A model for the sensitivity regulation in the primate outer retina is developed and validated using horizontal cell measurements from the literature. The main conclusion is that the phototransduction of the cones is the key factor regulating sensitivity. The model consists of a nonlinearity cascaded with three feedback control loops. The nonlinearity is caused by the hydrolysis of cGMP by activated phosphodiesterase. The first feedback loop is divisive, with calcium regulating the photocurrent in the cone outer segment. The second feedback loop is also divisive, with voltage-sensitive channels regulating the membrane voltage of the cone inner segment. The final feedback loop is subtractive, where the membrane voltage of the horizontal cell is subtracted from that of the cone before the cone drives the horizontal and bipolar cells. The model describes adequately the major characteristics of the horizontal cell responses to wide field, spectrally white stimuli. In particular, it shows (1) sensitivity and bandwidth control as a function of background intensity; (2) the major nonlinearities observed in the horizontal cells; and (3) the transition from linear responses toward contrast constancy (Weber’s law) for background illuminances ranging from 1–1000 td.

*τ*, transforming an input signal

_{y}*x*(

*t*) into an output signal

*y*(

*t*), is with the dot denoting time differentiation: Figure 1A shows how this filter is depicted in the models below. The solution of Equation 1 is which is the convolution of the input function

*x*(

*t*) with the impulse response of the filter, The low-frequency (DC-) gain of this filter equals 1, thus low frequencies are not changed by the filter.

*τ*. An example of such a process, where gain and time constant are covarying, is the voltage response to a current injected into a cell: When the membrane resistance is lowered, both the membrane time constant and the gain (voltage response to a given current) decrease. Below, such processes are described as the cascade of a separate gain

_{y}*τ*and a standard low-pass filter as in Figure 1A.

_{y}*y*(

*t*) as a function

*f*(

*x*(

*t*)) of the input

*x*(

*t*). The function f(·) itself is not a function of time, only of the present value of its input

*x*. A special case is the function

*f*(

*x*) =

*gx*, with g a constant describing a (linear) gain.

*I*) by the visual pigment (

*R*) via a G-protein (

*G*) to the production of activated phosphodiesterase (

*E**). In the first step, the visual pigment absorbs light, and is converted into an active form (

*R**). The active form is removed with a rate constant

*k*. It is assumed here that the light intensity is sufficiently low such that the concentration of

_{R}*R*is not significantly affected (new

*R*is produced fast enough, and there is no significant bleaching of the pigment). The reaction is then described by with

*I*the retinal illuminance in trolands, and

*A*a scaling constant. This can be rewritten as which can be recognized as a low-pass filter with time constant

_{R}*τ*and gain

_{R}*τ*.

_{R}A_{R}*R** activates a G-protein into its active form,

*G**, which subsequently forms an active complex

*E** with phosphodiesterase (PDE). Although the latter reaction can be modeled as a separate low-pass filter, its time constant is assumed to be so short that it can be neglected (Lamb & Pugh, 1992). It is again assumed that light intensities are such that neither

*G*nor PDE is significantly reduced. Therefore, the production of

*E** is described by with

*k*the rate of

_{E}*E** inactivation. This is also a low-pass filter, with time constant

*τ*and gain

_{E}*τ*. For simplicity, this gain and the one in Equation 8 are merged into the gain,

_{E}A_{E}*k*, describing the activity of

_{β}*E** (see below). Equations 8 and 9 can therefore be represented by the system model on the right side of Figure 2A.

*E** reduces the concentration of cGMP (symbolized by

*X*below), which is itself produced at a rate

*α*. The activity of PDE,

*β*, is described by a constant dark activity (here

*c*, often called

_{β}*β*

_{dark}in the literature) plus a term depending on the amount of activated PDE: with

*k*a constant. The concentration of cGMP is then described by At first sight, this looks like a complicated nonlinear equation, because the input,

_{β}*β*, is multiplied by the output,

*X*. However, it is possible to reformulate this equation as a static nonlinearity followed by a first-order low-pass filter with variable time constant. This can be seen be rewriting Equation 11 as In this equation, the input 1/

*β*is low-pass filtered into an output

*X*, with a time constant

*τ*depending on the input, and a gain

_{X}*α*. Assuming for the moment that the gain

*α*is constant (it is in fact under calcium control; see below), Equations 10 and 12 can be represented by the system model on the right side of Figure 2B. The importance of 1/

*β*for setting the sensitivity and time constant of the system has been discussed for rods (see Nikonov et al., 2000).

