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1 Department of Physiology and Biophysics, Dalhousie University, Halifax B3H 4H7, Nova Scotia, Canada; 2 Physiological Laboratory, University of Cambridge, Cambridge, CB2 3EG United Kingdom
Submitted 21 April 2003; accepted in final form 20 May 2003
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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). Drosophila photoreceptors provide a genetically tractable system in which individual cellular components, including ion channels and sensory transduction proteins can be studied (Hardie 2001; Weckström and Laughlin 1995). Combined with quantitative measures of photoreceptor performance, such as sensitivity, signal-to-noise ratio, and frequency response, as well as behavior (e.g., Schilstra and van Hateren 1999; van Hateren and Schilstra 1999), these cellular and molecular approaches provide a system where the contributions of individual cellular components to normal cell function may be determined (Juusola and Hardie 2001a).
Drosophila and other insect photoreceptors (e.g., Weckström et al. 1991) contain voltage-activated K+ channels (Hardie 1991) that contribute strongly to photoreceptor membrane behavior and may reflect the insects' visual ecology (Weckström and Laughlin 1995). Drosophila photoreceptors express an array of voltage-activated K+ channels including the rapidly inactivating Shaker K+ channel and at least two noninactivating K+ channels (Hardie 1991). The Shaker K+ channel in Drosophila was the first cloned K+ channel and has been extensively investigated (Hille 2001), but the roles of Shaker and its homologs remain unclear. Suggested functions include the attenuation of graded potentials and back-propagated action potentials in dendrites (Hoffman et al. 1997; Laurent 1990; Magee et al. 1998), modulation of firing frequencies of spiking neurons (Connor and Stevens 1971), and control of spike propagation reliability (Debanne et al. 1997).
Recent work has indicates that Shaker increases the dynamic range of phototransduction and improves the signal-to-noise ratio in Drosophila photoreceptors (Niven et al. 2003). However, the relationship between this voltage-activated ion channel and membrane potential is inherently nonlinear. To determine the nonlinear contributions of Shaker K+ channels we recorded photoreceptor membrane potentials in wild-type and Shaker mutant Drosophila during stimulation with wide-bandwidth light fluctuations, and measured the linear and second-order nonlinear components of transduction by a Volterra functional series expansion.
Wild-type photoreceptors demonstrated an early positive nonlinear component that amplified and accelerated the photoresponse, but this component was absent in mutant photoreceptors. Simulation of phototransduction with a nonlinear cascade model supported this conclusion and indicated that Shaker K+ channel adds a nonlinear effect early in the transduction process.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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The wild-type strain was red-eyed Drosophila melanogaster Oregon Red. The null mutation in the Shaker channel, Sh14, a missense mutation in the core region resulting in nonfunctional Shaker channels (Kaplan and Trout 1961), was also expressed in red-eyed flies. Both strains of flies were raised at 19°C in darkness.
Recording and stimulation
Flies were fixed with wax into a ceramic holder and a small window was cut into the surface of the compound eye (Juusola and Hardie 2001a). Intracellular recordings were made using quartz microelectrodes, with resistances between 150 and 220 M
, filled with 3M KCl. A second blunt microelectrode filled with fly Ringer was also placed in the flies' heads close to the eyes. All recordings were made using an intracellular amplifier (SEC-10L, NPI Electronic, Tamm, Germany) in current-clamp mode using switching frequencies of 8–20 kHz and low-pass filtered at 500 Hz (elliptic). Temperature was maintained at 25°C throughout the experiments to within 1°C accuracy using a custom-built thermocouple feedback to a Peltier device. Photoreceptors were only used if their membrane potential was more negative than –55 mV and they had at least a 45-mV saturating impulse response in dark-adapted conditions. Data acquisition, stimulus generation, and signal analysis were performed by a custom-built MATLAB interface (Juusola and Hardie 2001a).
