Refractory Sampling Links Efficiency and Costs of Sensory Encoding to Stimulus Statistics

Encoding efficiency decreases with broadening stimulus bandwidth

In each experiment, a photoreceptor was adapted to the same daylight level before starting unit-contrast GWN modulation (Fig. 4B), in which frequency range was low-pass filtered to 20, 50, 100, 200, and 500 Hz, while its amplitude range was flanked by contrasts of 1 (twice the mean) and −1 (darkness). This gave each stimulus the average contrast of ∼0.32. Theoretically, each of these light intensity time series is maximum entropy among stimuli with the same bandwidth and variance. Because light emission from the source can be modeled by Poisson statistics, we could calculate from photon fluctuation estimates their input information rates during repetitive stimulation (Fig. 2A–D; Table 1; see Materials and Methods). Then, the encoding efficiency of photoreceptors (their recorded outputs) could be assessed as the ratio between the corresponding output and input information rates (R/R_input) for each stimulus.

Intracellular recordings (Fig. 4C) showed that the narrow bandwidth stimulus (20 Hz cutoff) evoked the largest responses, whereas broadening the bandwidth reduced the responses. The effect was similar to that caused by increasing the playback velocity of GWN stimulation (Juusola and de Polavieja, 2003), implying that, when the stimulus bandwidth exceeded that of the responses (27.8 ± 1.0 Hz half-maximum cutoff, mean ± SD, 4 photoreceptors), more of its information allocated (i.e., was wasted on) frequencies too fast for the fly to see. This, however, did not affect voltage noise (Juusola et al., 1994) (Fig. 4D), which mostly reflects the average quantum bump (sample) size that integrates the mean responses (Juusola and Hardie, 2001a). Consequently, the mean response (signal; Fig. 4D) and SNR diminished with the broadening bandwidth (Fig. 4E), although, with too narrow a bandwidth, the stimulus lacked higher-frequency information that the fly eye can process and transmit (Zheng et al., 2006; Wardill et al., 2012). Therefore, these findings could much explain why photoreceptors' information transfer rate (Fig. 4F) first rose to ∼320 bits/s at 100 Hz stimulus cutoff (where GWN encoding was maximal) before falling to ∼190 bits/s at 500 Hz cutoff.

Information in GWN stimulation (R_input) increases steadily with broadening bandwidth (Fig. 4G, light gray). Interestingly, however, photoreceptors' encoding efficiency (R/R_input; black) neither reached a peak along their information capture of 50–100 Hz GWN (Fig. 4F, gray squares) nor plateaued thereabouts, as could happen if their rhabdomeres sampled light changes with a 100% photon-to-bump conversion rate (quantum efficiency). Instead, photoreceptors' encoding efficiency decayed exponentially with broadening stimulus bandwidth: from 80% to 90% for 20 Hz stimulation to ∼5% for 500 Hz GWN (Fig. 4G, black). These concurrent but opposing trends suggest that quantum efficiency of sampling adapted to the way light information was distributed within the stimulus bandwidth, further contributing to the photoreceptors' encoding efficiency.

Photoreceptors can sample more information from naturalistic stimuli than GWN

To determine whether the maximum information rate to 100 Hz GWN (Fig. 4F), having minimal temporal correlations, represented photoreceptors' capacity, or whether their information sampling could increase with input correlations, we examined encoding of naturalistic stimulation (NS).

Neighboring pixels in natural scenes most probably belong to the same object or background, reflecting similar light intensities, whereas object boundaries or edges reflect differently, separating the world into darker and lighter features (Field, 1987; Ratliff et al., 2010). A naturalistic light intensity time series, as a slice across these features, is not random but contain structured asymmetric contrast variations, which correlate strongly and drive photoreceptor output vigorously (van Hateren, 1997). But NS can also contain less dynamic sequences, such as intensities scanned over a smooth surface; these adapt responses to lower information rates (van Hateren and Snippe, 2001). Therefore, we chose a highly variable NS sequence (Fig. 2E, blue) that included large step-like transitions between longer dark and bright contrasts (van Hateren, 1997; (Zheng et al., 2009) for this test. To attribute potential improvements in encoding to the temporal structure (intensity correlations) of the light patterns (contrasts), NS stimulation was scaled to have the same mean daylight intensity as the GWN.

The given NS sequence (Fig. 4H) consistently evoked large voltage responses with higher SNRs (Fig. 4I) and information transfer rates (Fig. 4J) than the GWN stimuli. The recordings were performed sequentially and occasionally repeated in the same photoreceptor (e.g., Fig. 4F,J shows information rates of 7 complete recordings series from 5 cells). Because the responses were highly reproducible, it became evident that the highest information rate estimate (∼320 bits/s) to GWN stimuli did not represent the capacity, whereas, equally, the higher estimate (∼400 bits/s) to the NS was unlikely the maximum either. Presumably, there are other light intensity time series, which could evoke responses with even more information. Nonetheless, the findings indicated that Drosophila photoreceptors, overall, encoded the NS sequence more efficiently than broadband GWNs (200–500 Hz) of high R_input (Fig. 4K), but less efficiently than narrow-band GWN stimuli (20–100 Hz) of lower R_input (Fig. 4G). Specifically, encoding of ≤50 Hz GWN was submaximal because even their input information rates (Table 1) were below or about what photoreceptors can (or are expected to) sample from information-rich NS (Fig. 4J), whereas encoding of ≥200 Hz GWN was inefficient (Fig. 4K) because much of this information was inaccessible to photoreceptors, too fast to be sampled reliably.

To confirm that these encoding characteristics were independent of the ambient recording conditions, we also performed the experiments in flies with ∼5°C lower head temperatures (∼20°C) (Tables 2 and 3). Predictably, because of more sluggish phototransduction reactions in cooler photoreceptors (Juusola and Hardie, 2001b), responses were slower and their information transfer rates and encoding efficiencies lower (Juusola and Hardie, 2001b) than at the flies' preferred temperature (Sayeed and Benzer, 1996) of 25°C (Fig. 4). However, the photoreceptors' information capture from naturalistic stimulation was again higher than that from GWN stimuli by the same margins (NS: ∼300 bits/s; GWN: ∼220 bits/s).

