Neural constraints on learning

Journal name:
Nature
Volume:
512,
Pages:
423–426
Date published:
DOI:
doi:10.1038/nature13665
Received
Accepted
Published online

Learning, whether motor, sensory or cognitive, requires networks of neurons to generate new activity patterns. As some behaviours are easier to learn than others1, 2, we asked if some neural activity patterns are easier to generate than others. Here we investigate whether an existing network constrains the patterns that a subset of its neurons is capable of exhibiting, and if so, what principles define this constraint. We employed a closed-loop intracortical brain–computer interface learning paradigm in which Rhesus macaques (Macaca mulatta) controlled a computer cursor by modulating neural activity patterns in the primary motor cortex. Using the brain–computer interface paradigm, we could specify and alter how neural activity mapped to cursor velocity. At the start of each session, we observed the characteristic activity patterns of the recorded neural population. The activity of a neural population can be represented in a high-dimensional space (termed the neural space), wherein each dimension corresponds to the activity of one neuron. These characteristic activity patterns comprise a low-dimensional subspace (termed the intrinsic manifold) within the neural space. The intrinsic manifold presumably reflects constraints imposed by the underlying neural circuitry. Here we show that the animals could readily learn to proficiently control the cursor using neural activity patterns that were within the intrinsic manifold. However, animals were less able to learn to proficiently control the cursor using activity patterns that were outside of the intrinsic manifold. These results suggest that the existing structure of a network can shape learning. On a timescale of hours, it seems to be difficult to learn to generate neural activity patterns that are not consistent with the existing network structure. These findings offer a network-level explanation for the observation that we are more readily able to learn new skills when they are related to the skills that we already possess3, 4.

At a glance

Figures

  1. Using a brain-computer interface to study learning.
    Figure 1: Using a brain–computer interface to study learning.

    a, Monkeys moved the BCI cursor (blue circle) to acquire targets (green circle) by modulating their neural activity. The BCI mapping consisted of first mapping the population neural activity to the intrinsic manifold using factor analysis, then from the intrinsic manifold to cursor kinematics using a Kalman filter. This two-step procedure allowed us to perform outside-manifold perturbations (blue arrows) and within-manifold perturbations (red arrows). D, dimensions. b, A simplified, conceptual illustration using three electrodes. The firing rate (FR) observed on each electrode in a brief epoch define a point (green dots) in the neural space. The intrinsic manifold (yellow plane) characterizes the prominent patterns of co-modulation. Neural activity maps onto the control space (black line) to specify cursor velocity. c, Control spaces for an intuitive mapping (black arrow), within-manifold perturbation (red arrow) and outside-manifold perturbation (blue arrow). d, Neural activity (green dot) elicits different cursor velocities (open circles and inset) under different mappings. Arrow colours as in c.

  2. Better learning for within-manifold perturbations than outside-manifold perturbations.
    Figure 2: Better learning for within-manifold perturbations than outside-manifold perturbations.

    a, b, Task performance during one representative within-manifold perturbation session (a) and one representative outside-manifold perturbation session (b). Black trace, success rate; green trace, target acquisition time. Dashed vertical lines indicate when the BCI mapping changed. Grey vertical bands represent 50-trial bins used to determine initial (red and blue dots) and best (red and blue asterisks) performance with the perturbed mapping. c, Quantifying the amount of learning. Black dot, performance with the intuitive mappings; red and blue dots, performance (success rate and acquisition time are relative to performance with intuitive mapping) just after the perturbation was introduced for sessions in Fig. 2a and Fig. 2b; red and blue asterisks, best performance during those perturbation sessions; dashed line, maximum learning vector for the session in Fig. 2a. The amount of learning for each session is the length of the raw learning vector projected onto the maximum learning vector, normalized by the length of the maximum learning vector. This is the ratio of the length of the thin red line to the length of the dashed line. d, Amount of learning for all sessions. A value of 1 indicates complete learning of the relationship between neural activity and kinematics, and 0 indicates no learning. Learning is significantly better for within-manifold perturbations (red, n = 28 (monkey J), 14 (monkey L)) than for outside-manifold perturbations (blue, n = 39 (monkey J), 15 (monkey L)). Arrows indicate the sessions shown in Fig. 2a (red) and Fig. 2b (blue). Dashed lines, means of distributions; solid lines, mean ± standard error of the mean (s.e.m.). P values were obtained from two-tailed Student’s t-tests.

