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Introduction

Sensory systems face the challenge of reliably encoding the outside world as neural signals in the face of an ever-changing environment. A classical example is the light adaptation of the visual system over a vast range of intensities - the ability to ‘disregard’ redundant mean illumination so that contrast patterns can be encoded within the limited output range of neurons [1]–[3]. Insect compound eyes, which allow stable intracellular recordings from their visual neurons in the presence of intact circuitry and optical structures, make particularly useful models to study light adaptation [4]–[14], providing the opportunity to investigate how a sensory system extracts information from its surroundings, and how this function is optimized to environmental changes [2], [3], [9], [15], [16].

Photoreceptors constitute the sensory surface of insect compound eyes, the retina. In these cells, light patterns are encoded into graded membrane potentials for transmission through the first visual synapse [4], [5], [9], [10], [13]–[15], [17]. This starts a parallel flow of signals that is relayed several times [18], [19] before this neural image reaches the brain. The quality of the neural image at the photoreceptor level is critical to the animal's survival, as any higher-order processing by the brain ultimately relies on this representation of the visual scenery [1], [2].

In insect photoreceptors the absorption of a photon by rhodopsin leads to the initiation of ionic currents, and these currents elicit changes in the membrane potential [12]. The voltage signal is therefore co-processed by the phototransduction cascade and the membrane [11], both having their properties dynamically regulated [7], [8]. The resulting plastic, adaptive ‘gain control’ has evolved to work efficiently, despite several limiting factors. The following constraints are of particular relevance: (1) the noisiness of both the light input, such as photon shot-noise and optical blur, and the cellular machinery, such as chemical reaction dynamics and ion channel kinetics; (2) the vast range of light intensities to which the animal is exposed, threatening to saturate the small operational voltage range of a photoreceptor; and (3) the ambient temperature, which acutely affects the speed of intracellular reactions as most insects are poikilothermal. Our aim in this study is to quantify how locust photoreceptors encode visual information in vivo and how this process is affected by these three major parameters: noise, light background (BG) and temperature.

The photoreceptors of the desert locust (Schistocerca gregaria) offer several advantages in investigating how a sensory system reliably encodes information in a changing, noisy environment. First, the ecology and behavior of locusts is well-characterized [20]. In their natural habitat in Africa, these animals are active both by day and night. Therefore, locusts not only have to adapt to very different light BGs but they also face large temperature changes, making it biologically meaningful to investigate how these factors impact the way photoreceptors encode contrast signals. Secondly, their relatively large photoreceptors allow stable, long-lasting intracellular recordings [21]–[24]. One can therefore reliably repeat the experiment in the same cell at different temperatures and light BGs. Thus, we can unambiguously distinguish between variability attributable to the changing mean illumination and/or ambient temperature and variability attributable to cell-to-cell differences.

In this study, we quantify the response dynamics of locust photoreceptors to random (white-noise, WN) and naturalistic (1/f, NS) contrast stimuli at different light BGs and temperatures. We also investigate how their membrane properties change with these conditions by injecting current waveforms intracellularly. This combined approach allows us to elucidate the respective roles and intricate tuning of the phototransduction cascade and plasma membrane in shaping the voltage responses to light contrasts. We show that the temperature-dependence of different biochemical processes involved varies with light adaptation. Nevertheless, we also find that the locust photoreceptors are able to produce a remarkably invariable neural representation of naturalistic light patterns, irrespective of the prevailing light and temperature conditions. Based on our results, we reason that temporal input patterns continuously tune the interactions between the fast membrane reactions (bump waveform) and the slower intracellular reactions (bump latency distribution), enabling the speed of the voltage output to encode contrast values of the input. By accurately encoding the naturalistic contrast input into the rate of change of voltage responses (and so generating an invariable bandwidth for NS), the locust photoreceptors provide robust neural representations of the natural environment already at the first stage of neural information processing.

Materials and Methods

Preparation

Adult female locusts (Schistocerca gregaria) were reared in the Department of Zoology in the University of Cambridge. The culture contained 500–1,000 insects per 45×50×50 cm rearing units and was maintained under an 18 h:6 h light:dark cycle. The temperature during the light period was 37°C and during the dark period 25°C.