*X*) will open more cyclic nucleotide-gated (CNG) ion channels in the plasma membrane of the outer segment, resulting in an inward current consisting partly of Ca

^{2+}. The resulting increase in calcium concentration is counter-acted by a Na

^{+}/Ca

^{2+}-K

^{+}exchanger, and possibly by calcium buffering. The increased calcium concentration reduces the rate with which guanylate cyclase (GC) synthesizes cGMP, and thus counteracts the initial rise of cGMP. In a secondary loop, more important in cones than in rods (Rebrik & Korenbrot, 2004), Ca

^{2+}decreases the sensitivity of the CNG channels to cGMP.

^{2+}concentration, called

*C*here, is described by where

*η*is a scaling constant describing which proportion of the total photocurrent,

*I*

_{os}, is carried by calcium, and

*k*is the removal rate of calcium, presumably due to the exchanger. Thus

_{C}*C*is a low-pass filtered version of the photo-current, with time constant

*τ*and gain

_{C}*τ*. For the calculations, this gain will be merged into the scaling constant

_{c}ν*a*determining

_{C}*α*, as described below.

*I*

_{os}the photocurrent into the outer segment, and

*n*typically 3 (Koutalos & Yau, 1996). Scaling is incorporated into the scaling constant

_{X}*a*below, and into the scaling constant

_{C}*a*

_{is}determining the voltage response of the inner segment (see next section). The activity with which GC produces cGMP is, for the values of

*C*relevant for the fits, described by with

*a*a scaling constant, and

_{C}*n*typically taken to be 2 (Koutalos & Yau, 1996). The numerator of Equation 15 is scaled to 1 without loss of generality, because all scaling needed for the model fits is incorporated into the scaling constants

_{C}*a*and

_{C}*a*

_{is}.

*Q*(=1/

*β*) is divided by 1/

*α*, which is itself an expansive function of the calcium concentration, and therefore provides strong negative feedback. The broken line in the Figure represents the reduction of the channel sensitivity caused by calcium: either through the calcium concentration

*C*or, possibly more accurately, through the calcium current if the interactions are restricted to local domains. To keep the model as simple as possible, I am not modeling this interaction in detail. Instead, I assume it accounts for the fact that the standard value

*n*= 3 cannot produce acceptable fits of the model to the measurements. It produces major deficiencies in the sensitivities as a function of contrast and mean intensity. Leaving

_{X}*n*as a free parameter in the fits leads to much lower values, typically

_{X}*n*= 1. This low value of

_{X}*n*may be the result of the direct calcium feedback onto the CNG channels: Channel openings caused by an increase in

_{X}*X*are counteracted by the subsequent increase in calcium influx. It is well known in engineering that negative feedback loops can apparently linearize nonlinearities in the forward path. For example, a fast divisive feedback loop with a squaring operation in the feedback path will have output = input/output

^{2}, or output = input

^{1/3}, thus converting an actual

*n*= 3 into an apparent

_{X}*n*= 1. I will assume the value

_{X}*n*= 1 below.

_{X}*n*= 2 or 3, were significantly better for

_{C}*n*≈ 4. An apparent value of

_{C}*n*= 4 was recently reported and discussed for rods (Burns, Mendez, Chen, & Baylor, 2002). I will assume

_{C}*n*= 4 below.

_{C}*g*

_{is}, is given by a (nonlinear) function of the receptor potential

*V*

_{is}: where

*V*

_{is}is defined relative to the resting potential (i.e., the potential when

*I*

_{os}= 0). The voltage response to an abrupt change in

*I*

_{os}is given by the instantaneous conductance of the membrane,

*g*

_{i}, thus Finally, it is assumed that the instantaneous conductance approaches the steady-state conductance according to first-order kinetics: The precise form of

*g*

_{is}(

*V*

_{is}) is not known for primate cones, but I found that those aspects of horizontal cell responses that are presumably generated at the inner segment (in particular, response sagging, rebounds after pulses and steps, and reduced sensitivity at low frequencies) are well modeled by assuming where

*a*

_{is}is a scaling constant, and

*γ*is a constant that is approximately 0.7 according to the fits below.

