Table 1.
A non-exhaustive list of data showing similar qualitative behavior of photoreceptors across different taxa.
Figure 1.
Illustration of the Dynamical Adaptation (DA) model.
(A) The stimulus is convolved with two mono-lobed filters to produce the signals and
. These yield the neural response according to Eq. (2). The non-linear term in the equations, which involves the signal
, modulates gain (related to the area under the red curve) and time scale (related to the width of the red curve) in a history-dependent manner. Time scale and gain thus vary together, with small gains associated with short time scales and large gains associated with long time scales. (B) The
filter is broader than the
filter and, hence, can capture memory effects and mimic feedback. The parameter set is based upon the salamander data (see Methods and Table 2).
Figure 2.
Response to single and paired flashes in the dark—comparison of data and DA model predictions.
(A) Top: Traces of recorded hyperpolarizations in a red-sensitive turtle cone, induced by light flashes delivered in the dark. Integrated flash intensities range from 41 (lowest amplitude) to 6.7·105 photons/µm2 (highest amplitude) in intervals of factors of 2.1. (Data from Fig. 19 of Ref. [14].) Bottom: DA model predictions for corresponding intensities. The dotted line represents the response to a flash 100 times more intense than the largest experimental flash. (B) Peak hyperpolarization against flash intensity for the data displayed in Fig. 2A (open circles), for a separate experiment (closed circles; data from Fig. 7 of Ref. [15]), and for the DA model predictions displayed in Fig. 2A (solid, red line). (C) Peak delay (following the input flash) against flash intensity, extracted from Fig. 2A. (Symbols are as in Fig. 2B.) (D) Parametric plot of peak delay against peak hyperpolarization (normalized by the maximum hyperpolarization). Data points (open circles) summarize several experiments (from Fig. 10 in Ref. [13]). Parameters were chosen so as to minimize the root-mean-squared error in the voltage traces, and result in faster peaks at high intensities; alternative fitting criteria could better fit the peak timing. (E) The model predicts that responses to paired flashes add non-linearly. A conditioning flash (10 ms, 560 photons/µm2/s) is presented at
ms. A test flash of identical intensity to the conditioning flash is presented either before or after the conditioning flash, and the response is measured. The response to the conditioning flash (in the absence of any test flash) is represented as a thick, grey line, while the colored traces represent the paired flash response minus the conditioning response on its own. The test flash delivery times are indicated by small, vertical ticks of the corresponding colors. (F) Peak response to the test flash (normalized by the peak response to the conditioning flash alone) against the delay between conditioning and test flashes. Negative delays correspond to situations in which the test flash preceded the conditioning flash. Circles represent data (from Fig. 12 of Ref. [13], modified to undo a saturation correction performed there), while the solid, red line represents the DA model prediction. The DA model predictions for this figure are calculated using the parameter set BHL (see Table 2).
Figure 3.
Goodness-of-fit of the model output and robustness with respect to parameter variations.
(A) The three panel illustrate the relatively shallow way in which the goodness-of-fit, , is degraded from its maximum of 0.94 by varying one of the model parameters. Optimal parameters were obtained from the traces of Fig. 2A and are given in Table 2 (BHL parameter set). (B) Contour lines representing the degradation of the goodness-of-fit with respect to variations in pairs of parameters, as a way to illustrate the reliable parameter subspace. Successive contour lines correspond to 1% increments in goodness-to-fit degradation, with the widest line corresponding to a value of
equal to 80% of its maximum. The contour lines were derived from the Hermitian matrix computed for the optimal parameter values.
Figure 4.
Response to light steps in the dark—comparison of data and DA model predictions.
(A) DA model predictions (using the parameter set B, see Table 2). Step intensities range from 5·102 to 1·107
photons/µm2/s by factors of
. (B) Absolute value of peak hyperpolarization and steady-state hyperpolarization against step intensity. The peak response (top, black line) and the steady state response (closed circles) are from Fig. 7 of Ref. [18], and the DA model predictions (red, solid lines) use parameter set B (see Table 2). The gain was set to match the peak response.
Figure 5.
Response to flashes and steps on light backgrounds of different intensities—comparison of data and DA model predictions.