Encoding retains sensitivity to highly-structured local contrast changes

To what degree does photoreceptors' high information capture from NS reflect the immediate time order of light intensities (local contrast changes) rather than their global amplitude or frequency distributions? To explore this question, which is about efficiency of information sampling over different time scales, we added two different light intensity time series of equal means to the stimulation regimen. In addition to NS (Fig. 2E, blue), we now recorded responses to a stimulus (green) that had a randomized time order of NS intensities (decorrelated; spectrally “white”; Fig. 2G) but the same distribution. We also recorded responses to a Gaussian stimulus (Fig. 2F, orange), in which “pink” (1/f) frequency distribution (Fig. 2G) approximated that of NS. These three distinctive stimuli were presented successively to each tested photoreceptor. Our aim, through comparative quantitative analysis, was then to link differences in encoding (these stimuli) to differences in (their) information allocation.

We discovered that the responses (Fig. 5A) to the original NS (blue) were larger than those to the shuffled (green) or “pink” stimuli (orange). Again, all the responses had similar noise power (Fig. 5B), indicating equivalent average bump (sample) size, as expected after adaptation to the same mean intensity (compare GWN experiments above) (Juusola and Hardie, 2001a). But because the signal (average response) power and thus SNRs (Fig. 5C) were lower in responses to shuffled-NS and Gaussian-1/f stimuli, photoreceptors sampled less information from them. The respective information transfer rates were ∼74% (286 bits/s; Fig. 5D) and ∼78% (301 bits/s) of that to NS (386 bits/s), and these relationships remained the same also at 20°C (Tables 2 and 3). Hence, once photoreceptors adapted to stimulus repetition, the fine time course of light intensity fluctuations (local contrast changes) largely determined their high information capture, with sampling being less sensitive to the global amplitude distribution. Unsurprisingly, the encoding efficiency to shuffled-NS, which had the highest information content but minimal correlations between successive intensity values, much like GWN, was only 7%, half of that to NS (Fig. 5E). Shuffling reduced especially the conspicuous longer intensity fluctuations (phasic or “edge-like” contrast changes) that evoked the largest responses to NS. Instead, by now flipping (too) fast between intensities, much of this stimulus information became invisible to photoreceptors, akin to high-frequency GWN. More surprisingly, however, we found that the encoding efficiency to NS and Gaussian-1/f stimuli was approximately the same (∼15%), despite their different mean contrasts (Fig. 2E,F) and information contents (Fig. 2I). This implies that, while adapting to 1/f frequency distribution of the stimuli, photoreceptors retain sensitivity to the most representative contrast changes, lasting ∼≥30 ms (Fig. 5A–C; 2–30 Hz stimulus frequencies).

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Figure 5.

Drosophila R1–R6 photoreceptors' responses to naturalistic light intensity time series (NS, blue) are larger and carry more information than responses of the same cells to shuffled-NS (green) or Gaussian 1/f (orange) intensity series. A, Means (thick) and SD (thin) of intracellularly recorded voltage responses to the repeated stimuli. Each stimulus had the same mean light intensity (10⁶ photons/s). B, Signal (average response) power to NS is larger than to shuffled-NS or Gaussian-1/f stimuli, but their noise powers are similar. C, SNR of the responses is the highest to NS and the lowest to shuffled-NS. D, Accordingly, the information rate of the responses is the greatest to NS (**p = 2.0–3.8 × 10⁻²). Different lines indicate individual recordings from the same cells. E, Encoding efficiency is equally high for naturalistic and Gaussian-1/f stimuli but lower for shuffled-NS (***p = 8.9 × 10⁻⁴). D, E, Data are mean ± SD.

These findings, together with the results from the broadband GWN stimulation trials (Fig. 4J,K), imply that Drosophila photoreceptors can sample more visual information from the more-structured (“bursty” or “phasic”) high-contrast changes of appropriate duration than from decorrelated or symmetric (Gaussian) contrast distributions. Thus, GWN, shuffled-NS, or Gaussian-1/f stimulation, regardless of their bandwidth and amplitude modulation and the retina temperature, underestimates the potential information capacity of photoreceptors, which can encode more information from natural-like light intensity fluctuations.

Stochastic photoreceptor model encodes realistically

What is the biophysical basis for these encoding differences? Why did Gaussian or decorrelated light inputs of high information content (Fig. 2I) cause clearly submaximal neural output? To gain mechanistic insight into these open questions, we needed to understand how the different stimuli were sampled and processed at the level of microvilli. Therefore, we first applied the recently established stochastic Drosophila photoreceptor model (Song et al., 2012) to simulate responses for the stimuli used in the recordings. The model integrated quantum bumps from 30,000 microvilli to macroscopic LIC (Fig. 6A). This was converted to voltage responses through a HH-type cell body membrane model (Fig. 6B) (Vähäsöyrinki et al., 2006; Song et al., 2012), which also regulated the electromotive force for the light-sensitive current across all microvilli, similar to real cells. The bump latency and waveform dynamics were adjusted by those of the mean adapted photoreceptors (Juusola and Hardie, 2001a; Song et al., 2012), with free parameters fixed (Song et al., 2012).

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Figure 6.

