Fig 1.
Schematic of super-resolved tomographic reconstruction (STR) method.
(A) Response of a sample On parasol retinal ganglion cell to light-intensity steps without spatial structure. Top: Stimulus time course with the insets displaying a point-wise multiplication of the cell’s receptive field with the current stimulus. Bottom: Peri-stimulus time histogram (PSTH) of the cell’s response. For visualization purposes, the PSTH is repeated once. (B) Same cell as in (A), but for a grating stimulus with spatial structure. (C) Schematic depiction of using a spot stimulus (left) to probe the receptive field of a model On-type ganglion cell (center). Orange arrows on top of stimulus signify the shifts of the stimulus for probing the receptive field. Black circles represent the 1.5 σ ellipses of the subunits, red circle represents the 1.5 σ ellipse of the receptive field. Response of the model (right) shows which spot positions led to a strong response (black) and which to a weak response (white). (D) Same as (C), but for a stimulus with an added dark ring around the white spot. (E) 1D probing of model responses with a horizontal stripe (left) at different vertical positions in the receptive field of the same model. The response (right) depended on the vertical position of the stripe. (F) Same as (E), but for a Ricker stripe, which has added dark sidebands adjacent to the white center stripe. (G) Sinograms of the responses of the model to the plain stripes (left) and to the Ricker stripes (right) as measured from 36 stripe angles (steps of 5°) and 60 stripe positions (steps of 2/3 pixels). Dark shading denotes stronger responses. Black rectangles at 90° mark measurements shown in (E) and (F). Green line indicates sine trace of one subunit in the model’s layout. (H) Reconstructions of the sinograms in (G) using filtered back-projection (FBP). Red denotes positive values in the reconstruction, blue negative ones.
Fig 2.
Application of STR to model simulations with realistic settings.
Three sample layouts with 6 (top row), 10 (middle row), and 14 (bottom row) subunits are depicted. First column shows the subunit layouts, with black ellipses portraying the 1.5 σ ellipses of the subunits and red ellipses the 1.5 σ ellipses of the receptive fields. Second column is the receptive field. Third column contains the sinograms for spike rates, i.e., expected spike counts. Fourth column shows the reconstructions from the sinograms in the third column. Red denotes positive values, blue negative values. Fifth column holds the sinograms for measurements of stochastic spike counts. Each combination of the 36 stripe angles and 60 stripe positions was measured only once. Gaussian smoothing has been applied to these sinograms (described in more detail in the main text). Last column pictures reconstructions from the sinograms in the penultimate column.
Fig 3.
Optimal stimulus and analysis parameters.
(A) Sample model layout (left) and receptive field (right) used throughout this figure (layout outlines 1.5 σ ellipses of subunits and receptive field). (B) Illustration of the detection of hotspots in a reconstruction. Background image is FBP reconstruction with red and blue colors representing positive and negative values, respectively. Large dark-blue circle depicts area in which local maxima (yellow crosses) are identified. Local maxima are compared with 0.75 σ ellipses of the underlying subunits (black) to compute an F-score. (C) Sample sinograms (top row) and corresponding reconstructions (bottom row) of measurements with varying stimulus parameters. Surround factors s are 1, 2, and 5 from left to right, stripe width values w are 10, 5, 4. (D) Search for the optimal parameters in the parameter space of surround factor s and stripe width w. Brighter colors denote better average F-score for 1000 model instantiations with ten subunits each. Optimal parameters (s = 2.5, w = 5 pixels) are marked by a black dot. (E) Influence of smoothing the sinogram in position direction on a sample sinogram (top row) and the corresponding reconstructions (bottom row). Standard deviations σpos of the Gaussian filters are (from left to right) 0%, 1.5%, 3%, 4.5%, and 6% of the simulation area size. Smoothing in angle-direction is omitted for these plots (σang = 0°). (F) Like (D), but for search for optimal smoothing of the sinogram in the parameter space of standard deviations for stripe position smoothing σpos (optimum is 2.5%) and stripe angle smoothing σang (optimum is 5°).
Fig 4.