(A) Experimental responses (left) and DA model predictions (right) for brief flashes of light presented at time 0 on backgrounds of increasing intensity. Data intensities and traces are extracted from Fig. 4 in Ref. [16], where the authors varied flash intensities so as to match peak response among the different background intensities. Model predictions are computed using the corresponding parameter set DN (see Table 2). The dotted curve follows Eq. 8. (B) Experimental responses (left) and DA model predictions (right) for steps of light on three backgrounds of increasing intensity. Data intensities and traces are extracted from Fig. 1 in Ref. [16]. Model predictions are computed using the corresponding parameter set DN (see Table 2). Background light intensity is indicated as in (A). (C) DA model predictions of responses to 100 ms bright (+) and dark (−) flashes equal to the background intensity, delivered at time 0. Flashes are delivered on top of a light background of 2.6·105 photons/µm2/s (
, for a comparison with the strength of the non-linearity). (D) Family of peak responses to steps against step intensity. Each curve of the family corresponds to a different background of light; the background intensity increases to the right. The abscissa measures the total light intensity, so the zero crossing of each curve yields the corresponding background intensity. Negative ordinate values (upward segments of the curves) result from light steps, while positive ones (downward segments of the curves) result from dark steps. Data (open circles) are extracted from Fig. 8 in Ref. [18]. DA model predictions (solid lines) use parameter set B (see Table 2). (E) DA model predictions of responses to 10 ms flashes with large contrasts, delivered at time 0, varying from 0.5 to 215 times the background intensity by factors of 4; the individual values are listed against the curves. The background light intensity,
, was set to
. The parameter set B was used to compute the traces in both D and E (see Table 2). (F) Step response sensitivity (normalized by the response in the dark) against background intensity. Data points (open circles) are from Fig. 11 in Ref. [18]. The model prediction (red line) is computed using parameter set B (see Table 2). Both data and model satisfy the Weber-Fechner law over seven decades. (G) Delay (following the input flash) of peak responses to a fixed flash against background light intensity. Data are extracted from Fig. 5A (closed, black circles) and from Fig. 12 in Ref. [15] (open, black circles). DA model predictions use parameter set DN (red circles, see Table 2).
Table 2.
Four different parameter sets used to fit data.
Figure 6.
Analyses of salamander cone data with a Linear-Non-linear (LN) model and with the DA model.
(A) A white-noise flickering light stimulus (top) is presented to a salamander cone while its membrane potential (bottom) is measured. (B) Enlarged cone response trace and comparison of the experimental curve, the LN model prediction, and the DA model prediction. The LN model prediction deviates from the recorded trace with a RMS residual of 0.277 (in units of response s.d., or 0.533 mV); equivalently, a value of 0.927
. The DA model has a RMS residual of 0.243 (in units of response s.d., or 0.468 mV); equivalently, a
value of 0.943
. The DA model follows the experimental output more closely, especially at peaks and troughs where discrepancies with the LN model are most prominent (signaled by black triangles). (C) Schematic illustration of the LN model. The stimulus is convolved with a best-fit linear filter, obtained by reverse correlation of the response to the stimulus. A static, non-linear function is then evaluated, with the output of the convolution as its argument, to produce the predicted response. (D–F) Scatter plots of the experimental cone response against the model predictions. In (D), the cone outut is compared to a linear prediction (denoted
in panel C). The LN non-linearity is read off from this plot. In (E), the cone output is compared to the full LN model prediction. In (F), the cone output is compared to the DA model prediction. Low intensity (green) and high intensity (red) points are highlighted as those for which the preceding 300 ms of stimulus is in the brightest and dimmest 10%, respectively. The slope obtained from all points taken together is 1 (black line), and tick marks are 1 s. d. For the linear prediction, the slopes of the two subsets of points are 1.36 (green line, low light) and 0.76 (orange line, bright light). For the LN predictions, the slopes of the green and orange lines are 1.18 and 0.87, respectively. These differences are statistically significant (
0.01, see Methods). For the DA predictions, the slopes of the green and orange lines are 0.98 and 0.94, respectively. This discrepancy is not statistically significant (
0.1, see Methods). Thus, the DA model prediction replicates the experimental output more precisely than the LN model prediction. Overall
values in the three cases (D), (E), and (F) are 0.918, 0.927, and 0.943, respectively; thus, the LN non-linearity accounts for 11% of the missing variance, while the DA model accounts for 30% of the missing variance. (G) Plot of the instantaneous gain as a function of the average light intensity in the preceding 300 ms. The instantaneous gain was calculated, at each time, as the slope of the linear fit in an experimental response-versus-linear prediction scatter plot. (H) Variation of the response time scale as a function of the preceding light intensity. Instantaneous gain values along the time trace were split into ten percentile groups and, for each group, the time of maximum cross-correlation between input and experimental response was calculated (see Methods). The resulting value is plotted against the instantaneous gain value of the percentile. The slope of the best linear fit is 9.6±3.6 ms. (I) Variation of the shape and, specifically, time scale of the instantaneous best linear filter as a function of the preceding light intensity. Three linear filters, computed for the highest, middle, and lowest 10% of instantaneous gain values, are plotted. The data show that both the gain and time scale vary dynamically with light intensity.
