Inter-to-intra cluster distance ratio

To compute the inter-to-intra cluster distance ratio for working memory activity in the full space (Figure 2—figure supplement 5), we bootstrapped 250 unmixed single-trial working memory activity for seven target locations. First, to compute the mean inter-cluster distance, we computed the pairwise distances between all cluster means, and then computed the grand mean of all the pairwise cluster distances. Inter-cluster distance could be intuitively understood as the measure of separation between clusters. Second, to compute the mean intra-cluster distance, we first computed the intra-cluster distance in each cluster (mean pairwise distance among all the data points in one cluster), and then computed the grand mean of the intra-cluster distance among all the clusters. Intra-cluster distance could be intuitively understood as the trial-by-trial variability among all the target conditions. The inter-to-intra cluster distance ratio is then a concept similar to the signal-to-noise ratio of the working memory activity in the state space. We repeated this procedure for 1,000 times to get a distribution of the inter-to-intra cluster distance ratio, and presented the 5^th to 95^th percentile of this distribution in the boxplot in Figure 2—figure supplement 5. The same procedure was repeated for the motor preparation activity.

To compute the inter-to-intra cluster distance ratio in subspaces (Figure 4c,d), we first projected the full-space activity into the desired subspaces, and then repeated the procedure mentioned above.