  3. Alternative explanations do not explain the difference in learnability between the two types of perturbation.
    Figure 3: Alternative explanations do not explain the difference in learnability between the two types of perturbation.

    a, Performance impairment immediately following within-manifold and outside-manifold perturbations. b, Mean principal angles between intuitive and perturbed mappings. c, Mean required change in preferred direction (PD) for individual neural units. For all panels: red, within-manifold perturbations; blue, outside-manifold perturbations; dashed lines, means of distributions; solid lines, mean ± s.e.m.; P values are for two-tailed Student’s t-tests; same number of sessions as in Fig. 2d.

  4. Properties of the intrinsic manifold.
    Figure 4: Properties of the intrinsic manifold.

    a, Cross-validated log-likelihoods (LL; arbitrary units) of the population activity for different days. The peaks (open circles) indicate the estimated intrinsic dimensionality (EID). Vertical bars indicate the standard error of log-likelihoods, computed across four cross-validation folds. A ten-dimensional intrinsic manifold was used for all experiments (solid circles). b, EID across all days and both monkeys (mean ± s.e.m., 9.81 ± 0.31). c, Difference between the LL for the ten-dimensional (10D) model and the EID model. Units are the number of standard errors of LL for the EID model. For 89% (78 of 88) of the days, the LL for the ten-dimensional model was within one standard error of the EID model. All sessions were less than two standard errors away. d, Cumulative shared variance explained by the ten-dimensional intrinsic manifold used during the experiment. Coloured curves correspond to the experimental days shown in Fig. 4a. The black curve shows the mean ± s.e.m. across all days (n = 88; monkey J, 58; monkey L, 30).

  5. Performance during baseline blocks.
    Extended Data Fig. 1: Performance during baseline blocks.

    a, Histograms of success rate during the baseline blocks on days when the perturbation would later be within-manifold (red) and outside-manifold (blue) for monkey J (top) and monkey L (bottom). For days with multiple perturbation sessions, the data are coloured according to the first perturbation type. Dashed lines, means of distributions; solid lines, mean ± s.e.m. b, Histograms of target acquisition time during baseline blocks. Number of days for panels a and b: within-manifold perturbations, n = 27 (monkey J), 14 (monkey L); outside-manifold perturbations, n = 31 (monkey J), 14 (monkey L). c, Sample cursor trajectories to all eight targets. At the beginning of each day, the monkeys used the intuitive mapping for 250–400 trials. The monkeys were able to use these mappings to control the cursor proficiently from the outset (as measured by success rate and acquisition time). On all sessions, the success rates were near 100%, and the acquisition times were between 800 and 1,000 ms. No performance metrics during the baseline blocks were significantly different between within-manifold perturbation sessions and outside-manifold perturbation sessions (P > 0.05; success rate, Wilcoxon rank-sum test; acquisition time, two-tailed Student's t-test).

  6. Changes in success rate and acquisition time during perturbation blocks.
    Extended Data Fig. 2: Changes in success rate and acquisition time during perturbation blocks.

    In Fig. 2d, we quantified the amount of learning in each session using a single metric that combined improvements in success rate and acquisition time. Here, we consider each metric separately. In each comparison, better performance is to the right. a, Change in success rate from the first 50-trial bin in the perturbation block to the bin with the best performance. The change in success rate was significantly greater for within-manifold perturbations than for outside-manifold perturbations for monkey J (top, P < 10−3, t-test). For monkey L (bottom), the change in success rate was greater for within-manifold perturbations than for outside-manifold perturbations, and the difference approached significance (P = 0.088, t-test). b, Change in acquisition time from the first 50-trial bin in the perturbation block to the bin with the best performance. For both monkeys, the change in acquisition time for within-manifold perturbations was significantly greater than for outside-manifold perturbations (monkey J (top), P < 10−4, t-test; monkey L (bottom), P = 0.0014, t-test). Note that a negative acquisition time change indicates performance improvement (that is, targets were acquired faster). Number of within-manifold perturbations, n = 28 (monkey J), 14 (monkey L); outside-manifold perturbations, n = 39 (monkey J), 15 (monkey L).