Stochastic Drosophila photoreceptor model encodes light information much like a real photoreceptor. A, The model generates LIC bumps, mimicking 30,000 stochastically operating microvilli. B, Bumps sum up a macroscopic LIC that charges up voltage responses, V_m, on an HH-type photoreceptor membrane model, having capacitance, C_m, voltage-gated K⁺ conductances (g_ksh, g_dr, g_novel), and K⁺ and Cl⁻ leaks (g_Cl and g_Kleak) (Vähäsöyrinki et al., 2006; Song et al., 2012). C, 20 Hz GWN evokes the largest simulated responses; 500 Hz the smallest. Plots show the means (colored) and individual responses (thin gray). The simulations used the same band-limited unit-contrast GWN stimuli as the recordings in Figure 4C, but at the light level of 10⁵ photon/s. This is because the model lacks the intracellular screening pigments (a pupil mechanism), which reduce light input by ∼10-fold (see Fig. 12). D, Signals (thin traces) adapt but noise (thick) is unchanged by GWN modulation; power spectra calculated from the time series difference between individual responses and their mean (signal) for each stimulus. Because simulations lack recording noise, muscle activity, and intracellular pupil, there is less low-frequency noise power than in recordings (compare Fig. 4D). E, SNR of the model output is the highest for 20 Hz GWN. Broadening the GWN bandwidth reduces the maximum but widens the bandwidth of reliable responses (SNR > 1). F, The simulations' information transfer rates vary with the GWN bandwidth; from 190 to 320 bits/s. G, Encoding efficiency (R/R_input) of simulated photoreceptor output decreases with broadening GWN bandwidth, much like in the real recordings (compare Fig. 4G). H, NS evokes large simulated responses; mean (blue) and 10 responses (cyan). I, SNR of the model output to NS (cyan area) exceeds that to GWN for all tested bandwidths. J, Information transfer to NS is >20% higher than to the maximum GWN (100 Hz cutoff), having the same relative ratios as the recordings (compare Fig. 4J). K, Photoreceptors encode more efficiently NS than 500 Hz GWN, similar to the recordings (compare Fig. 4K).

The simulated voltage responses (Fig. 6C) to the band-limited GWN stimuli closely matched the real recordings (Fig. 4C), showing similar dynamics with the estimated noise power, SNR, and information transfer rate (Fig. 6D–F), suggesting equivalent encoding. Differences were minor and predictable, mostly attributable to the missing recording noise, microsaccadic eye movements, long-term adaptation, and intracellular pupil mechanism that dynamically limits photon influx to microvilli (Song et al., 2012), making the simulations proportionally less noisy at lower frequencies (Figs. 4D and 6D). The simulations further lacked information fed via feedback synapses and gap junctions from the neighboring cells (Zheng et al., 2006; Wardill et al., 2012). These extra inputs likely broadened the bandwidth in real recordings, particularly to 20 Hz GWN (Figs. 4H and 6H, thin red lines).

Nonetheless, most importantly for this analysis, the model output to naturalistic stimulation (Figs. 6H and 7A, blue) differed as predicted from those to GWN, shuffled-NS (Fig. 7A, green) and Gaussian-1/f stimuli (Fig. 7A, orange), closely following the behavior of the real recordings in vivo (Figs. 4 and 5). Aptly, the simulated responses to NS had a higher signal power, SNR and information transfer rate than to the other stimuli (Fig. 6I,J: GWN; Fig. 7B–D: shuffled-NS and Gaussian-1/f), whereas the noise powers of the simulations showed virtually identical frequency distributions, as expected for the same mean intensity stimuli (Juusola and Hardie, 2001a) (Figs. 6D and 7B). Furthermore, for the tested high-input information stimuli (Fig. 2I), the simulations had the highest encoding efficiency with NS (Fig. 7E), which evoked the largest response fluctuations. Tables 2 and 3 show these correspondences at 20°C.

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Figure 7.

Simulated voltage responses of Drosophila R1–R6 photoreceptors to naturalistic light intensity time series (NS) are larger and carry more information than those to shuffled-NS or Gaussian 1/f intensity series of the same mean (10⁵ photons/s). A, Means (thick) and SD (thin) of the simulated responses to the repeated stimuli. B, Signal (average response) power to NS is larger than to shuffled-NS or Gaussian-1/f stimuli, but their noise powers are similar. C, SNR of the responses is highest for NS and lowest for shuffled-NS. D, Accordingly, the information rate of the responses is the greatest to NS. E, Encoding efficiency is the highest to NS and lowest to shuffled-NS. The simulated responses behaved very similar to the real recordings (compare Fig. 5). A different Gaussian-1/f stimulus sequence (of the same statistics) was used in real recordings (compare Fig. 5A), but this made little difference to signaling performance (as predicted by the stochastic adaptive photoreceptor model).

Information increases with sample rate modulation

Because the stochastic photoreceptor model sampled and processed light information much like its real counterparts, we could assess with simulations the contribution of microvilli refractoriness to the observed differences in encoding. We did this systematically by comparing, for GWN stimuli and NS, the bump (sample) counts of the stochastic model (Fig. 8B) to the bump counts of a deterministic model (Fig. 8A). In the stochastic model, refractory microvilli cannot produce bumps; whereas in the deterministic model, microvilli had no refractoriness, converting practically every absorbed photon to a bump. In the simulations, the inputs (photon counts) to the two models were identical and the bumps (outputs) generated by their 30,000 microvilli had the same average shape and latency distribution (see Materials and Methods). Therefore, the synchronized ratios between the model outputs (their bump counts) should provide us unique but representative dynamic quantum efficiency estimates for each stimulus (Fig. 8C).

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Figure 8.

Stochastic microvilli refractoriness reduces sensitivity (sample numbers) through a large fall in quantum efficiency (bump-to-photon conversion ratio). A, Sample (bump) count of a deterministic photoreceptor model to unit-contrast GWN (20, 50, 100, 200, and 500 Hz cutoffs) and NS of the same mean brightness (10⁵ photon/s). Every absorbed photon causes a bump, except those hitting the same microvillus before the previous bump has ended. Here, only ∼7% failed. B, In the steady-state-adapted stochastic model, microvilli refractoriness reduces sample (bump) counts to GWN and NS by ∼65% (dotted red lines); compare the lower sample counts with the same stimuli (A). Photon absorbed in the same microvillus before the previous bump ends or during refractory period are lost. C, Quantum efficiency was estimated as the ratio between the sample counts of the stochastic (B) and deterministic (A) photoreceptor models at each moment of time; it shows coordinated fluctuations for each stimulus, which represent a dynamic account of successful microvilli activations. D, Sample (bump) rate modulation of stochastic photoreceptor model (B) was quantified as SD of the sample counts to each stimulus. Mean ± SD to each stimulus was calculated from three data sections (50% overlapping 500-ms-long windows) that matched those used in the information rate estimation (see Materials and Methods). E, Sample rate modulation (as SD) to the given stimuli correlated clearly with their corresponding information rate estimates (from Fig. 6F,J).

