Shared variance component analysis

The SVCA method gives an asymptotically unbiased lower-bound estimate for the amount of a neural population’s variance reliably encoding a latent signal. A mathematical proof of this is given in the appendix; here, we describe how the algorithm was implemented for the current study.

We first split the population into two spatially segregated populations. To do so, we divided the XY plane into 16 nonoverlapping strips of width 60 μm and assigned the neurons in the even strips to one group and the neurons in the odd strips to the other group, regardless of the neuron’s depth. Thus, there did not exist neuron pairs in the two sets that had the same XY position but a different depth, avoiding a potential confound that a neuron could be predicted from its own out-of-focus fluorescence.

Neural population activity was binned at 1.2- to 1.3-s resolution (see above), and each neuron’s mean activity was subtracted from its firing trace. We divided the recording into training and test time points (alternating periods of 72 s each), thereby obtaining four neural activity matrices: F→train, F→test, G→train, and G→test of size N→neurons×N→time points, where F→ and G→ represent activity of the two cell sets. We compute the covariance matrix between the two cell sets on the training time points asS^k=u→k⊤ F→testG→test⊤v→k/Ntime pointsTo obtain the fraction of reliable variance (Fig. 1L), we normalize this reliable variance by the arithmetic mean of the variances of the test set data for each cell set on the corresponding projections, Sk, tot=(u→k⊤F→testF→test⊤u→k+v→k⊤G→testG→test⊤v→k)/2.