Fig. 4: Canonical components of the neural population represent task variables in continuous time.

a, An illustration of the CCA for finding a reduced-dimensional space in the task space and the corresponding subspace in the neural activity space. The canonical components define the maximally correlated subspaces between the task variables and the neural (neur) activity. The 51-dimensional task space was made from 6 task variables by passing each variable through a set of basis functions (Methods). In brief, the basis functions were pulse-shaped temporal delay filters for press, reward and choice events. For continuously evolving task variables (the waiting time, the reward ratio and two-dimensional ___location), the basis functions were a set of instantaneous power functions. Overall, 51 predictors were made using these six task variables. The neural space was made using all simultaneously recorded neurons. b, Left: the weight of the contribution of each task variable in the first ten canonical components, sorted in the descending order of the correlation between the projection of each component in the task and neural spaces. The indices of the components representing waiting time, reward ratio, reward and choice are color coded for easier association. Right: neural representation of four task variables: reward, choice, waiting time and reward ratio for the same sample session on the left. The component that was associated with each of these four task variables was identified as the component for which the absolute value of the weights was highest, compared with the weights for the other task variables. c, Cross-validated Pearson correlation coefficient between the reward predictors, waiting time (left) and reward ratio (right; *P < 0.005), with either individual neurons or clusters of five or more neurons (Extended Data Fig. 6) that are maximally correlated with each reward predictor, compared with the correlation coefficient between the reward predictors and the canonical components (WRFDR; left: \(P\) ≪ \({10}^{-3}\) for clusters and 0.003 for neurons; right: P > 0.1). Each data point is associated with one session (n = 30).