Dissection: the pronotum was carefully removed, the head cut off and its back sealed with beeswax to prevent it from drying. The antennae and mouthparts were delicately removed to avoid muscular saccades. The head was then fixed with beeswax to the open end of a conical holder, mounted on top of a ceramic recording platform. Two openings were cut on the head with a sharp razor edge. The first one, a size of a few ommatidia, was made on the dorsal cornea of the left eye and sealed with Vaseline to prevent the eye from drying. The second, a larger one on the top of the head, was used to implant the indifferent electrode. Despite the dissection, the health of the preparation was excellent, providing with very stable recordings. If it were correctly sealed, the head could survive many hours - when left overnight the preparation was still alive; it responded electrically to light the following day. All the experiments were realized during the mid-afternoon, when the animals were in their ‘day state’ [25]–[27].

Temperature control

The hollow copper core of the holder was shielded within a ceramic insulator and fitted tightly onto a Peltier element. Heat sink paste was used to enhance heat conduction. Underneath the Peltier element, a large copper rod embedded in ice functioned as a heat sink. The temperature of the head was measured with a thermocouple, mounted in the copper core next to the head. A custom-designed power source, controlled by the feedback from the thermocouple, was used to drive the Peltier element. The room temperature was monitored with a separate thermocouple. Control measurements from the head revealed that its temperature depended linearly on the temperature of the copper holder at a given room temperature. The actual temperature of the head was estimated from a reliable calibration, using the measured temperature values of the thermocouple at the constant room temperature of 19°C (constantly monitored and controlled by air conditioning). All the experiments were realized with an actual temperature of the head ranging from 13 to 25°C. Although the behaviorally relevant temperatures for Schistocerca gregaria certainly extend to higher temperatures, stabilizing the preparation temperature with such a differential from the room temperature proved technically difficult. This range of temperature was nevertheless sufficient to accurately estimate Q₁₀ values (the rate of change as a consequence of increasing the temperature by 10°C) for various parameters.

Microelectrodes and cell selection

The microelectrodes were pulled with a horizontal laser puller (P-2000; Sutter Instrument Company) from filamented quartz glass capillaries (Sutter, with an inner and outer diameter of 0.5 and 1.0 mm, respectively). Electrodes were back-filled with 3 M KCl, having resistance between 80 and 180 MΩ in the tissue. Microelectrodes were mounted on a manual micromanipulator (HB3000R; Huxley Bertram) and entered the compound eye through the previously prepared small hole. A blunt reference microelectrode, filled with locust Ringer's containing in mM: 10 TES buffer, 140 NaCl, 10 KCl, 4 CaCl₂, 4 NaHCO₃, 6 Na₂HPO₄, adjusted to pH 6.8 with NaOH/HCl [28], entered the locust's head through the other opening.

Membrane potentials of green-sensitive R1–R6 photoreceptor cells [29], [30] were recorded with a switched-clamp amplifier SEC-10L (NPI Electronic) operating in the compensated current-clamp mode. A successful photoreceptor penetration was seen as a 60–80 mV drop in the electrode potential followed by vigorous responses to dim pulses. Before the experiments, the cells were allowed to dark-adapt and seal properly. Only data from photoreceptors with saturating impulse responses ≥40 mV and dark resting potential ≤−60 mV were used in the analysis. In this article, we exhibit our findings using two exemplary photoreceptors. Similar results were obtained from other photoreceptors (n = 15) that endured long-lasting recordings. These data are presented in the Supporting Information. A first photoreceptor is used throughout Materials and Methods to illustrate the way data was analyzed (Figs. 1 to 3) at a constant temperature (19°C). The second one is used throughout the article (Figs. 4 to 12). Because of its exceptional stability, we were able to use this cell in many separate experiments and so to explore how light adaptation occurs over a vast range of background intensities and temperatures (from 17 to 23°C). For these experiments we used both white-noise (WN) and naturalistic stimulation (NS), and were able to further investigate how the membrane properties of the cell varied at each experimental condition. Additionally, we made recordings from many other photoreceptors (>30 of outstanding quality) over a smaller range of experimental conditions. These recordings were consistent with the general framework presented here. Because we believe that intrinsic functional variability between photoreceptors could be an important feature of locust vision (see Discussion), we do not show averaged quantities. Data from these cells is shown as Q₁₀ values in Table 1 and detailed further in Table S1.

Figure 1. Signal and noise analysis of the voltage responses to a white-noise (WN) light stimulus.