At bright illumination, many stochastically operating microvilli failed to respond to photons because photons arrived/were absorbed during the refractory period. This fall in quantum efficiency (Fig. 8C; bump-to-photon ratio) reduced dramatically the overall sample count (Fig. 8B), here by ∼65%. Importantly, the simulations revealed that the diminishing information transfer rates with broadening GWN bandwidth (Figs. 4F and 6F) resulted from diminishing bump fluctuations, henceforth called sample rate modulation. This was caused by the diminishing contrast visible to Drosophila and further modulated by the fall and fluctuations in quantum efficiency (Fig. 8C). Nonetheless, the average sample rate to the different bandwidth GWN stimuli remained the same (Fig. 8B, red dotted line). Thus, with the bumps to different GWN being of the same average size (Juusola et al., 1994; Juusola and Hardie, 2001a) (Fig. 6D), the smallest output modulation to 500 Hz GWN simply comprised the fewest bumps (Fig. 8B, wine). As the photoreceptor membrane affects the signal and noise equally during bump integration (Juusola and de Polavieja, 2003; Song et al., 2012) and data processing theorem (Shannon, 1948), the smallest sample rate modulation must have directly contributed to the lowest information transfer rate (Figs. 4F and 6F). At the opposite end of this scale, naturalistic stimulation (blue), which incorporated larger contrast changes in bursty sequences, used microvilli more efficiently. It evoked the largest sample rate modulation (Fig. 8B), causing the highest SNR (Figs. 4H and 6H) and rate of information transfer (Figs. 4I and 6I) in the photoreceptor output.

These observations were quantified by basic statistical metrics. The relationship between the size of sample rate modulation, as measured by SD, and GWN bandwidth (Fig. 8D) traced photoreceptors' rate of information transfer (Figs. 4F and 6F) to the same inputs. The weakest correlation was with 20 Hz GWN (Fig. 8E). Whilst this low-frequency stimulus evoked large sample rate modulation, with its range approaching that of NS, its high SNR allocated only a fraction of photoreceptors' full frequency range. This penalized it against the broader bandwidth responses, with integration over its narrower bandwidth (Eq. 2) giving a proportionally lower information rate.

These results support earlier suggestions that photoreceptors' adaptation to different light intensity levels largely reflects a divisive steady-state nonlinearity (van Hateren and Snippe, 2001) and propose refractory information sampling as an important mechanism for it. A photoreceptor's sensitivity is set by its quantum efficiency (mean bump/photon conversion rate). At dim conditions, sensitivity is high (∼100% quantum efficiency) because there are many more available microvilli than incoming photons are being sampled (to bumps). Whereas, at very bright conditions, sensitivity is low (≪50% quantum efficiency) because many microvilli are refractory and most incoming photons are not sampled (to bumps). Markedly, these effects are further augmented by subsequent adaptation in average sample size; the bumps are large in dim and small in bright illumination (Wong et al., 1982; Juusola and Hardie, 2001a). Dynamic changes in their waveforms also contribute, but to a lesser extent, to photoreceptors' information transfer (Song et al., 2012).

Refractoriness accentuate intensity changes in image pixels

We further discovered how stochastic refractoriness of light-activated microvilli, through exerting a memory of past events in bump integration (Song et al., 2012), accentuated certain stimulus features relative to others. To elucidate this phasic nonlinearity and how it contributed to information sampling, we first highlight simulated and recorded examples where fluctuations in refractoriness, as measured by quantum efficiency, shaped responses (Fig. 9A) to 20 Hz GWN (left), 100 Hz GWN (middle), and NS (right). In these plots, sensitization is indicated in green and desensitization in orange.

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Figure 9.

Fluctuations in quantum efficiency accentuate certain stimulus features relative to others. More microvilli recover from refractoriness (and are available to generate bumps) during dark than light contrasts. The resulting increase in quantum efficiency (QE) enhances bump production rate to a subsequent light increment (positive contrast: C), sensitizing photoreceptor output (R). A, Gray panels (left, middle, right) represent how quantum efficiency shapes voltage responses to 20 Hz and 100 Hz GWN and NS, respectively. In each case, the recorded voltage responses (top first row) are compared with the corresponding simulations (second row) of the stochastic photoreceptor model. The quantum efficiencies were estimated as in Figure 8. Signals sensitized by increase in quantum efficiency are shown in green; desensitized ones in orange. Left, Following a larger dark contrast (black area), increase in quantum efficiency QE₁ enhances the response R₁ to the first contrast C₁, whereas subsequent quantum efficiency QE₂ reduces, desensitizing the second response R₂ to an equal-sized contrast C₂. Middle, Contrast C₃ is ∼3 times C₄, but with more dark contrasts (black areas) preceding C₄, QE₃ = QE₄, responses R₃ and R₄ reach the same size. Right, Dark contrasts are more representative during NS, causing more dynamic quantum efficiency fluctuations. Contrast C₆ is ∼2 times of C₅. Dark contrasts (black areas) before C₅ are larger than those before C₆, fluctuating quantum efficiency (QE_5–6) so that responses R₅ and R₆ become approximately equal. B, SD of quantum efficiency is larger during NS than 50–500 Hz GWN stimuli, but not for 20 Hz GWN. C, SD of quantum efficiency during different stimuli correlates with the corresponding information transfer rate of photoreceptor output. Average SD to each stimulus was calculated from three data sections (50% overlapping 500-ms-long windows) that matched those used in the information rate estimation (compare Fig. 8D,E).