Figure 7.
Response to periodic inputs—DA model predictions.
Parameter set B was used for theory curves in this figure (see Table 2). (A) Traces of model responses (thin lines) to 25%, 50%, and 100% contrast sinusoidal inputs with frequencies 0.1, 1.25, 2.5, and 5 Hz, superimposed on a background light intensity of 3.6·105 photons/µm2/s. The thick line represents the input. The abscissae are scaled so as to allow for two periods. Horizontal lines for each trace represent the potential when the input sinusoid has 0 amplitude. (B) ‘Effective gain’, calculated as the ratio of the trough-to-peak amplitude of the response to the through-to-peak amplitude of the input. Different colors correspond to different background intensities, varying from 360
to 7.2·105
by factors of ∼2. At low input contrast (thin lines), the DA model behaves as a band-pass filter, while at high input contrast (thick lines) it behaves nearly as a low-pass filter.
Figure 8.
Response and dynamical adaptation with respect to natural fluctuating inputs—DA model fit and predictions.
(A) Top: Sample of a light intensity trace, from the natural time series in Ref. [50]. Bottom: The corresponding DA model response trace (red), superimposed upon goldfish cone recording (black) from Ref. [49]. The agreement between the two traces is quantified by an values of 0.934. The parameters
,
, and
were fitted to the data; otherwise parameters from set B were used (see Table 2).(B) Top: Different clip from the same light intensity trace as in (A), from the natural time series in Ref. [50]. Bottom: The corresponding model response trace. (C) DA model predictions of responses to small (100-photon) flashes superimposed on the fluctuating natural light intensity. The flash is presented at time
. Thin pink curves represent individual flash responses, while the thick red curve is the average over all such responses. The weakest (1st percentile) and strongest (99th percentile) peak responses are measured as −0.0238 mV and −0.5524 mV, respectively, i. e., they differ by a factor greater than 20. The dotted thick red curve is the flash response in the presence of a constant background matched to the mean of the fluctuating input. The dotted red curve peaks at
0.081 mV, while the solid red curve peaks at
0.135 mV. Thus, individual flash responses vary greatly as a function of background history, and their mean is offset with respect to the constant-background case. (D) Response to a fluctuating input with time-varying contrast. Top: Superimposition of several input traces. The standard deviation of the flicker switches, suddenly and periodically, from its natural value to zero, with a period of 2 s, while its mean remains constant. Bottom: Trace of the mean DA model response to the time-varying flicker. (The thick red line represents an average over multiple natural stimuli. The pink area represents the standard error of this average.) Each switch is signaled by an over- or under-shoot in the mean response, depending upon the direction of the switch. The ‘steady-state’ mean response is greater (more hyperpolarized) in the constant-background half-period than in the fluctuating-background half-period. Parameter set B was used for all theory curves in (B), (C), and (D) (see Table 2).
Figure 9.
Schematic illustration of the phototransduction cascade.
Initially, photons are absorbed by rhodopsin molecules. This triggers a sequence of biochemical reactions resulting in photoreceptor hyperpolarization and calcium influx. We highlight the reaction step in which activated phosphodiesterase (red) increases the rate of conversion of cyclic GMP to GMP (green). The documented calcium feedback mechanisms (blue) include positive and negative regulations of reaction rates (denoted by the symbols + and − respectively).
Figure 10.
Behavior of the DA model for different parameter values.
