A photoreceptor’s encoding efficiency,h, is the ratio between the information rates of the voltage output,Routput, and the effective light input (photon absorptions),Rinput, that drove it:
h¼Routput Rinput
(A2.2)
As we already had determined the maximum photoreceptor output information rates,Routput
(Equation A2.1;Figure 2C,Figure 2—figure supplement 1CandFigure 4C), only the
corresponding information rates of the effective light stimuli,Rinput, needed to be worked out. Because the output simulations’ maximum information rates matched well the corresponding mean rates of the real recordings (Appendix 2—figure 6A), we had extrapolated successfully each effective light intensity (photon absorption) time series that drove the voltage response. Therefore, we could now estimate the rate of information transfer of the effective light input by making the following assumptions:
. Photon emission from the light source (LED) follows Poisson statistics; this may or may not be
true (see the discussion below).
. But if true, the effective photons, which survived photomechanical adaptations (Appendix 2—
figure 2) and were absorbed by a photoreceptor and used for calculatingRinput, should also
follow Poisson statistics (Song et al., 2012;Song and Juusola, 2014;Juusola et al., 2015;
Song et al., 2016).
Photons are thought to be emitted by the light source, such as the LEDs, at random, exhibiting detectable statistical fluctuations (shot noise). Such dynamics can be modelled by Poisson statistics (Song and Juusola, 2014). Therefore, as each light stimulus trace differs from any other, with their mean equaling their variance, we could estimate through simulations their average signals and noise, and signal-to-noise ratios,SNRinput(f). The corresponding information transfer rates,Rinput, could then be estimated by Shannon’s equation
(Equation A2.1). For each tested stimulus pattern, this was done by using the same amount of simulated input data as with the output data (2,000 points x 20 repetitions) to control
estimation bias. More details and examples about Poisson stimulus simulation procedures are given in (Song and Juusola, 2014).
. Notice that currently there are no manmade sensors more efficient than the biological photo-
receptors themselves for measuring the photon emissions from the LED light source. There- fore, we had no good direct methods to measure the LED’s photon rate changes at the same level of accuracy as the photoreceptor output that it evoked. Accordingly, calculatingmutual informationdirectly between the less accurate light input estimate and the more accurate photoreceptor output would be both impractical and erroneous.
For the simulated inputs and outputs, the data processing theorem (Shannon, 1948) dictates thatRinputRoutput; thush1 (100%). If not, then one or both estimates are biased or incorrect; information cannot be created out of nothing. However, for the efficiency estimates based on the real recordings, it is quite possible thatRoutput>Rinput, and thush>1 (>100%),
because R1-R6s receive extra information from the network (Appendix 2—figure 5and
Appendix 2—figure 6) that is missing from theRinputestimates of an average R1-R6
photoreceptor’s photon absorptions (cf.Appendix 2—figure 4B).
We recognize that there are methodological limitations and unknowns, which may affect the accuracy and consistency of these estimates:
. Experimental and theoretical evidence suggests that photon output of some light sources
might be sub-Poisson (Teich et al., 1984); meaning, not maximally random. If this were true for our LED, then our approach would slightly underestimateRinput, used in the experiments,
and consequently overestimateDrosophilaphotoreceptors’ encoding efficiency.
. Shannon’s equation can bias information transfer rate estimates for any corresponding light
input and photoreceptor output differently. This is because the signal and noise components of the input and output may deviate from the expected Gaussian by different amounts. Even
though we used systematically the same amount of data for both estimates (202,000 data points), in the cases where light distribution is skewed (bursty stimuli) but the photoreceptor output is more Gaussian, it is possible that Shannon’s equation would underestimate input but not (or less so) output information, causing us to overestimate efficiency.
. Small data chunks limit analyses. In the past, we have compared information transfer rate
estimates, as obtained by Shannon’s equation to those estimated through the triple extrapo- lation method (Juusola and de Polavieja, 2003), which is directly derived from Shannon’s information theory. For ergodic data of different distributions, and when appropriately applied, both methods provided similar estimates (Figure 2—figure supplement 4)
(Juusola and de Polavieja, 2003;Song and Juusola, 2014). However, the triple extrapola-
tion method works best with large sets of data; preferably containing30 responses to the
same stimulus (Juusola and de Polavieja, 2003). In the current study, because of the practi- cal limitations (to map a photoreceptor’s whole encoding space within a reasonable time), all the selected recordings and simulations consisted responses to 20 stimulus repetitions. This data size was deemed insufficient for an accurate estimate comparison between the two methods and was not done here. In the analyses, to provide fair comparison between simula- tions and recordings in all tested conditions, all the data chunks (for the recordings, simula-
tions and stimuli) were exactly the same size (20 2,000 points) and they were processed
systematically in the same way (apart from the two exceptions we discuss next). Therefore, the data-size bias should be under control and the results comparable within these limits.
. Implementation of Shannon’s equation (Equation A2.1) in digital computers typically
requires windowing of the data chunks (for signal and noise) before calculating their power spectra though Fast Fourier Transfer (FFT). Windowing combats spectral leakage (smearing), but this affects especially low-frequency signals, in which information content is low. So this trade-off can be considered reasonable, and its effect on most performance estimates is mar- ginal. But here as the input and output information transfer is calculated separately, window- ing affects more 20 Hz GWN light input than its corresponding photoreceptor output. This is because windowing clips lower frequency power from 20 Hz GWN input, whereas in the sim- ulated and real voltage responses much of this power is nonlinearly translated (through adap- tation) to higher frequencies, including those over 20 Hz. The simulated light input, of course, carries no information on frequencies > 20 Hz, but now it has also lost in windowing some of its low-frequency modulation, which the photoreceptors could translate into high- frequency voltage modulation (note, photomechanical phase enhancement can further con- tribute to this nonlinearity, see Appendix 3). For the two lowest intensity levels only, we judge that because of this methodological bias, the efficiency estimates for the 20 Hz GWN input-output data became unrealistic by a small margin of 10–40 bits/s, implying that
Routput>Rinput. Therefore, for data to these two stimuli only, we applied box-car windowing (instead of the normal Blackman-Harris type), to retain its low frequency information content, and so to reduce this bias.
Because of all these possible error and bias sources, aDrosophilaphotoreceptor’s encoding efficiency (h) estimates given in this publication must be considered as upper bounds. Nonetheless, for real photoreceptors, it is realistic to expect their maxima to approach 100%, and in some cells (likely R6s) be beyond, for the tested low-frequency stimuli (20 Hz). This is because of the extra information from the network, which is missing from the simulated mean photon absorption estimates. (Note that in thein vivoexperiments, the light source emits at each moment 100–10,000-times more photons than what can be absorbed by the tested photoreceptor. Thus, the light source’sRinputemittedalways exceeds a photoreceptor’s
Rinputabsorbed).
Overall, the maximumhvalues are slightly higher but consistent with our previous
(conservative) estimates of 90–95%, in which the light input intensity to microvilli was inferred by comparing the wild-type photoreceptor performance to that of white-eye mutants, lacking the intracellular pupil. Therefore, we conclude (again conservatively) that the error margin of