Conceptual sketch of spike-triggered non-negative matrix factorization.

(A) The stimulus ensemble (top) is a matrix of pixels by spikes. Each column is the effective spike-triggered stimulus of a given spike (bottom). The stimulus ensemble serves as the input to the matrix factorization (dimensions not to scale). (B) Using semi-NMF, the stimulus ensemble is decomposed into two smaller matrices, the non-negative spatial modules and their corresponding weights. Together, these represent a lower-dimensional approximation of the stimulus ensemble. (C) The models can be reshaped into two-dimensional spatial layouts. Some modules exhibit localized structure and are identified via their spatial autocorrelation as subunits, whereas others (here displayed with reduced saturation) capture noise.

STNMF with Accelerated Fast HALS (AF-HALS) converges faster than the previous implementation of STNMF.

Reconstruction error of STNMF implemented with AF-HALS (black) and STNMF based on the active-set method used in Liu et al. (2017; gray) for a sample salamander ganglion cell. Arrows denote the number of iterations. Insets allow visual inspection of the recovered modules at different states during the iterations. Although the active-set method calculates more accurately at each step, the reconstruction process is similar after ten iterations in both methods (blue and red insets). The active-set method does not reach the low error that AF-HALS accomplishes within a few seconds. The visualized subunits of the active-set method after 45 minutes (purple inset) arrive at the state that AF-HALS reached within five seconds (green inset). Top right: Distributions of iteration speeds (measured as inverse of each iteration duration) of the active-set method (gray) and of AF-HALS (black) for the sample cell.

A single run of NNSVD-LRC-initialized STNMF yields subunits consistent with recurring subunits from 100 different, randomly initialized runs.

(A) Contour outlines of subunits of 100 randomly initialized STNMF runs for two sample cells (fast-OFF type) from salamander retina. Subunits of different runs are regarded as identical if they exhibit substantial spatial overlap (Jaccard index > 0.5; see Methods). Subunits that emerged in more than half of the runs are colored, with saturation corresponding to the level of recurrence (see colorbar) and marked with a percentage of recurrence. (B) Comparison of subunit layouts recovered with NNSVD-LRC (gray) and with randomly initialized runs and a 50%-recurrence-criterion (yellow). Each yellow outline is the contour of the mean of the recurring subunits in (A). For the two sample cells, a single NNSVD-LRC-based STNMF run found all recurring subunits of the randomly initialized runs. (C) Distributions over 40 analyzed fast-OFF cells of the fraction of recurring subunits recovered by a single run with NNSVD-LRC (dark gray) or by 100 runs with random (light gray) initialization. Recovered subunits are determined via the Jaccard index (see above and Methods). NNSVD-LRC-based STNMF reliably finds more of the recurring subunits than the individual randomly initialized runs (averaged over 100 runs for each cell).

Consensus analysis for determining the suitable weight of sparsity regularization.

(A) Module sparsity (fraction of zeros in the modules) of the decomposition with increasing sparsity regularization for a sample salamander cell. The first step in the analysis is finding the window (dotted frame) in which regularization starts to dominate the decomposition, that is, module sparsity approaches unity. Within that range, the consensus of decompositions is probed. (B) Reordered consensus matrix showing the pairwise agreement between spike pairs across 30 decompositions. High consensus is indicated by the ten clusters on the diagonal corresponding to ten reliably recovered subunits as visualized in the second subunit layout inset of (C). (C) Stability curve for the same cell as obtained from consensus matrices at different regularization strength. The diagonal clustering structure in (B) is measured with a scalar value, the cophenetic correlation coefficient (CPCC). The correlation increases with increasing sparsity regularization, supported by refined subunit outlines (first and second inset). After a peak in stability, subunits become too sparse (third inset) until they eventually vanish when regularization becomes too strong. (D) The stability curves for a wide range of regularization parameters of a subset of cells (dark) compared to the curves of all 40 fast-OFF cells (yellow) from a salamander retina. Dotted frame indicates the range of interest as specified in (C).

Subunits of AF-HALS-based STNMF match previous findings and bipolar cell receptive fields.

Subunit layouts of salamander cells (yellow) are compared to Liu et al. (2017; gray). Subunits are represented as 1.5-sigma ellipses of Gaussian fits in gray, and as yellow contour lines (see Methods). (A) Decomposition for two example cell receptive fields. Receptive field outlines are 1.5-sigma ellipses of Gaussian fits. The subunit layouts resemble the previously estimated subunits. (B) Comparison of subunit mosaics of a population of fast-OFF cells, colors like in (A). Most of the previously identified subunits as well as some additional ones are recovered. (C) Examples of subunit layouts for adjacent fast-OFF ganglion cells with overlapping (shared) subunit, as obtained with both methods. (D) Comparison of recovered subunits with receptive field of a simultaneously recorded bipolar cell. One of the subunits (yellow outlines) matches the bipolar cell receptive field (red outline). Panels (A gray), (B left), (C left), and (D rightmost) are adapted from Liu et al. (2017) licensed under CC BY 4.0.

