A transient high-dimensional geometry affords stable conjunctive subspaces for efficient action selection

Abstract

Flexible action selection requires cognitive control mechanisms capable of mapping the same inputs to different output actions depending on the context. From a neural state-space perspective, this requires a control representation that separates similar input neural states by context. Additionally, for action selection to be robust and time-invariant, information must be stable in time, enabling efficient readout. Here, using EEG decoding methods, we investigate how the geometry and dynamics of control representations constrain flexible action selection in the human brain. Participants performed a context-dependent action selection task. A forced response procedure probed action selection different states in neural trajectories. The result shows that before successful responses, there is a transient expansion of representational dimensionality that separated conjunctive subspaces. Further, the dynamics stabilizes in the same time window, with entry into this stable, high-dimensional state predictive of individual trial performance. These results establish the neural geometry and dynamics the human brain needs for flexible control over behavior.

Introduction

Integration of our abstract knowledge, goals, contexts, or internal states to select pathways for specific actions is at the basis of intelligent, adaptive behavior. For example, most of the time, we might respond to notifications of a text thread from a friend immediately. However, on occasion, the same inputs might be routed to different actions (e.g., pressing mute) depending on the prevailing context (e.g., when in a meeting with a colleague). Therefore, as this case illustrates, flexible behavior depends on our capacity to map similar or even identical inputs to many alternative and distinct actions, as the situation and our internal goals demand. This capacity is termed cognitive control. A neural representation that enables this type of context-sensitive, nonlinear mapping is a control representation^1,2,3. Thus, understanding the nature of control representations is fundamental for constraining theory and advancing our understanding of adaptive and flexible action selection.

Encoding of control representations is constrained by the dynamic geometrical properties of neural trajectories that unfold nonlinearly during task execution. To achieve reliable and flexible context-sensitivity for cognitive control, one critical challenge for neural systems is to assemble control representations that maintain selectivity, separability, and stability of task-critical information in a large and dynamic neural state space^4,5,6,7,8. Specifically, for action selection, a control representation should (i) be selective to task-critical information about the input, output, or goal, (ii) encode highly similar neural input states in a format that separates task-critical dimensions for different contexts, and (iii) provide temporally stable readout subspaces to downstream neurons that can bridge variable gaps in time. In other words, control representations with robust context sensitivity must achieve dynamic geometrical properties that enhance separability while preserving the stability of neural trajectories for the selective extraction of task-critical information.

The separability of a neural population encoding control representations can be modulated by its geometric organization. For instance, representational dimensionality–the minimum number of axes needed to account for the variability of neural patterns that span across the task input conditions– is known to balance a computational tradeoff between separability and generalizability of representations in terms of available readouts^{9,10,11,12,13}. Specifically, a higher-dimensional representational geometry allows the system to encode even similar inputs into more separable or orthogonal neural patterns, thus reducing the overlap of the patterns (Fig. 1A). Thus, representations encoded in a high-dimensional format could be beneficial for cognitive control, in that changing contextual contingencies often requires distinguishing highly correlated inputs^1,14.

Fig. 1: Representational dimensionality, dynamics, and selectivity.

Schematic illustration of representational dimensionality, dynamics, and selectivity of the conjunctive control representations. Each panel plots the response of a toy population of three units to input conditions varying in shape (square or circle) and color (red or black). Axes represent the firing rates of single units, collectively defining a neural activity space for the population. Each point within this activity space represents the population response to a given input (identified by colored shapes). Distance between points reflects how distinct responses are, and the jittered cloud of points reflects the trial-by-trial variability in responses to a given input. A The geometric format of the population neural responses is defined by response patterns arranged in 3 dimensions. A linear readout is implemented by a decision hyperplane (yellow) that divides the readout subspace into different classes (e.g., different shapes of inputs). The high-dimensional representation (traced by solid black lines), where no cluster of responses are aligned to each other, allows a wider variety of input conditions to be linearly separable. In addition, responses projected to a readout subspace defined by a linear hyperplane tend to dissociate input conditions (e.g., red-square and black-square) that are on the same side of the decision boundary, increasing the separability of neural responses overall. B Due to the time-varying nature of neural activity, the neural state space is changing and reshaping its underlying geometry over time. The dynamic neural trajectories potentially require changes in the weights for optimal linear hyperplanes for downstream readouts. C Units within the population show a heterogeneous turning profile or nonlinear mixed selectivity. To illustrate, the tuning profile for one unit (r₂) is plotted along the corresponding axis. The bars plot the activity of r₂ to each of the four conditions of input and depict a non-linear mixed selective pattern. Note that similar geometric properties are expected at the level of a single unit (i.e., mixed-selectivity) or population of neurons (i.e., integrative subspaces). The event-file representations could be conceptualized as one form of mixed selectivity where a unit or groups of units are exclusively tuned to a specific combination of task-critical factors more than other pairs.

However, achieving stability for temporal invariance of readout while enhancing separability for context-sensitivity via high-dimensional geometry, often poses a computational challenge. For example, recurrent neural networks that use chaotic dynamics or reservoir computing can achieve exceptionally high context sensitivity, yet they are temporally unstable^{15,16,17,18,19}. Such fully dynamic coding makes reliable readout of task-critical information difficult because the selectivity and geometry of neural populations could change over time²⁰. This makes computed solutions inefficient and not generalizable to other event timings¹⁴.

The problem of time-invariance can be mitigated by learning stable, temporally low-dimensional, attractor-based dynamics where trajectories span subspaces or manifolds that offer a fixed, common set of weights for readout^21,22. Indeed, neural computations of task-critical information often involve partially dynamic yet structured neural trajectories^{23,24,25,26,27,28,29}, reaching stable subspaces to represent information despite time-varying dynamics in the overall neural state space^{20,30,31,32,33}. Thus, the fundamental question is what kind of stable subspaces could provide context-sensitivity with enhanced separability that benefits cognitive control problems.

A possible solution to enhance separability while maintaining stability for robust cognitive control is to selectively encode stable conjunctive subspaces that integrate task-critical information within a high-dimensional representational geometry. The expansion of dimensionality affords the integration of low-dimensional task features such as the abstract context and the settings for inputs and outputs. This solves the context-sensitivity problem and enhances separability by flexibly encoding dissociable context-dependent mappings. Further, achieving stable dynamics (e.g., fixed points) within conjunctive subspaces provides the temporal invariance that facilitates more stable readout even while neural trajectories are highly dynamic in the overall neural state space.

In line with this hypothesis, several theories and computational models of cognitive control suggest that robust selection of context-dependent actions requires encoding and retrieval of conjunctive representations, which bind key task features such as goals, rules, stimuli, responses, and outcomes^{34,35,36,37,38,39,40}. Supporting these theories, a line of studies decoding EEG signals in humans during the performance of response selection tasks has found consistent evidence of conjunctive representations supporting context-dependent action selection^{41,42,43,44,45}.

Notably, conjunctive subspaces can be easily implemented as a neural population code that exhibits nonlinear mixed-selectivity^{11,14,30,46,47,48,49,50,51}, Fig. 1C or larger-scale task-contingent interactions of neural assemblies that effectively expand representational dimensionality of population codes^36,52,53,54. Within such a high-dimensional, expressive geometry, readouts of task-critical information, including conjunctions, are expected to be enhanced.

Finally, neural network models with nonlinear recurrent dynamics solving context-dependent tasks show stable dynamics that implement subspaces or manifolds across task-critical dimensions, increasing the adaptability of learned representations^{16,21,55,56,57,58}. Therefore, if conjunctive representations are encoded in stable subspaces on the fly, cognitive control could operate by directly implementing readouts of context-dependent input-output mappings that are minimally required by the task, recruiting stable yet separable neural patterns (Fig. 1A–C). However, no direct links between such dynamic geometric properties of control representations and controlled behavior have been established in humans.

