Fig 1.
Spike-triggered analysis of contrast adaptation.
Analyses for three sample cells, representative for Type I (top), Type II (middle), and Type III (bottom), under high contrast (purple) and low contrast (orange). (A) Firing rate histograms, averaged over all trials of high contrast (0–20 s) and low contrast (20–120 s). Note that the firing rates were obtained as trial averages with a different white-noise sequence in each trial. (B) Spike-triggered averages (STAs) for each contrast. (C) Corresponding nonlinearities for each contrast. Note that the slightly non-monotonic shape of the nonlinearity for the second sample cell reflects On-Off response characteristics of this cell [23]. (D) Eigenvalue spectrum obtained by STC analysis. The eigenvalues corresponding to the most and second-most informative features are marked by green and light blue circles, respectively. (E) Scatter plot of spike-triggered stimuli, projected onto the most informative stimulus feature k1 and the second-most informative feature k2. For clarity, only 10% of all analyzed spikes are shown in this plot.
Fig 2.
Information-theoretic analysis of relevant stimulus features.
(A) Scatter plots of information rates carried by the features corresponding to the most positive (v1) and the two most negative (v19, v20) eigenvalues under high-contrast stimulation. (B) Histogram over all Type I and Type III cells of the difference in information rates for v1 and v19. (C) Distributions of information rates for the three most informative stimulus features, k1, k2, and k3, respectively, extracted from the STC analyses. The bar graphs denote the median by a circled black dot, the range between first and third quartile by a thick bar, and the entire range of information values by a thin line. (D) Histograms of the fraction of information captured jointly by k1 and k2, relative to the total information per spike. This analysis was performed only for a subset of cells where recordings with repeated white-noise sequences were available for measuring the total information. (E) Average firing rates for each cell to alternating bright (“ON response”) and dark (“OFF response”) illumination. The gray line depicts equality.
Fig 3.
STA fits with STC-derived features for three sample cells.
(A-C) Fits of high-contrast and low-contrast STAs with features derived from the high-contrast STC analysis. The three sample cells shown in (A), (B), and (C) are the same as in Fig 1. The STC-derived features k1 and k2 (left column) are used as a basis for fitting the high-contrast STA (middle column) and the low-contrast STA (right column). (D-F) Same as (A-C), but using k1 and k2 as derived from the low-contrast STC analysis.
Fig 4.
Population analysis of STA fits with STC-derived features.
(A) Assessment of fit quality by the coefficient of determination R2 for fitting high-contrast and low-contrast STAs with the high-contrast-derived features. The gray line marks identity. (B) Weights of k1 and k2 obtained from fitting the high-contrast STAs (left) and low-contrast STAs (right) by high-contrast-derived features. The colored lines mark the unit circle, which is a bound for the weights. (C) Same as (A), but using for each cell all features obtained as significant from the high-contrast STC analysis. (D-F) Same as (A-C), but based on the low-contrast-derived features for fitting the STAs. (G) Comparison of how much of the contrast-induced variation of the STAs were captured by the two-feature basis obtained from either high contrast or low contrast. This was quantified by the coefficient of determination, R2, for the difference of the fitted STAs.
Fig 5.
STC analysis of contrast adaptation models.
(A) Layout of the models. (i) Spike-feedback model [35,36]. The model consists of a linear filter, a threshold operation to determine spikes, and an exponentially decaying feedback signal that is subtracted from the feedforward filter output after a spike has occurred. (ii) LNK model [37]. Adaptation is modeled by a kinetic model with a resting state R, an active state A, and two inactive states I1 and I2. Possible transitions between the states are indicated by arrows, together with the applied transition rates. The input to the kinetic model, u(t), is derived from a linear-nonlinear cascade and modulates the transition from the resting to the active state as well as from the first to the second inactive state. (B) STAs obtained from the two models under high- and low-contrast stimulation as well as corresponding STC spectra and features k1 and k2. (C) STA fits with features obtained from high-contrast STC analysis. The corresponding R2 values are indicated in the plots. The inset shows the weights for k1 and k2 obtained from the fits, with the gray line indicating the unit circle and data points near the circle corresponding to good fits. (D) Same as (C), but for STA fits with features obtained from the low-contrast STC analysis.
Fig 6.
Analysis of contrast-dependent filter changes in Type II cells after separation of On and Off pathways.