The simulations in left and middle panels of Figure 9A show how, after longer darker stimulus periods (bottom, dark contrasts, black), transient increases in quantum efficiency (middle; QE₁ and QE₄) sensitized photoreceptor output (top; local voltage responses: R₁ and R₄) to light increments (bottom; light contrasts: C₁ and C₄). Equally in the same traces, brighter periods of stimulation reduced quantum efficiency (QE₂ and QE₃) to same-sized (C₂) or larger (C₃) light increments, desensitizing photoreceptor output (R₂ and R₃). In other words, after darker periods, fewer microvilli were refractory and a photoreceptor sampled more bumps, integrating larger macroscopic responses to comparable light increments than after brighter periods. As expected, fluctuations in stochastic microvilli refractoriness were prominent to NS (right), in which dark contrasts are more representative (Ratliff et al., 2010). Thus, these features (e.g., C_5–6), by strongly modulating the number of microvilli that participates in the response, increased phasic nonlinearities in quantum efficiency (QE_5–6) and photoreceptor output (R_5–6), comparable with what we see in the real recordings (top) (compare Juusola, 1993; Song et al., 2012). The simulations also revealed how photoreceptors' information transfer rate increases with quantum efficiency fluctuations (Fig. 9B,C). These basic correlative analyses further imply that quantum efficiency fluctuations modulate bump integration (sample rate modulation: Fig. 8D,E), supported by strong relationships between the magnitude of these fluctuations, simulation bandwidth, and corresponding information transfer rates.

These results imply that stimulus-dependent changes in microvilli refractoriness are likely the main reason why Drosophila photoreceptors sample visual information from different stimulus statistics differently. However, the weakness of our observations and interpretations is that these much depend upon the accuracy of the dynamic quantum efficiency ratio, estimated between the two models: with and without the refractory microvilli. Therefore, to reduce potential bias in our conclusions, we next compared the normalized outputs (bump counts) of the models (Fig. 10A,B) directly. The normalization of each model output was done for each stimulus; by dividing the respective bump count (at each 1 ms time bin) by its maximum (across the time bins).

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Figure 10.

A, Stochastic refractoriness enhances relative sample rate modulation to contrast changes. Compare the normalized bump counts of the stochastic (means: wine and blue) and deterministic no refractory period (dark yellow area) models to unit-contrast 500 Hz GWN (above) and NS (below). Arrows indicate enhanced phasic components to sudden contrast changes; SDs (orange and cyan). The model outputs are time-aligned. B, Stochastic refractory period in Drosophila photoreceptor model enhances neural representations of sudden light changes (i.e., intensity changes in image pixels). This is clearly seen when comparing normalized bump counts of the stochastic model (colored lines; mean ± SD) with that of a mock model (green histogram), in which refractory periods have taken randomly from the same distribution. C, Stochastic refractoriness reduces information in photoreceptor output to NS only by ∼12% (403 bits/s) with respect to the model without the refractoriness (A, 455 bits/s). The loss of information for the other tested stimuli was similar: 20–100 Hz GWN (7 ± 5%); 200–500 Hz GWN (17 ± 1%), NS-shuffled (13%), and Gaussian-1/f (13%). This limited information loss is attributable to enhanced responses to contrasts, encoded by relatively large changes in sample numbers. In a deterministic model, where the bumps were randomly assigned refractory periods from the same distribution, more information was lost (17%; 379 bits/s). Again, responses to the other stimuli showed comparable losses: 20–100 Hz GWN (11 ± 10%); 200–500 Hz GWN (23 ± 3%), NS-shuffled (18%), and Gaussian-1/f (17%). D, Mean microvilli usage is ∼40%, but NS drove stochastically operating microvilli population more dynamically than GWN stimuli, which lack longer dark or large bursty contrasts.

The normalized sample counts of the stochastic simulations (colored traces) appeared clearly more differentiated than the deterministic ones (dark yellow histograms) (Fig. 10A), which essentially depict normalized counts of the photons reaching the microvilli. Most obviously, as expected, the normalized bump counts of the stochastic model to NS (right) accentuated contrast fluctuations more than those of the deterministic model (SD_stochastic > SD_{deterministic}; p = 2.3 × 10⁻³; n = 20 nonoverlapping 50-ms-long samples), but these effects were noticeable even in the bump counts to 500 Hz GWN stimulus (left). Refractoriness thus caused sample rates to react more rapidly and strongly (arrows) to sudden light increments and decrements, exerting a short period of amplification, relative to the foregoing rate. Specifically, the early on-responses to light increments comprised more samples, and thus had better SNRs than subsequent responses, for which fewer microvilli were activated. Likewise, the early off-responses to light decrements were larger as more microvilli remained quiescent than in subsequent responses because initially many were still refractory. The polarity of the rate change is not critical; the bigger and faster the increment or decrement, the more information it can potentially carry. Consequently, a photoreceptor's information transfer is higher at large dim-to-bright or bright-to-dim stimulus transitions and then reduces in correlation with the adaptation to the stimulus, as quantified by previous experiments (compare Juusola and de Polavieja, 2003, their Fig. 4).

These differences were equally distinct compared with sample rate modulation of a closely related deterministic model, in which bump refractory periods were random (thereby abolishing any memory) but from the same distribution (Fig. 10B), indicating that refractory periods contain information of their past bumps. Again, the phasic accentuation of bump counts was visually most distinctive to NS (right: SD_stochastic > SD_random; p = 1.2 × 10⁻²; n = 20 nonoverlapping 50-ms-long samples), especially to long and large contrast fluctuations, but it was also evident to larger transient changes in GWN stimulation (left: SD_stochastic > SD_random; p = 1.6 × 10⁻³, n = 20). Notably, the enhancement was independent of stochastically changing bump shapes, which also carry memory (Song et al., 2012), as only the bump times were counted here. Because of this accrued information and the sample rate modulation being still large (Fig. 8B), the stochastic microvilli refractoriness reduced information in photoreceptor output only by ∼12% (Fig. 10C). If, instead, the bump refractoriness was assigned randomly, the information transfer fell by ∼17%. Thus, in bright stimulation, microvilli refractoriness stores some information (or loses less) when it exchanges sensitivity (reduction in sample numbers) for contrast resolution (relative increase in sample rate modulation), sculpting more phasic responses. This nonlinear neural enhancement of intensity changes in image pixels (Burton and Laughlin, 2003; Brinkworth et al., 2008; Zheng et al., 2009) likely improves motion (Joesch et al., 2010) and features (Marr and Hildreth, 1980) detection downstream.