A, A pseudorandom light intensity pattern superimposed on a constant light background provided a WN contrast stimulus that was presented 30 times to the cell. The evoked responses are averaged to give the voltage signal and the remaining differences are the noise traces (A, scale bars: 500 ms, 5 mV). B, The corresponding power spectra are calculated for each of the five light BGs. Note that 〈|S(f)|²〉, 〈|N(f)|²〉, and 〈|r(f)|²〉 are displayed using the same scale, in mV² Hz⁻¹. 〈|C(f)|²〉 is in c² Hz⁻¹. C, These changes can be further quantified by computing the signal-to-noise ratio spectrum, SNR(f), and the cross-spectrum between the signal and the stimulus. These two spectra are the starting points to quantify the properties of the photoreceptor voltage responses (Figs. 2 and 3), the SNR(f) being used for the analysis of the coding properties (Fig. 2) and the cross-spectrum for the analysis of the transfer properties (Fig. 3). ‘Power’ on the ordinate scale of the cross-spectrum means here c mV Hz⁻¹.

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Figure 2. Voltage responses to a WN light stimulus: analysis of the coding properties of the photoreceptor.

Analysis of the coding properties of the photoreceptor, based on the SNR(f) (Fig. 1C, right). A, The total signal power increases ~40 times from BG-4 to BG0. The variance calculated in the time domain, σ_S², (not shown) is virtually identical, verifying the calculations. B, The total noise power is reduced ~2 times from BG-4 to BG0. Here again it is identical to the noise variance calculated in the time domain, σ_N², (not shown). C, Information in the frequency domain is calculated from SNR(f) at each frequency as log₂[SNR(f)+1]. All the information resides in a frequency range below 100 Hz. This information is integrated to give the information transfer rate (Shannon's formula), D, which increases ~11 times from BG-4 to BG0. The ratio of the signal and noise variances, SNR_t (not shown) scales well with the information transfer rate. This highlights that the information transfer rate is a measure of the number of the ‘coding states’ used by the cell during a second. These states are the different voltage levels confined within the used voltage range (which is ~ signal as σ_S²>>σ_N²) and separated one from another by the ‘resolution’ of the system (noise). From information transfer rate estimates we define three relevant backgrounds: BG-3, named as ‘dim’ (~100 bits/s); BG-2 as ‘mid’ (~200 bits/s) and BG0 as ‘bright’ (~300 bit.s⁻¹). E, Linear coherence, γ_lin, is calculated from SNR(f). At dim BGs the stimulus is itself noisy (attributable to the photon shot-noise), and so is the cell's behavior. At bright BGs the cell's response (assuming linearity, see Materials and Methods) is remarkably noise-free (γ_lin>99% at BG0) up to ~30 Hz.

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Figure 3. Voltage responses to a WN light stimulus: analysis of the transfer properties of the photoreceptor.

Analysis of the transfer properties of the photoreceptor, based on the cross-spectrum between the stimulus and the signal (Fig. 1C, left). A, Gain is the norm of the frequency response (see Materials and Methods). It displays the range and extent of stimulus frequencies the cell amplify linearly. B, Areas (integrals) under gain curves at different BGs, and D, corresponding 3 dB cut-off frequencies. The amplification increases with the light BGs whereas the cut-off frequency remains virtually unchanged. C, Phase of the frequency response and the minimum phase, calculated from the gain curves, exhibits a phase-lag. E, Impulse response K₁ is calculated from the frequency response function (real parts seen as gain, A, and phase, C). It approximates the linear filtering properties of the system. Brightening increases its area, scaling closely with the gain power (not shown), and reduces its onset-delay, F, as well as its time-to-peak (the delay between onset and peak is virtually constant ~20 ms). The dead-times estimated from the phase-shift observed in C (not shown) and from the impulse response (F) behave very similarly, vindicating the analysis. G, Noise-free coherence, γ_NF, indicates the frequency range where a photoreceptor, if operating linearly, would reproduce exactly the same response at each stimulus presentation. γ_NF departs from unity at certain frequencies, reflecting selective nonlinearities, which enhance particular features of the stimulus. The bandwidths of the coherences, H, are defined as the frequency beyond which γ<0.5. The bandwidths increase with brightening BGs, reflecting the photoreceptor's ability to follow the stimulus on a shorter time-scale (γ_lin). This increased precision takes place in a frequency range where the photoreceptor encodes linearly the WN stimulus (γ_NF>γ_lin at each BG).

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Figure 4. Light background and temperature are critical parameters for the visual coding in locust photoreceptors.