Photoreceptors were stimulated with a pseudorandom light contrast x(t), generated by a high-intensity green light-emitting diode with peak wavelength 525 nm (Marl Optosource) at a mean intensity of 3 x 106 photons/s. Intensity was extrapolated from the average number of single photon responses to a dim, steady light background counted in the same photoreceptor during 10 s. Background intensity was increased or reduced by neutral density filters between the light source and the eye. All light intensity measurements were converted to dimensionless contrast units with a mean contrast of 0.32. Light intensity and photoreceptor membrane potential were sampled at 1-ms intervals during the presentation of pseudorandomly fluctuating sequences of 100 s duration.
Nonlinear system identification
The original sampling resolution of 1 ms was reduced to 2 ms by combining adjacent points to give records of 50,000 data pairs (light intensity in contrast units and receptor potential in mV). For each record, the first 40,000 data pairs were analyzed as the input and output of an unknown nonlinear dynamic system with light intensity as the input x(t), as a function of time t, and the receptor potential as output y(t), represented by the first 3 terms of a Volterra functional series
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Several methods have been developed for kernel estimation (French and Marmarelis 1999). Earlier methods relied on stimulating the unknown system with Gaussian white noise, or a close approximation, but more recent methods avoid this requirement. The parallel cascade method (Korenberg 1991) is based on the principle that a wide range of nonlinear dynamic systems can be approximated by a parallel cascade of simple nonlinear systems (branches), each consisting of a linear filter followed by a static nonlinearity (Fig. 1). Branches are added one by one, with each chosen to minimize the squared error remaining after subtracting the sum of all the previous branch outputs from the actual system output. Addition of branches proceeds until some predetermined criterion of error level or number of branches is reached.
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The choice of linear filters is arbitrary. Filters chosen by cross-correlation between input and output can produce very rapid convergence (French and Marmarelis 1999; Korenberg 1991). We used an alternative approach, where the filters are formed from Gaussian distributed random numbers (French et al. 2001). This method is relatively slow, and often requires 
105 cascade branches, but it is completely general and makes no initial assumptions about the forms of the kernels or the nature of the input or output signals. Modern computing power allows this inefficient, but general and objective method to be easily applied to a wide range of data.
Once each filter is chosen, the polynomial function is calculated by linear regression. Kernels are then updated by adding the filter function multiplied by the appropriate polynomial coefficients. An important feature of the method is that it can be applied to systems containing relatively high-order nonlinearities by extending the polynomial function. However, it is not necessary to construct all of the higher-order kernels. Another important point is that the final number of parameters used to fit the data are given by the kernel values themselves, so a large number of cascade branches does not increase the number of fitted parameters, only their accuracy.
After kernel estimation, percentage mean square error (MSE) values (French and Marmarelis 1999) were calculated for the zero and first-order kernels alone and for the combined, zero-, first-, and second-order kernels from
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The kernel estimates were then substituted into Eq.1 and used to predict the output of the nonlinear system to the input signal of the remaining 10,000 data pairs of each record. Therefore all predictions were based on data that had not been used for system identification.
Simulation
Nonlinear responses in Drosophila photoreceptors were simulated by a NLN (nonlinear static–linear dynamic–nonlinear static) cascade model (French et al. 1993). The two nonlinear components were third-order polynomial functions and the linear component was the Wong and Knight photoreceptor model (Wong et al. 1980)
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are parameters to be fitted. To remove redundant parameters, only one constant term was included in the model, as an offset in the output of the linear component. Similarly, the first nonlinear component had a fixed first-order coefficient of unity. The numbers of unknown parameters were therefore: 2 (first polynomial) + 3 (Wong and Knight model plus offset) + 3 (second polynomial) = 8. The NLN cascade model was fitted to the same 40,000 data pairs as used for the kernel analysis. Fitting was performed in 2 stages, using simulated annealing to obtain an approximate fit and then the Levenberg–Marquardt method with the simulated annealing results as the starting condition (Press et al. 1990). The fitting algorithms always converged satisfactorily. Mean square errors were again calculated using Eq.2. Data analysis and graphical presentations were performed by custom-written software using the C++ programming language and PC-compatible desktop computers.
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RESULTS |
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; Niven et al. 2003). However, that experimental approach could not separate uncorrelated noise from adaptive nonlinearities, and may thus have given biased estimates of the difference between wild-type and Sh14 photoreceptors. To study the nonlinear dynamic properties of wild-type and Sh14 photoreceptors, we estimated both linear and second-order nonlinear components of voltage responses using nonaveraged data.