Encoding efficiency depends upon microvilli recovery

The mean microvilli usage over time, including both refractory and activated microvilli, was ∼40% for both GWN and naturalistic stimuli (Fig. 10D), here mostly reflecting the same mean light intensity. With the fall in quantum efficiency, the photoreceptor output adapted close to the midpoint, which theoretically enables near-maximal responsiveness to light increments and decrements (Song et al., 2012) while being much protected from saturation. Conspicuously, naturalistic stimulation, with its larger and bursty contrast changes, was driving the microvillar population far more dynamically than GWN stimuli, which lack these amplitude and phase correlations.

To summarize, our results from the Drosophila photoreceptor models (Figs. 7, 8, 9, and 10; for daylight conditions) suggest that the longer dark contrasts of naturalistic light intensity time series facilitate recovery of refractory microvilli. Because NS further contains large light contrasts, photoreceptors can sample up larger and information-richer responses than from GWN, shuffled-NS, or Gaussian-1/f stimuli, all of which have less these features. Dark contrasts thus increase encoding efficiency by improving quantum efficiency and microvilli usage over time.

Importantly, however, stochastic refractoriness shapes nonlinearly photoreceptor output to all stimulus statistics. It exerts sensitivity control on information sampling in two parallel ways. On one hand, it exerts divisive steady-state nonlinearity by adapting the mean bump/photon conversion ratio (quantum efficiency) to current light conditions. On the other hand, it exerts phasic nonlinearity on bump integration by enhancing sudden stimulus fluctuations in the expense on background, the mean bump production rate. With the overall bump count reduced (by the fall in quantum efficiency), refractory microvilli inevitably sacrifice some information. However, this reduction in information transfer is relatively small, as bump fluctuations in each time moment still contain several hundreds of samples, in which integration generates a smooth response of very high SNR.

We next examined how three related physical factors determine photoreceptors' signaling performance through bump (sample) rate modulation: (1) the number of sampling units (fewer vs more microvilli), (2) light intensity (fewer vs more photons), and (3) speed of sampling (slower vs faster microvilli).

Performance improves with increasing microvilli numbers

What happens to signaling performance if photoreceptors had fewer microvilli? Rhabdomeric photoreceptors of insects show structural and functional adaptations, including different microvilli numbers, associated with special habitats and lifestyles (Gonzalez-Bellido et al., 2011). Moreover, insects go through a metamorphosis whereupon their larvae and adults have quite differently structured photoreceptors (Frolov et al., 2012). These adaptations seemingly support unique behaviors in prevailing light conditions of often quite different visual environments. Many questions about visual behaviors or capabilities thus remain but with limited quantitative evidence how these relate to photoreceptors' ability to sample light information. For this purpose, we analyzed by simulations how microvilli numbers may contribute to insect vision.

A photoreceptor's signaling performance depends upon sample rate modulation, originating from light statistics (Juusola and Hardie, 2001a; Juusola and de Polavieja, 2003) and microvilli availability (Howard et al., 1987; Song et al., 2012). The more light entering the photoreceptor and the fewer microvilli available to encode it, the more saturation could compromise the performance. To start untangling the contributions of these factors in encoding, we changed systematically the bandwidth of unit-contrast GWN stimulus and microvilli numbers in model simulations. Crucially, we further tested how well these models encoded naturalistic stimulus of the same daylight level. The stochastic phototransduction in each microvillus was modeled by that of Drosophila, generating current bumps that integrated macroscopic LIC responses.

The results showed that responses increase with microvilli numbers for all stimulus bandwidths (Fig. 11A), improving the SNR of the photoreceptor output (Fig. 11B). Overall, microvilli resisted complete saturation amazingly well; as always, some were stochastically returning to the pool of available ones. In particular, if the stimulus contrast was predominantly within the visible low-frequency range (i.e., occurred slower than a photoreceptor's integration time), as few as 300 stochastically operating microvilli could generate responses to bright narrow band GWN (20 Hz cutoff) and NS with >10 times more signal than noise. However, the more microvilli the models contained, the finer stimulus changes these could resolve as different, broadening the output range, shown for 100 Hz GWN stimulus (Fig. 11C).

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Figure 11.

Stochastic simulations show how adding more microvilli (sampling units) enlarges photoreceptor output, its bandwidth, and information to GWN and NS at daylight level (10⁵ photon/s). Thus, photoreceptors' bump (sample) rate modulation to light fluctuations increases with stochastically operating microvilli population. A, Having more microvilli increases responses to stimulus despite their bandwidth; shown for, 20, 100, and 500 Hz GWN and NS that has “pink” (1/f) power spectra. Macroscopic LIC: mean and 50 responses (light gray). If contrast modulation is too fast, photoreceptors sample little of it; LIC responses appear saturated. B, The SNR of LIC depends on the number of microvilli and is higher for responses to stimuli with large low-frequency modulation (contrast). C, SNR to the same GWN stimulus increases with the microvilli population, which thus can generate larger sample rate modulation to contrast changes. D, A photoreceptor's information transfer increases by increasing the number of microvilli or the low-frequency contrast modulation (visible to the fly: NS and 20 Hz GWN). Different microvilli numbers lead to different information transfers for different GWN stimuli, suggesting evolutionary transition points where the cost of maintaining microvilli may outweigh any information gains of having faster vision. NS evokes responses with high information rate. E, Photoreceptors encode 20 Hz GWN, which has low information content (R_input), most efficiently; regardless of the microvilli numbers.

Remarkably, the model with only 300 microvilli could encode nearly 100 bits/s from NS and 20 Hz GWN (Fig. 11D), yet it performed poorly with high-frequency stimuli (∼20 bit/s), generating minuscule noisy responses (Fig. 11A). Nonetheless, the increasing microvilli numbers and information capture from the different GWN stimuli formed characteristic relationships of diminishing returns, suggesting transition points between the affordable slow and the more expensive fast vision for diurnal insects. For instance, the models with 300–3000 microvilli encode approximately the same amount of information from NS and 20 Hz GWN, and only with >3000 microvilli, encoding of NS become more profitable. Moreover, having 10 times more microvilli (from 3000 to 30,000) only improves low-frequency (20 Hz cutoff) contrast detection by ∼30% (from 188 to 267 bits/s) but should increase maintenance costs greatly. Consequently, slow diurnal insects or larvae need not invest in a myriad of microvilli to see slow contrast changes well (compare Frolov et al., 2012). However, to encode more fast information, a fly photoreceptor may invest in more microvilli; here, to have at least 9000 of the modeled kind.

