Spectral sensitivity of guinea pig rods
Dark-adapted guinea pig rods were targeted for electrical recording based on their smaller inner segment diameters relative to cones (Yin et al., 2006
). Rod identification was further confirmed by measuring spectral sensitivity in 9 cells (). Sensitivity at 430, 500, 570, and 660 nm was calculated as the inverse of the intensity evoking a half-saturating response. The mean spectral sensitivities were fit with a standard photopigment nomogram (Baylor et al., 1987
), with corrections for photopigment self-screening calculated from reported transverse optical density (Parry and Bowmaker, 2002
) and rod dimensions (Yin et al., 2006
). The best fit nomogram for the average spectral sensitivity function peaked at 496 nm, in agreement with previous estimates from microspectrophotometry (Jacobs and Deegan, 1994
) and from electrical recordings in downstream retinal neurons (Yin et al., 2006
Figure 1 Guinea pig rod spectral sensitivity. Points plot the mean sensitivity of 9 rods for 430, 500, 570, and 660 nm wavelength stimuli. Standard deviations are smaller than symbol diameters. The curve is equation 6 from Baylor et al. (1987) after correction (more ...)
Electrical coupling reduces variability in response amplitudes by averaging the membrane potential across neighboring rods (Fain, 1975
; Hornstein et al., 2005
). To study the effects of coupling on response variability we recorded rod voltage in perforated patch mode and measured responses to brief, full-field flashes at a series of intensities evoking ~1–6 R*
(mean number of photoisomerizations per rod). The large response variability in some rods indicated weak or no coupling (), while the small variability in other rods was consistent with strong coupling ().
Figure 2 Coupling reduces the variability of dim flash responses. A,B: Rod voltage responses recorded in two rods to a series of dim flashes evoking ~1 R*. Tick marks indicate flash timing. Bandwidth, DC–5 Hz. C,E: Response mean (C) and variance (E) of (more ...)
To quantify the signal averaging effect of coupling, we computed the ensemble mean and variance for 50–100 repetitions at each flash intensity (). Both the peak amplitude of the mean (μ) and peak amplitude of the variance (σ2
) were proportional to R*
(), as expected if photon responses summed linearly and amplitude variability was dominated by quantal fluctuations in the number of photoisomerizations. The average single photon response amplitude, calculated from the value of μ at R*
=1, was 0.61 ± 0.16 mV (mean ± S.D., 14 rods). The waveform of the photon response could be described by equation 19
= 3 and tp
= 180 ms.
Response variability depends on the number of rods coupled, as well as on the ratio of gap junctional conductance to rod membrane conductance. As a metric to quantify the degree of coupling we calculated N, the number of “perfectly” coupled rods (Gj=∞) that would lower response variability to the observed value. N was calculated from the mean and variance via: N = k12/k2, where k1 = μ/R* and k2 = σ2/R*. For uncoupled rods (N=1), k12 = k2. The values for k1 and k2 were obtained from the slopes of the lines passing through the origin that best fit the functions μ vs. R* and σ2
vs. R* ().
The measured variability in some rods ( and ○ in D,F) was consistent with the absence of electrical coupling (k12
), while in other rods ( and
in D,F), the observed reduction in variability (k12
) was consistent with signal averaging within a coupled network. In 14 rods, the coupling metric N
ranged from 0.7–11.6, averaging 4.8 ± 4.5 ().
We determined the number of rods physically coupled to one another by recording from individual rods with whole-cell electrodes containing the tracer Neurobiotin and counting the rods subsequently labeled. Occasionally tracer pools also contained cones, as identified by peanut agglutinin binding and by the absence of a narrowing of the cell diameter between the inner segment and cell body (data not shown). Out of 37 rod injections, only 5 showed tracer coupling to cones, and these pools were not included in the subsequent analysis.
Of the remaining 32 rod injections, 8 (25%) showed no tracer coupling () while 24 (75%) were coupled to one or more neighboring rods (). The total number of rods in a tracer coupled pool ranged from 1–21, averaging 4.0 ± 4.7 (). Many rod pairs had multiple apparent sites of contact, including rod spherules, axons, cell bodies, and inner segments. Some rods appeared to be laterally separated by several micrometers from all the other rods in the pool at one plane within the photoreceptor layer (), but were found to be in close apposition at a different plane. For most injections the labeling was restricted to a small clearly demarcated cluster of rods. However, in 2 injections labeling intensity decreased with distance from the injected rod until it was no longer detectable. Thus while rods generally appeared to be coupled in discrete pools, a continuous network of rods could not be ruled out in some instances.