A, Light-induced depolarization, at 19°C, is clearly seen in 1 s-long recordings of the membrane potential of a photoreceptor adapted to different light conditions – to darkness and to three different light BGs. Brightening reduces voltage noise, as seen from the corresponding probability distributions (right; scale bar: 500 ms). B, Voltage responses of a dark-adapted photoreceptor to a 10 ms-long light pulse of saturating intensity at 17, 19, 21, and 23°C (scale bars: 100 ms, 10 mV) show that warming accelerates voltage responses to light but has little impact on their amplitude (~40 mV). The mean potentials have been set to the same value for clarity. The arrow indicates a fast depolarizing transient [29], similar to the ones reported in Calliphora [76] and Drosophila [7] photoreceptors.

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Figure 5. Analysis of the voltage responses to a light WN stimulus at different BGs and temperatures.

Changes in signal, A, and noise, B, power with brightening and warming lead to an increase in information transfer rate, C. Warming increases 3 dB cut-off frequencies of the signal, D, noise, E, and gain function, F. Dead-time in the voltage response, as seen with the onset time of the impulse response, G, is also reduced with both warming and brightening. H, Cut-off frequency of the noise-free coherence, γ_NF, i.e. the frequency beyond which γ_NF<0.5, and I, cut-off frequency of the linear coherence, γ_lin, are presented using the same scale, highlighting that for every experimental condition γ_NF>γ_lin. J, WN stimulus can be characterized by its temporal pattern (scale bars: 300 ms, 1 contrast unit), by its probability distribution and by its power spectrum.

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Figure 6. Bump noise analysis of the voltage responses to a WN light stimulus.

A, Single photon response recorded in a dark-adapted photoreceptor at 26°C and super-imposed Γ-distribution (n = 4, τ = 8 ms). An initial estimate of these parameters is necessary to guide the fitting algorithm. These first-guess parameters can be estimated by calculating the power spectrum of a single bump and fitting a Lorentzian to obtained curve (see Materials and Methods). For this bump the parameters estimated in the frequency space gave n = 4, τ = 9 ms. The Γ-distribution accurately describes the bump shape. The mean residual of the fit is 0.085 mV² (estimated between 0 and 90 ms, i.e. where the bump is actually happening), smaller than the fluctuations of membrane potential in bump-free zone (variance ~0.1 mV²). B, At a given temperature we estimate the noise spectra of the voltage responses at the three adapting BGs and in darkness. The dark noise is virtually the same over the temperature range; it is subtracted from the total noise at each BG to give the light-induced noise power spectra. By fitting a single Lorentzian to these spectra we obtain parameters for the bump waveform (see Materials and Methods). C, Normalized bump shapes for different BGs at 19°C and for different temperatures at the dim BG illustrate that both brightening and warming accelerate the bumps. D, This is further quantified by estimating the bump durations (or time-to-peak; not shown as it displays identical behavior).

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Figure 7. Bump latency distributions.

A, Latency distribution is calculated by deconvolving the bump shape from the corresponding impulse response (see Materials and Methods) at each experimental condition. B, Estimated latency distributions are shown for the same conditions as in Figure 6. As their time-to-peak decreases with brightening or warming, bumps appear sooner. C, The bumps are also more precise (synchronized), as it can be seen in the decrease of the width of the (normalized) latency distributions.

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Figure 8. Voltage responses to a NS light pattern at 19°C.

A, In separate experiments, a NS light pattern is repeatedly presented to a photoreceptor as dim and bright intensity variations. B, The superimposed traces show the corresponding voltage responses to the 20^th stimulus presentations. For the bright NS the cell dedicates a larger voltage range for encoding the stimulus. C, Averaging over the 100 individual traces gives the corresponding signals, normalized to exhibit the differences in their timing. With the dim NS the voltage output of the photoreceptor follows the light input with a delay greater than the one with the bright NS by ~1 ms. Nevertheless, the voltage responses can follow the same stimulus pattern, suggesting that the photoreceptor is utilizing the same frequency range at different BGs. This is confirmed by the analysis of the responses power spectra at different points during the repeated stimulation (3^rd and 30^th traces), D. Whilst the amplification is higher for bright NS, as was already apparent in A, the range of frequencies encoded is virtually the same. This is quantified by calculating the 3 dB cut-off frequency, f_{3 dB}, which equals to 14 Hz in all cases. Comparing the spectra of the responses to the 3^rd (left) and 30^th (right) stimulation shows no additional adaptive trend, suggesting that the system adapts rapidly (after the 1^st stimulation) to a relatively invariable coding state (see Text S1 and Fig. S2). The signal power spectra (not shown) look very similar to the power spectra of the responses, as expected from the high signal-to-noise ratio (Fig. 9).