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Kernel estimates for photoreceptors
Preliminary measurements indicated that both first-order and second-order kernels extended for <60 ms along the time axes. Therefore all estimation and modeling was conducted using 30 lags of 2 ms. Parallel cascade fitting was used with cascade branches being added until the square of the residual error [z(t) in Fig. 1] did not change after trying 
100 new branches. Data are presented for three representative photoreceptors from each of the wild-type and Sh14 flies. The number of cascade branches required was approximately 20,000 for wild-type receptors and 100,000 for Sh14 photoreceptors (Table 1), with a mean cascade error of 25.4% for wild-type and 41.2% for Sh14. These values reflect the relatively higher noise levels in Sh14 photoreceptors (Niven et al. 2003).
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The MSE (Eq. 2) was calculated for the Volterra series (Eq. 1) using only K0 and K1 (MSE-K1 in Table 1) and using all the kernels (MSE-K2 in Table 1). Addition of the second-order kernel reduced MSE by about 2% in wild-type flies, and about 6% in Sh14 flies, indicating that there are nonlinear contributions to the responses of both animals, but that normal Sh14 K+ channels confer more linear responses. MSE-K2 values closely mirrored the cascade errors in all cases (note that these 2 errors were computed differently).
Kernel forms
Figure 3 shows kernel estimates from 3 representative wild-type photoreceptors. First-order kernels K1(u) had a repeatable form that was similar to flash responses in light-adapted Drosophila photoreceptors (Juusola and Hardie 2001a) with an initial delay of about 5 ms and a peak response at about 15 ms. Second-order kernels K2(u, v) also had a repeatable form, with a positive peak on the diagonal at about 12 ms and a negative, inhibitory peak at about 20 ms that spread away from either side of the diagonal by more than 5 ms. Plots of second-order kernel diagonals K2(u, u) are also shown in Fig. 3 (bottom traces). Note that the positive peak always occurred about 2 ms before the first-order kernel peak, and that the inhibitory peaks, which were clearly evident in the perspective plots, made little contribution to the kernel diagonal.
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First-order kernels of Shaker mutant photoreceptors had very similar amplitudes and forms to those from wild-type flies (Fig. 4). However, the second-order kernels were dramatically different, with complete loss of the early positive peak and much clearer inhibitory peaks, which now dominated the diagonals (Fig. 4, bottom traces).
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Predictions of photoreceptor output from the Volterra series (Eq. 1) were closely similar to the experimental values by eye. Examples for 2 s of the response from a wild-type receptor are shown in Fig. 5, both with and without the second-order kernel contributions. It was impossible to see the nearly 2% improvement in error contributed by K2(u, v) by eye.
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NLN cascade model of phototransduction
Volterra series contain large numbers of parameters (496 in the present case). Attempts to convert these estimates to models with smaller numbers of parameters have concentrated on cascades of simple linear dynamic and nonlinear static components (French and Korenberg 1989; Korenberg and Hunter 1986). A model containing 2 static nonlinearities surrounding a linear filter has been used before to reproduce the responses of photoreceptors to a wide range of stimuli (French et al. 1993) and the gamma function model (or Wong and Knight model) of photoreceptor function (Wong et al. 1980) has also been used in cascade models (Weckström et al. 1995). We combined these ideas to create the NLN model (Fig. 5 and Eq. 3) having only 8 variable parameters (see METHODS) to simulate the Drosophila photoreceptor responses.
Error values for the NLN model (MSE–NLN) were always intermediate between MSE-K1 and MSE-K2 (Table 1), indicating that the NLN model captured some, but not all, of the nonlinear photoreceptor behavior identified by the Volterra series. Expanding the polynomial functions to fourth-order did not reduce the error (data not shown). Fitted parameters for the same 3 wild-type and Shaker mutant photoreceptors are shown in Table 2 and plotted graphically in Fig. 7. Compared with wild-type flies, Shaker mutant parameters for the NLN model were different in two respects: 1) a weaker second-order term made the polynomial function of the first static nonlinear component more linear, and 2) the response of the linear component was slower and of smaller amplitude.