Figure 3 Rod tracer coupling. A,B: Combined confocal fluorescence and DIC images of the outer nuclear layer after single rod injections with Neurobiotin (green). Scale bar, 10 μm. A: Rod is not tracer coupled to neighbors. B: Four tracer-coupled rods. (more ...)
Gap junctional conductance
To measure gap junctional conductance, we made whole-cell voltage-clamp recordings from pairs of neighboring rods. While holding one rod at a constant voltage, changes in that rod’s membrane current were measured in response to voltage pulses applied to one of its neighbors (). Changes in membrane currents were linear with voltage (), reflecting the ohmic behavior of the gap junctions for brief voltage pulses. The slope of the current-voltage relationship gives the junctional conductance Gj. As expected for junctions between homologous cell types, conductance was symmetric; Gj was independent of which rod in the pair was the driver and which the follower.
Not all rod pairs showed measurable coupling conductances; in 11 out of 22 paired rod recordings Gj was below the resolvable limit given measurement noise. The remaining 11 rod pairs had junctional conductances ranging from 195–580 pS, with an average of 386 ± 112 pS ().
This estimate of Gj
assumed that the only current path between rods was via a gap junction directly between the recorded rods. However, current could also flow via indirect paths through intervening mutually coupled rods and thus increase the apparent junctional conductance. For a pair of recorded rods connected directly to one another and also connected indirectly via one intervening rod () the fraction of the total measured current due to the indirect path would be (3 + β)−1
(see Materials and Methods). In guinea pig, with β = 2.7 (see Guinea pig rod network model), the indirect current would be 18% of the measured current. For a pair of coupled rods that are also connected via two indirect paths (), the proportion of indirect current would increase to (2 + β/2)−1
, or 30% of the measured current. If we assume hexagonal rod packing (), there can be at most two such indirect paths with single intervening rods (). In an infinite fully connected hexagonal network, there are an infinite number of indirect current paths with varying numbers of intervening rods. However, the proliferation of current paths to ground makes each indirect path more leaky such that the overall proportion of indirect current is never greater than (2 + β/2)−1
. In practice, given the modest connectivity observed in tracer-coupling pools, the fraction of indirect current is expected to be small for most of the pairs from which we recorded. Rods may also be connected via intervening cones. However, the rod-evoked light responses recorded in dark adapted cones in primate retina (Hornstein et al., 2005
) and guinea pig retina (data not shown) are an order of magnitude smaller than those measured directly in rods. Consequently, network analysis indicates that any signaling from rod to rod via intervening cones is negligible.
Figure 5 Hexagonal network modeling. A: DIC photomicrograph of the inner segment layer illustrating the roughly hexagonal packing of the guinea pig rod mosaic. Scale bar, 10 μm. B: Schematic of a resistive rod network model: a field of hexagonally packed (more ...)
Given that our recording electrodes were on the rod inner segments while gap junctions between rods are reported as far away as the rod synaptic spherule (Raviola and Gilula, 1973
; Tsukamoto et al., 2001
), the electrotonic distance between the inner segment and spherule might reduce the apparent junctional conductance. Smith et al. (1986)
estimated ~15% voltage loss across the axon connecting the inner segment and spherule in cat rods. Adjusting for species differences in axon diameter and length, we calculate the voltage loss in guinea pig rods to be ~12%. Hence junctional conductance estimates could be underestimated by as much as 24%. However, to the extent that gap junctions are present between rod inner segments, (Uga et al., 1970
; Cohen, 1989
), the electrotonic distance errors in Gj
could be negligible. Overall, errors in the estimated junctional conductance due to incomplete space clamping or indirect current flow between rod pairs are expected to be small and opposing.
Guinea pig rod network model
A resistive circuit model of the rod network was constructed to consolidate the measurements of fluctuation analysis, junctional conductance, and tracer coupling, and to enable further psychophysical modeling. For the purposes of modeling, rod-cone coupling was ignored under the assumption that it would contribute negligibly to rod network behavior.