Photoreceptors' encoding efficiency increased with microvilli numbers for all the tested light stimuli (Fig. 11E). This implied that, with more microvilli available, a smaller fraction of them were refractory, and the photoreceptors could generate larger sample rate modulations to given light intensity fluctuations. Specifically, we found that efficiency was consistently the greatest for 20 Hz GWN of the lowest input information content, approaching 100%. Thus, increasing microvilli numbers beyond 30,000 could not improve neutral representations of low-frequency contrasts any further but instead be metabolically wasteful. It is noteworthy that this stimulus, because of its low-frequency content, contained longer dark contrasts than the other GWN. These further helped to recover refractory microvilli, and by that increase quantum efficiency of sampling.

Performance improves with increasing light modulation

The easiest way to increase sample rate modulation, and thus photoreceptors' information transfer rate, is to increase stimulus intensity. However, mechanistic interpretation of such dynamics and its limiting factors is nontrivial. In dim illumination, Poisson statistics of photon emission make stimulation noisy (photon shot-noise), whereas the quantum efficiency of fly photoreceptors is virtually 100%, with only few microvilli refractory, by chance. This should cause a photoreceptor's encoding efficiency to be high but information transfer rate (R) low, as it follows the low information content in light input (R_input). But when the light intensity is increased (and so R_input), encoding efficiency should reduce with quantum efficiency (Song et al., 2012) as more microvilli become refractory. Ultimately, a photoreceptor's information rate to the brightening stimulation should increase with its increasing sample rate modulation until potential saturation, when its most microvilli are most of the time refractory. Saturation, however, much depends upon stimulus statistics. In particular, it should affect high-frequency GWN, which lack longer dark contrasts. Responses to NS, which has these features, are more immune to saturation effects (Fig. 11), even at very bright mean intensities, as we have shown previously (Song et al., 2012).

We tested these theoretical concepts experimentally by recoding voltage responses of photoreceptors in wild-type and white-eye Drosophila to 200 GWN stimuli at different light levels, covering nearly 4 log intensities. The white-eye eyes lack screening pigments and intracellular pupil that give the wild-type eye its red appearance and reduce light influx into rhabdomeres. Therefore, white-eye photoreceptors should show signs of saturation at lower light levels than wild-type ones. In comparable simulations, the average bump shapes and their latency distributions were incorporated into the stochastic model from measured values at each light level (Juusola and Hardie, 2001a), as described recently (Song et al., 2012). Notably, the average bumps are larger and slower at dim intensities and smaller and faster at brighter illumination, whereas their latency distributions differ little within the tested light intensity range (Juusola and Hardie, 2001a). By simulating bump production over all 30,000 microvilli, the model was used to predict macroscopic voltage responses during repeated GWN stimulation at each light intensity level. We then compared the signaling performance and dynamics of the real recording series with those of simulated responses.

We found that both recordings and simulations showed similar encoding dynamics (Fig. 12), explainable by theoretical concepts of stochastic adaptive sampling. Increase in light intensity level (and thus modulation) increased sample rate modulation to GWN stimulation, predicting photoreceptors' macroscopic response amplitude (Juusola et al., 1994; Juusola and Hardie, 2001a) and its eventual saturation (Howard et al., 1987). The simulations could also replicate the corresponding relative changes in contrast gain (Fig. 12A,B), SNR (Fig. 12D), and information transfer rate (Fig. 12E), although in the simulations, as in real recordings (Juusola and Hardie, 2001a; Song et al., 2012), adapting bump shapes further adjusted its broadening frequency response (Fig. 12A,B). Moreover, by recording both from wild-type and white-eye photoreceptors, we could further estimate how far the eye's pigmentation protected microvilli from saturation, extending the efficient encoding range for bright GWN. The difference between the stimulus intensities evoking comparable maximum information transfer rates in these photoreceptors (Fig. 12E) suggested that the screening pigments reduced the effective photon input to microvilli by ∼10-fold and thus extended the encoding range the same amount. Because the model photoreceptors lack this mechanism, daylight inputs in the simulations were systematically reduced by 10-fold to minimize potential bias to the corresponding recordings throughout this study (see Materials and Methods).

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Figure 12.

Photoreceptor output, its bandwidth, SNR, and information rate increases with the brightness of unit-contrast GWN stimulation. Bump (sample) rate modulation increases with increasing light modulation until saturation, and thus with it information in photoreceptor output. The stochastic model (right) predicts the experimental results (left, middle). A, Responses of Drosophila R1–R6 photoreceptors increase with brightening light modulation, regardless of the stimulus bandwidth (here, GWN with 200 and 500 Hz cutoffs). Left column, R1–R6 WT red-eyed Drosophila. Middle, WT white-eyed Drosophila. Right, Stochastic model simulations. Mean and 10 responses (light gray). Responses in white-eyed Drosophila saturate at the brightest stimulation because its photoreceptors lack screening pigments and the intracellular pupil, which reduce light input in WT microvilli. Consequently, at bright light levels, the majority of microvilli are in a refractory state, and the bright fast GWN modulation drives them inefficiently. B, The contrast gain of photoreceptor output increases with light intensity until bump (sample) rate modulation from microvilli saturate (at two brightest intensities). C, The phase of a photoreceptor's linear frequency response accelerates with increasing light intensity. D, SNR in photoreceptor output to the same GWN stimulus increases with light intensity until microvilli saturate (at two brightest intensities). In white-eyed Drosophila, this is most dramatic likely because their photoreceptors experience higher photon influx, including any scattered light. E, Photoreceptors' information transfer rate increases with light intensity and narrowing GWN bandwidth (the contrast modulation in the stimulus visible to the fly), until bump (sample) rate changes from microvilli saturate (at two brightest intensities). Different stimulus bandwidth (compare Fig. 4F) causes different rates. But the overall light-induced increase in a photoreceptor's information transfer occurs regardless of stimulus bandwidth. F, Photoreceptors' encoding efficiency to GWN stimuli reduced with increasing light intensity, caused by reduction in quantum efficiency (bump/photon conversion ratio).