  7. After-effects during washout blocks.
    Extended Data Fig. 3: After-effects during washout blocks.

    After 600 (monkey J) or 400 (monkey L) trials using the perturbed mapping, we re-introduced the intuitive mapping to observe any after-effects of learning. We measured the after-effect as the size of the performance impairment at the beginning of the washout block in the same way that we measured the performance impairment at the beginning of the perturbation block. A larger after-effect indicates more learning had occurred in response to the perturbation. For monkey J (left), the after-effect was significantly larger for within-manifold perturbations (red) than for outside-manifold perturbations (blue) (Wilcoxon rank-sum test, P < 10−3). For monkey L (right), the trend is in the same direction as monkey J, but the effect did not achieve significance (Wilcoxon rank-sum test, P > 0.05). These data are consistent with the hypothesis that relatively little learning occurred during the outside-manifold perturbations in comparison to the within-manifold perturbations. Number of within-manifold perturbations, n = 27 (monkey J), 14 (monkey L); outside-manifold perturbations, n = 33 (monkey J), 15 (monkey L).

  8. Learning did not improve over sessions.
    Extended Data Fig. 4: Learning did not improve over sessions.

    It might have been that, over the course of weeks and months, the animals improved at learning to use perturbed mappings, either one type or both types together. This did not occur. Within-manifold perturbations showed more learning than outside-manifold perturbations across the duration of experiments. Animals did not get better at learning to use either type of perturbation separately (red and blue regression lines, F-test, P > 0.05 for all relationships) nor when considering all sessions together (black regression line, F-test for linear regression, P > 0.05). Same number of sessions as in Extended Data Fig. 2. Each point corresponds to one session.

  9. Hand speeds during BCI control and hand control.
    Extended Data Fig. 5: Hand speeds during BCI control and hand control.

    We loosely restrained the monkeys’ arms to the chair’s armrests during experiments. The monkeys minimally moved their hands, but the movements did not approach the limits of the restraints. a, Average hand speeds across all trials in all sessions for the baseline blocks (left column), within-manifold perturbation blocks (middle column), and outside-manifold perturbation blocks (right column) for monkey J (top row) and monkey L (bottom row). b, Average hand speed during a typical point-to-point reaching task (monkey L). Thus, the hand movements for the BCI tasks are substantially smaller than for the reaching task.

  10. Accounting for within-class differences in learning.
    Extended Data Fig. 6: Accounting for within-class differences in learning.

    a, Relation between amount of learning and initial impairment in performance for monkey J (top) and monkey L (bottom). Each point corresponds to one session. Lines are linear regressions for the within-manifold perturbations and outside-manifold perturbations. *Slope significantly different than 0 (F-test for linear regression, P < 0.05). b, Relation between amount of learning and mean principal angles between control spaces for perturbed and intuitive mappings. c, Relation between amount of learning and mean required preferred direction (PD) change. Same number of sessions as in Extended Data Fig. 2. Figure 3 showed that the properties of the perturbed mappings (other than whether their control spaces were within or outside the intrinsic manifold) could not account for differences in learning between the two types of perturbation. However, as is evident in Fig. 2d, within each type of perturbation, there was a range in the amount of learning, including some outside-manifold perturbations that were learnable5, 7. In this figure, we examined whether learning within each perturbation type could be accounted for by considering other properties of the perturbed mapping. We regressed the amount of learning within each perturbation type against the various properties we considered in Fig. 3. Panel a shows the initial performance impairment could explain a portion of the variability of learning within both classes of perturbation for monkey J. That monkey showed more learning on sessions when the initial performance impairment was larger. For monkey L, the initial performance impairment could account for a portion of the within-class variation in learning only for outside-manifold perturbations; this monkey showed less learning when the initial performance impairment was larger. We speculate that monkey J was motivated by more difficult perturbations while monkey L could be frustrated by more difficult perturbations. Panel b shows that the mean principal angles between control planes were related to learning within each class of perturbation for monkey L only. Larger mean principal angles between the control planes led to less learning. Panel c shows that the required PD changes were not related to learning for either type of perturbation for both monkeys. This makes the important point that we were unable to account for the amount of learning by studying each neural unit individually.