(A) Average F-score (over 1000 instantiations) versus number of subunits for the default parameters (blue, optimal for ten subunits) and for parameters adjusted for each number of subunits (red). (B) Optimal stripe width w and optimal surround factor s, depending on number of subunits. (C) Optimal position smoothing σpos and optimal angle smoothing σang. w and σpos in (B) and (C) were obtained by scaling with the number of subunits (rounded to an accuracy of 0.2 and 0.1, respectively), and s and σang were kept constant. (D) Subunit detection performance (F-score) depending on the ratio of the number of stripe positions versus angles for fixed total numbers of stripe presentations (green: 2160, default; red: half of the default; blue: double the default) and for different numbers of subunits N, using corresponding optimal parameters. Black cross marks the default scenario of ten subunits, 60 stripe positions, and 36 stripe angles. (E) Same as (D) but with the absolute number of stripe positions on the x-axis, up to a maximum of 100 positions. (F) Same as (D) but with the absolute number of stripe angles on the x-axis, up to 100 angles. (G) Subunit detection performance depending on measurement time for different numbers of subunits N, each with optimal parameters. Dashed horizontal line marks a threshold of 0.8. (H) Measurement time required to pass the 0.8 threshold in (G) depending on the number of subunits. (I) Exemplary subunit layout (left) consisting of 50 subunits, reconstructed with noise-free rate responses (center) and spike responses (right). Ricker stripes were presented at 89 positions (required minimum number according to scaling described in main text) and with 202 angles, equating to 3 hours of stimulation. Stripe and analysis parameters were scaled as before, but with a reduced angle smoothing of σang = 2°. (J) Left: Sample subunit layout with an added temporal filter to simulate responses to spatiotemporal stimuli. Center: Spatial filters from spike-triggered clustering with locally normalized L1 regularization of the sample layout’s responses to 30 minutes of coarse binary white noise. Stimulus pixels had a size of 4x4 simulated screen pixels. Red pixels denote positive values, blue pixels negative values, with each filter normalized to its absolute maximum. Right: Spike-triggered clustering with simulated responses to ten hours of fine white noise (2x2 screen pixels).
Fig 5.
Robustness of STR to model variations.
Each row demonstrates the effect on STR of one variation of the model via a sample simulation. Layout of the rows is the same as in Fig 2. Top row shows a model with increased subunit overlap (see Methods for details), apparent from the 1.5 σ subunit ellipses. In the second row, the Gaussian-shaped subunits were replaced by subunits with a cosine profile. For comparability, the ellipses in the subunit layout depiction are 1.5 σ ellipses of Gaussians fitted to the cosine subunits. Third row contains a replacement of the rectified-linear nonlinearity with a rectified-quadratic nonlinearity. Weights of the subunits in the fourth row were not all equal as in the default model, but chosen according to a large spatial Gaussian. In this example, the strongest subunit weight was roughly eight times that of the weakest weight. In the bottom row, a base activity of three expected spikes was added to all responses.
Fig 6.
Variations to the model structure.
(A) Receptive field and subunits (left) of a model with two superimposed subunit layouts (black and green ellipses, respectively), each consisting of ten subunits. Measurements with Ricker stripes (center) yields reconstructions depicted for rate responses and spiking responses (right). Arrows highlight subunits/hotspots described in main text. (B) Same as (A), but for a model with two layouts of differently sized subunits, one containing four subunits (left, black ellipses) and one containing 16 subunits (green ellipses). Measurements with wide Ricker stripes (top row) lead to different reconstructions (right) than measurements with narrow stripes (bottom row). (C) Same as (A), but for a model with one On (black ellipses) and one Off (green ellipses) subunit layout. The depicted receptive field is for On-type stimulation. Measurements with stripes of different polarity (top and bottom row) lead to different reconstructions. (D) Depiction of an LNLNLN model with Gaussian photoreceptors (blue ellipses, representing 1.5 σ contours). Green lines display the connection weights between photoreceptors and subunits by their width.
Fig 7.
Shortcomings of FBP as a reconstruction method for subunit layouts.
(A) Sample stimuli of a vertical (left) and a horizontal (right) Ricker stripe hitting the center of an exemplary elliptical subunit (here in green). (B) Rate responses of a model consisting of only that one subunit depicted as a sinogram. (C) Resulting reconstruction via FBP from the sinogram in (B). Red denotes positive values, blue negative values.
Fig 8.
Experimental application of STR.
(A) Autocorrelation functions, receptive fields (RFs) displayed as 1.5 σ ellipses of Gaussian fits (one distant cell not included), temporal STAs (normalized to unit Euclidean norm), and nonlinearities (scaled to equal maximum) of all identified Off (top) and On (bottom) parasol cells. (B) Excerpt of a sample stimulus projected onto the retina during a flash. (C) Unprocessed sinogram of a sample Off parasol cell. (D) Same sinogram as in (C), but processed by correcting for the receptive field position and applying a Gaussian filter. (E) Overview of the results of four sample cells. Left column depicts spatial STAs, middle column illustrates PSTHs (yellow background designates presentation of the Ricker stripes), right column shows reconstructions from the processed sinograms via FBP. Red colors in reconstructions denote positive values, blue colors denote negative values. Spatial scales of STAs and reconstructions are equal, but reconstruction has higher resolution. PSTHs were computed irrespective of the angle and position of the stripes with a bin size of 10 ms. Sample cell in top row is same cell as in (C) and (D). Bottom two rows contain the results of an analysis of the offset responses of cells. (F) Reconstructions of the sample Off parasol cell from the top row of (E) from separate analyses of the first and second halves of the measurement. (G) Same as (F), but for the On parasol cell from (E). (H) Distribution of the number of hotspots identified in the reconstructions across all recorded Off and On parasol ganglion cells. Colored ticks at the bottom mark the medians for the two cell types. (I) Distribution of the average distance of a hotspot to its nearest neighbor in each ganglion cell’s reconstruction. Ticks at bottom mark the medians.