Figure 1. The proposed generative model and inverse problem.

Observed (green) and latent (orange) variables pertinent to the $j^{\mathrm{𝗍𝗁}}$ neuron are indicated, according to the proposed model for estimating the signal (blue) and noise (red) correlations from two-photon calcium fluorescence observations. Calcium fluorescence traces $\left(y_{t,l}^{(j)}\right)$ of $L$ trials are observed, in which the repeated external stimulus $\left({\mathbf{𝐬}}_t\right)$ is known. The underlying spiking activity $\left(n_{t,l}^{(j)}\right)$ , trial-to-trial variability and other intrinsic/extrinsic neural covariates that are not time-locked with the external stimulus $\left(x_{t,l}^{(j)}\right)$ , and the stimulus kernel $\left({\mathbf{𝐝}}_j\right)$ are latent. Our main contribution is to solve the inverse problem: recovering the underlying latent signal $\left(\mathbf{𝐒}\right)$ and noise $\left(\mathbf{𝐍}\right)$ correlations directly from the fluorescence observations, without requiring intermediate spike deconvolution.