Spline parameterization for unsupervised decoding (SPUD)

We used the SPUD algorithm described in³⁹. We fitted the manifolds with piecewise linear curves. We chose to fit a curve L(y) with ten knots to the data points xi embedded in the two-dimensional spaces by Isomap. Initially, the positions of knots were determined by K-means clustering centroids of the data points. Each knot was connected to the other knot with the highest data point density in between to form the initial curve. Then, positions of the knots were iteratively optimized to minimize (Σi∣∣(L(y)−xi∣∣)∣L(y)∣, where ∣∣(L(y)−xi)∣∣ is the Euclidean distance between the ith data point and the nearest point on the curve, and |L(y)| is the length of the curve.

We picked a random origin on the curve and assigned coordinates from 0 to 1 to the point on the curve. The coordinate of each data point x_i was decoded as the coordinate of its nearest point on the curve. We shifted or flipped the coordinates of the data points to minimize the mean squared error between the decoded coordinates and the rescaled actual time in the movie (rescaled to (0,1]). The decoded time for a given data point was set to the resulting coordinate scaled up to (0, 35) seconds.

For cross-validation, we randomly picked 80% of the instantaneous population activity as the training set (distributed across all the weeks), and the remaining 20% as the test set. We fit the spline to data from the training set and evaluated the decoding performance using data from the test set.

Note that the neural manifold for shuffled data often did not have a perfect ring structure (Supplementary Fig. ^7a). The SPUD would fail without carefully choosing the positions of initial knots. For a fair quantitative comparison between original and shuffled data (Supplementary Fig. ^7b), we chose ten trial-averaged projected instantaneous population activity evenly distributed in time as the initial knots for the shuffled data analysis.