*V*

_{is}in response to a change in

*I*

_{os}is in reality not as instantaneous as suggested by Equation 17, because the membrane capacitance has to be charged. This can be represented by an additional low-pass filter with time constant

*τ*

_{m}, assumed to be approximately constant. The gain of this filter is merged into the scaling constant

*a*

_{is}. The above equations lead to the system model shown in Figure 4. Although the signal transfer from the cone inner segment to the cone pedicle may produce some additional filtering (Hsu, Tsukamoto, Smith, & Sterling, 1998), it is assumed here that the signal arriving at the cone pedicle is that of the inner segment. An alternative interpretation is that any additional filtering can be thought to be incorporated into the parameters

*γ*,

*a*

_{is},

*τ*

_{is}, and

*τ*

_{m}used here for the model of the inner segment.

*V*

_{s}for transmitter release

*I*

_{t}is determined by the difference of the voltage of the inner segment,

*V*

_{is}, and the voltage

*V*

_{h}of the horizontal cell multi-plied by a gain

*g*

_{h}. Thus In the calculations below,

*g*

_{h}is fixed to 1, because it can be merged with the forward gain

*g*

_{s}. Going around the loop there are three low-pass filters, which together with the gain

*g*

_{s}determine the characteristics of the resonant oscillations observed in horizontal cells. The minimum number of low-pass filters required to obtain oscillations is two, but I found that the shape of the resonance peak and the associated oscillations is better described with three filters than with two or more than three. Although the filters can be arranged in any order, I tentatively consider the filters with relatively short time constants,

*τ*

_{1}and

*τ*

_{2}, as being involved in the processes of transmitter release (e.g., related to the rate of presynaptic calcium extrusion; Morgans, El Far, Berntson, Wässle, & Taylor, 1998), synaptic diffusion (e.g., related to the rate of glutamate removal; Gaal, Roska, Picaud, Wu, Marc, et al., 1998), or postsynaptic transduction. The longer time constant

*τ*

_{h}may then be interpreted as the effective time constant of the horizontal cell. The properties of the feedback loop, in particular the total gain, will depend on the spatial properties of the stimulus with respect to the (broad) receptive field of the horizontal cell. Because the spatial extent of all stimuli used in the measurements considered in this article was kept constant (5° diameter), I assume a fixed total gain for each cell. The influence of the spatial layout of the stimulus will be considered in a forth-coming article.

*g*

_{s}, produces acceptable fits to all horizontal cell measurements considered below. There are two nonlinearities that improved the fits sufficiently to justify their inclusion into the model. For high-contrast steps at high illuminance, there is an indication of a transiently reduced gain

*g*

_{s}(see Comparison with measurements, Figure 8). To model this, I included a nonlinearity similar to the one used by Kamermans, Kraaij, and Spekreijse (2001): The effective transmitter release as a function of

*V*

_{s}is, for the values of

*V*

_{s}relevant for the fits, described by a Boltzman function: with

*g*

_{t},

*V*

_{k}, and

*V*

_{n}constants. The more negative

*V*

_{s}becomes, the more the transmitter release will shut down. Because the stimuli considered in this article are spectrally white and have a broad spatial extent (5°), the modulation of

*V*

_{s}is generally small (e.g., see Figure 6B, panel 11). Therefore, the effect of the nonlinearity of Equation 21 is limited. This would be different, however, when the stimulus has a narrow spatial extent, resulting in a larger difference between the response of the excited cones and the horizontal cell, and thus in a larger modulation of

*V*

_{s}. Similarly, a non-white stimulus can increase the modulation of

*V*

_{s}as well. The nonlinearity of Equation 21 is then more important (Kamermans & Spekreijse, 1999).

