(A) Cartoon illustrating the structure of single-trial population responses along different directions in the state-space. The thick and thin blue lines represent respectively the trial-averaged and single-trial responses. Single-trial responses start in the vicinity of the trial-averaged activity . Both the network and single-cell mechanisms dynamically shape the structure of the single-trial responses, here represented as graded blue ellipses. The red and black lines represent respectively the amplified and random directions considered in the analyses. (B) Time course of the variability computed along the amplified direction (solid trace) and along a random direction (dashed trace) for one example stimulus and one example session (287 simultaneously recorded neurons). In this and all the subsequent panels the amplified direction is defined as the population activity vector at the time when the norm of the trial-averaged activity of the pseudo-population pooled over sessions and animals reaches its maximum value (thick dashed line). Thin dashed lines denote stimulus offset. Shaded areas correspond to the standard deviation computed over 20 bootstrap subsamplings of 19 trials out of 20. (C, E, and G) Variability amplification (VA) computed for the amplified and random directions on the calcium activity data (panel C), on trajectories generated using the network model (panel E) and the single-cell model (panel G); (see Materials and methods, Section 'Single-trial analysis of population OFF responses'), for one example stimulus (same as in B). The network and single-cell models were first fitted to the trial-averaged responses to individual stimuli, independently across all recordings sessions (13 sessions, 180 ± 72 neurons). 100 single-trial responses were then generated by simulating the fitted models on 100 initial conditions drawn from a random distribution with mean and covariance matrix equal to the covariance matrix computed from the single-trial initial states of the original responses (across neurons for the single-cell model, across PC dimensions for the recurrent model). Results did not change by drawing the initial conditions from a distribution with mean and isotropic covariance matrix (i.e. proportional to the identity matrix, as assumed for the theoretical analysis in Materials and methods, Section 'Single-trial analysis of population OFF responses'). In the three panels, the values of VA were computed over 50 subsamplings of 90% of the cells (or PC dimensions for the recurrent model) and 50 shuffles. Error bars represent the standard deviation over multiple subsamplings, after averaging over all sessions and shuffles. Significance levels were evaluated by first computing the difference in VA between amplified and random directions () and then computing the p-value on the difference between and across subsamplings (two-sided independent t-test). For the network model, the VA is higher for the amplified direction than for a random direction, and this effect is significantly stronger for the real than for the shuffled responses. Instead, for the single-cell model the values of the VA computed on the real responses are not significantly different from the ones computed on the shuffled responses. (D, F, and H) Values of the VA computed as in panels C, E, and G pooled across all 16 stimuli. Error bars represent the standard error across stimuli. Significance levels were evaluated computing the p-value on the difference between and across stimuli (two-sided Wilcoxon signed-rank test). The fits of the network and single-cell models of panels E, G, F, and H were generated using ridge regression ( for both models).