(A) Separation of spike-triggered stimuli into On and Off pathway shown for a sample cell of Type II. Left: Scatter plot of spike-triggered stimuli for high-contrast stimulation, projected onto k1 and k2. Red and blue data points mark the clusters corresponding to the On and Off pathways, respectively. Center: Filters kON and kOFF, obtained by computing the STAs for the two clusters separately, as well as the full STA, obtained from all spikes. Right: Nonlinearities obtained from the separated On and Off clusters (red and blue, respectively) as well as for the entire set of spikes (thick line). (B) Same as (A), but for low-contrast stimulation. (C) STA fits with the kON and kOFF features, obtained under high contrast. Fit of high-contrast STA (i) and low-contrast STA (ii) for the sample cell as well as coefficients of determination (iii) and corresponding weights obtained for the two features (iv) for all recorded cells of Type II. Note that, in contrast to k1 and k2, kON and kOFF are generally not orthogonal to each other. Thus, the sum of squared weights obtained from the fit is not bounded by unity. The inset in (i) shows kON and kOFF. The green data points in (iii) and (iv) mark the sample cell. (D) Same as (C), but using only spikes from the Off-pathway cluster and their features k1 and k2, derived from the high-contrast STC analysis. The insets in (i) show the corresponding eigenvalue spectrum as well as k1 and k2.
Fig 7.
Estimation of spike-timing dynamics.
(A) Stimulus segments of high contrast (left) and low contrast (right) and corresponding spike trains of a sample ganglion cell (here Type III). The cell’s filter output, obtained by convolving the stimuli with the cell’s corresponding STAs, is shown below. Local maxima are marked by green dots. Two segments of the filter output trace and corresponding spikes are shown enlarged to illustrate the collection of relative spike times in a window around local maxima. (B) Histograms of relative spike timing for peak values of the filter output in four different ranges. The boundaries of these ranges are specified in the legend. Histograms are normalized to their peak values. Note that the negative values of relative spike timing do not imply acausal effects. Rather, the computation of the activation level by filtering the stimulus with the STA includes an average delay between the relevant stimulus and the elicited spikes, in the range of around 100 ms (see, for example, the STA peak times in Fig 1B). The relative spike timing observed here can thus be viewed as a correction to this average delay. (C) The characteristic spike-time shift, computed as the location of the maximum of the relative spike-timing histogram, plotted versus the mean filter output peak size in each applied filter output range. The characteristic spike-time shifts become more negative with increasing filter output peak size, showing that stronger activation leads to relatively earlier spikes. The data points corresponding to the histograms depicted in (B) are marked by dots in the respective colors. Solid lines display linear fits to the data. (D) Population averages of the absolute values of the slopes (∆Spike-time shift/∆Activation level) obtained from the linear fits as illustrated in (C) for Type I (blue), Type II (green), and Type III (red) cells. Error bars denote standard errors. Differences between the populations are marked as significant (t-test at 5% significance level, Bonferroni corrected) by asterisks and otherwise by “n.s.” (E) Population average of the curves shown in (C) for Type I (blue), Type II (green), and Type III (red) cells. Error bars denote standard errors. The plot shows that Type III cells span a much larger range of delays than the other two types.
Fig 8.
Contrast-dependent filter changes in a model with spike-timing dynamics.
(A) Layout of the model. The stimulus feeds into a linear-nonlinear model, resulting in a firing rate, from which spikes are obtained by a Poisson process. Individual spikes are then shifted in time (as indicated by the red arrows), with the size of the shift depending on the activation level, which is given by the firing rate. (B) STAs obtained from model simulations for low and high contrast. (C) Eigenvalue spectrum obtained by STC analysis for high and low contrast, with the two most significant eigenvalues marked by green and light blue. (D) Scatter plot of spike-triggered stimuli for high-contrast stimulation of the model, projected onto k1 and k2. For clarity, only 0.2% of all analyzed spikes are shown in this plot. The inset shows the features k1 (green) and k2 (light blue), corresponding to the eigenvalues of the same color shown in (C). (E) Same as (D), but for low-contrast stimulation. (F) STA fits with features obtained from high-contrast STC analysis. Fit of high-contrast STA (i) and low-contrast STA (ii) as well as corresponding coefficients of determination (iii) and weights obtained for the two features (iv). (G) Same as (F), using the low-contrast-derived features.