  11. Offline analyses of intrinsic manifold properties.
    Extended Data Fig. 7: Offline analyses of intrinsic manifold properties.

    a, The intrinsic dimensionalities for all sessions for monkey J (left) and monkey L (right). For both monkeys, the intrinsic dimensionalities were not significantly different between days when we performed within-manifold perturbations and days when we performed outside-manifold perturbations (t-test, P > 0.05). Dashed lines, means of distributions; solid lines, mean ± s.e.m. Same number of days as in Extended Data Fig. 1. b, Relation between intrinsic dimensionality and the number of data points used to compute intrinsic dimensionality. For each of 5 days (one curve per day), we computed the intrinsic dimensionality using 25%, 50%, 75% and 100% of the total number of data points recorded during the calibration block. As the number of data points increased, our estimate of the intrinsic dimensionality increased in a saturating manner. c, Tuning of the raw factors. These plots exhibit the factors that were shuffled during within-manifold perturbations. We show for one typical day the average factors ( ) corresponding to the ten dimensions of the intrinsic manifold over a time interval of 700 ms beginning 300 ms after the start of every trial. Within each row, the coloured bars indicate the mean ± standard deviation of the factors for each target. The line in each circular inset indicates the axis of ‘preferred’ and ‘null’ directions of the factor. The length of the axis indicates the relative depth of modulation. The tuning is along an axis (rather than in a single direction) because the sign of a given factor is arbitrary. d, Tuning of the orthonormalized factors. Same session and plotting format as c. The orthonormalized dimensions are ordered by the amount of shared variance explained, which can be seen by the variance of the factors across all targets. Note that the axes of greatest variation are separated by approximately 90° for orthonormalized dimensions 1 and 2. This property was typical across days. The retrospective estimate of intrinsic dimensionality (Fig. 4 and Extended Data Fig. 7a) may depend on the richness of the behavioural task, the size of the training set (Extended Data Fig. 7b), the number of neurons, the dimensionality reduction method and the criterion for assessing dimensionality. Thus, the estimated intrinsic dimensionality should only be interpreted in the context of these choices, rather than in absolute terms. The key to the success of this experiment was capturing the prominent patterns by which the neural units co-modulate. As shown in Fig. 4d, the top several dimensions capture the majority of the shared variance. Thus, we believe that our main results are robust to the precise number of dimensions used during the experiment. Namely, the effects would have been similar as long as we had identified at least a small handful of dimensions. Given the relative simplicity of the BCI and observation tasks, our estimated intrinsic dimensionality is probably an underestimate (that is, a richer task may have revealed a larger set of co-modulation patterns that the circuit is capable of expressing). Even so, our results suggest that the intrinsic manifold estimated in the present study already captures some of the key constraints imposed by the underlying neural circuitry. The probable underestimate of the ‘true’ intrinsic dimensionality may explain why a few nominal outside-manifold perturbations were readily learnable (Fig. 2d). It is worth noting that improperly estimating the intrinsic dimensionality would only have weakened the main result. If we had overestimated the dimensionality, then some of the ostensible within-manifold perturbations would actually have been outside-manifold perturbations. In this case, the amount of learning would tend to be erroneously low for nominal within-manifold perturbations. If we had underestimated the dimensionality, then some of the ostensible outside-manifold perturbations would actually have been within-manifold perturbations. In this case, the amount of learning would tend to be erroneously high for outside-manifold perturbations. Both types of estimation error would have decreased the measured difference in the amount of learning between within-manifold perturbation and outside-manifold perturbations.