A roughly hexagonal packing pattern of rods was observed in the inner and outer segment layers (). Since gap-junctional coupling requires close membrane apposition, and rods do not send processes to distant targets (Raviola and Gilula, 1973
; Kolb, 1977
), the model assumed that each rod could make direct gap junctional connections with at most six nearest neighbors. Rods were modeled as nodes with membrane resistances to ground (Rm
) and current sources representing phototransduction. Some nodes were connected laterally through junctional resistances (Rj
) to represent coupling ().
The model assumed a value for Rm
of 1.5 GΩ based on the instantaneous input resistances measured in our whole-cell recordings. Since input resistance is reduced by rod coupling, Rm
was based on our highest measured input resistances. For comparison, Rm
was estimated as 2.5 GΩ for guinea pig rods in bright light (Demontis et al., 1999
). It was assumed that the average measured Gj
of 386 pS was slightly overestimated due to indirect coupling through other rods (). Hence the assumed value of Rj
, where Rj
, was rounded up to 3 GΩ.
For a resistive network of a given connectivity, the signal averaging behavior of the network depends only on the ratio β = Rj
. For guinea pig, with the resistances determined above, β = 2. Equations 9
were used to calculate signal transfer ratios and to predict the degree of signal averaging as quantified by N
. For an infinite network of coupled rods with full hexagonal connectivity, the model predicts a value of N
= 9.1. By restricting the connectivity of the model to just two surrounding layers of rods (19 rods total), the predicted N
for the central rod was reduced by only 10%. Thus, with β = 2, signal averaging is effectively restricted to only the first few layers of surrounding neighbors, even for an infinite pool of connected rods.
Effects of capacitance on guinea pig rod network model
The inclusion of membrane capacitance into the network model had a negligible effect. In whole cell recordings, rod membrane capacitance ranged from 5–10 pF (data not shown). Taking Ca = Cm = 10 pF for all a, and assuming Rm = 1.5 GΩ and Rj = 3 GΩ, we calculated the transfer matrix H for f = 0–10 KHz.
While capacitative filtering is expected to draw out the response time course and hence reduce the peak amplitude of the voltage responses in coupled cells, the magnitude of this effect was found to be negligible, reducing the peak amplitude of the signal transfer by <0.5%. Capacitance altered the times to the peak of the response in coupled cells as well, but this effect was also negligible.
Effects of voltage-activated conductances on guinea pig rod network model
Light-evoked hyperpolarization in guinea pig rods evokes changes in voltage-dependent conductances, resulting in membrane depolarization (Demontis et al., 1999
). These conductance changes are expected to reduce the spread of photon signals within the rod network in a frequency-dependent manner (Detwiler et al., 1980
). For small perturbations in membrane potential, this effect can be modeled by adding a shunted inductance in parallel with Rm
() (Detwiler et al., 1980
). In turtle and guinea pig rods recorded at room temperature (Detwiler et al., 1980
; Demontis et al., 1999
) the observed changes in membrane potential evoked by injecting small steps in current were consistent with an equivalent inductance L
of 1–2 GH in series with a shunting resistance RL
of 0.6–4.0 GΩ. The voltage transfer was thus band-passed (equation 14
), with a peak at ~1–2.5 Hz.
At mammalian body temperature however, we expect faster channel activation and a shift in the band-pass of signal transfer to higher frequencies (Demontis et al., 1999
). We measured these effects in 8 guinea pig rods recorded at 35 °C by applying a −1 mV voltage pulse from a holding potential of −50 mV, or by injecting a 1 pA hyperpolarizing current pulse (). We observed voltage-activated conductances consistent with a modeled inductance of ~250 MH in series with a shunt resistance of ~2.5 GΩ. The voltage transfer under these conditions peaks at about 8 Hz, significantly faster than the dynamics of the guinea pig photocurrent, which has 95% of its power below 5 Hz. Consequently the voltage-activated conductances quickly counteract the rod photoresponse, reducing rod-rod transfer by an additional 15% compared to the transfer in a purely resistive network. This reduction in signal transfer is equivalent to an increase in β from 2 to ~2.7.
