All experiments were conducted in accordance with EU Directive 2010/63/EU for animal experiments. Cockroaches, Periplaneta americana, and crickets, Gryllus bimaculatus, were obtained from Blades Biological Ltd. (Edenbridge, Kent, UK) and maintained at 25°C under inverse 12–12 h illumination conditions, with experiments performed on dark-adapted insects during daytime. Adult backswimmers (Notonecta glauca) were collected locally in Oulu (Finland) or purchased from Blades Biological Ltd. Photoreceptors were always allowed to adapt to dark conditions for periods of several minutes before recordings. Some recordings from N. glauca and G. bimaculatus were used previously (Frolov and Weckström, 2014; Immonen et al., 2014a). Ommatidia were dissociated as described previously (Krause et al., 2008; Immonen et al., 2014b). Whole-cell recordings from dissociated ommatidia were performed at room temperature (20–22°C) as described previously (Hardie et al., 1991; Krause et al., 2008). In brief, an Axopatch 1-D patch-clamp amplifier and pClamp 9.2 software (Axon Instruments/Molecular Devices, CA, USA) were used for data acquisition and analysis. Patch electrodes were fabricated from thin-walled borosilicate glass (World Precision Instruments, Sarasota, FL, USA). Electrodes had a resistance of 5–15 MΩ. Bath solution contained (in mM): 120 NaCl, 5 KCl, 4 MgCl₂, 1.5 CaCl₂, 10 N-Tris-(hydroxymethyl)-methyl-2-amino-ethanesulfoncic acid (TES), 25 proline and 5 alanine, pH 7.15. Patch pipette solution contained (in mM): 140 KCl, 10 TES, 2 MgCl₂, 4 Mg-ATP, 0.4 Na-GTP, and 1 NAD, pH 7.15. All chemicals were purchased from Sigma Aldrich Inc. (St. Louis, USA). The liquid junction potential (LJP) between bath and intracellular solution was −4 mV. A holding potential of −74 mV (including LJP) was used for voltage-clamp recordings. The series resistance was compensated by at least 80%, with access resistance after compensation typically not exceeding 15 MΩ. Recordings were performed from green-sensitive photoreceptors. A 60 s naturalistic contrast sequence from the van Hateren natural image database was used as the input signal to drive the light stimulus (van der Schaaf and van Hateren, 1996).

Membrane current and membrane potential were initially sampled at a rate of 1200 Hz (0.833 ms sample interval). Preliminary measurements found negligible power in the input or output signals above 50 Hz, so all data files were ten-point averaged to give a resolution of 8.33 ms.

Coherence functions, γ²(f), where f is frequency, for each input-output set were obtained from the spectra of the input, S_xx(f), output, S_yy(f), and cross-spectra, S_xy(f) (Bendat and Piersol, 1980): (1)γ2(f)= < |Sxy(f)|2>/(<Sxx(f)><Syy(f)>) where < > indicate ensemble averages. Linear information capacity, R, was estimated from (Juusola and French, 1997): (2)R=∫log2(1/(1−γ2(f)))df

Signal entropy was estimated as described previously (Pfeiffer et al., 2012). Signals were normalized and digitized so that the maximum amplitude range could be represented by 10-bit numbers or 1024 different amplitude levels. Entropy was obtained by context-independent data compression of regularly sampled continuous signals. Each of the 1024 numerical values representing the digitized signal was treated as an independent symbol in a linear sequence, or message. Data compression was performed by repeatedly replacing pairs of symbols that occurred with greatest frequency by new symbols, until no further compression was achieved. The entropy, E, was then given by: (3)E=(N log2M)/10 where N is number of symbols in the compressed message and M is the number of different symbols in the message and the division by 10 compensates for digitization (Cover and Thomas, 1991).

The same model system (Figure 1) was used to simulate both photoreceptor current and potential. The model was based on the log-normal model of Payne and Howard (1981), shown in the center box of Figure 1, but preceded by a nonlinear component that reduces the overall gain of the system with time from the start of stimulation by including an additional amplitude parameter, α, whose effect declines exponentially with time constant, η. Since the initial gain change was usually accompanied by a small change in mean current or potential, we included an addition to the mean, μ, that decays by the same time constant, η. The final stage of the model consists of a static (memory-less) nonlinear change in amplitude and mean approximated by a second order polynomial function with parameters a, b, and c (Figure 1). The overall gain of the model, including conversion from light intensity to membrane potential or current is assumed to occur in the final stage, but the polynomial displays were normalized to unit input and output for graphical display.

Fitting the model to the data was performed on 9000 input-output data pairs by simulated annealing (Kirkpatrick et al., 1983; Press et al., 1990), brute force and Levenberg–Marquardt (Marquardt, 1963) methods to minimize the mean square error (MSE) between receptor current or receptor potential output, y(t), and the simulated output, y_s(t): (4)MSE=100[(y(t)−ys(t))2]/([y2(t)]−[y(t)]2) where [] indicate time averages (French and Marmarelis, 1999). All software for model fitting, entropy and information capacity estimation was custom written in multi-threaded C++ and operated on standard desktop personal computers.

Membrane current and membrane potential changes during the 60 s of naturalistic light stimulation could be fitted by the model (Figure 2), even at the earliest stimulation times when the gain of the photoreceptors was clearly decreasing. This is an important feature of the model. Error (MSE, Equation 4) values at the completion of fitting ranged from 2.1 to 62.9%, with 16 of the 47 MSE values at or below 10%. MSE values were always higher for receptor current than receptor potential, and error levels were similar for all three species. The highest error values were only observed under the dimmest light conditions. Gain changes after the start of light stimulation (first component of Figure 1) were larger (amplitude parameter α) and more rapid (time constant parameter η) at higher maximum light intensities (Figure 3). These effects were seen in both membrane current and potential recordings, and the fitted gain change parameters agreed for the two types of recordings.

In contrast to the initial gain changes, fitted parameters of the log-normal filter (center component of Figure 1, time constant τ and width parameter σ) did not vary strongly with light level or with species (Figures 4, 5). As a Periplaneta example shows (Figure 4), the peak response shifted by less than a factor of two over the light intensity range of 1:1000. The log-normal filter parameters varied with the different species used, being most rapid for Notonecta and slowest for Periplaneta (Figure 4 insets). Mean parameter values (τ and σ) for receptor potential models were approximately constant at different light levels (Figure 5). Mean parameters for receptor current showed some slowing and broadening of the response at the lowest light levels, but there were not enough data to test for statistical significance. The smaller sets of data for Notonecta and Gryllus agreed well with the mean Periplaneta data, but again showed faster responses, especially for Notonecta, and more clearly at higher light levels.

The final static nonlinearity was modeled by a second-order polynomial function of the output from the log-normal filter (last component of Figure 1). Nonlinear functions are shown for the three species over the range of light levels, but with the full ranges of the input and output signals to each function normalized to unity, in order to show the effects of the nonlinearities (Figure 6). There was a clear general transition from positive, expansive functions at low light intensities to saturating, compressive functions as light intensity increased in both current and potential for all species. Negative and positive overshoots of the functions were presumably caused by the limited number of polynomial terms in the estimates, suggesting that the responses tend to exhibit threshold behavior at the lowest light intensities and strong saturation at the highest intensities.

The photoreceptor models did not add uncorrelated or correlated noise to the transduced signal, which allowed some separation of the relative contributions of noise and nonlinearity to limiting information transmission by the experimental photoreceptors. Information capacity between input naturalistic light stimulus and output membrane current and membrane potential were estimated from the coherence function (Equation 3). Similar measurements were then made by feeding the same input sequence into the best-fitting model (Figure 1) for each recording. Mean values of these data are shown for the different light intensities used in the Periplaneta experiments (Figure 7, upper). Total signal entropy of the input time sequence, resulting membrane potential, membrane current, and corresponding model outputs were measured by data compression (Pfeiffer et al., 2012). Mean values of these data are also shown for the different intensities in the Periplaneta experiments (Figure 7, lower).

Information capacities were low for the experimental data, both for membrane potential and current, with no definite trend vs. light intensity. The fitted models gave higher values, particularly at low intensities. Information capacity can be reduced by uncorrelated noise or by nonlinearity, but the models were purely parametric and did not add uncorrelated noise. Since a linear, noise-free system has infinite information capacity, it follows that the reduced capacity of the models was entirely due to nonlinearity.

Signal entropy also increased at lower light intensities, for both experimental data and modeled responses, and approached the constant value for the input signal entropy in some cases. Input entropy was close to, but below the maximum theoretical entropy that could be produced by this estimation technique (dashed line, Figure 7), indicating that the naturalistic signal exercised the receptors over their full response ranges.

Suggested mechanisms of gain change in insect photoreceptors include optical phenomena, such as changes in the acceptance angle due to rhabdomere or screening pigment migration (Immonen et al., 2014a), changes in the phototransduction cascade itself, and membrane electrochemistry, particularly shunting (Laughlin, 1989). The time course of the gain change that we observed (up to 30 s—Figure 3) suggests a relatively slow process like pigment migration rather than more rapid membrane phenomena.

Although the log-normal filter became faster at brighter levels (Figures 4, 5) these changes were not large, and might not even be statistically significant if more data were available. This relative refractoriness may reflect the wide dynamic and amplitude ranges of the naturalistic stimulus; since the model was required to fit the responses over the whole period of stimulation, and may therefore represent an average description of the photoreceptor dynamics over these wide stimulation ranges.

In contrast to the slow initial gain change, the nonlinear function at the end of the model cascade was static. While some responses were approximately linear, we observed both expansive and compressive behavior as the light intensity increased. The apparent expansion may represent some form of threshold behavior at low light levels. Compressive saturation of electrical responses is well-known in insect photoreceptors, with at least one mechanism being the shunting of transduction current by voltage activated ion channels as the cell depolarizes (Weckström et al., 1988, 1995; French et al., 1993). Saturating nonlinearities in receptor current at higher light intensities (Figure 6), suggest that some nonlinearities occur before ion channels are opened. However, current data must always be treated with caution because of the difficulties of achieving accurate voltage clamp of cells with complex membrane geometry, such as photoreceptors, especially at higher current amplitudes. The present experiments used only a second-order approximation to the final nonlinearity, which limits its interpretation. Extension to higher order nonlinearities would be possible, but require much longer experiments to justify the increased number of fitting parameters.

While hypotheses of possible links between fitted parameters and physiochemical processes are interesting and may suggest further experiments, it must be emphasized that the present mathematical models were not designed to emulate specific biological mechanisms.

Linear information capacities of the experimental receptor current and receptor potential were lowest under the dimmest and brightest conditions (Figure 7). These results are not unexpected, because information capacity would be reduced by noise at the lowest light levels and by nonlinearity at the brightest levels. Similar maxima of information capacity at intermediate light intensities were found in the stick insect Carausius morosus (Frolov et al., 2012), the common backswimmer N. glauca (Immonen et al., 2014a), the water strider Gerris lacustris (Frolov and Weckström, 2014), and the lesser water boatman Corixa punctata (Frolov, 2015). This suggestion is also supported by the model values of information capacity. Since noise was absent from the models, information capacity was only limited by nonlinearity, which was maximal under the brightest conditions. Consequently, information was greatest at the lowest light intensity levels (Figure 7).

Entropy measurements can include both transduced signal and uncorrelated noise, but they are not dependent on linearity. If the models of receptor current and receptor potential are assumed to represent real photoreceptor behavior, the higher values of entropy seen in the experimental measurements than the model simulations (Figure 7) must represent contributions from uncorrelated noise. In this case, the additional noise in the real cells added about 20 Bits/s of entropy to the signal.

A nonlinear dynamic system does not necessarily lose information as long as the receiving system is designed to receive a distorted version of the input signal. However, a nonlinear system can easily lose information that can never be recovered at the output. A trivial example would be a squaring operation that produces a positive output for both positive and negative inputs, so that information about input sign is irretrievably lost. Interpreting the entropy data on this basis indicates that the model transmitted about 80% of the input signal entropy at low light levels, when it was behaving approximately linearly, but lost at least 50% of the input entropy when it became more nonlinear at high light levels. Inspection of the raw data confirms this interpretation (Figure 2). While the average amplitude of the naturalistic stimulus remained constant, the amplitude of the photoreceptor response dropped sharply during the first few seconds. This nonlinear change means that a receiver of the photoreceptor output could not reliably recover the absolute amplitude of the input signal. Information about the amplitude of input signal fluctuation was permanently lost.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.