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DISCUSSION |
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; Hoffman et al. 1997).
Fast, voltage-dependent ion channels in Drosophila photoreceptors change the membrane impedance and modify the frequency tuning of phototransduction (Juusola and Hardie 2001a; Niven et al. 2003). Flies with the Sh14 mutation differ from wild-type in the range of functional ion channels expressed in their photo-insensitive plasma membranes. The differences between the Volterra kernels obtained from wild-type and Sh14 photoreceptors probably reflect differences in these light-induced membrane impedance changes. How well can the known membrane properties and dynamics of wild-type and Sh14 photoreceptors explain the observed differences in the kernels?
Parallel cascade estimation gave consistent results for the Volterra kernels of phototransduction in both wild-type and Sh14 Drosophila photoreceptors. First-order kernels, representing the dominant linear contribution, were indistinguishable between wild-type and Sh14 flies, indicating that the basic phototransduction machinery and light current in Shaker mutants is intact. However, clear differences were seen in the forms of the second-order kernels from wild-type and mutant flies, with the Sh14 photoreceptors lacking a major amplifying component that normally peaks during the rising phase of the response (Figs. 4 and 5), suggesting that Shaker K+ channels mediate an amplification of the graded voltage signal that is absent from the Sh14 photoreceptors. This agrees with recent suggestions based on photoreceptor contrast gain functions, that Shaker K+ channels normally amplify the voltage response in Drosophila photoreceptors (Niven et al. 2003).
The time course of the second-order kernel in wild-type photoreceptors corresponds to the timing of the Shaker A-current IA. The time constant of IA, measured in dissociated wild-type photoreceptors by whole cell patch clamp, was 1.6 ± 0.3 ms at the resting membrane potential, and is likely to be faster in the depolarized state of our experiments at 25°C because the time-to-peak of IA at 20 mV above rest was 1.3 ± 0.2 ms at 20°C (Hevers and Hardie 1995). We must also consider that the kernel results were not obtained from voltage or current pulse stimuli, but from white noise light stimulation where the membrane current response is delayed by the dead time of phototransduction (French 1980; Juusola and Hardie 2001b). It seems clear that the Shaker conductance, reflected in K2, amplifies and accelerates the voltage response to light stimulation.
Photoreceptors with a Shaker null mutation also operate at much lower mean light-induced potentials than wild-type photoreceptors (Fig. 2). The lack of Shaker K+ channels leads to a compensatory increase in leak conductance, as indicated by the reduced membrane impedance and higher dark resting potential of Sh14 photoreceptors (Niven et al. 2003), which probably causes the reduced response to light. The second-order kernels of Sh14 photoreceptors caused a significant gain reduction during the time course of the transduction seen in the first-order kernels, reflecting this voltage-dependent shunting.
The NLN cascade simulation, although clearly not adequate to account for all the nonlinear behavior, also suggested that Shaker K+ channels increase the amplitude and speed of the phototransduction response (Fig. 7). The relatively small positive deviation in the first nonlinear component at negative contrasts has been reported before in another fly (Calliphora) photoreceptor (Weckström et al. 1995) and tentatively linked to fast voltage-activated K+ channels. The reduction of this deviation in Sh14 photoreceptors indicates that there are differences between these two photoreceptor membranes in addition to the lack of Shaker K+ channels in Calliphora. This difference probably arises from compensatory leak conductances (Niven et al. 2003), which partially offset the loss of Shaker K+ channels in Drosophila, but are not equally present in Calliphora photoreceptors.
Kernel estimations in mutant flies gave much higher error levels than those in wild-type, supporting the idea that Sh14 photoreceptors have lower signal-to-noise ratios (Niven et al. 2003). Therefore it is difficult to estimate the relative contributions of nonlinearities and noise to changes in the coherence function because both reduce it (Bendat and Piersol 1980). However, here we have an independent measure of nonlinear contribution to the response by observing the reduction in MSE produced by adding the second-order kernel. Sh14 photoreceptors showed a greater reduction in MSE than that of their wild-type counterparts on addition of the second-order kernel (Table 1), indicating that Shaker K+ channels partially linearize the normal photoreceptor voltage response. This linearizing effect of Shaker K+ channels is supported by the forms of the 2 static nonlinear components in the NLN model (Fig. 7), where the form of the first nonlinear component in the wild-type flies would partly compensate for the opposite form of the third component, which was seen in all flies and is probably caused by membrane shunting.