In the current study, we sought to characterize the nature of control representations in terms of their geometric properties and dynamics during context-dependent action selection in humans. Specifically, we hypothesized that temporally stable conjunctive subspaces that are expressed in a high-dimensional representational geometry provide enhanced separability that facilitates the readout of specific controlled actions. Conversely, responses made before the computation of a stable conjunctive state within a high-dimensional geometry cannot take full advantage of greater separability and will be more likely to produce errors.

To test this hypothesis, we combined the rule-based action selection task with a psychophysical response-deadline procedure and several time-resolved decoding techniques for EEG (Fig. 2; see “Methods” for details). After varying time intervals, participants were cued to make context-dependent actions immediately. This procedure compelled responses based on the current state of neural representations, thereby capturing the readout of task-critical information at distinct stages during the progression of representational neural trajectories. This allowed us to evaluate the dimensionality, dynamics, and selectivity of task representations governing behavior (Fig. 1).

Fig. 2: Task design and the procedure of decoding analyses.

A Sequence of trial events in the rule-selection task with the response deadline. In the variable or sampled SOA phase, an audio trigger signal indicated the start of a response deadline time window. B Spatial translation of different rules (rows) mapping different stimuli (columns) to responses (arrows), yielding 12 independent conjunctions. C Schematic of the time-resolved representational similarity analysis. For each sample time (t), a scalp-distributed pattern of EEG was used to decode the specific rule/stimulus/response configuration of a required action. The decoder produced sets of classification probabilities for each of the possible action constellations. The profile of classification probabilities reflects the similarity structure of the underlying representations, where action constellations with shared features are more likely to be confused. For each trial and timepoint, the classification probabilities were regressed onto model vectors as predictors that reflect the different, possible representations. In each model matrix, the shading of squares indicates the theoretically predicted classification probabilities (darker shading means higher probabilities) in all possible pairs of constellations. The coefficients associated with each predictor (i.e., t-values) reflect the unique variance explained by each of the constituent features and their conjunction. D Schematic of the time-resolved binary classification method used to estimate the representational dimensionality⁵¹. For each time point (t), a pattern of EEG associated with unique action constellations or input conditions (c_1–12) takes a position in the multidimensional neural space spanning in r_1-e dimensions. By assigning new binary class labels (e.g., A or B) to the input conditions, we can generate arbitrary binary groupings given the task conditions. Because the higher dimensional geometry of neural responses generally affords more arbitrary linear separations by being more expressive, the count of successfully implementable binary classifications of newly defined groupings scales with the representational dimensionality. To adapt the method to EEG by lowering the cutoff threshold for classification, we used an exclusive cutoff method. In this method, classifications must exceed the cutoff threshold in all the different input conditions assigned to each trial, as opposed to the averaged performance across c_1–12, to be marked as successful.

To preview our results, we discovered that successful action selection recruits a transient expansion of representational dimensionality, whose peak is coincident with temporal stability in a conjunctive subspace established just prior to response execution. Collapsed dimensionality led to incorrect and/or slow responses. In addition, earlier development of temporally stable conjunctive subspaces predicted a unique variance in the quality of trial-to-trial action selection. These results provide evidence that highly separable yet temporally stable dynamics in the coding of conjunctive control representations are important for efficient goal-contingent action selection.

Participants performed this rule-based selection task in three task phases in order: the no-deadline phase, the variable stimulus-onset asynchrony (SOA) phase, and the sampled SOA phase. In the no-deadline phase, after 15 practice blocks, participants performed the rule-based action selection task and were only instructed to respond as accurately and quickly as possible, balancing both equally. The no-deadline phase provided reference RTs, which were used to adjust the timing of SOA intervals for subsequent phases as described in the Estimation of SOA functions (Supplementary Fig. S1A).

Results

Behavior in the no-deadline, variable SOA, and sampled SOA task

During the no-deadline task phase, participants learned to perform the task rapidly and accurately (M = 620 ms, SE = 69 ms for RTs and M = 0.07, SE = 0.03 for error; Supplementary Fig. S1A), following ex-gaussian distributions (Supplementary Fig. S1B).

Next, participants performed the same task with a response deadline and adaptive stimulus onset asynchrony (SOA) intervals between the onset of the stimulus and the response deadline. We term this phase the variable SOA task. As a function of increasing SOA, the rate of correct responses (i.e., accurate responses that occurred within response deadline) significantly increased then plateaued (Fig. 3A), t(1,41) = 23.31, beta = 5.10, 95 % CI [4.67 5.53] for linear effect, and t(1,41) = − 17.97, beta = − 2.35, 95 % CI [− 2.60 − 2.94] for quadratic effect. Focusing on accurate and within-deadline responses, participants responded faster as SOA increased (M = 248 ms, SD = 17.76 ms). Both complete omission of responses (M = 1.83 %, SD = 1.58 %) and premature responses emitted before the trigger signal (M = 2.70 %, SD = 2.70 %) were rare. Further, as the SOA intervals increased, the omission of responses (M = 1.83 %, SD = 1.58%) occurred significantly less, and premature responses occurred more (Supplementary Table 1).

Fig. 3: Psychophysical curve of on-time response accuracies via SOA manipulation.

A Empirically observed changes in the probability of correct and on-time responses as a function of the subjects-specific SOA intervals. Each line plots an individual subject. B Predictions by the model for each subject of an exponential function approaching the limit using the subjects-specific estimates of parameters. C Distributions of the parameters of the model with an exponential function to the limit: β is the rate parameter, which indexes the speed at which accuracy grows to asymptote; δ is the intercept reflecting the discrete point when accuracy departs from 0, and λ is the asymptotic accuracy level reflecting the overall probability of successful responses. D Empirically observed probability of correct responses during sampled SOA phase where we simultaneously measured EEG. The subjects-specific SOA intervals estimated from earlier phases were used to force an expected number of errors for short SOA (p = 0.5), medium SOA (p = 0.75), and long SOA (p = 0.9) conditions, respectively. C, D The box plots show the 25^th, 50^th, and 75^th percentile as a box, and the minima/maxima of whiskers correspond to smallest/largest values within 1.5 times the interquartile range below/above the 25^th/75^th percentile.

Inappropriate responses consisted of response selection errors or responses that were too late for the deadline, and both types of errors drove adaptive SOA adjustments. On average, 28.86% (SD = 7.23%) of responses occurred after the deadline. Excluding complete omission trials (i.e., no response at all) and across all SOA conditions, response commission errors occurred on 10.04% (SD = 6.28%) of trials within the deadline and 29.43% (SD = 11.29%) after the deadline (see Supplementary Fig. S2 for the distributions of trials of different response types).

Commission response errors significantly decreased as a function of SOA, t(1,41) = − 11.27, beta = − 3.93, 95% CI [− 4.62 − 3.25] for linear effect, and t(1,41) = − 0.16, beta = − 0.25, 95% CI [− 0.34 − 0.29] for quadratic effect for responses that occurred within the deadline; t(1,41) = − 2.47, beta = − 0.62, 95% CI [− 1.12 − 0.13] for linear effect, and t(1,41) = 3.11, beta = 0.52, 95% CI [0.19 0.85] for quadratic effect for responses that occurred after the deadline. A majority of late (i.e., after the deadline) responses occurred in shorter SOA conditions and occurred significantly less as participants prepared for a longer time, t(1,41) = − 32.87, beta = − 6.48, 95% CI [− 6.87 − 6.09] for linear effect, and t(1,41) = 11.78, beta = 1.42, 95% CI [1.19 1.66] for quadratic effect. Thus, longer SOAs led participants to implement on-time and accurate action selection.

To force response errors at the expected rates, we modeled observed psychometric functions from the adaptive SOA procedure with an exponential rise approaching a plateau as SOA increases⁵⁹ see Estimation of SOA functions for detail). This allowed us to describe changes in performance in a continuous manner (Fig. 3A) and estimate individuals-specific SOA intervals that are expected to produce at specific error rates (50, 70, and 90%; Fig. 3B, C).