STNMF recovers subunits in all four major cell types of the marmoset retina.

Data is shown from ON and OFF parasol cells (one retina; mid-periphery) and ON and OFF midget cells (one retina; periphery). Note that the scale bars differ between the four sample cells. (A) Receptive fields, STNMF modules, and subunit layouts of sample ON and OFF parasol cells and ON and OFF midget cells. Modules identified as localized subunits by their autocorrelation (see Methods) are marked by colored frames and depicted together in a colored subunit layout of contours superimposed on the receptive field contour (gray; top right). The outlines are numbered in descending order according to the mean STNMF weight. (B) Receptive field mosaic of the four main cell populations (colored). Gray outlines correspond to the receptive fields of the corresponding type with reversed polarity of preferred contrast. The temporal dynamics of the receptive fields are depicted by the temporal filters of the spike-triggered average (insets; scale bar 200 ms). The sample cells of (A) are marked in red. (C) Subunit mosaics corresponding to (B). Sample cells marked in red. For some cells, in particular several OFF midget cells, no subunits could be reliably identified. Scale bars (300 µm) correspond to both (B) and (C).

Stability curves align well among cells of same functional type.

(A) Stability defined by the cophenetic correlation coefficient (CPCC) for different sparsity regularization strength for ON and OFF parasol cells (one retina) and ON and OFF midget cells (one retina). Consensus for cells without nonlinear integration in their receptive fields may be undefined as indicated by incomplete stability curves. The dark curves represent the medians (excluding not defined values) of the corresponding population. Stability curves show the expected increase and plateau. (B-D) Consensus comparison of ON and OFF midget cells. (B) Superimposed stability curves of the different cell types, showing slight but systematic differences. (C) Stability curves projected onto the first two principal components of all curves from the two populations. (D) Inter- and across-type Euclidean distances in the two-dimensional PCA space, showing significant differences of distances across versus within cell types. Central line and box represent the median and the interquartile range (IQR; 1st to 3rd quartile), respectively. Whiskers extend to most extreme values within 1.5 IQR, and dots indicate outliers.

Subunits match anatomical properties of bipolar cells.

(A) Subunit diameters of the four major cell types collected from three marmoset retinas. (Box plots like in Figure 7D.) Parasol and midget subunits are comparable in size to diffuse bipolar and midget bipolar cells, respectively. Bipolar cell sizes were previously reported in the macaque retina according to physiological measurements (Dacey et al. 2000) and to calculations based on cone inputs (Boycott and Wässle 1991). (B-C) Examples of the subunits of adjacent parasol (B) and midget (C) cells. Markers represent centers of mass. While the parasol subunits can display substantial overlap from adjacent cells, the midget cell subunit layouts are typically distinct with no or few overlapping subunits. (D) Histograms of subunit pair overlap. Overlap between subunit pairs within ON (top), OFF (middle), and across ON and OFF (bottom) parasol and midget cells is measured with the Jaccard index. Only overlap pairs with an index greater than zero are visualized. The dashed line indicates the lower bound of what we consider here as substantial overlap, suggesting shared bipolar cell inputs. Consistent with the examples in (C), we observed few overlapping pairs for midget cells, but a considerable amount (B) for OFF parasol cells.

ON- and OFF-type subunit mosaics are aligned, while receptive field mosaics show anti-alignment.

(A) Contour mosaics of ON- and OFF-type parasol (top) and ON- and OFF-type midget (bottom) ganglion cell receptive fields, each from one retina, mid-periphery and periphery, respectively. Centers of mass (black markers) inside the region of interest (ROI, shaded area) are considered for analysis (colored markers inside black frame, right). One center-of-mass mosaic is shifted relative to the other, as schematically indicated (right); n = numbers of enclosed subunits. (B) Topographical map visualizing the inter-mosaic coordination energy (IMCE) at different mosaic shifts around the original position in the center. (C) Dependence of the IMCE on radial distance by averaging across angles in (B), displaying mean (black) and standard deviation (shaded gray). IMCE topographical map and radial average curves are z-scored, and the shift distance r is normalized to the median homotypic nearest-neighbor distance of the ON-type mosaic. (D-F) Same as (A-C), but for the subunit mosaics of the cell populations analyzed in (A-C). The red rectangles in (D) correspond to the red rectangles of the corresponding cell populations in (A). The decreasing IMCE curves in (F) indicate that corresponding ON and OFF subunit mosaics tend to be aligned.