Extended Data Fig. 9: At a population level, the distributions over temporal discounts and reversal points are statistically independent.

However, in principle, the joint distribution can be decoded without assuming factorization over time and magnitude. a, Estimated reversal points as a function of estimated temporal discount factors, for photo-identified and putative DANs. Only neurons with reversal points in the range of reward magnitudes given in the experiment (1 μl–8 μl) were included. The p-value refers to the chi-squared test, considering the null hypothesis that the joint distribution over the temporal discounts and reversal points is equal to the product of the marginals, considering 16 degrees of freedom, n=122, χ²=27.86. b, We simulate a population of n=100 units with a diverse set of temporal discount (γ) and reversal points uniformly sampled, and decode the reward distribution over magnitude and time (not assuming these features are independent), from the responses at the cue. Importantly, these simulations were done for the cue that predicts a variable amount after a delay of 3 s. We first apply the inverse Laplace for each reversal point and get the temporal evolution of reversal points (middle). Then, for each time we decode the distribution over reward magnitudes (lower), by dividing the probability for each time point by one minus of probability of reward=0. c, The same simulations but for all cues, as described in Fig. 4. Data underlying the figure can be found in the Supplementary Data.