As a result, during the sampled SOA task while EEG was recorded, correct responses were observed at calibrated rates (Fig. 3D). Further, we observed a similar behavioral pattern during the EEG experiment to the variable SOA task in terms of RTs, omission errors and commission errors as a function of short, medium and long SOA (Supplementary table 1). A majority of late responses occurred again in shorter SOA conditions 43.34% (SD = 10.31%) for short SOA, 20.39% (SD = 6.98%) for medium SOA, and 3.67% (SD = 2.41%) for long SOA. Excluding complete omission trials, commission errors occurred within the deadline at 20.37% (SD = 9.90%) for short SOA, 12.21% (SD = 5.70%) for medium SOA, and 3.93% (SD = 2.57%) for long SOA. These rates increased to 29.98% (SD = 10.18%) for short SOA, 21.67% (SD = 8.66%) for medium SOA, and 18.05% (SD = 11.30%) for long SOA for responses that occurred after the deadline (Supplementary Table 1).

Again, the commission response errors significantly decreased as a function of SOA, t(1,41) = − 15.30, beta = − 1.36, 95% CI [1.18 1.53] for linear effect, and t(1,41) = − 7.50, beta = − 0.29, 95% CI [− 0.36 − 0.21] for quadratic effect for responses that occurred within the deadline; t(1,41) = − 6.33, beta = − 0.47, 95% CI [− 0.61 − 0.32] for linear effect, and t(1,41) = 2.24, beta = 0.10, 95% CI [0.01 0.20] for quadratic effect for responses that occurred after the deadline. The effect of SOA on RT, the occurrence of late responses, and premature responses were also similar to the pattern observed in the variable SOA task (Supplementary Table 1). Overall, the quality and success of action selection were strongly influenced and regulated by SOA between the onset of trial to the deadline signal.

Task representational selectivity

We first characterized time-dependent changes in encoding of various task representations during controlled actions while being forced to read task information so a response could be made within the deadline determined by the three SOAs (Fig. 3D). By analyzing the representational similarity structure, we decoded the representations of the rule, stimulus, response and conjunction of a prepared action from the moment-to-moment patterns of EEG activity (Fig. 2C).

Consistent with our previous reports using a similar task (Kikumoto & Mayr), we found that a conjunctive representation of the action rule, stimulus, and response developed before response execution over and above the individual representations of those same constituent task features (Fig. 4; see Supplementary Figs. S4, S5 for the stimulus-aligned and the trigger-signal aligned results). The trajectories remained identical when the number of trials for correct, incorrect, and too-late trials was subsampled during decoding analyses (Supplementary Fig. S3). In short and medium SOA conditions, the strength of task representations except stimulus features was higher in trials that led to accurate and within-deadline responses, t(1,41) = 4.72, beta = 0.04, 95% CI [0.05 0.22] for rule, t(1,41) = − 0.68, beta = − 0.01, 95% CI [− 0.02 0.01] for stimulus, t(1,41) = 5.97, beta = 0.05, 95% CI [0.03 0.06] for response, and t(1,41) = 3.10, beta = 0.02, 95% CI [0.01 0.04] for conjunction (see Supplementary Table 2 for the stimulus-aligned and the trigger-signal aligned results). However, no significant increase in decoding strength was observed between these two SOA conditions (Supplementary Table 3), suggesting that merely increasing SOA intervals (i.e., pre-deadline response selection time) does not immediately translate to improved decoding performance.

Fig. 4: Response-aligned time-course of decoding of task representations.

Average, single-trial RSA coefficients (t-values) associated with each of the basis set task features (rule, stimulus, and response) and their conjunction that are aligned to the onset of trial-to-trial responses. The left, middle, and right columns correspond to the short, medium, and long SOA conditions, respectively. The two lines in each panel show the average scores in trials comparing correct vs. too late or incorrect responses. The patterns were aligned to the onset of responses in each trial. Shaded regions specify the 95% within-subject standard errors.

We next related decoding strength with behavior. Higher decodability of each task representation was independently correlated with a higher probability of making accurate response selections within the deadline (Fig. 5 top panel) on a trial-wise basis. A similar predictive effect of response accuracy was observed in trials where responses occurred within the deadline (Fig. 5 bottom panel). Notably, the effect of a rule and conjunctive representation on successful responses shift to earlier time points (− 280 ms before the onset of a trigger for deadline) in medium SOA trials compared to short SOA trials (Supplementary Fig. S7). Indeed, predictability of response failures by earlier (i.e., − 400 ms prior to the trigger onset) signals encoding conjunctive representations was enhanced for longer SOAs, t(1,41) = 2.85, beta = 0.004, 95% CI [0.01 0.05] for linear effect and t(1,41) = 2.85, beta = 0.004, 95% CI [0.01 0.05] for quadratic effect, controlling for all main and interaction effects of SOAs and other task factors (see also the Predicting Accurate Responses within Deadline Separately in Separate SOAs in Supplementary information for detail). Further analyses revealed that responses that are too late or incorrect both decrease the strength of conjunctions to a similar degree (Supplementary Fig. S6), whereas the cued response representations were absent when committed responses were incorrect. Thus, more reliable encoding of the key task variables was critical for response selection under time pressure. These observations replicate and extend prior findings that the independent contribution of conjunctive representations is unique for the prepared action.

Fig. 5: Time-course of the impact of decoding of task representations on successful response selection.

The time course of the effect of decoded task representation strength on response selection performance. The z-values were obtained from multilevel, logistic regression models predicting the variability in trial-to-trial accurate and on-time responses from the strength of task representations at each moment. The signals are aligned to the onset of trial-to-trial responses. The top panel shows the result using all responses except complete omissions and premature responses, whereas the bottom panel shows the result with responses made within the deadline. Positive z-values indicate more successful responses as the strength of decoded representations increases. The colored straight lines at the bottom denote the significant time points using a nonparametric permutation test (cluster-forming threshold, p < 0.01, cluster-significance threshold, p < 0.01, two-tailed).

Task representational dimensionality

How does the representational dimensionality of a control representation change over the course of neural trajectories? How does this relate to successful action selection? And, what are the functional links to the quality of the component task representations (i.e., stimulus, rule, and response) and the conjunctions (Fig. 2C)? To this end, we sought to measure the time-dependent changes in the representational dimensionality (i.e., shattering dimension) of task representations. To estimate the dimensionality of the neural responses, we adapted the binary classification method used in physiological non-human primate recording data⁵¹ for substantially noisier human scalp EEG data by adding constraints on selecting successful separations. This exclusive cutoff method is detailed in the Dimensionality Analysis via the Binary Classification Method.

The results of the simulation verified that the binary classification method with exclusive cutoff could be applied to noisy data where the separability of patterns is generally low to study the geometry of neural responses (Supplementary Fig. S8). Further, we tested how the selectivity of patterns that are tuned for basis features of the current task in a pure-selective vs. mixed-selective (i.e., conjunctive) manner contributes to the representational dimensionality. Note that, to focus on the geometric properties provided by the integration of information irrespective of the scale of the neural activity (e.g., at the level of single neurons or manifolds), we sampled these patterns directly from RSA models (Fig. 1C) to generate patterns that differ in the degree of integration of task-relevant information. We confirmed that in patterns of neural activity that is exclusively tuned to one specific instance of the action constellation like an event file (i.e., excluding diverse mixing of task features), as assumed in the conjunction RSA model (Fig. 2C), the dimension of neural patterns expanded substantially (Supplementary Fig. S9; see Dimensionality Analysis via Simulation in the Supplementary Information for detail).