(A) and (B) Families of responses to a flash in different light backgrounds. We use the BHL parameter set as default parameter set, with changes in and
as indicated. The flash intensities take the values
,
,
,
,
, with the background light intensities ranging from
to
by factors of 10, respectively. (A) and (B) represent identical curves; traces were normalized by their peak values in (B) so that shapes can be compared. As
increases from
to
, the shape of the saturated curves remain unchanged, but the onset of the non-linearity and the amplitude of the curve are affected. As
increases, the second, overshoot lobe becomes shallow, and hence more difficult to observe (especially in the potential presence of noise). (C) Comparison of the three sets of model parameters used to fit data. Responses to a flash superimposed upon a light background are displayed for different background intensities. In each panel, the five curves correspond to background intensities increasing from from
to
by factors of 10; the associated flashes occur at time 0, last for 1 ms, and have unit Weber contrast, i.e., have equal intensity to that of the background. We note that the value of gamma in the BHL panel is higher than in other panels likely because it was fit only to flash responses in the dark, so that amplitude shifts at high background were not included in the fitted data. (D) DA model responses to a weak flash against a dim background (green,
), a weak flash against a bright background (red,
), and an intense flash against a bright background (blue,
). Note that, in the presence of a bright background, zero crossings always occur at the same point and, despite the 400-fold difference in flash strength, the intense-flash response is only 5-fold greater than the weak-flash response.
Figure 11.
Behavior of DA model responses in the presence of Gaussian fluctuating inputs. All calculations in this figure use parameter set B.
(A) Top: A sample Gaussian input with correlation time of 200 ms. Middle: Model responses for three different contrasts of the fluctuating input. The mean input intensity is given by . Bottom: After normalization of the response by the input's standard deviation, one can see signatures of the model's non-linearity as the curves do not collapse unto a single curve. (B) Mean response of a model photoreceptor presented with Gaussian flickers with three different contrasts: analytical and numerical results. Larger contrasts yield less hyperpolarized responses, on average. Black dots: numerical result; red curve: analytical result. (C) Mean flash responses of a model photoreceptor in the presence of Gaussian flickering backgrounds with different variances. Top: Average flash responses were calculated numerically by running simulations with two different stimuli: one with Gaussian flicker only, the other with Gaussian flicker and superimposed flashes. The average flash response was obtained as the difference between the two outcomes, averaged over flicker instantiations. Bottom: The fractional difference between flash responses in the presence of Gaussian flicker with three different contrasts. Solid lines: numerical result (±1 SEM error bars); dotted lines: analytical result. (D) Fractional change in average flash response as a function of flicker correlation time. From Eqs. (44, 45, 46), the magnitude of the average flash response depends upon the correlation time of the random flicker. Black dots: numerical result; red curve: analytical result. (E) Responses of a model photoreceptor in the presence of flicker with time-varying contrast. We fed the DA model Gaussian flicker with standard deviation alternating between 35% and 5%, with a 1 second period (top). Average flash responses were calculated at different times during the period (middle), as was done in (C). Sample average flash responses are displayed in the inset panel, while the main panel shows the variation of the flash response amplitude across one period. Black curve: numerical result (±1 SEM error bars); red curve: analytical result. The average response (bottom) was also calculated numerically (black, ±1 SEM) and analytically (red). Note the overshoots of the average response following contrast switches (see the text for an explanation).
Figure 12.
Illustration of the non-linear transformation of fluctuating inputs.
(A) A concave functional form, as that of the gain prefactor in the flash response (Eq. (49)), increases the mean in the transformation. That is, downward fluctuations in contribute more to increasing
than upward fluctuations to decreasing it. Therefore,
. The flash response gain in the presence of a fluctuating background is thus larger than that in the presence of a constant, mean-matched background. (B) A convex functional form decreases the mean in the transformation, so that
. Depending upon the strength of the background,
, and the relative structure of the two kernels,
and
, the mean response to a fluctuating input may be either suppressed or enhanced compared to the flicker-free case. The DA model (see, e. g., Eq. (55)) takes on the convex form shown here as
in the case with
(and
), thus decreasing the mean response relative to that in the presence of a constant input. In the case in which
and
differ in their timing, the functional form becomes concave as in (A) and the mean response to a fluctuating input is enhanced. The modulation of the mean response therefore depends sensitively upon stimulus and kernels parameters.