*a*

_{I}, decreases the gain

*g*

_{t}and increases the time constants

*τ*

_{2}(or, equivalently,

*τ*

_{1}) and

*τ*

_{h}with decreasing intensity. The effective gain and time constants are then

*g*

_{t}/

*a*

_{I},

*a*

_{I}

*τ*

_{2}, and

*a*

_{I}

*τ*

_{h}. Decreasing intensity covaries with increasing

*V*

_{is}; Good fits were obtained by making

*a*depend on

_{I}*V*’

_{is}, a low-pass filtered version of

*V*

_{is}, as , with

*V*and

_{I}*μ*constants, and

*V*’

_{is}obtained by low-pass filtering

*V*

_{is}with a time constant

*τ*of 250 ms. Similar results were obtained by assuming that

_{a}*a*depends on

_{I}*V*

_{h}rather than

*V*

_{is}. The changes in gain and time constants obtained from the fits were modest, typically 5–20% for 10-fold steps of the background intensity. Figure 5B shows the system model associated with both of the above nonlinearities.

*τ*and

_{R}*τ*, yielding a signal proportional to the concentration of activated PDE (panel 2). The rate of cGMP hydrolysis,

_{E}*β*, is elevated by

*c*relative to the

_{β}*E** curve (Equation 10), hardly visible in the graph at these illuminances (panel 3). The inverse of

*β*(panel 4) is the signal that acts as the input to the calcium feedback loop. It is regulated by 1/

*α*(panel 8), a low-pass filtered version of the loop output. This signal is slightly delayed relative to 1/

*β*because of the low-pass filtering, and therefore boosts high temporal frequencies (panel 5). Furthermore, the feedback effectively reduces the dynamic range needed by the signal (cf. panels 6 and 4). The feedback loop of the inner segment (panel 10) reduces low frequencies (panel 9). In panel 9 the membrane voltage of the cone is shown relative to its membrane potential in the dark (i.e.,

*V*

_{is}-

*V*

_{is,dark}). Similarly, panel 14 shows the membrane voltage of the horizontal cell relative to its dark value (i.e.,

*V*

_{h}-

*V*

_{h,dark}). It should be noted that the scaling of

*V*

_{is}relative to

*V*

_{h}is not fully determined by the measure-ments on

*V*

_{h}considered here, because it depends on fixing

*g*

_{h}= 1 in Equation 20. In the cone-horizontal cell feedback loop (panels 11–14), the membrane potential of the horizontal cell is subtracted from that of the cone. The resulting

*V*

_{s}=

*V*

_{is}-

*V*

_{h}is small, only in the order of a mV. It is transient, because the signal in the horizontal cell is delayed, due to the low-pass filtering, relative to that in the cone. The resulting oscillations (panels 11, 12) arrive, however, strongly attenuated in the horizontal cell (panel 14). Note that the signal presumably going to the bipolar cells (bc in Figure 6A) is considerably more transient (panel 13) than that of the horizontal cell. How much of this shows up in the bipolar cells depends on the amount of low-pass filtering and further processing occurring in the bipolar cells.

*r*

_{max}were used for simultaneous fitting, the RMS-deviations were scaled by 1/

*r*

^{0.25}

_{max}to prevent the largest responses to completely dominate the fit. This was a pragmatic choice that still assigns more weight to large responses because they are less affected by noise and provide more information on the nonlinearities in the model. The measured data from horizontal cells in Figures 7–14 were obtained from the graphs in the on-line versions of Smith et al. (2001) and Lee et al. (2003), using specialized software (g3data).

*τ*, the output

*y*(

*n*) to an input

*x*(

*n*), with a time step Δ

*t*, is given by the following ARMA filter (Brown, 2000): with In the calculations I used Δ

*t*= 100 μs; with this time step, the model computes approximately 100 times faster than the real cone (3 GHz PC, Intel Fortran compiler on Linux). I verified that results were indistinguishable from those obtained with time steps of 10 μs or 200 μs. For time steps significantly longer than 200 μs, the implicit extra delay of one time step in the high-gain feedback loops of Figure 6A leads to spurious oscillations.