We applied our validated (see supplement) exclusive cutoff pairwise classification method in a time-resolved manner in order to characterize the dynamic changes of representational dimensionality while context-dependent action selection unfolds. Consistent with our hypothesis that the benefits of high dimensional representational geometry on controlled actions by enhancing separability, we found that the dimensionality transiently peaked before response execution and then collapsed rapidly (Fig. 6). Consistent with the previous works in non-human primates, we found that the dimensionality is overall significantly higher when participants made correct and on-time responses (Fig. 6; t(1,41) = 9.09, beta = 0.48, 95% CI [0.38 0.59] for stimulus-aligned and t(1,41) = 7.39, beta = 0.42, 95% CI [0.30 0.53] for response-aligned). Importantly, the difference appeared even before the trigger for response-deadline was presented, t(1,41) = 5.43, beta = 0.37, 95% CI [0.24 0.51], and persisted until the end of the deadline, t(1,41) = 6.14, beta = 0.45, 95% CI [0.31 0.60]. Further, the same effect was observed in the short SOA condition where the number of correct and too late or incorrect responses were matched in principle, t(1,41) = 3.59, beta = 0.26, 95% CI [0.12 0.41] for response-aligned. Both the temporal trajectories and the condition difference of dimensionality were preserved for higher cutoff threshold values (Supplementary Fig. S10B, D). Thus, consistent with our hypothesis, representational dimensionality changes dynamically over the course of action selection, and its development strongly correlates with more successful implementation of context-dependent actions.

Fig. 6: Time-course of task representational dimensionality.

Average, dimension score during action selection. The dimensionality is estimated as the proportion of implementable binary separation, n_c/T_c, out of all possible arbitrary binary pairs. The left, middle, and right column corresponds to the short, medium, and long SOA conditions respectively. The top, middle, and bottom rows correspond to the results using signals aligned to the onset of stimulus, trigger, or response in each trial. The two lines in each panel show the average scores in trials with correct vs. too late or incorrect responses. Shaded regions specify within-subject standard errors.

How does the representational dimensionality relate to the quality of action representations? To investigate this, we summarized the result of a binary classification analysis with exclusive cutoff (p = 0.51) separately for decile bins of RSA scores of each task factor across trials. Based on the results of the simulation, we expect a positive relationship between the strength of the signal and the representational dimensionality (n_c/T_c) where the steepness of functions relating to these two measures depends on the tuning profile and the noise level of underlying neural activities (Fig. 6). Consistently, we found that higher RSA scores of the rule, stimulus, response, and conjunction overall expand the representational dimensionality (Fig. 7). The degree of enhancement was strongest for conjunctive representations, supporting that RSA-derived conjunctions indeed reflect the mixture of task-critical information. The effect of conjunctions on the dimensionality was preserved for higher cutoff threshold values (Supplementary Fig. S10A, D). Taken together, these results confirm that higher dimensional representational geometry that accompanies the encoding of event-file-like conjunctive representations enhances separability for linear readouts, which strongly predicts the successful implementation of goal-contingent action selection.

Fig. 7: Relationship between the dimensionality and selectivity of task representations.

Changes in the representational dimensionality as a function of RSA scores of each of the basis set task features (rule, stimulus, and response) and their conjunction. The dimension scores (n_c/T_c) are calculated separately in decile bins of RSA scores of each feature within subjects, which correspond to individual points in each panel. The t-value is the statistic of a linear regression model fitted to the results.

Task representational dynamics

By focusing on the geometry of neural responses in a time-resolved manner, we found that encoding of conjunctive control representations recruits a transient high-dimensional geometry, which peaks prior to response execution, that enhances the separability of neural patterns. Yet, the utility of such heightened separability partially depends on the underlying dynamics. This is because, to implement optimal readouts of information, the neural trajectories must be either predictable or stabilized within the projected task-relevant subspaces. To assess the stability of underlying neural trajectories, we combined the cross-temporal generalization analysis⁶⁰ with single-trial RSA.

We first performed an exploratory analysis using our previously published data using a similar task without the response deadline⁴¹. The result showed a similar increase of the representational dimensionality that peaked slightly before response execution, which scaled with the strength of conjunctive representations (Fig. 6 bottom and Supplementary Fig. S11 top). This result indicates that a response deadline itself is not a necessary condition to induce expansion of representational dimensionality. In addition, we found enhanced separability was established when the conjunctive subspaces became temporarily stable, which is indicated by the decision hyperplanes becoming more generalizable to other time points (Supplementary Fig. S11 bottom). In particular, early forward generalization of the decision hyperplane (i.e., upper-left portion of the decoding matrix) preceding the onset of responses suggests that there is an attractor-like stabilization of neural trajectories that will retain readouts of neural states to be generalized for future states.

How the convergence of neural dynamics influences successful action selection is difficult to assess without any constraints on the encoding of neural trajectories for task computations. For example, slow responses may reflect failures of readout of task representations or strategic withholding for the stabilization of dynamics. In addition, the passage of time is often confounded with temporal stabilization of dynamics as in the cases in typical delayed maintenance tasks. The current study’s use of the response-deadline procedure circumvents these issues by forcing participants to use the product of dynamic computations at premature or developed states.

By assessing the temporal generalizability of linear hyperplanes that separate conjunctions defined by the combination of task basis features, we found that neural trajectories in the conjunctive subspace became relatively stable toward the moment that responses are executed (Fig. 8). Importantly, the gradual expansion of representational dimensionality peaked while the underlying dynamics were stable when accurate and on-time responses were achieved. In contrast, such temporally stable dynamics emerged at much later time windows when responses were incorrect or too late (see Supplementary Figs. S12,S13 for the results for other task factors).

Fig. 8: Representational dynamics in the subspace of conjunctive control representations.

Temporal generalization of linear hyperplanes that separate conjunctions of task basis features summarized as a decoding matrix. The X-axis denotes time points where signals were sampled to train decoders and then submitted to single-trial RSA. The Y-axis denotes time points where the test sets were derived. Thus, the diagonal components of the matrix correspond to trajectories of time-resolved decoding analyses, as shown in Fig. 4. The off-diagonal components show the results of temporal generalization analysis. The regions surrounded by a white contour denote the significant clusters using a nonparametric permutation test (cluster-forming threshold, p < 0.01, cluster-significance threshold, p < 0.01, two-tailed). The top panel shows changes in the representational dimensionality relative to the onset of responses (Fig. 7 bottom row) merging short and medium SOA conditions. The middle panel plots temporal generalization for correct responses, and the bottom panel plots temporal generalization for incorrect or too-late responses. Shaded regions of the top panel specify within-subject standard errors.

To test the importance of early development of stable dynamics, we predicted single-trial response successes by a forward generalization of hyperplanes of all task-relevant features while controlling for the predictability of the dynamic portion of the variability in encoded task representations. Critically, we found that more temporally stable dynamics in subspaces encoding conjunctive representations, over and above the dynamic coding of all task factors and the conjunction, uniquely predict a higher probability of selecting responses accurately within the deadline, t(1,41) = 0.21, beta = 0.01, 95% CI [− 0.03 0.04] for rule, t(1,41) = 2.65, beta = 0.05, 95% CI [0.02 0.07] for stimulus, t(1,41) = 0.71, beta = 0.14, 95% CI [− 0.02 0.04] for response, and t(1,41) = 2.60, beta = 0.05, 95% CI [0.01 0.09] for conjunction, using time window from − 400 ms to 0 ms relative to the onset of responses (see Supplementary Tables 4, 5 for the statistics of all other predictors in the model and the results using both forward and backward generalizations with different sets of time windows). Note that the dynamic component of stimulus encoding did not predict response failures (Fig. 5 and Supplementary Table 5). Thus, the effect is likely driven by shifting early dynamic encoding of stimulus further past because of delayed responses (Supplementary Fig. S13). The unique contribution of conjunctive representations was detected when the signals are aligned to the onset of stimuli, t(1,41) = 2.15, beta = 0.03, 95% CI [0.01 0.05] for rule, t(1,41) = 1.23, beta = 0.02, 95% CI [− 0.01 0.04] for stimulus, t(1,41) = 3.99, beta = 0.06, 95% CI [0.03 0.08] for response, and t(1,41) = 4.18, beta = 0.07, 95% CI [0.04 0.10] for conjunction, using time window from 0 ms to 400 ms relative to the onset of stimuli.