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Author information

  1. These authors contributed equally to this work.

    • Byron M. Yu &
    • Aaron P. Batista

Affiliations

  1. Department of Bioengineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, USA

    • Patrick T. Sadtler,
    • Kristin M. Quick,
    • Elizabeth C. Tyler-Kabara &
    • Aaron P. Batista
  2. Center for the Neural Basis of Cognition, Pittsburgh, Pennsylvania 15213, USA

    • Patrick T. Sadtler,
    • Kristin M. Quick,
    • Matthew D. Golub,
    • Steven M. Chase,
    • Byron M. Yu &
    • Aaron P. Batista
  3. Systems Neuroscience Institute, University of Pittsburgh, Pittsburgh Pennsylvania 15261, USA

    • Patrick T. Sadtler,
    • Kristin M. Quick &
    • Aaron P. Batista
  4. Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA

    • Matthew D. Golub &
    • Byron M. Yu
  5. Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA

    • Steven M. Chase &
    • Byron M. Yu
  6. Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA

    • Stephen I. Ryu
  7. Department of Neurosurgery, Palo Alto Medical Foundation, Palo Alto, California 94301, USA

    • Stephen I. Ryu
  8. Department of Physical Medicine and Rehabilitation, University of Pittsburgh, Pittsburgh, Pennsylvania 15213, USA

    • Elizabeth C. Tyler-Kabara
  9. Department of Neurological Surgery, University of Pittsburgh, Pittsburgh, Pennsylvania 15213, USA

    • Elizabeth C. Tyler-Kabara

Contributions

P.T.S., K.M.Q., M.D.G., S.M.C., B.M.Y. and A.P.B. designed the experiments. S.I.R. and E.C.T.-K. implanted the arrays. P.T.S. collected and analysed the data. P.T.S., B.M.Y. and A.P.B. wrote the paper.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

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Author details

Extended data figures and tables

Extended Data Figures

  1. Extended Data Figure 1: Performance during baseline blocks. (142 KB)

    a, Histograms of success rate during the baseline blocks on days when the perturbation would later be within-manifold (red) and outside-manifold (blue) for monkey J (top) and monkey L (bottom). For days with multiple perturbation sessions, the data are coloured according to the first perturbation type. Dashed lines, means of distributions; solid lines, mean ± s.e.m. b, Histograms of target acquisition time during baseline blocks. Number of days for panels a and b: within-manifold perturbations, n = 27 (monkey J), 14 (monkey L); outside-manifold perturbations, n = 31 (monkey J), 14 (monkey L). c, Sample cursor trajectories to all eight targets. At the beginning of each day, the monkeys used the intuitive mapping for 250–400 trials. The monkeys were able to use these mappings to control the cursor proficiently from the outset (as measured by success rate and acquisition time). On all sessions, the success rates were near 100%, and the acquisition times were between 800 and 1,000 ms. No performance metrics during the baseline blocks were significantly different between within-manifold perturbation sessions and outside-manifold perturbation sessions (P > 0.05; success rate, Wilcoxon rank-sum test; acquisition time, two-tailed Student's t-test).

  2. Extended Data Figure 2: Changes in success rate and acquisition time during perturbation blocks. (164 KB)

    In Fig. 2d, we quantified the amount of learning in each session using a single metric that combined improvements in success rate and acquisition time. Here, we consider each metric separately. In each comparison, better performance is to the right. a, Change in success rate from the first 50-trial bin in the perturbation block to the bin with the best performance. The change in success rate was significantly greater for within-manifold perturbations than for outside-manifold perturbations for monkey J (top, P < 10−3, t-test). For monkey L (bottom), the change in success rate was greater for within-manifold perturbations than for outside-manifold perturbations, and the difference approached significance (P = 0.088, t-test). b, Change in acquisition time from the first 50-trial bin in the perturbation block to the bin with the best performance. For both monkeys, the change in acquisition time for within-manifold perturbations was significantly greater than for outside-manifold perturbations (monkey J (top), P < 10−4, t-test; monkey L (bottom), P = 0.0014, t-test). Note that a negative acquisition time change indicates performance improvement (that is, targets were acquired faster). Number of within-manifold perturbations, n = 28 (monkey J), 14 (monkey L); outside-manifold perturbations, n = 39 (monkey J), 15 (monkey L).