What is the physiological role of the Shaker conductance in photoreceptors at bright light levels, which depolarize wild-type photoreceptors by approximately 20–25 mV above the dark resting potential (Em = –68.1 ± 3.2 mV, n = 21; Niven et al., 2003)? The majority of Shaker K+ channels should be inactivated and therefore contribute relatively little to the photoreceptor membrane response. However, our results indicate that Shaker K+ channels make significant contributions at depolarized potentials, both amplifying and accelerating the photoreceptor responses. This may be explained by the voltage separation of the steady-state activation and inactivation curves of the Shaker and delayed rectifier K+ channels (Niven et al. 2003). Over a voltage range from –58 to –46 mV, depolarization decreases total steady-state K+ conductance because Shaker channels are inactivating and relatively few delayed rectifier channels are activating. This reduction in total K+ conductance amplifies the voltage signal. This model is supported by the loss of amplification and acceleration in the Sh14 mutant photoreceptors, which show a membrane shunt at high light intensities that is probably generated by activation of delayed rectifier K+ channels. This effect would be masked in wild-type cells by Shaker-mediated amplification.
Are the effects of the Sh14 mutation entirely attributable to loss of functional Shaker channels in the photo-insensitive membrane? Sh14 has a missense mutation in the core region of the Shaker channel (Lichtinghagen et al. 1990). The effects of this mutation are not restricted to photoreceptors but have also been reported in motor neurons and muscles in both adults (Ganetzky and Wu 1982) and embryos (Broadie and Bate 1993; Haugland and Wu 1990). Sh14 affects circuits underlying flight and an escape response circuit (Engel and Wu 1992) as well as producing uncoordinated walking behavior (Ganetzky and Wu 1982) in adult flies. These effects all seem to be caused by loss of functional potassium channels in the neuromuscular system, rather than any pleiotropic effects. However, this does not exclude the possibility that elimination of functional Shaker channels has other effects on photoreceptor physiology. Additionally, if experience-dependent plasticity is important in developing normal photoreceptor function, it is possible that behavioral differences in Sh14 flies could in turn lead to different photoreceptor properties. However, this is unlikely to be a factor in the present study because all flies were reared in darkness.
An animal's lifestyle and behavior control the range and speed of natural stimuli it encounters, placing constraints and demands on the quality and processing capacity of its visual system. Laughlin (1996) suggested that slowly flying insects may use rapidly inactivating potassium conductances to create ohmic photoreceptor membranes under various light conditions. This metabolically inexpensive design, called matched filtering, would both maintain a high gain for relevant slow stimulus frequencies and filter out high-frequency phototransduction noise. However, our results suggest that the situation is not so simple, at least for the photoreceptors of relatively slow moving and turning Drosophila. Although Drosophila has poor spatial resolution with a small number of facets and blurred optics (Land 1997) its photoreceptors produce fast responses to constant variance light contrast stimuli. We found that the rapidly inactivating Shaker channels, instead of making the photoreceptor membranes ohmic act as nonlinear filters to accelerate and amplify phototransduction responses, and hence enhance the bandwidth of reliable signaling. The NLN model suggests that this amplification is an early nonlinearity that partly offsets later nonlinearities of opposite form.
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DISCLOSURES |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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FOOTNOTES |
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Address for reprint requests and other correspondence: A. S. French, Department of Physiology and Biophysics, Dalhousie University, Halifax, Nova Scotia B3H 4H7, Canada. (E-mail: andrew.french{at}dal.ca).
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REFERENCES |
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cjs(t)j, where cj are the polynomial coefficients]. Structure of single branch is shown below. Each branch m of cascade has different parameters g(u) and cj to produce a set of outputs ym(t). Sum of cascades w(t) is subtracted from unknown system output to give residual signal z(t). Addition of each branch to cascade is chosen to minimize mean squared value of z(t).