Taken together, these results suggest that effective controlled actions involve the recruitment of temporally stable conjunctive subspaces. The earlier establishment of these stable conjunctive representations is linked to more efficient controlled behavior, in addition to the benefits derived from their dynamic computations. The stability of these dynamics is attained simultaneously with the transient expansion of the representational dimensionality of neural trajectories, which promotes more reliable readouts while facilitating the separation of critical information for the task at hand.

Discussion

Flexible, goal-directed behavior depends on reliably integrating specific action pathways with abstract knowledge about goals and contexts. At the heart of this integrative capacity are control representations that select, organize, and maintain task-critical information using neural codes suited to the computational demands of behavior. Control representations must be context-sensitive, separating highly correlated inputs into distinct neural states for efficient readout. Yet, they must also be robust in the face of the inherent temporal variability of noisy, biological computation, allowing for stable readout for flexible behavior. In this study, we investigated how control representations are expressed within the complex, time-varying geometry of neural state space. In particular, we tested the hypothesis that temporally stable conjunctive spaces expressed within a high-dimensional geometry influenced controlled action selection, thus enabling both separability and stability required for cognitive control problems. To this end, we characterized the temporal development of representational dimensionality and subspaces for individual task features relative to behavioral events, adapted the classification-based dimensionality estimation for the noisy neural data in humans, and explicitly related the dimensionality to the encoding of conjunctive representations.

First, we found that the relevant contexts or action rules are integrated with other key task information as conjunctive control representations, replicating prior work⁴¹. Extending our previous findings, our procedure with variable SOA intervals and a response deadline demonstrated that the encoding of integrated task features was crucial for the efficient execution of controlled actions independent of the decision to make a response. Moreover, we found that reaching a neural state that allows for the temporally stable readout of conjunctions precedes successful responding, regardless of the enforced variability in the timing of response execution. The encoding of conjunctive and response representations was weak or absent when responses were delayed or incorrect, possibly indicating impaired readout of responses (Supplementary Fig. S6).

Second, we found that neural representational (i.e., embedding) dimensionality transiently changed within each episode of action selection. Specifically, during episodes that produce successful, controlled actions, dimensionality rapidly expanded and peaked just prior to response execution, leading to a sudden drop. When this expansion had not yet happened, participants failed to make correct or on-time responses. Furthermore, this higher-dimensional geometry particularly enhanced the conjunctive control representations. These findings align with our simulation results and are in line with theoretical and empirical evidence suggesting that neural mechanisms involved in integrating task-critical information, even by encoding exclusive conjunctions alone, expand the embedding dimension of the representation. During this transient period of dimensionality expansion, the separability of the neural patterns encoding task states is increased, making task-critical features accessible to a linear readout.

Finally, we found that the complex neural trajectories encoding control representations contained both dynamic and temporally stable components. In particular, we found evidence of conjunctive subspaces that exhibit temporal stability in the period just prior to response execution, despite the ongoing dynamics of neural trajectories in the entire state space. The timing of accessing the stable conjunctive state explained unique variance in achieving successful response selection beyond that explained by the strength of the relevant task variables, including conjunctions, decoded from the dynamic trajectories alone. The contributions of both dynamic and stable components were observed in conjunctive subspaces uniquely, suggesting efficient dynamics for task-tailored solutions reflecting required goal states⁶¹.

Thus, our overall results support the hypothesis that encoding a stable conjunctive subspace within a high-dimensional representational geometry provides enhanced separability and temporal stability that are ideal for solving cognitive control problems. Several recent theories and their associated computational models of cognitive control^{34,35,36,37,38,62} directly emphasized that the integration of task-relevant features is an important computational step toward flexible outputs, which contrasts with the more traditional view of cognitive control recruiting a hierarchical, multi-step process (Refs. ^{3, 63, 64}). Our findings impose additional empirical constraints on these theories, as they must account for an integrative subspace that facilitates the separation of task-critical information in a temporally stable manner, in concert with a dimensionality expansion. Below, we further elaborate on why such features of control representations may benefit the system and the implications of these observations for theory and future work.

Many cognitive control theories assume integration or association of goals with other task-relevant features at some level, but not all emphasize these as the control representation needed to solve context-dependent behavioral problems. For example, the widely influential⁶⁵ guided activation model proposed a solution to the Stroop task via lateral inhibition without units that encode separate conjunctions between task demands and other features. This leads to the view that control is achieved by goal representations maintained in a factorized, low-dimensional (i.e., compositional) format, which biases neural activity in action selection stages. Our results indicate that the encoding of conjunctions is also critical for implementing context-sensitivity, by directly associating goal contexts into stimulus and response bindings^34,38,40,66. They further indicate that encoding stable conjunctions can be understood as a mechanism that recruits additional embedding dimensions to bind goals, contexts, and action rules with other compositional task features.

In this sense, conjunctions may serve the computational function of enabling linear separability of the low dimensional inputs, thereby guiding cognitive control. Indeed, the embedding dimensionality of neural activity adapts to the complexity of the computational demands of the task, often enhancing the separability of compositional input features^{5,67,68,69,70,71}. Similarly, to afford linear separability of low dimensional inputs, recurrent neural networks with chaotic dynamics also show temporally dimensional expansion prior to predictable readout¹⁸. In cognitive control tasks requiring context-dependent mappings, conjunctions play a crucial computational role in disambiguating similar neural states induced by low-dimensional inputs, by projecting them onto a higher-dimensional geometry. Supporting this interpretation, we observed a robust reliance on conjunctions for successful context-dependent actions (Fig. 5) and discovered a significant correlation between the strength of conjunctions and the dimensionality of neural activity (Fig. 7 and Supplementary Fig. S10). Thus, this viewpoint explains how encoding of the conjunctive state itself enhances the separability of control representations in the service of context sensitivity by inducing higher embedding dimensions. This connection between the dimensionality to a theoretically important control representation has not been established explicitly in humans.

Further, we observed that the transient expansion of the embedding dimension increased beyond the dimensionality strictly required to solve the task, supporting more diverse linear separations of task variables than needed. Such an expansion of the embedding dimension, which consistently occurred just prior to response execution, should be beneficial for creating additional subspaces by pushing neural patterns closer to orthogonality⁵¹. However, the observed transient, higher-dimensional geometry is not solely explained by the encoding of conjunctions, as suggested by the mismatch between the time courses of decoded conjunctive representations and dimensional expansion (Fig. 4 and Supplementary Fig. S11), and the lower average level of dimensionality even in trials with strong conjunctive representations (Fig. 7). One possible advantage of a larger embedding dimension at the moment of transition to response execution, is to accommodate both the response and the conjunctive subspaces, thereby achieving better dissociation of neural trajectories. Similarly, several recent studies in non-human animals have observed the dynamic use of orthogonal subspaces to separate different task-critical information, potentially reducing interference of behaviorally relevant variables^{72,73,74,75,76,77}. An important area for future research is to clarify the neural mechanisms that generate transient dimension expansion and their relationship to stable subspaces of task representations.

Our results also emphasize that control mechanisms implement a temporally stable yet integrative subspace within dynamic neural trajectories. Such solutions with stable subspaces not only enable reliable readout of information but also provide a few advantages in solving cognitive control problems. In neural networks, optimizing for context sensitivity often induces highly dynamic neural trajectories as solutions^15,18. A computational consequence of this time-varying activity is that the readout must learn distinct weights for each moment. In contrast, encoding a stable conjunctive subspace within high dimensional geometry allows for a fixed, common set of weights that can produce time-invariant readout, while affording the context-sensitivity required by cognitive control. Such stable subspaces provide flexible and reliable readout over variable temporal delays until the completion of the tasks^16,20. This temporal invariance provides the ability to decode task-critical information (e.g., to select a response) at arbitrary timings without explicitly encoding time, and it also offers robustness to the time-varying nature of noisy computation afforded by biological circuits. However, we note that the periods of stability we observe here were brief, arising in the moments prior to the response. So, whether they would bridge longer variable delays is a matter for future research. This caveat notwithstanding, however, a stable state within conjunctive subspaces also promotes learning a single set of readout weights despite the dynamic nature of the overall neural trajectory. This significantly reduces the complexity of learning downstream readouts, facilitating more efficient learning of the task.

