Figure 2:

(a) Each of the three classes of methods was applied to the simultaneously-recorded fluorescence of a population of neurons (y1,y2,…yq) to extract latent variables. Top, Approach 1: a standard dimensionality reduction method (e.g., LDS) applied directly to calcium imaging recordings, extracting corresponding low-dimensional latent variables at each time point (illustrated here with two dimensions, z₁ and z₂). Middle, Approach 2: deconvolution is applied separately to each neuron’s fluorescence trace to estimate its underlying spiking activity (s1,s2,…,sq). A standard dimensionality reduction method (e.g., LDS) is then applied to the estimated spiking activity to extract latent variables (z1 and z2). Bottom, Approach 3: A unified method (e.g., CILDS) that takes calcium imaging recordings as input and performs deconvolution and dimensionality reduction simultaneously to extract the latent variables (z1 and z2). (b) Cartoon depicting the intuition behind the difference between Approaches 2 and 3. Center column: a latent variable z (representing, for example, common input) is used to generate spike trains which, in turn, are used to generate fluorescence traces. Left column: Deconvolution is performed neuron by neuron (Approach 2, deconv-LDS), then an LDS is applied to the estimated spiking activity to extract latent variables. Right column: A unified method (Approach 3, CILDS) is applied to all neurons together to dissociate the calcium transients from the underlying shared spiking activity among neurons (i.e., the estimated latent variable). This is done by jointly performing deconvolution and dimensionality reduction, as illustrated by the the double arrows. Note that the estimated spiking activity is depicted here as spike trains for visual clarity, even though they are in fact continuous-valued time courses.