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Figure 9. Analysis of the voltage responses to a light NS stimulus at different BGs and temperatures.

Responses to a NS light stimulus change with warming and brightening. Behaviors of signal power, A, noise power, B and information transfer rate, C, as estimated with the triple extrapolation method, resemble those of the WN experiment (Fig. 5). D, signal 3 dB cut-off frequency remains virtually unchanged over all the BG-temperature conditions, differing dramatically from the WN experiment, whereas, the cut-off frequencies of noise power, E, and gain, F, behave much as in WN stimulation. Onset time of the impulse response, G, and the cut-off frequencies of the linear, H, and noise-free, I, coherences show similar evolution as seen with WN stimulation. The temporal pattern of the NS stimulus, J, displays long-term correlations (with no characteristic time constant), leading to a typical 1/f power spectrum trend and a probability distribution that completely departs from Gaussian (scale bars: 300 ms, 3 a.u. of light intensity).

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Figure 10. Membrane properties deduced from the current steps experiment.

Voltage responses to the injected current steps were used to investigate the transmission properties of the photoreceptor membrane (Fig. S1). A, Membrane time-constant, τ, is greatly reduced from the dark-adapted state by dim light adaptation, but it reduces only slightly further with brightening BGs; i.e. the membrane ‘switches’ from a dark to a light-adapted state. B, Membrane resistance, R, displays a complex behavior in the light BG–temperature plane that correlates with the duration of the bumps, estimated from the noise power spectra (Fig. 6D). C, Mean membrane potential MMP shows that the light-induced depolarization increases by ~15 mV from dark to bright BG, yet it is virtually temperature insensitive.

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Figure 11. Dynamical properties of photoreceptor membrane investigated by WN currents.

A, Noise-free coherence, γ_NF, shows that the membrane can linearly translate WN current input into voltage output up to very high frequencies (500 Hz). B, Linear coherence, γ_lin, shows only very small noise contamination over the whole frequency range. C, Impedance curves, Z(f), show that photoreceptor membrane acts as a low-pass filter. Data for A–C was recorded at 19°C in dark and at the three light BGs; the results at the other temperatures show nearly identical behavior. From the impedance functions we estimated the resistance, R, and the 3 dB cut-off frequency, f_3dB. D, resistance estimate from WN stimulation strongly resembles the resistance estimated from the current steps experiment. E, Cut-off frequency is virtually constant and much higher than the cut-off frequency of the voltage responses to light. F, Plotting the normalized impedance (i.e. current-to-voltage gain) and the light-to-voltage gain clearly shows that, at the level of the photoreceptor soma, the membrane is not matched to filter out high-frequency phototransduction noise (shown for 19°C – mid BG, all conditions giving similar results).

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Figure 12. The photoreceptors enhance transient features of the stimulus and flatten the probability density of the transmitted frequencies.

The reliability of temporal patterns in the photoreceptor responses is analyzed by comparing the average response, A (i.e. signal), to the time-dependent variability of the voltage responses, B (i.e. noise SD), evoked by a NS sequence. The probability distributions of these functions are shown in right. Noise SD is non-uniform across the stimulation pattern, calculated for every time-point across the voltage traces to the last 90 presentations of the NS light pattern (the first 10 showing an adapting trend), at the bright BG at 19°C. At every time-point (left) the spread of voltage values of the responses follows an individual distribution, varying from skewed to Gaussian; however, their overall probability distribution approximates a Gaussian (right). The changes in noise SD are then compared to the SNR, C, estimated by calculating the signal SD and the noise SD over 5 consecutive time points (using a 10-point window gives similar results). Notice that the amplitude of the rate of change in the signal, i.e. the absolute value of its time derivative (red trace), behaves similarly as the SNR, indicating that the locust photoreceptors encode most efficiently fast voltage changes. D, By ignoring their temporal order, 1000 values for (noise SD and signal) and (noise SD and rate of change of signal) are displayed as functions of voltage and rate of voltage change, respectively. The noise SD depends mostly on the rate of voltage change (linear fit slope = 0.08 ms, R = 0.26) and little on the instantaneous voltage value (linear fit slope = 0, R = −0.08). Notice that the noise SD does not only depend on the absolute value but also on the sign of the derivative. This could imply that there is an asymmetrical step in the phototransduction cascade, possibly arising from a process that involves 2 different time-constant for the transition between 2 different states (e.g. phosphorylated/non-phosphorylated). Such asymmetry would naturally occur if the 2 transitions involved 2 different enzymes. Alternatively, fast membrane dynamics or synaptic feedbacks could enhance depolarizing and hyperpolarizing response patterns asymmetrically. E, The normalized power spectra of the NS stimulus (ordinate units c² Hz⁻¹) and of one stretch of the photoreceptor response (as in Fig. 8, at bright BG, ordinate units mV² Hz⁻¹) illustrates how the cell enhances selected stimulus frequencies, whitening its output and increasing the entropy of transmitted signals.

