Model and fitting procedure. A. Phototransduction cascade and differential equations that describe operation of key components. Abbreviations are as follows: Stim: stimulus; R: receptor (photopigment) activity; P: phosphodiesterase activity; G: cGMP concentration; S: rate of cGMP synthesis; I: membrane current; C: calcium concentration. B. Fits (red) of a model with four free parameters (γ, σ, η, KGC) to the responses of a mouse rod (see Matlab code for parameter values). All measured responses are from the same rod, and all model responses use the same values for the free parameters. Responses to different flash strengths or flash timing have been displaced vertically in the left two panels. The weakest flash in the far left panel produced on average ∼3 activated rhodopsin molecules, and each successive flash was twice as bright. The bottom trace in the middle panel was to a light step alone, while the other traces in this panel each had 5 superimposed flashes. The horizontal scale bar for the inset in B is 1 sec.

Photoreceptor model and fits. A. Comparison of measured responses (black) with predictions of full (red) and linear model (blue) to variable mean noise stimulus (gray). Full model responses use consensus parameters from fitting responses of multiple cells of each type simultaneously, with the dark current and sensitivity allowed to vary between cells (see Methods). The linear model was generated from fitting the low-contrast responses of the full model (see Methods). Insets expand regions in gray boxes. B. Fraction of variance explained for the full model fit to each cell individually (y-axis) plotted against that for the consensus model that has fixed parameters across cells except for the dark current and sensitivity.

Steps in model inversion. A test current was generated from a variable mean noise stimulus using the full model (far right). Step 1 (right) converts this current into changes in cGMP and calcium using Eqs. 4 and 5 (see Figure 1). Step 2 converts the time course of the cGMP and calcium into that of the PDE activity using Eqs. 3 and 6. Finally, step 3 converts the PDE to the stimulus using Eqs. 1 and 2. The estimated stimulus is identical to the initial stimulus because there is no added noise and the inversion process is exact.

Test of model inversion based on measured responses. A. Stimulus (gray in top panel), measured response (black in lower panels) and estimated stimulus (blue in top panels) calculated by using the measured response as input to the inverse model as in Figure 3. Estimates are able to recover both the periodic changes in mean intensity and the more rapid superimposed stimulus modulations (insets). B. Variance explained for stimulus estimates based on the average response across multiple stimulus trials compared to that based on individual responses. Since the model captures only the deterministic part of the response, noise in the individual responses lowers the accuracy of the estimates and causes the points to fall below the unity line. This effect is modest but systematic.

Light-adaptation clamp procedure. Starting with an initial stimulus (left), we generate a target or desired response (second panel from right). In this case, the target was chosen to be the response of a linear phototransduction model; for a sinusoidal input, a linear model produces a sinusoidal output. We use the (linear) target response as the input to the inverse phototransduction model and identify the stimulus required to elicit that response (red in third panel from left). Substantial stimulus modifications are required for the full model to produce a sinusoidal output. Finally, we confirm that the modified stimulus works as designed in direct recordings, in this case from a primate cone (right panel).

Compensating for increment/decrement asymmetries in responses to sinusoidal stimuli. A. Photoreceptor responses (bottom, black) to original sinusoidal stimuli (top, black) and modified stimuli (red). Thin lines in the bottom panels are best-fitting sinusoids for reference. Original and modified stimuli are shown above the recorded responses. B. Mean-squared-error between measured responses and best fitting sinusoids for modified stimuli plotted against that for original stimuli for each recorded cell. Sinusoidal fits had lower error for the modified stimuli compared to the original stimuli.

Compensating for adaptation produced by changes in mean light level. A. Responses to a brief light flash delivered before and during a step in light intensity. For the original stimuli, flashes delivered before and during the step are identical, and the resulting responses decrease in amplitude ∼2-fold (summarized on x-axis of bottom panels). Red traces show responses to stimuli designed to compensate for the adaptation produced by the change in light intensity (following approach in Figure 5). B. Summary of gain changes (amplitude of response during the step divided by that of response prior to step) for responses to modified (y-axis) and original (x-axis) stimuli.

Manipulating kinetics of photoreceptor responses. A. Responses of primate cones and mouse rods to brief flashes (black) and stimuli designed to slow down responses slightly (left) and more substantially (right) (red). Thin traces show the target responses used to generate the modified stimuli. B. As in A for stimuli designed to speed up responses. C. Measured change in time-to-peak plotted against predicted change for a variety of manipulations of kinetics as in A and B.

Cone and post-cone adaptation. A. Responses of a cone (top), horizontal cell (middle) and On parasol RGC (bottom) to the step and flash protocol for both original and modified stimuli. B. Summary of experiments like those in A, plotting the change in gain for the modified stimuli against that for the original stimuli.

Tests of the ability of the model to generalize across stimuli and across cells. A. Generalization across stimuli. Responses of primate and mouse rods were fit to the variable mean noise stimulus only and model performance was evaluated for responses to the flash family and flashes and steps, using the fraction of variance explained to evaluate the fits (see Figure 1). This generalization performance to stimuli not included in the fit (y-axis) is compared to performance when the model is fit to all stimuli (x-axis). B. Generalization across cells. Models were fit to the variable mean noise stimulus. The performance of each cell was evaluated for a model in which all cells were fit (x-axis), or in which the test cell was excluded from the fit (y-axis).

Errors in stimulus estimation. Power spectra of stimulus and residual (stimulus - estimate) for example primate and mouse rods and cones. The stimulus and residual power spectra diverge strongly at low frequencies where the estimates are close to the actual stimulus, and converge at higher frequencies when the estimate becomes poor. The convergence point depends on photoreceptor type, as expected for the different kinetics of rod and cone responses.