Our results add to the converging evidence that costs on behavior directly emerge from the dynamics and geometry of task representations. Various psychological and neural factors, such as the objectives and states of learning, task demands, and interactions of multiple cognitive systems, could influence these properties.

For instance, gaining expertize in a fixed task environment has been theorized to reflect more efficient encoding or retrieval of instances or task-specific memory that minimally specifies required goal states³⁷. Given that stabilization and expansion of dimensions are not achieved instantaneously, how efficiently the system reaches highly separable geometry and a stable, conjunctive state may be directly optimized through experience. The frequently observed co-existence of initially dynamic and later stabilizing subspaces of task representations may reflect the efficient transformation of task information in a goal-contingent manner^78,79,80. In that case, the benefits and costs in response selection, such as repetition priming^66,81,82, pre-cueing preparation effects⁸³, or multi-tasking costs⁸⁴, could also depend on how the dynamics and geometry unfold in the relevant control episodes^41,44.

A limitation of the current study is that our analysis of EEG data cannot inform our understanding of the scale and locus of the neural computations underlying conjunctive subspaces and their readout. This ambiguity arises from the difficulty in distinguishing between connectivity and activity of neural assemblies at the level of EEG. The signals we observe may reflect local computations within specific brain regions and networks^{7,46,51,77,85} or macroscale brain states resulting from distributed interactions among multiple neural populations and networks^{52,54,86,87,88}. Thus, future research should aim to distinguish these alternatives, perhaps through complementary methods in humans and animals that combine neural dynamics with spatially resolved data⁸⁹. However, we note that our representational similarity analysis using a GLM with competing models makes it unlikely that conjunctions arise from stimulus, rule, and response information co-occurring in separate pools of domain-specific electrodes⁴¹.

Another limitation is how generalizable our findings are beyond the current task, where context-sensitivity is explicitly required to solve the task. Our claim is that enhanced, stable separability in task representations – which is partially provided by the encoding of stable conjunctive subspaces – is useful for performing any task requiring context sensitivity. This kind of context specificity (i.e., mapping the identical inputs to different responses depending on the contexts) is a feature of many if not most, standard cognitive control tasks such as the Stroop task, task-switching tasks, the Wisconsin Card Sorting task, S-R compatibility tasks, the Flanker task, the MSIT, etc. However, it is also possible that the geometric and dynamical complexity of task representations adapts flexibly to task demands in a brain region-specific manner (Refs. ^{90, 91}). The generalizability of our findings, such as stable conjunctive subspaces and transient dimensionality expansion, should be directly empirically tested in future studies.

In conclusion, we characterized the geometric properties and dynamics of control representations that facilitate separability and stability for reliable and flexible action selection. Specifically, when people successfully perform context-dependent action selection, representational dimensionality expands, coinciding with temporal stability in subspaces defined by conjunctions of key task features. Early entry into this stable state and enhanced separability are crucial for efficient behavior. These findings prompt a new mechanistic focus on cognitive control, geared toward understanding the conditions and processes that implement and bring about these representational states. They underscore the potential value of adopting both a geometric and a dynamic state-space perspective to gain deeper insights into the control representations that govern our adaptable behavior.

Methods

Participants

Forty-two participants (27 female; mean age: 22 years) were recruited and gave informed consent. This followed procedures approved by the Human Subjects Committee at the RIKEN. The sex and gender of participants were determined based on self-report. No sex and gender analysis was carried out because our hypotheses were not dependent on it. No statistical method was used to predetermine the sample size. All participants had normal or corrected-to-normal vision and had no history of neurological or psychiatric disorders. After preprocessing the EEG data, one participant was removed and was not analyzed further due to excessive amounts of artifacts (i.e., more than 25% of trials; see EEG recordings and preprocessing for details).

Behavioral procedure

Throughout all phases of the experiment (see below), participants performed a rule-based action selection task (⁸²; Fig. 2A), which required participants to behave according to 12 unique stimulus-response (S-R) mappings (Fig. 2B). The twelve S-R mappings were defined by the combination of three spatial transformation rules (vertical, horizontal and diagonal) over four stimuli (a dot at top-left, top-right, bottom-left, and bottom-right of the white frame). The rules described how to select one of four responses that were arranged in a 2 × 2 square matrix (4, 5, 1, and 2 on the number pad) based on the stimulus. Specifically, the diagonal rule required the participant to choose the response that was diagonally adjacent to the stimulus location. For example, if a dot was presented in the left-bottom of the frame and the vertical rule was cued, this specified the left-top response as correct (Fig. 2A). Accordingly, the vertical rule specified the vertically adjacent response, and the diagonal rule specified the diagonally adjacent response (Fig. 2B).

Thus, for each trial, participants selected one of the four responses based on a random cue for one of the three action rules and one of the four stimuli, yielding one of 12 unique combinations of rule, stimulus, and response. One experimental block lasted 18 s, during which participants were instructed to complete as many trials as possible. Trials that were initiated within the 18 s block duration but extended beyond it were allowed to finish. There were 25, 35, and 185 experimental blocks for each task phase. Before participants started the no-deadline task, they practiced the same baseline task (i.e., without deadlines) for 15 blocks.

Participants performed this rule-based selection task in three task phases in order: the no-deadline phase, the variable stimulus-onset asynchrony (SOA) phase, and the sampled SOA phase. In the no-deadline phase, after 15 practice blocks, participants performed the rule-based action selection task and were only instructed to respond as accurately and quickly as possible, balancing both equally. The no-deadline phase provided reference RTs, which were used to adjust the timing of SOA intervals for subsequent phases as described in the Estimation of SOA functions (Supplementary Fig. S1A).

In the variable SOA task, participants were presented with an imperative auditory stimulus as the trigger signal, at a point following the stimulus and rule presentation (Fig. 2A). They were required to make their response within 350 ms of hearing the trigger signal. The trigger signal (a pure tone in 500 Hz and 250 ms duration) was presented after one of 12 unique, subject-specific SOA intervals, where SOA was defined as the time interval between the stimulus-rule presentation and the trigger sound. One of these 12 unique SOAs was sampled randomly for each trial. The determination of the timing of the SOA is described below. Importantly, here, participants were instructed and regularly encouraged to make prompt responses within the deadline following the trigger signal, even if they were not ready to respond.

In the sampled SOA phase, we measured EEG while participants performed the same task but with three SOA intervals that were estimated from SOA functions, defined from the variable SOA phase, to produce response errors (including responses that are incorrect and/or too late) at 50, 70 and 90% of trials. The SOA intervals of each error probability condition are denoted as short, medium, and long SOA, respectively. One of these 3 unique SOAs was sampled at random for each trial.

Estimation of SOA functions

To sample responses at fixed error probabilities during the EEG session, we estimated individual SOA psychophysical functions by systematically modulating SOA intervals before the deadline (“trigger”) signal (Fig. 2A). We first obtained the distribution of RTs of correct responses in the no-deadline phase excluding trials where RTs were slower than 5 SD. The RT distribution was modeled by the ex-Gaussian distribution using the exgauss toolbox in Matlab 2019B (⁹²; Supplementary Fig. S1B). Then, in the variable SOA phase, participants performed the same task with 12 SOA intervals ranging from − 450 ms to 200 ms (− 450, − 400, − 350, − 300, − 250, − 200, − 100,− 50, 0, 100, 200 ms) referenced to their ex-Gaussian μ parameters. Negative SOA intervals were dropped during estimation. As a result, we obtained individuals’ probabilities of producing accurate and on-time responses as a function of SOAs averaged over trials (Fig. 3A).