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Table 1. Q₁₀ (17–27°C) for normalized data.

doi:10.1371/journal.pone.0002173.t001

Recording procedures

The stimulus generation, data acquisition, and signal analysis was performed by a custom written program (BIOSYST, © M. Juusola, 1997–2008) based on the MATLAB programming language (Mathworks) using an interface package for National Instruments boards (MATDAQ, © H.P.C. Robinson, 1997–2008). More details on data acquisition and analysis are given in Juusola and Hardie [7] and Juusola and de Polavieja [6].

Light stimulation

Light stimuli were provided with a green high-intensity light-emitting diode (Marl Optosource) driven by a custom-built LED driver. The light output of the LED was monitored continuously with a pin diode circuit. The LED light output was attenuated by neutral density filters (Kodak Wratten) to provide five illumination levels, or adapting backgrounds; each one log-unit apart, indicated as BG0 (10⁷ photoconversions s⁻¹), BG-1, BG-2, BG-3, and BG-4. The light output range was calibrated by counting the number of single photon responses, bumps, [22] during prolonged dim illumination [31]. A Cardan arm system allowed free movements of the light source at a constant distance (85 mm) from the eye's surface with the light source subtending an angle of ~2°, comparable to the reported values for the angular sensitivity of locust photoreceptors (from ~1.2° when light-adapted to ~2.6° when dark-adapted, [24]).

White-noise stimuli (WN) were generated using MATLAB functions. These pseudorandom contrast modulations had Gaussian amplitude distributions and were spectrally flat up to a chosen cut-off frequency (an example can be seen below, in Fig. 5J). WN stimuli with different cut-off frequencies were used in preliminary experiments (from 10 Hz to 10 kHz, at BG0 in the same cell), causing similar changes in the responsiveness and information transfer of the photoreceptors as reported earlier with blowfly photoreceptors [6]. We used 1s-long WN light stimuli and selected 200 Hz as the cut-off frequency, as this covered the range of frequencies locust photoreceptors could see (Fig. 1) without allocating much power on light patterns that are too fast for these cells to follow. 1 s-long naturalistic stimulus (NS) sequences were extracted from patterns downloaded from the van Hateren database [10]. They had a characteristic 1/f-type spectrum, a non-Gaussian distribution (an example can be seen below, in Fig. 9J), and were presented to the eye at 1 kHz. Four different NS patterns were first used to control that the results were independent of some peculiarities in the intensity variations but rather depended on the global statistics of the stimuli. The total power of the chosen NS pattern (the one that elicited the largest responses) and the total power of the WN stimulus pattern were very similar (differed by ~4%). Therefore, any observed difference could be attributed to differences in the statistical properties of the stimuli.

Preliminary results (Figs. 1–3) indicated that three adapting BGs were representative of three different working regimes of the photoreceptor. These BGs are named as ‘dim’ (BG-3), ‘mid’ (BG-2), and ‘bright’ (BG0). Light contrast (c) was defined as a change in the light intensity (ΔY) divided by the mean light BG (Y_mean):
(1)
For white-noise contrast modulation, ΔY was defined as the SD of the stimulus modulation. For naturalistic stimulation, the read-out values of the pin diode circuit monitoring the LED output were used without any calibration. These data are therefore presented using arbitrary units (a. u.) instead of contrast units. For direct comparison with WN, the NS data could be re-scaled in term of contrast units, but as its probability distribution departs completely from Gaussian, the SD of such a distribution has little significance. Alternatively, the NS values could be normalized by setting the lowest bound (when the diode is in effect off) to 0 and its highest (where the diode saturates) to 1. Either procedure left the results virtually unchanged. In the experiments, the cells were adapted to a selected light BG for >20 s before the WN or NS patterns were presented. Notice that because the WN patterns were superimposed on a constant light BG, whereas the NS patterns - that included longer dark periods - were not, the mean of the WN stimulus is higher than that of the NS stimulus, despite both having the same power.