  3. Extended Data Figure 3: After-effects during washout blocks. (90 KB)

    After 600 (monkey J) or 400 (monkey L) trials using the perturbed mapping, we re-introduced the intuitive mapping to observe any after-effects of learning. We measured the after-effect as the size of the performance impairment at the beginning of the washout block in the same way that we measured the performance impairment at the beginning of the perturbation block. A larger after-effect indicates more learning had occurred in response to the perturbation. For monkey J (left), the after-effect was significantly larger for within-manifold perturbations (red) than for outside-manifold perturbations (blue) (Wilcoxon rank-sum test, P < 10−3). For monkey L (right), the trend is in the same direction as monkey J, but the effect did not achieve significance (Wilcoxon rank-sum test, P > 0.05). These data are consistent with the hypothesis that relatively little learning occurred during the outside-manifold perturbations in comparison to the within-manifold perturbations. Number of within-manifold perturbations, n = 27 (monkey J), 14 (monkey L); outside-manifold perturbations, n = 33 (monkey J), 15 (monkey L).

  4. Extended Data Figure 4: Learning did not improve over sessions. (122 KB)

    It might have been that, over the course of weeks and months, the animals improved at learning to use perturbed mappings, either one type or both types together. This did not occur. Within-manifold perturbations showed more learning than outside-manifold perturbations across the duration of experiments. Animals did not get better at learning to use either type of perturbation separately (red and blue regression lines, F-test, P > 0.05 for all relationships) nor when considering all sessions together (black regression line, F-test for linear regression, P > 0.05). Same number of sessions as in Extended Data Fig. 2. Each point corresponds to one session.

  5. Extended Data Figure 5: Hand speeds during BCI control and hand control. (108 KB)

    We loosely restrained the monkeys’ arms to the chair’s armrests during experiments. The monkeys minimally moved their hands, but the movements did not approach the limits of the restraints. a, Average hand speeds across all trials in all sessions for the baseline blocks (left column), within-manifold perturbation blocks (middle column), and outside-manifold perturbation blocks (right column) for monkey J (top row) and monkey L (bottom row). b, Average hand speed during a typical point-to-point reaching task (monkey L). Thus, the hand movements for the BCI tasks are substantially smaller than for the reaching task.

  6. Extended Data Figure 6: Accounting for within-class differences in learning. (170 KB)

    a, Relation between amount of learning and initial impairment in performance for monkey J (top) and monkey L (bottom). Each point corresponds to one session. Lines are linear regressions for the within-manifold perturbations and outside-manifold perturbations. *Slope significantly different than 0 (F-test for linear regression, P < 0.05). b, Relation between amount of learning and mean principal angles between control spaces for perturbed and intuitive mappings. c, Relation between amount of learning and mean required preferred direction (PD) change. Same number of sessions as in Extended Data Fig. 2. Figure 3 showed that the properties of the perturbed mappings (other than whether their control spaces were within or outside the intrinsic manifold) could not account for differences in learning between the two types of perturbation. However, as is evident in Fig. 2d, within each type of perturbation, there was a range in the amount of learning, including some outside-manifold perturbations that were learnable5, 7. In this figure, we examined whether learning within each perturbation type could be accounted for by considering other properties of the perturbed mapping. We regressed the amount of learning within each perturbation type against the various properties we considered in Fig. 3. Panel a shows the initial performance impairment could explain a portion of the variability of learning within both classes of perturbation for monkey J. That monkey showed more learning on sessions when the initial performance impairment was larger. For monkey L, the initial performance impairment could account for a portion of the within-class variation in learning only for outside-manifold perturbations; this monkey showed less learning when the initial performance impairment was larger. We speculate that monkey J was motivated by more difficult perturbations while monkey L could be frustrated by more difficult perturbations. Panel b shows that the mean principal angles between control planes were related to learning within each class of perturbation for monkey L only. Larger mean principal angles between the control planes led to less learning. Panel c shows that the required PD changes were not related to learning for either type of perturbation for both monkeys. This makes the important point that we were unable to account for the amount of learning by studying each neural unit individually.