Symbol | Description | Units | Generic value | Range |
---|---|---|---|---|

τ_{R} | time constant of R* inactivation | ms | 3.4 | 0.5 – 6.5 |

τ_{E} | time constant of E* inactivation | ms | 8.7 | 3.0 – 16.8 |

c_{β} | rate constant of cGMP hydrolysis in darkness | (ms)^{−1} | 2.8·10^{−3} | 2.0·10^{−3} – 4·10^{−3} |

k_{β} | rate constant of cGMP hydrolysis | (ms)^{−1} | 1.6·10^{−4} | 4.9·10^{−5} − 3.9·10^{−4} |

β | cGMP hydrolysis rate | (ms)^{−1} | — | — |

τ_{X} | time constant of cGMP turnover | ms | — | — |

X | scaled cGMP concentration | au | — | — |

n_{X} | apparent Hill coefficient of CNG activation | — | 1 | fixed |

I_{os} | scaled photocurrent of outer segment | au | — | — |

τ_{C} | time constant of
Ca^{2+} extrusion | ms | 3 | 2 – 6.3 |

C | scaled Ca^{2+} concentration | au | — | — |

a_{C} | scaling constant of GC activation | au | 9·10^{−2} | 3.5·10^{−2} – 2.1·10^{−1} |

n_{C} | apparent Hill coefficient of GC activation | — | 4 | fixed |

α | GC activity | au | — | — |

τ_{m} | capacitive membrane time constant | ms | 4 | fixed |

V_{is} | membrane voltage of inner segment | mV | — | — |

γ | parameter of membrane nonlinearity | — | 0.7 | 0.49 – 0.73 |

α_{is} | scaling constant of membrane nonlinearity | au | 7·10^{−2} | 1.9·10^{−2} – 1.7·10^{−1} |

τ_{is} | time constant of membrane nonlinearity | ms | 90 | 23 – 139 |

V_{s} | effective membrane voltage of cone pedicle after subtractive feedback | mV | — | — |

g_{t} | parameter of transmitter activation curve | au | 125 | 71 – 185 |

V_{k} | parameter of transmitter activation curve | mV | −10 | fixed |

V_{n} | parameter of transmitter activation curve | mV | 3 | fixed |

I_{t} | transmitter activation | au | — | — |

V_{I} | parameter of gain factor a_{I} | mV | 20 | 20 – 50 |

μ | parameter of gain factor a_{I} | — | 0.7 | 0.17 – 0.73 |

α_{a} | time constant for gain factor a_{I} | ms | 250 | fixed |

a_{I} | gain factor | — | — | — |

τ_{1} | time constant of cone — horizontal cell loop | ms | 4 | fixed |

τ_{2} | time constant of cone — horizontal cell loop | ms | 4 | 2.5 – 4 |

τ_{h} | time constant of cone — horizontal cell loop | ms | 20 | 20 – 35 |

V_{h} | membrane voltage of horizontal cell | mV | — | — |

^{2}, of the pupil of the eye times the scene luminance, expressed in candela/m

^{2}. At a luminance of 100 cd/m

^{2}, the pupil has a diameter of approximately 3 mm, thus then 100 cd/m

^{2}corresponds to 640 td. Such a luminance corresponds roughly to the mean luminance outdoors on a dull cloudy day, and is an order of magnitude higher than the typical mean luminance indoors.

*I*

_{os}. The response sagging during the step responses at 100 td, and the response rebounds after pulses and steps, are mainly due to the properties of the model components representing the inner segment. The high-frequency oscillations are due to the cone-horizontal cell feedback loop.

*V*

_{s}becomes strongly negative, and thus brings the transmitter release, Equation 21, into a part of the curve with decreased slope. This decreases the small-signal gain, and this decreased gain in the feedback loop leads to a less steep rise of the response and less prominent oscillations. The fact that the oscillations in the model response are not as strongly reduced as in the measured response suggests that there are additional nonlinearities present, possibly due to voltage-sensitive channels in the horizontal cell membrane active at large hyperpolarizations.

*τ*typically 3 ms. Forcing

_{C}*τ*to be much larger worsened the fits considerably, and led to strongly biphasic cone photocurrents and membrane voltages in response to pulses and steps. This biphasic behavior is not consistent with the horizontal cell measurements considered here, which show only a mild rebound (Figure 7), with dynamics differing from those predicted from a slow calcium feedback. In rods, the photocurrent is generally not biphasic, unless the calcium dynamics are manipulated (Torre, Matthews, & Lamb, 1986). Although there are reports in the literature of strongly biphasic photocurrents and membrane voltages in primate cones (Schnapf, Nunn, Meister, & Baylor, 1990; Schneeweis & Schnapf, 1999), these findings may well be a consequence of a disturbed calcium dynamics due to the experimental techniques used. Such a disturbance of the cones is unlikely to have occurred in the horizontal cell measurements considered here, because cones were not directly manipulated, and the preparation left the retina mostly intact (Dacey, 1999; Smith et al., 2001).

