Extended Data Figure 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.