  7. Extended Data Figure 7: Offline analyses of intrinsic manifold properties. (260 KB)

    a, The intrinsic dimensionalities for all sessions for monkey J (left) and monkey L (right). For both monkeys, the intrinsic dimensionalities were not significantly different between days when we performed within-manifold perturbations and days when we performed outside-manifold perturbations (t-test, P > 0.05). Dashed lines, means of distributions; solid lines, mean ± s.e.m. Same number of days as in Extended Data Fig. 1. b, Relation between intrinsic dimensionality and the number of data points used to compute intrinsic dimensionality. For each of 5 days (one curve per day), we computed the intrinsic dimensionality using 25%, 50%, 75% and 100% of the total number of data points recorded during the calibration block. As the number of data points increased, our estimate of the intrinsic dimensionality increased in a saturating manner. c, Tuning of the raw factors. These plots exhibit the factors that were shuffled during within-manifold perturbations. We show for one typical day the average factors ( ) corresponding to the ten dimensions of the intrinsic manifold over a time interval of 700 ms beginning 300 ms after the start of every trial. Within each row, the coloured bars indicate the mean ± standard deviation of the factors for each target. The line in each circular inset indicates the axis of ‘preferred’ and ‘null’ directions of the factor. The length of the axis indicates the relative depth of modulation. The tuning is along an axis (rather than in a single direction) because the sign of a given factor is arbitrary. d, Tuning of the orthonormalized factors. Same session and plotting format as c. The orthonormalized dimensions are ordered by the amount of shared variance explained, which can be seen by the variance of the factors across all targets. Note that the axes of greatest variation are separated by approximately 90° for orthonormalized dimensions 1 and 2. This property was typical across days. The retrospective estimate of intrinsic dimensionality (Fig. 4 and Extended Data Fig. 7a) may depend on the richness of the behavioural task, the size of the training set (Extended Data Fig. 7b), the number of neurons, the dimensionality reduction method and the criterion for assessing dimensionality. Thus, the estimated intrinsic dimensionality should only be interpreted in the context of these choices, rather than in absolute terms. The key to the success of this experiment was capturing the prominent patterns by which the neural units co-modulate. As shown in Fig. 4d, the top several dimensions capture the majority of the shared variance. Thus, we believe that our main results are robust to the precise number of dimensions used during the experiment. Namely, the effects would have been similar as long as we had identified at least a small handful of dimensions. Given the relative simplicity of the BCI and observation tasks, our estimated intrinsic dimensionality is probably an underestimate (that is, a richer task may have revealed a larger set of co-modulation patterns that the circuit is capable of expressing). Even so, our results suggest that the intrinsic manifold estimated in the present study already captures some of the key constraints imposed by the underlying neural circuitry. The probable underestimate of the ‘true’ intrinsic dimensionality may explain why a few nominal outside-manifold perturbations were readily learnable (Fig. 2d). It is worth noting that improperly estimating the intrinsic dimensionality would only have weakened the main result. If we had overestimated the dimensionality, then some of the ostensible within-manifold perturbations would actually have been outside-manifold perturbations. In this case, the amount of learning would tend to be erroneously low for nominal within-manifold perturbations. If we had underestimated the dimensionality, then some of the ostensible outside-manifold perturbations would actually have been within-manifold perturbations. In this case, the amount of learning would tend to be erroneously high for outside-manifold perturbations. Both types of estimation error would have decreased the measured difference in the amount of learning between within-manifold perturbation and outside-manifold perturbations.

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