Following⁵⁹, we estimated the changes in the rate of successful response selection in variable SOAs with an exponential function approaching a limit:

$$p({correct})=\lambda (1-{e}^{-\beta \left(t-\varOmega \right)}),{time} \, > \, \varOmega \,{else} \,0$$

(1)

In the equation above, t is the predicted p(correct) or the probability of correct responses at SOA interval t; λ is the asymptotic accuracy level reflecting the overall probability of successful responses; β is the rate parameter, which indexes the speed at which accuracy grows to asymptote; and Ω is the intercept reflecting the discrete point when accuracy departs from 0. The parameters were estimated by a nonlinear least square solver via a lsqcurvefit function in Matlab, which performs nonlinear curve fitting using a least square solution with the lower bound of 0, 0, and the minimum value of tested SOA intervals; and the upper bound of 1, 1, and the maximum value of tested SOA intervals. Note that the sole purpose of the estimation of SOA functions is to induce sufficient number of response errors consistently across participants, thus we do not intend to propose the SOA model reflects underlying cognitive processes. Using this approach, we were able to predict the probabilities of correct responses as a function of continuous SOAs (Fig. 3B, C) to produce expected distributions of errors (Fig. 3D).

EEG recordings and preprocessing

The continuous EEG was recorded using a Brain Products actiCHamp recording system (Brain Products GmbH). Recordings were obtained from a broad set of scalp sites (Fp1, Fp2, F7, F3, F4, F8, FC5, FC1, FC2, FC6, T7, C3, C4, T8, CP5, CP1, CP2, CP6, P7, P3, P4, P8, PO9, PO10, O1, O2, Fz, FCz, Cz, Pz, and Oz), from the left and right mastoids, and from electrodes lateral to the external canthi and below the left eye. The electrodes (TP9 and TP10) were placed on the mastoids beneath the cap as reference sites. Using additional passive electrodes, a horizontal electrooculogram (EOG) was recorded from laterally placed electrodes to detect horizontal eye movements with the separate ground electrode placed on the left side of the forehead. We verified that eyeblinks were visible and recorded at frontal electrodes such as Fp1 and Fp2. The EEG was filtered online with 0.1 Hz high-pass and 1000 Hz low-pass filters and digitized at 1000 Hz. The left mastoid was used as a reference for all recording sites, and data were re-referenced off-line to the average of left and right mastoids. The EEG data were obtained during the sampled SOA task after participants-specific SOA intervals were estimated.

The scalp EEG and EOG were amplified with an SA Instrumentation amplifier with a bandpass of 0.01–45 Hz, and signals were downsampled at 250 Hz using EEGLab⁹³. EEG data was first segmented into 22 s intervals to include all trials within a block and to remove edge artifacts in later analysis steps. After time-frequency decomposition was performed, these epochs were further segmented into smaller epochs for each trial. For stimulus-aligned epochs, we used the time interval of − 200 ms to 800 ms relative to the onset of stimuli, which encompasses all SOAs of all participants (Fig. 3A). For response-aligned epochs, we used the time interval of − 800 ms to 200 ms relative to the onset of responses. For epochs aligned to the response deadline signal (i.e., SOA-aligned), we re-epoched intervals from the stimulus-aligned epochs using the time interval of − 400 ms to 350 ms relative to the onset of a signal reflecting participants specific SOA intervals. The trial-to-trial epochs that included blinks (> 250 μv, window size = 200 ms, window step = 50 ms), large eye movements (> 1°, window size = 200 ms, window step = 10 ms), blocking of signals (range = − 0.01 to 0.01 μv, window size = 200 ms) were excluded from subsequent analyses. Any epochs that showed EEG activities larger than 5 s.d were excluded.

Time-frequency analysis

Temporal-spectral profiles of single-trial EEG data were computed via complex wavelet analysis⁹⁴ by applying time-frequency analysis to preprocessed EEG data epoched for the entire block (> 22 seconds to exclude the edge artifacts). The power spectrum was convolved with a series of complex Morlet wavelets $( \left(2 \right) \, {e}^{i2 \pi {ft}}{e}^{-(t/{\sigma })^{2}/2})$, where t is time, f is frequency increased from 2 to 40 Hz in 35 logarithmically spaced steps, and σ defines the width of each frequency band, set according to n/2$\pi$f, where n increased from 3 to 10. We used logarithmic scaling to keep the width across the frequency band approximately equal, and the incremental number of wavelet cycles was used to balance temporal and frequency precision as a function of frequency of the wavelet. After convolution was performed in the frequency domain, we took an inverse of the Fourier transform, resulting in complex signals in the time domain. A frequency band-specific estimate at each time point was defined as the squared magnitude of the convolved signal for instantaneous power.

Representational similarity analysis

We performed a time-resolved multivariate pattern classification analysis to decode action-relevant information following our previously reported method based on the representational similarity analysis with a few modifications^41,95. As the first step, at every sample in trial-to-trial epochs, separate linear decoders were trained to classify twelve possible action constellations or each of S-R mapping (Fig. 2B). We used a penalized linear discriminant analysis using the caret package in R version 4.3.1^96,97 to solve this multiclass classification problem. This step produced a graded profile of classification probabilities for each action constellation over time, reflecting the similarity of underlying multivariate neural patterns between actions. For example, in trials where the horizontal rule and the left-top stimulus were presented, the goal of decoders is to assign a higher probability for that specific S-R condition over others (Fig. 2C). The same target label is used for all time points in the same trials.

Decoders were trained with the instantaneous power of rhythmic EEG activity, which was averaged within the predefined ranges of frequency values (2-3 Hz for the delta-band, 4–7 Hz for the theta-band, 8–12 Hz for the alpha-band, 13–30 Hz for the beta-band, 31–35 Hz for the gamma-band), corresponding to 155 features (5 frequency-bands X 31 electrodes) to learn. Prior to the training of decoders, trials where responses occurred before the deadline period (i.e., before the trigger signal) and trials where responses were completely omitted were excluded. Further, all trials in the first block (of the sampled SOA task phase) were excluded.

Within individuals and frequency bands, the data points were z-transformed across electrodes to remove effects that scaled all electrodes uniformly. We used a k-fold repeated, cross-validation procedure to evaluate the decoding results⁹⁸ by randomly partitioning single-trial EEG data into five independent folds with an equal number of observations of each action constellation. After all folds served as the test sets, each cross-validation cycle was repeated ten times, in which each step generated a new set of randomized folds. The number of observations in each target class (e.g., action constellation) was equated by randomly dropping excess trials for certain conditions. In addition, the effect of subsampling the number of trials for correct, incorrect, and too-late responses was assessed in separate analyses (Supplementary Fig. S3). Resulting classification probabilities were averaged across all cross-validated results with the best-tuned hyperparameter to regularize the coefficients for the linear discriminant analysis.

We next performed representational similarity analysis (RSA) on the graded classification probabilities to assess the underlying similarity structure of task variables. Each RSA model matrix uniquely represents a potential, underlying representation (i.e., rules, stimuli, responses, and conjunctions), which makes unique predictions of similarity patterns for different action constellations. To obtain time-resolved estimates of each RSA model, we regressed the vector of logit-transformed classification probabilities onto RSA model vectors for independent decoding results.

Importantly, to estimate the unique variance explained by competing models, we regressed all model vectors simultaneously. The features used for training decoders range in spatial- and frequency-domain, which leaves the possibility of identifying clusters of features that consistently contribute to the decoding of any task features. However, by using competing models, it is unlikely that conjunctions arise from linearly additive effects of co-occurring stimulus, rule, and response information. We also included subject-specific regressors of z-scored, average RTs, and probabilities of on-time and correct responses in each action constellation to reduce potential biases in decoding. These coefficients, expressed in their corresponding t-values, reflect the quality of action representations at the level of single trials, which was later related to variability in behavior and dimensionality measures. We excluded t-values that exceeded 5 SDs from means for each sample point, which excluded less than 1% of the entire samples.

These analysis steps were repeated separately using EEG signals that are stimulus-aligned and response-aligned. The results aligned to the onset of the deadline signal are obtained by sorting stimulus-aligned results to idiosyncratic SOA timings.