The filtering of the network is computed to have only a small effect on the time course of transferred signals, reducing the time-to-peak of the photovoltage in the neighboring cell by ~3%, and reducing the integration time by ~7%. While the effects on time course accumulate as the signal gets transferred from one rod to the next, the cumulative effect is small because signal spread is effectively restricted to only the first two layers of surrounding neighbors. As was shown above for the purely resistive network, by restricting the connectivity of the hexagonal model to just two surrounding layers of rods, the predicted N for the central rod was reduced by only 5% compared to that of an infinite hexagonal network. Thus, with β = 2.7, the temporal changes across the network are modest. These results were confirmed using a circuit simulator.
Taking voltage-activated conductances into account reduces the calculated value of N
expected in a fully connected hexagonal network by 25%, from 9.1 to 6.8. This is the maximal expected value; with less than full connectivity () N
would be less than 6.8. However, the values of N
derived experimentally from responses to dim flashes () were observed to be as large as 11.6, suggesting that at least in some cases the coupling conductance or connectivity might be greater than that modeled here, or else that the counteracting effect of voltage-activated conductances were less. Although the packing of inner segments is roughly hexagonal (), rods typically also have points of contact with additional neighbors in the nuclear layer (Hornstein 2005
, Fig 9B & 9C). Changing the hexagonal connectivity model to a square model in which each rod contacts 8 neighbors increases the calculated N
from 6.8 to 9.9 for β = 2.7. The discrepancy between the maximal modeled vs
. observed values may further reflect the difficulty of precisely estimating N
from fluctuation measurements when N
is large and the amplitude of the light-evoked variance is small compared to the baseline fluctuations in variance.
Note that while the predicted value of N is highly sensitive to the value assumed for β, the signal transfer ratio w is only ~60% as sensitive, and the psychophysical effects considered below are even less sensitive.
Primate rod network model
In macaque monkey, rods form tracer coupled pools of 1–10 rods and exhibit signal averaging of N
= 1–5.9 (Hornstein et al., 2005
). Although junctional conductances have not been measured directly in primate rods, we asked whether an Rj
of 3 GΩ as measured in guinea pig would be consistent with the tracer coupling and signal averaging found in primate. Taking Rm
= 1.2 GΩ in primate rods (Schneeweis and Schnapf, 1995
) and Rj
= 3 GΩ gives β = 2.5.
Assuming from the tracer coupling data a maximal pool size of 10 rods, a hexagonal network was constructed with 1 central rod connected directly to an inner layer of 6 rods, and indirectly to 3 additional rods in a second layer. The predicted N = 5.6 was close to the maximum value of N = 5.9 measured in primate rod recordings. Thus the tracer coupling and signal averaging observed in primate rods are consistent with a model in which each rod connects to between zero and six neighbors with a junctional resistance of 3 GΩ.
Because the value of Rm
= 1.2 GΩ in primate rods was derived by taking the ratio of the peak photon response amplitude of the photovoltage to the photocurrent (Schneeweis and Schnapf, 1995
), any effects of voltage-dependent conductances on β were already accounted for. To ascertain whether voltage-dependent conductances play a role in shaping photon responses in primate, we measured membrane current in response to −1 mV pulses in 6 rods in macaque monkey at 35°C (). Results were very similar to those in guinea pig except that the peak frequency was shifted to 6 Hz. Changes in the response kinetics were calculated to be small, reducing the time-to-peak in the neighboring rod relative to the central rod by ~8%, and reducing the integration time by ~6%. These temporal effects were ignored in the modeling described below.
Previous modeling of the effect of rod coupling on signal detection in primates (Hornstein et al., 2005
) was based on fluctuation analysis and the simplifying assumption of perfect coupling between connected rods. Since the average N
calculated from fluctuation analysis was 2.3 rods, the model had assumed a network consisting of pairs of rods coupled via Rj
= 0 (i.e. N
= 2, ). Here we enhanced the model by replacing perfect coupling with the more realistic junctional resistance of 3 GΩ. While the pattern of rod connectivity was found to be quite variable, for simplicity we assumed a uniform pattern of coupled rods whose network properties reflected the average behavior measured experimentally. Assuming a repeated arrangement of four rods linked in a ring () with β = 2.5, the model predicts N
= 2.3, precisely the average value found experimentally in primate. Compared to the previous two-rod model, the four-rod resistive network is also a better match for the primate tracer data, where the average tracer-coupled pool size was 3.4 rods.