_{C}*β*. The low-intensity part of the sinusoidal stimulus produces a small

*β*, which is subsequently blown up by 1/

*β*to the high peak in the response. The peak height is limited by the minimum value of

*β*,

*c*in Equation 10. The high-intensity parts of the sinusoid produce a large

_{β}*β*, which is subsequently compressed by 1/

*β*, which acts then as a compressive nonlinearity. The detailed shape of the distortion at 0.61 Hz is also determined by the calcium feedback loop, which in effect relinearizes the response to some extent: the high peak (low intensity, large 1/

*β*) produces high levels of calcium, reducing the gain of the forward path (divisive gain control in Figure 3). This brings the response considerably closer to the steady-state level (dashed line). On the other hand, the low response (due to high intensity, small 1/

*β*) produces low levels of calcium, increasing the gain of the forward path, also resulting in a response closer to the steady state than would have resulted without the calcium feedback. Because the former effect is stronger than the latter, the distortion is reduced.

*β*gradually disappears, because the first two low-pass filters,

*τ*

_{R}and

*τ*

_{E}, reduce the depth of modulation of

*β*and thus 1/

*β*. At 4.88 and 9.76 Hz, another distortion becomes clearly visible: The falling flank of the response becomes steeper than the rising flank. This distortion is mainly due to the low-pass filters in the calcium feedback loop: The maximum reduction due to the control signal 1/

*α*(Figure 6A) is only reached right after the peak in 1/

*β*, which results in a steep falling flank right after that peak.

*V*ty should decline inversely with the back-ground illuminance: The response to 1 td at a background of 100 td (contrast 0.01) should be 10 times smaller than the response to 1 td at a background of 10 td (contrast 0.1). Intermediate stages between linear and Weber are possible as well.

_{I}*a*t identical to those obtained using a linearized, small-signal version of the model (see Supplementary material. The small-signal analysis also shows that the cGMP hydrolysis by PDE (i.e., 1/

_{C}*β*) is the key factor determining the illuminance dependence of responsivity and cut-off frequency.

*τ*

_{R}, is known or, in cones, at least suspected to depend on the calcium concentration (Pugh & Lamb, 2000). The membrane properties, such as input resistance and time constant, of both cone and horizontal cell are likely to change depending on membrane potential, and thus to change depending on stimulus contrast and background illuminance. For a more precise fitting of the model to the measurements, many of such parameter changes would probably need to be included. However, it appears that such changes can be considered as second-order effects for the present purpose. The model, as it stands, gives an adequate description and explanation of the major response properties, and it is simple and accurate enough to serve as a useful preprocessing stage for models of upstream processing in the retina and beyond.

*τ*may increase, the calcium dynamics may become slower and more complicated, the production rate of cGMP,

_{R}*α*as described in Equation 15, may need an extra term

*α*

_{min}(Nikonov et al., 2000) at large calcium concentrations, and the filtering properties of the inner segment may change as well when it is steadily depolarized. Several of these processes could be added to the model under guidance of existing experimental data, but ideally an extended model should be validated to a set of experiments similar to those used here, per-formed at a still larger range of illuminances.

*τ*

_{C}≈3 ms) and powerful (

*n*≈ 4), whereas its forward gain is effectively linearized (

_{C}*n*≈ 1), possibly due to the calcium gain control on the CNG channels. For the intensity range considered, the model did not require a calcium-dependent change in the inactivation rate of the cone visual pigment.

_{X}*β*nonlinearity of the cone.

*gt*

_{1}of their model is reflected in the changing

*τ*in the present model. The model for the inner segment used here behaves, for small signals, approximately as a lead-lag filter (see Supplementary material, similarly to the lead-lag filter used by Smith et al. (2001). The second-order resonator introduced in their model behaves not unlike the implicit resonance of the cone-horizontal cell feedback loop introduced here. Finally, the feedforward gain control of Lee et al. (2003) is to some extent related to the calcium feedback gain control applied here. Nevertheless, the pre-sent model supersedes the earlier attempt, because it en-compasses responses over the entire range of background levels of the measurements at once, including all major nonlinearities, and is physiologically realistic.

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