fig. Voxelwise effect sizes for searchlight hexadirectional modulation analysis within ventromedial prefrontal cortex.

Searchlight maps (radius = 5 mm) reflect Cohen’s d values of pattern similarity differences between aligned and misaligned trial pairs across participants (n = 18 independent participants). These unthresholded maps are used to characterize the spatial distribution and magnitude of the hexadirectional effect within an anatomically defined mask of medial prefrontal cortex, comprising Brodmann areas 10, 11, 14, 24, 25, and 32. Warmer colors indicate greater similarity for aligned relative to misaligned trial pairs. Only voxels with positive effect sizes are displayed.

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S11 Fig. Whole-brain searchlight map of hexadirectional modulation.

Parametric t-statistic maps show effects of hexadirectional modulation derived from an exploratory whole-brain searchlight analysis (radius = 5 mm). These maps are used to display regions showing evidence of hexadirectional modulation throughout the brain, so voxels are thresholded at p < .01 (one-tailed; uncorrected), with a minimum cluster extent of k ≥ 20 contiguous voxels for visualization. Warm colors reflect greater similarity for aligned relative to misaligned trial pairs. No voxels survived false discovery rate correction at a threshold of q < .05. Note that this analysis was exploratory as the sample size was chosen to test grid-like coding within independently defined, theoretically motivated regions of interest rather than to localize effects across the whole brain.

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S12 Fig. Group-level univariate hexadirectional modulation.

Parametric t-statistic maps show the consistency across participants of the subject-level F-test assessing any linear combination of the cos(6θ) and sin(6θ) parametric modulators. Voxels were displayed at a false discovery rate (FDR)-corrected threshold of q < .05 with a minimum cluster extent of k = 10 contiguous voxels for visualization. Warm colors indicate stronger evidence for consistent hexadirectional modulation across participants. This univariate analysis revealed hexadirectional modulation in ventromedial prefrontal cortex (vmPFC). Multivariate analysis on a sphere (radius = 5 mm) centered on the peak vmPFC coordinate (MNI [3.5, 61.5, −18.5]; t = 8.8971, Z = 5.3596) showed that the pattern similarity difference between aligned and misaligned trial pairs was positive across participants (Δr = 0.0048, 95% CI [0.0001, 0.0096], d = 0.4618, p = .0354). Patterns in the subject-level Z maps exhibited positive similarity to the publicly shared group-level Z map from Constantinescu and colleagues (2016) (mean cosine similarity = 0.1633, t₁₇ = 5.150, p < 0.0001).

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S13 Fig. Distribution of stimulus properties across trajectory angles.

(A) Proportion of highest-rated emotion categories across the four 15° bins of the trajectory angle (θ mod 60°) for starting and ending face and phrase stimuli. Bars show the relative frequency of categories appearing within each bin. (B) Distributions of valence and arousal ratings across the same four angle bins for starting and ending face and phrase stimuli. Each point represents one stimulus. (C) Heatmaps showing the number of trials associated with each face identity or phrase setting across the four angle bins.

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S14 Fig. Distributions of trajectory angles and distances.

(A) Distribution of trajectory angles across four 15° bins (θ mod 60°), for affective space derived from valence-arousal ratings. (B) Distribution of trajectory distances. (C) Trajectory distances plotted across four 15° bins of the trajectory angle. Each point represents an individual trajectory.

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S15 Fig. Comparison of task-related factors between aligned and misaligned trial pairs in analysis on ventromedial prefrontal cortex.

(A) Proportion of trial pairs sharing the same start category, end category, face identity or phrase setting, and behavioral choice. (B) Mean pairwise differences in start valence, start arousal, end valence, end arousal, trajectory distance, valence change, and arousal change. Values were calculated at the participant level after averaging within cross-validation folds. Boxplots show the median and interquartile range; whiskers extend to 1.5 × the interquartile range. Black circles and error bars denote the mean and 95% confidence interval. Each point represents one subject and points from the same subjects are connected by gray lines. All numerical data are provided in S7 Data.

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S16 Fig. vmPFC pattern expression predicts behavioral consistency.

Thin blue lines show logistic regression curves fit separately for each participant. The thick black line shows the fixed-effect prediction from the mixed-effects logistic regression model. The dotted horizontal line indicates chance.

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S17 Fig. Grid-like coding effect in ventromedial prefrontal cortex using trajectory angles derived from individual ratings.

Each point represents the pattern similarity difference between aligned-aligned and aligned-misaligned trial pairs for one subject (n = 18 independent subjects) and points from the same subjects are connected by gray lines. Boxplots show the median and interquartile range; whiskers extend to 1.5× the interquartile range. Black circles and error bars denote the mean and 95% confidence interval. We did not observe a reliable 6-fold effect in vmPFC using trajectory angles obtained from participants’ own ratings (Δr = 0.0041, 95% CI [−0.0008, 0.0090], d = 0.4034, p = .0617; 6-fold vs. other folds: Δr = 0.0041, 95% CI [−0.0005, 0.0086], d = 0.4293, p = .0527). Numerical data are provided in S8 Data.

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S1 Data. Behavioral consistency across valence and arousal distances.

Individual participant-level proportions of in-scanner judgments consistent with group-average post-scan ratings across quartiles of absolute distance in the judged dimension. Related to Fig 1D.

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S3 Data. Grid-like coding effects for affective space after controlling for confounds and across alternative analyses.

Individual participant-level pattern similarity measures from covariate-adjusted analyses, balanced subsampling, cross-modal generalization, and analyses based on a multidimensional scaling solution. Related to Fig 3B, 3C, 3E, and 3F.

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S4 Data. Relationship between grid-like coding, distance-related signals, and behavioral consistency.

Individual participant-level pattern similarity differences, encoding model performance, and proportions of consistent behavioral judgments across the corresponding analysis conditions. Related to Fig 4B–4E.

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