Human psychophysical model
Our rod-rod coupling measurements were incorporated into a psychophysical model in order to evaluate how coupling affects visual performance. For a given intensity and spatial pattern of retinal illumination, this model computed 1) the Poisson statistics of photon capture and resulting phototransduction in individual rods, 2) the spread of signal and noise across the rod network for a given coupling connectivity, 3) the signal and noise in rod-bipolar cells after accounting for nonlinearities at the rod output synapse, and 4) psychophysical detection performance, assuming that detection depends on dark/light comparison by a detector stage that linearly sums the output of bipolar cells, with input from a total of 10,000 rods (see Materials and Methods).
We found that a uniformly illuminated pool of 10,000 uncoupled rods yields a detection threshold of 9.7 photoisomerizations over the entire pool, closely matching earlier modeling (Hornstein et al., 2005
) and the psychophysically measured threshold of 10 photoisomerizations (Sharpe, 1990
). Under the perfect coupling assumption, where the detection pool was divided into 5000 pairs of coupled rods with Rj
= 0 (), detection threshold increased by 62% (Hornstein et al., 2005
). This elevation in threshold can be attributed to less effective noise filtering at the rod-bipolar cell synapse of coupled rods (Field and Rieke, 2002
). However, assuming instead a detection pool divided into 2500 discrete four-rod networks with realistic junctional resistance (), the detection threshold for uniform illumination is calculated to be 11.0 photoisomerizations, just 13% greater than the threshold for uncoupled rods. Thus although the perfect and resistive coupling models were set to have equivalent signal-to-noise ratios (SNR) within their respective rod pools (i.e. N
≈ 2), the resistive model performed significantly better in the psychophysical detection task. Replacing perfect coupling with realistic physiological junctional resistances mitigates most of coupling’s putative detrimental effects by allowing the noise filtering nonlinearity of the rod synapse to function more effectively.
As the diameter of the model stimulus decreases so as to illuminate less than the full 10,000 rods in the detector pool, performance improves for resistively coupled networks relative to the uncoupled network (). This improvement reflects the benefit of rod coupling in mitigating synaptic saturation as stimulus size decreases and photon density increases. For stimulus diameters < 0.11 degrees, performance of the coupled network surpasses the performance of the uncoupled network. The smallest retinal image practically achievable, i.e.
the retinal image resulting from a point source of light, is limited by the optics of the eye. This point-spread function, which varies with pupil size, was calculated as the Fourier transform of the modulation transfer function of the human eye (Artal and Navarro, 1994
; Guirao et al., 1999
). The transform is valid at the large pupil diameters (6–8 mm) typical for dark-adapted eyes (Spring and Stiles, 1948
). For computational tractability, the light intensity was set to zero when the number of photons delivered per rod dropped to <0.05% of the total number of photons in the stimulus. For a 6 mm pupil diameter, the calculated detection threshold for a point source was found to be 30% lower in the coupled rod network compared to uncoupled rods; for an 8 mm pupil it was 17% lower ( arrows). Thus at absolute threshold, electrical coupling is expected to have beneficial effects for detection of small stimuli, offsetting the modest detrimental effects expected for large stimuli.
Figure 7 Coupling effects on human visual detection. A: Dark-adapted detection threshold for coupled rod network TC relative to uncoupled rod network TU, as a function of stimulus diameter on the retina. Points plot model calculations with β=2.5. For small (more ...)
The extension of the dynamic range of the rod to rod bipolar synapse due to coupling is also expected to be beneficial for contrast detection against background illumination. As background intensity increases, the intensity of a just detectable test flash also increases, resulting in increasing synaptic saturation. Using the psychophysically derived incremental threshold data from a human rod monochromat (Sharpe et al., 1992
), we calculated the SNR at the detector for stimuli that uniformly illuminated the detection pool (). The calculation was limited to backgrounds < 3 R*
/s, the range over which rods and rod bipolar cells maintain their dark sensitivity (Baylor et al., 1984b
; Kraft et al., 1993
; Dunn et al., 2006
). The SNR of the coupled network surpassed that of the uncoupled network at background light intensities > 0.17 R*
/s, and exceeded it by 14% at the highest light levels modeled. Thus the benefits of coupling on incremental sensitivity also roughly balance the detrimental effects for detection at absolute threshold.