Current stimulation

To investigate how membrane properties of locust photoreceptors are modulated during light adaptation, we injected pulses or pseudorandomly modulated current into photoreceptors via the recording microelectrode. Electrode capacitance was carefully compensated before the current injection experiments. The use of a switched-clamp amplifier allowed us to record and monitor the true intracellular voltage and current during current injections and light stimulation [32].

Data acquisition

Current and voltage responses were low-pass filtered at 1 kHz (KEMO VBF/23 low pass elliptic filter). These signals were sampled at 10 kHz for NS - 1 kHz was sufficient for WN signals as the corresponding light stimuli are cut-off at 200 Hz - then digitized with a 12 bit A/D converter (PCI-MIO-16E-4; National Instruments), and stored on the hard disk of a PC. The sampling was synchronized to the computer-generated stimuli and records of light and current stimulus, and voltage responses were stored during each recording cycle. To allow a fair comparison between WN and NS, the voltage response was re-sampled from 10 to 1 kHz. We checked that the results of the calculations were virtually independent on the sampling rate by re-sampling the data at 0.5, 1 and 2 kHz and repeating the analysis. The recording system, including the microelectrode, had a frequency response with a 3-dB high-frequency cut-off at 10 kHz or higher and therefore had negligible effect on the results. The noise level, estimated from measuring voltage fluctuations (SD) when the electrode was in the eye tissue, was <0.2 mV. Each experiment proceeded from the dimmest to the brightest adapting BG, at a given temperature.

Data analysis

Most of the data analysis was conducted as explained in Juusola and Hardie [7] and in Juusola and de Polavieja [6]. Here we summarize the different stages of the analysis, using the example of a photoreceptor's voltage responses to light WN at 19°C. This allows us to define the relevant parameters used throughout this article and to highlight their biological significance, and by doing so to present the underlying assumptions and approximations of this study. We describe how the bump and membrane properties were investigated, and give a brief description of the triple extrapolation method used for estimating the information transfer rate of voltage responses to NS. We further define how the probability distributions were calculated to gauge the system's stationarity, and how Q₁₀ values were estimated to quantify temperature-dependent changes.

Processing in the time domain: signal and noise analysis.

Repeated presentations of the stimulus (WN or NS) evoked slightly different voltage responses. (WN data consisted of 31 responses, 101 responses were recorded for NS stimulation). In both cases, we rejected the first trace from the analysis as they systemically showed strong adaptive behavior. For each recording series, the averaged response is the ‘signal’, whereas the ‘noise’ is the difference between individual traces and the ‘signal’ (Fig. 1A). Hence for an experiment using n trials (with n = 30, WN, or n = 100, NS) there is one ‘signal’ trace and n ‘noise’ traces. In the simple case where linearity and additivity can be assumed (reasonable for WN, as discussed later on in the article), the noise term constitutes a random parameter that independently ‘contaminates’ each trial. In the case of NS, the noise term represents the probability distribution of all the possible response traces. The variance of the signal, σ_S², and noise, σ_N², were then calculated from the corresponding signal and noise traces. Additionally, we calculated the noise using the following method that prevents signal and noise from being correlated [33]. n-1 trials of an experiment consisting of n trials were used to compute the mean and the remaining one to compute the noise. This procedure is repeated for each possible set of n-1 responses, giving n uncorrelated noise traces. These two methods gave similar noise estimates.

We checked the distributions of signal and noise at the different experimental conditions. For the WN stimulation the distributions are very close to Gaussian at most conditions, although the noise distribution is skewed toward depolarization at very dim BGs, attributable to individual bumps, and the signal distribution is slightly skewed away from depolarization at the brightest BGs, attributable to saturation. When stimulating with NS, the distribution of the signal departs clearly from a Gaussian distribution, resembling the distribution of the response given below in Figure 11, whereas the distribution of the noise is Gaussian. However, in line with data obtained from photoreceptors of the flies Calliphora [6] and Drosophila [14] the variance of the responses differs at different moments of stimulation. This suggests that the ‘noise’ may play a role in the coding and the transfer of the visual information. Fourier analysis of signal and noise neglects the potential importance of ‘noise’, being an inherent limitation of this type of approach (see Discussion).