Dimensionality analysis via binary classification method

To assess the representational dimensionality (i.e., shattering dimension) of action representations, we used a binary (pairwise) classification method^51,67. A particular point in the time series of EEG activities obtained in unique action contexts takes a certain location in trajectories that traverse a multi-dimensional space. The binary classification method evaluates the geometry of representations by counting how many different input conditions, which are defined by the combination of orthogonal action features, could be reliably separated (i.e., shattered) by linear readouts. Specifically, the count of linear separations can be obtained by training a set of linear classifiers that learn to decode all possible arbitrary binary groupings given the input conditions. The number of implementable linear separations is known to scale with the dimensionality of underlying neural activity, which has been closely linked to the nonlinear mixed selectivity-like tuning profile of information.

To evaluate the representational dimensionality with EEG data, we performed a binary classification analysis with cross-validation in a time-resolved manner⁵¹. showed that the count of implementable arbitrary binary classifications is robustly related to the dimensionality of the neural responses. Intuitively, distinct neural activity patterns with a low amount of noise corresponding to the c different input (i.e., task) conditions lead to higher discriminability of patterns, hence, the higher dimensionality of neural state space. The authors identified the point (c*) where the fraction of implementable classification (n_c/T_c) undergoes a phase transition point where the probability of successful binary classifications starts sharply dropping as the sampled input condition c increases. The critical value (c*) is defined as the maximal value of input conditions (c) for which almost all of the binary classifications out of all possible ways of binary groupings (T_c) could be implemented. This requires an arbitrary cutoff threshold of classification accuracy that is sufficiently above the chance level (e.g., p = 0.75–0.80 in the original study) and another cutoff threshold for the proportion of n_c/T_c to determine a sharp dropoff point.

Our goal was to assess the relative differences in dimensionality scores across conditions (e.g., correct vs. incorrect/too-late responses) that scale with the overall separability of neural patterns of EEG that contain a non-trivial level of noise. This requires a lower cutoff threshold of classification accuracy without losing sensitivity to changes in n_c/T_c driven by expansion or contraction of the dimensionality of neural patterns. To this end, we adapted the original binary classification method introduced in ref. ⁵¹ to EEG data by adding constraints on how we select implementable binary classifications (see also the Dimensionality Analysis via Simulation in Supplementary information).

Specifically, we trained separate decoders for all possible 4096 (i.e., 2¹² − 2, excluding two cases where all labels are identical for all c = 12 task conditions) sets of arbitrary binary separations defined by 12 unique action constellations or the input conditions (Fig. 2B). Resulting decoding accuracies were counted as reliable linear separations when the following two conditions were satisfied: 1) the decoding accuracies exceeded the arbitrarily chosen cutoff threshold of decoding accuracy of 0.51 (Figs. 6, 7) or higher (Supplementary Fig. S10) where the chance level is 0.50 and 2) above-threshold decoding accuracies were observed in each of the unique conditions of the input space exclusively (exclusive cutoff method) or in an aggregated manner averaging across all conditions (overall cutoff method). Specifically, the decoding accuracies were first averaged separately for each action constellation, then a cutoff was applied. This second criterion was applied to increase the sensitivity to detect changes in dimensionality scores driven by the non-linear mixing of information even when the cutoff threshold accuracy is significantly lowered to the chance level (Fig. 6). Note that the effect of lowered cutoff threshold values was tested in the simulated data (Fig. 6 and Supplementary Fig. S8). These results showed showed this approach avoids an inflationary bias on the estimated number of dimensions when using the binary classification method with a low cutoff threshold of decoding accuracy to counteract substantially noisy data (e.g., EEG).

As a result, we obtained the proportion of reliable binary separations out of all possible groupings (i.e., n_c/T_c = count of implementable binary classifications / (2¹² − 2)), which approximates the dimensionality of neural responses to the input space (i.e., dimension = log₂(n_c/T_c)). When comparing experimental conditions, the aforementioned exclusive cutoff was applied separately in conditions of interest (e.g., SOA or types of responses). The other settings of decoding analyses, such as the algorithm of classification, the steps of cross-validation, and the exclusion criteria of trials, were identical to the RSA analysis except for EEG signals being averaged over 20 ms non-overlapping consecutive time windows. The results of stimulus-aligned, response-aligned and go-signal-aligned data were generated in a similar manner as the RSA step.

To test the effect of changing values of the cutoff threshold, we conducted control analyses on both simulated (Supplementary Figs. S8, S9) and empirical data (Supplementary Fig. S10). We found that increasing cutoff threshold values did not affect differences in dimensionality across critical conditions in empirical data, including their temporal changes, the effect of response accuracy types, and the relationships to decoded task variables (Supplementary Fig. S10). Furthermore, the chosen cutoff threshold replicated the reported main result when applied to the data in ref. ⁴², where an analogous task without the response deadline was used (Supplementary Fig. S11; see Dimensionality Analysis via Simulation and Modulation of Cutoff Threshold Values for detail).

Temporal-generalization analysis

To assess the temporal stability of linear readout or linear hyperplanes learned by decoders, we used the temporal generalization analysis⁶⁰. Firstly, we trained a set of decoders to classify action constellations, following the identical procedure as the time-resolved RSA analysis, using EEG patterns at the specific time point. These time-specific decoders were applied to the held-out data obtained at identical (matched) or different (generalized) time intervals. This generates a series of decoding results in a two-dimensional matrix form where the on-diagonal entries correspond to the results in which the time points of the train and test dataset match, whereas the off-diagonal entries are the results of the generalization of linear hyperplanes over other time. Assessing the pattern of temporal generalization reveals the dynamic nature of underlying neural computation such as the temporal stability of linear readouts of task critical information. The resulting decoding results, corresponding to every pairing of training and testing non-overlapping time intervals (window size = 12 ms), were submitted to the regression-based RSA procedure to decode rules, stimulus, response, and conjunctions simultaneously. The results of stimulus-aligned, response-aligned and trigger-aligned data were generated in a similar manner as in the RSA steps.

Multilevel modeling and permutation tests

We used multilevel models to further assess the statistical significance of representational dimensionality and decoded representations via RSA and to relate these scores to trial-to-trial variability in behavior (e.g., probability of making on-time responses). We tested statistical significance by either computing time-averaged scores over a priori selected time intervals or time-resolved scores combined with non-parametric permutation tests. For models predicting accuracies (i.e., correct responses made within the deadline period), we used a multilevel logistic regression, excluding trials with responses that were omitted and/or occurred after the deadline. For models including the dimension scores, only random intercepts are included because the exclusive cutoff method requires aggregation of multiple trials (see Dimensionality Analysis via Binary Classification Method for details). Unless otherwise indicated, subject-specific intercepts and slopes were included as the random effects based on trials and participants as independent levels.

For models using time-averaged scores, the intervals were defined as 0 to 500 ms from the onset of stimuli for stimulus-aligned results, − 500 to 0 ms from the onset of response for response-aligned results, − 400 to 0 ms, and 0 to 350 ms relative to the onset of response-deadline (a trigger signal) for SOA-aligned results. Separate models were constructed using RSA scores averaged over these intervals as dependent variables (Supplementary Table 2 and Supplementary Table 3) or as predictors (Fig. 5 and Supplementary Fig. S7).

For models using time-resolved scores, we performed non-parametric permutation tests on the results of multilevel models⁹⁹. First, we performed a series of regression analyses in the continuous temporal space of interest and identified samples that exceeded the threshold for cluster identification (cluster-forming threshold, p < 0.01). For example, for the temporal generalization analysis (Fig. 8 and Supplementary Figs. S11, S12, and S13), we performed tests in two-dimensional space defined by the training and test time intervals. Then, empirical cluster-level statistics were obtained by taking the sum of t-values in each identified cluster with consecutive temporal regions. Finally, nonparametric statistical tests were performed by computing a cluster-level P-value (cluster-significance threshold, p < 0.01, two-tailed) from the distributions of cluster-level statistics, which were obtained by Monte Carlo iterations of regression analysis with shuffled condition labels.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.