The simulations further characterized how encoding efficiency reduces with increasing mean light intensity, as shown for 500 Hz GWN stimulation (Fig. 12F), validating the prediction above. Markedly, as the stochastic photoreceptor model's information transfer rate approached saturation, encoding efficiency fell proportionally the most: from ∼10% at 1.5 × 10⁵ photons/s to ∼5% at 5.1 × 10⁵ photons/s (arrows).

Performance improves with faster sampling

Alternatively, sample rate modulation could be increased by quickening the processing speed (Gonzalez-Bellido et al., 2011) and microvilli refractoriness (Song et al., 2012). To quantify this effect on GWN encoding, we recorded voltage responses of killer fly (C. attenuata) R1-R6 photoreceptors to a unit-contrast stimulus with 300 Hz cutoff (Fig. 13A; wine). We also simulated their outputs with a stochastic model, in which parameters were obtained from the best recordings. Coenosia is a small fast-flying aerial predator of the same muscoid family as Drosophila but separated by 120 million years of evolution (Gonzalez-Bellido et al., 2011). Its fast photoreceptors have presumably been adapted to prevent blurred vision during acrobatic hunts. However, similar to the slower fruit fly photoreceptors, each has ∼30,000 microvilli. In Coenosia, though, their processing speed and recovery are ∼3 times faster (Song et al., 2012). This enabled Coenosia photoreceptors to encode >5 times more information (Fig. 13B) from the same GWN stimulus than Drosophila photoreceptors (Fig. 13C,D).

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Figure 13.

Photoreceptors' information transfer rate and encoding efficiency to GWN and NS are different in flies with fast and slow vision. Both fast-flying Coenosia and slow-flying Drosophila R1–R6 photoreceptors have 30,000 microvilli, but the former sample light changes and recover from them faster, resulting in higher information capture. A, Voltage responses of a Coenosia photoreceptor (brown) and respective stochastic model simulations (red) to unit-contrast GWN stimulation with 300 Hz cutoff; light level: ∼10⁶ photons/s. B, Information transfer of recorded and simulated Coenosia photoreceptors to GWN stimulus (A) and NS; these cells can capture from the GWN up to ∼80% of the information they capture from NS. Recordings: mean ± SD, p = 9.5 × 10⁻⁷; simulations: mean ± estimation error. C, Coenosia photoreceptors encoded ∼30% of information in NS; encoding it ∼1.7 times more efficiently than 300 Hz GWN; GWN R_input ∼ 4600 bits/s and NS R_input ∼ 3300 bits/s. D, Voltage responses of a Drosophila photoreceptor (brown) and respective stochastic model simulations (red) to the same unit-contrast GWN stimulation as in A. E, Information transfer rates of recorded and simulated Drosophila photoreceptors to GWN (D) and NS; these cells capture less information from the GWN than NS. Recordings: mean ± SD, p = 1.2 × 10⁻⁴; simulations: mean ± estimation error. Furthermore, Drosophila photoreceptors encode less efficiently both the stimuli than Coenosia photoreceptors. F, Drosophila photoreceptors encoded NS ∼2.5 times more efficiently than 300 Hz GWN. B, E, Lines indicate recordings from same photoreceptors. In every cell, NS evoked higher information transfer. Here simulated Coenosia photoreceptor output carries proportionally more information than the average recordings because it is based on the best GWN and NS recordings (gray arrows). Simulations lack recording noise, muscle activity, and intracellular pupil, which reduce information in recordings. Data at 20°C as in Tables 2 and 3, not 25°C as in Figs. 3⇑⇑⇑⇑⇑⇑⇑⇑–12.

In each photoreceptor, we also measured its encoding performance to the same natural light intensity time series. Predictably, both Coenosia and Drosophila photoreceptors extracted more information from the natural stimulus than from GWN, but very different amounts in relative terms (∼1.3 and 2.2 times more, respectively). Thus, encoding efficiency (Fig. 13C,F) for the given stimuli (300 Hz GWN and NS) varies greatly in the photoreceptors of different fly species.

Last, using biophysical analyses, we examined how refractory microvilli affect the metabolic cost of visual information to different stimulus statistics.

Refractory microvilli reduces energetic encoding costs

Information transfer by neurons is inherently linked to their energy budget. For graded potential neurons, such as photoreceptors, maintaining ion gradients needed for generating continuous voltage responses consumes ATP (Laughlin et al., 1998). Because we used a full stochastic photoreceptor model, in which macroscopic response (LIC) drove the generation of signals on the voltage-sensitive plasma membrane (of HH type), we could estimate the relationship between the rate of transmitted information and its price in ATP generation directly for different stimulus statistics.

We discovered that refractoriness lowers the metabolic cost of information. Every bump costs, and maintaining a higher membrane potential consumes more ATP (Laughlin et al., 1998). Without microvilli refractoriness, thousands of bumps at any one moment would integrate (Fig. 8A), yielding a mean potential approximately twice as high (30.4 mV vs 16.1 mV), costing ∼5.6 × 10⁹ ATP molecules/s (Table 4). Remarkably, with refractoriness the price falls by >40% to ∼3.3 × 10⁹ ATP molecules/s, and the price for transmitting one bit of information by ∼35%. For naturalistic stimulation, we estimate that a Drosophila R1-R6 photoreceptor can encode at least 400 bits/s (likely more, if the stimulus is sped up) (Juusola and de Polavieja, 2003), and consequently the estimated metabolic cost of information is ∼10 million ATP molecules/bit. These calculations further suggest that information encoding of naturalistic stimuli is less expensive than encoding of broadband (>200 Hz cutoff) GWN stimulation (compare Laughlin et al., 1998; Niven et al., 2007). Therefore, GWN stimulation underestimates neural performance and can overestimate energy consumption.