Calculation of the spectra.

We calculated the corresponding power spectra for the mean stimulus, the signal, and every noise and response traces (Fig. 1B). They were divided into 50% overlapping stretches and windowed with a Blackman-Harris 4-term window [34]; then a fast Fourier transform algorithm was used to calculate their power spectra. Noise and response spectra were then averaged to improve these estimates (Bendat and Piersol, 1971). 〈|C(f)|²〉, 〈|S(f)|²〉, 〈|N(f)|²〉, and 〈|r(f)|²〉 are the stimulus (contrast, C), signal (S), noise (N), and response (r) power spectra, respectively, where || denotes the norm and 〈 〉 the average over the different stretches. From the spectra the different 3 dB-cut off frequencies, f_{3 dB}, are calculated as the bandwidth at half height. Alternative ways of calculating the f_{3 dB}, e.g. using the value where half the area under the curve is reached, gave virtually identical results. The coding properties are deducible from the SNR(f) (see below; Fig. 1C right), whereas the transfer properties can be derived from the cross spectrum between the stimulus and the signal (Fig. 1C left) as will be explained below.

The SNR spectrum: coding properties.

For the WN stimulation, signal-to-noise ratio SNR(f) was calculated from the signal and noise power spectra, 〈|S(f)|²〉 and 〈|N(f)|²〉, respectively, as their ratio (Fig. 1C right). From 〈|S(f)|²〉, 〈|N(f)|²〉 and SNR(f) several parameters about the coding efficiency of the cell can be calculated (Fig. 2). Figures 2A and 2B show that brightening increases the signal power but reduces the noise power, respectively. In this article, when we mention power we mean the value integrated over the corresponding power spectrum. This result is further confirmed in the time domain from the independent measurements of the signal, σ_S², and noise, σ_N², variances, and their ratio, SNR_t. We also estimated the information transfer rate from SNR(f) using the Shannon formula, which is applicable for this special case when both signal and noise distributions approximate a Gaussian [35]:
(2)
where the lower limit of the integral was set to 2 Hz, because of the finite size of the recording (1 s), and the upper limit was set to 100 Hz, because of the unreliability of the signal at higher frequencies (Fig. 2C). Note that since , intuitively the information transfer rate measures the number of ‘coding states’ the cell uses. Two voltage states must be separated by at least N for being distinguishable and the useable voltage range r consists of r/N such states. The information transfer rate of the responses scales closely with the SNR over the tested luminances (not shown), vindicating the analysis.

From the SNR(f) we also calculated the linear coherence γ_lin [33]:
(3)
Notice that ; assuming that the system behaves linearly (see below for a test of this assumption), the more γ_lin departs from unity the noisier the response at the given frequency (Fig. 2E).

The cross-spectrum of signal and stimulus: transfer properties.

The cross-spectrum (Fig. 1C left) is calculated from the Fourier transforms of the signal and stimulus. It can be used for building several estimators that give insight about how a photoreceptor transforms the light stimulus into a voltage signal (cf. Fig. 3, below). Here we consider the cell as a filter; knowing its input (the controlled contrast stimulus) and output (the recorded voltage changes), we characterize its transfer properties. From the previous analysis it is clear that the noise is very small compared to the signal (SNR_t~70 at bright BG). In such practically noise-free (NF) conditions, we can use the noise-free coherence function, γ_NF, to estimate the system's linearity [36]:
(4)
where * denotes complex conjugate. γ_NF is essentially the normalized signal and stimulus cross-spectrum. Assuming noise-free transmission, if γ_NF is unity the system behaves linearly. This assumption is true in locust photoreceptors in most light conditions as seen in Figures 2E and 3G over a wide range of stimulus frequencies (see also [24]). This range, roughly between 4 and 60 Hz, is also where most information is carried (Fig. 2C). Therefore, for WN stimulation we can consider a photoreceptor as a linear filter and calculate its frequency response, T(f), as:
(5)
T(f) is a complex-valued function and therefore can be expressed in term of its norm G(f), the gain of the contrast-to-voltage transformation shown in Figure 3A, and its phase P(f), the phase-shift between the input and the output shown in Figure 3C, explicitly: