A model of spatiotemporal signal processing by the cone–horizontal cell circuit in the primate outer retina is developed and validated using measurements on the H1 horizontal cell from the literature. The model extends an earlier temporal model that mainly addressed the regulation of sensitivity by the cones. Three elements are added to the earlier model to describe the full spatiotemporal processing by horizontal cells. First, the feedback gain from horizontal cells to cones is made adaptive, depending on field size. Second, the spatial filtering by the horizontal dendritic tree is modeled as a two-component spatial filter. Third, an adaptive temporal low-pass filter is added, also depending on field size. The resulting model adequately describes all available measurements on spatiotemporal processing in macaque H1 cells. The adaptive feedback gain is argued to contribute to negative afterimages and chromatic adaptation in human vision.

*τ*. The gains associated with the physiological substrate of these filters are combined as much as possible with other gains to minimize the number of free parameters. Dimensional conversion constants are omitted to keep the equations as concise as possible. Throughout the article, the term (light) intensity is used when referring to the retinal illuminance expressed in trolands.

*I*excites cone pigment

*R,*producing

*R**.

*R**, which has a lifetime

*τ*

_{R}, excites a G protein (transducin), which rapidly forms an active complex with phosphodiesterase, yielding

*E**.

*E**, which has a lifetime

*τ*

_{E}, hydrolyzes cGMP with a rate

*β,*with

*β*consisting of a dark rate

*c*

_{β}in addition to a factor proportional to

*E**. An increase in

*E** thus produces a decrease in cGMP, closing ion channels in the cone outer segment that were held open by cGMP. The current flowing into the outer segment,

*I*

_{os}, subsequently drops. Because part of this current consists of Ca

^{2+}, the Ca

^{2+}concentration in the cell declines, which reduces the Ca

^{2+}inhibition of guanylylcyclase to produce cGMP. The original decrease in cGMP is thus counteracted, and these processes form, therefore, a negative feedback loop.

*β,*which is followed by a low-pass filter with time constant

*τ*

_{X}= 1/

*β*. The 1/

*β*nonlinearity and

*τ*

_{X}are directly dependent on the level of

*E** and, thus, on light intensity, and they are the main factors responsible for the fact that cones become less sensitive and faster when the ambient light intensity becomes higher (Nikonov, Lamb, & Pugh, 2000; van Hateren, 2005). In addition, the calcium feedback has an important role in fine-tuning sensitivity and speed (see the supplementary material in van Hateren, 2005, for an extensive analysis).

*I*

_{h}(Hestrin, 1987) produces band-pass filtering (guinea pig rods: Demontis, Longoni, Barcaro, & Cervetto, 1999; modeled cat cones: Hennig, Funke, & Wörgötter, 2002; tiger salamander rods: Mao, MacLeish, & Victor, 2003). Although the influence of these ion channels on macaque cones under physiological conditions is not well known, I assume here (as in van Hateren, 2005) that they are responsible for the low-frequency falloff of the temporal frequency curves (e.g., Figure 5; see further van Hateren, 2005). The analysis in the appendix of Detwiler, Hodgkin, and McNaughton (1980) is adopted to avoid introducing many poorly known parameters, which leads to the rightmost feedback loop in Figure 1A. This loop represents the voltage-sensitive and filtering properties of the inner segment, axon, and cone pedicle (van Hateren, 2005). The resulting membrane voltage arriving at the synapse,

*V*

_{is}, is slightly band-pass filtered by this feedback loop compared with the photocurrent generated by the outer segment (

*I*

_{os}).

*V*

_{h}, is subtracted from the membrane potential of the cone,

*V*

_{is}. The result,

*V*

_{s}, effectively drives the synapse through a nonlinear function

*I*

_{t}(

*V*

_{s}), representing the activation function of the synapse. Subsequent low-pass filters

*τ*

_{1},

*τ*

_{2}, and

*τ*

_{h}then produce

*V*

_{h}. The factor

*a*

_{I}(Figure 1B; see further van Hateren, 2005) is a factor that slightly adjusts two of the time constants and the gain of

*I*

_{t}over the range of background intensities (1–1,000 td) for which the model was validated. For the results presented in this article, with all measurements obtained at high background intensities,

*a*

_{I}was kept fixed at a value of 1.

*V*

_{is}is defined relative to the voltage that corresponds to

*I*

_{os}= 0. This was convenient for describing the nonlinear dynamics of the feedback loop of the inner segment. However, for describing the feedback loop involving the horizontal cell, in particular the elaborated loop presented here, it is more convenient to define the membrane voltages of cone pedicle and horizontal cell relative to their dark values. Hence,

*V*

_{iz}=

*V*

_{is}−

*V*

_{is,dark}is the voltage of the cone pedicle thus defined. In the dark,

*V*

_{iz}= 0 by definition, and the horizontal cell voltage,

*V*

_{h}, is then zero as well. The latter is guaranteed by choosing the synaptic activation function,

*I*

_{t}(

*V*

_{s}), such that

*I*

_{t}(0) = 0. This choice can be made without loss of generality because it only results in redefining the zero points of the currents and voltages involved. Inspection of the feedback loop in Figure 1C shows that all voltages and currents in the loop must be at their zero points when

*V*

_{iz}= 0.

*V*

_{n}determines the width of the midsection of the curve,

*s*is a factor that determines the skewness (asymmetry) of the function relative to

*I*

_{t}= 0, and

*g*

_{t}is the dark gain, that is, the slope of the function at

*V*

_{s}= 0. Note that

*I*

_{t}(0) = 0, as required above. The panel in the lower left of Figure 1C shows

*I*

_{t}(

*V*

_{s}) for typical parameter values.

*V*

_{s}will be mostly less than zero, apart from transients, strong surrounds, or strong chromaticity. The activation is therefore mainly situated in the toe of the curve.

*g*

_{h}, is assumed to be adaptive. Secondly, the spatial integration by the horizontal cell is described by two components at different spatial scales (Packer & Dacey, 2002, 2005). Finally, there is an adaptive time constant,

*τ*

_{p}, immediately before the feedback loop. I will discuss below these additions in more detail and justify their inclusion into the model. This is done by changing or omitting, in turn, each of the three additions from the full model of Figure 1C. Thus, the effect of each addition is analyzed, with the other two additions being fully operational.

*τ*

_{1},

*τ*

_{2}, and

*τ*

_{h}) in addition to the low-pass filters of the cone (van Hateren, 2005).

*λ*(Jack, Noble, & Tsien, 1983). Although the coupling and, therefore,

*λ*depend on the background light level in many vertebrate retinas, this appears not to be the case for macaque H1 cells (Packer & Dacey, 2002, their Figure 11). I will therefore assume that

*λ*is a constant. The horizontal cell network then acts as a fixed spatial low-pass filter on its input (the two-dimensional array of cones). Stimuli of small spatial extent are thus more reduced in amplitude than wide stimuli because of spatial blurring.

*g*

_{h}. As is clear from the figure, such a model cannot simultaneously fit the measurements at 2° and 10° stimulus diameters. Moreover, the phase characteristic of the 2° model fit deviates strongly from the measured phase in the 20- to 40-Hz region (not shown).

*V*

_{h}by changing the surround illumination. A static nonlinearity for the feedback gain (

*g*

_{h}) driven by

*V*

_{h}can therefore be ruled out as an explanation for the response characteristics discussed above.

*g*

_{h}, the feedback gain exerted by each horizontal cell dendrite onto the presynaptic elements in its own triad, is not fixed but adaptive (see the Discussion section for possible physiological substrates). The measured membrane potential of the horizontal cell,

*V*

_{h}, is then different from the effective signal produced by the dendrite,

*V*

_{d}, which is the actual signal subtracted from the potential at the cone pedicle,

*V*

_{p}. The result,

*V*

_{s}, drives the transmitter release of the triad. Depending on the physiological substrate of the feedback mechanism (which is still uncertain, see the Discussion section),

*V*

_{d}may be interpreted as, for example, the extracellular voltage surrounding the dendrite or the local H

^{+}concentration.

*g*

_{h}is driven by the output

*I*

_{t}of the synaptic activation function of the triad, via a low-pass filter

*τ*

_{itd}and a nonlinearity NL

_{itd}. The time constant

*τ*

_{itd}must be longer than a few seconds because, otherwise, its effect would have been detected (as gain changes dependent on

*V*

_{h}) in the spot-annulus experiment of Lee et al. (1999). These experiments, with an annulus modulating

*V*

_{h}(and thus, indirectly,

*I*

_{t}) at 0.6 Hz, showed no corresponding gain changes for stimuli presented in the central spot, and therefore,

*τ*

_{itd}must be significantly longer than 1/0.6 s. Simulations confirm that a

*τ*

_{itd}shorter than a few seconds does not produce adequate fits to these measurements (see further the Spot-annulus experiment section).

*τ*

_{itd}because the experiments of Smith et al. (2001) and Packer and Dacey (2002, 2005) were all done with a static field diameter and, therefore, provide no specific information on

*τ*

_{itd}. I will assume

*τ*

_{itd}= 10 s below, a value that is sufficiently long to obtain steady-state conditions for the measurements considered here but is otherwise rather arbitrary.

_{itd}is also poorly constrained because only a few field diameters were used in the experiments (typically 2°, 5°, and 10°). I found that a simple nonlinearity of the form

*c*

_{h}and

*I*

_{h}are constants, was able to adequately describe the available measurements with different field sizes. The argument of

*g*

_{h}is here formally

*I*

_{t}, as it would be in the steady state (when the

*τ*

_{itd}filter has unit gain), but more generally, its argument would be the result of

*I*

_{t}low-pass filtered by

*τ*

_{itd}. An example of

*g*

_{h}with typical parameter values is shown in Figure 2C.

*g*

_{h}is illustrated in Figure 2D, which shows calculations with the full model of Figure 1C, but with

*g*

_{h}clamped to particular values. Whereas the

*V*

_{h}for 10° and 2° field diameters are quite different at low frequencies because of the cable properties of the horizontal cell, this is partly compensated in the feedback signal

*V*

_{d}by an increased

*g*

_{h}for the smaller field diameter. The result is that the values of

*V*

_{d}for 10° and 2° field diameters are not very different at low frequencies, and the loop gain for 2° is less reduced compared with the gain for 10° than would have been the case without the change in

*g*

_{h}. Consequently,

*V*

_{h}for 2° still shows a considerable enhancement of frequencies up to 30–40 Hz compared with the cone response, in accordance with the measurements on H1 cells.

*λ*

_{S}, is presumably mainly determined by the H1 cell's own dendritic tree, whereas the component with the longer scale,

*λ*

_{L}, presumably reflects the electrical coupling between H1 cells (Packer & Dacey, 2005). I found that it is indeed necessary to include such a two-component spatial receptive field into the model to explain the measured spatial frequency response.

*λ*(according to the left diagram in Figure 3B). The best fit with all other components of the model of Figure 1C included shows a clear discrepancy with the measurements (red trace: fit; symbols: measurements from Packer & Dacey, 2002). For the sake of an efficient computation (see the Model implementation section), the spatial filtering was implemented as a filter separable from the temporal filtering of the H1 cell. For the two-dimensional cable equation, this is quite accurate on a time scale slower than the time constant of the H1 cell but is only an approximation on faster time scales. Provisional calculations using the two-dimensional cable equation indicate that the lack of spatiotemporal inseparability does not affect the conclusions presented in this article.

*h*(

*r*), approximates an exponential; that is,

*r*is the distance from the origin,

*λ*is the space constant, and

*c*

_{ λ}is a normalization constant such that the filter has unit response to a homogeneous field. It should be noted that the two-dimensional cable equation has a modified Bessel function

*K*

_{0}as its (steady-state) spatial point spread function (Jack et al., 1983) and that an exponential only approximates this at large distances. Whereas the

*K*

_{0}function gives 1/(1 + (2

*πf*

_{s}

*λ*)

^{2}) as the corresponding spatial modulation transfer function, with

*f*

_{s}spatial frequency, an exponential has 1/(1 + (2

*πf*

_{s}

*λ*)

^{3}) as its spatial modulation transfer function. The analysis performed in Packer and Dacey (2002) on the spatial frequency curves used the latter. However, I found that performing the same analysis on the same data gives rather similar results for either choice of filter: In both cases, a superposition of two filters is needed to explain the measurements. Here, I will use filters based on Equation 3.

*λ*

_{S}(

*λ*

_{L}) is the short (long) space constant and

*w*

_{S}(with 0 ≤

*w*

_{S}≤ 1) is the relative weighting of the

*λ*

_{S}component. Note that the normalization of the two contributing filters ensures that

*h*(

*r*) has a unit response to a homogeneous field.

*f*

_{s}= 0) and thereby strongly changing the working point of

*I*

_{t}(

*V*

_{s}), this was done in the following way. First, the calculation was performed assuming only the filter with

*λ*

_{S}(with unit weight, as in Equation 3), while keeping all other parameters fixed to their fitted values. The resulting response was multiplied by

*w*

_{S}and shown as the blue lines marked

*λ*

_{S}in Figure 3C. Second, the calculation was performed assuming only the filter with

*λ*

_{L}, and the result was multiplied by 1 −

*w*

_{S}(blue lines marked

*λ*

_{L}). Although it is clear that there is no linear superposition in the midrange of frequencies, it can be seen that, as expected,

*λ*

_{L}is mainly responsible for the peak of the spatial frequency response at low frequencies and that

*λ*

_{S}is mainly responsible for the tail at high frequencies.

*τ*

_{p}, immediately before the feedback loop (see the fits in Figure 4B). The change in

*τ*

_{p}is assumed to be driven by the output

*I*

_{t}of the synaptic activation function, via a slow low-pass filter,

*τ*

_{itp}, and a nonlinearity, NL

_{itp}(Figure 4C). For the same reasons as discussed above for

*τ*

_{itd},

*τ*

_{itp}must be slower than a few seconds but has no upper bound based on the measurements considered here. I will assume

*τ*

_{itp}= 10 s below. The measurements can be well fitted by assuming a nonlinearity, NL

_{itp}, of the same form as NL

_{itd}:

*c*

_{p},

*I*

_{p}, and

*τ*

_{p,max}are constants. An example of

*τ*

_{p}with typical parameter values is shown in Figure 4C. For a field size of 2°,

*V*

_{d}is smaller (closer to zero, thus less close to

*V*

_{iz}and

*V*

_{p}) than for a field size of 10° (which is less reduced by the cable properties of the horizontal cell). Therefore,

*V*

_{s}is more negative (closer to

*V*

_{iz}and

*V*

_{p}) for 2° than for 10°; hence,

*I*

_{t}is more negative, and therefore,

*τ*

_{p}is larger for 2° than for 10°.

*y*(

*n*) to an input

*x*(

*n*) is then given by

*τ*is the time constant of the low-pass filter and Δ

*t*is the time step. For the calculation, Δ

*t*= 0.1 ms turned out to be sufficiently short for accurate results. This was verified, firstly, by varying Δ

*t,*and, secondly, by comparing the results with those obtained by numerical integration in Matlab. An ARMA implementation (in Fortran) is about two orders of magnitude faster than solving the system of differential equations by conventional numerical integration and, therefore, crucial for obtaining fits within a reasonable amount of time (see the Computation section).

*p*(

*n*) into a spatial output

*q*(

*n*), is given by

*n*increasing, and

*λ*is the space constant of the filter and Δ

*s*is the step size. For most calculations, Δ

*s*= 0.3° (with field size 10°) turned out to be sufficiently small for accurate results; for Figure 6, it was necessary to use Δ

*s*= 0.1° (with field size 15°), and for Figure 8, Δ

*s*= 0.05° (with field size 5°) was needed. The latter values of Δ

*s*are close to the cone spacing expected at the approximate eccentricity of the measured cells. Accuracy was checked by varying Δ

*s*and field size.

*f*

_{2}and

*f*

_{3}are combined into

*g*

_{2}, and only the present input sample,

*p*(

*n*), is used and not the previous input sample,

*p*(

*n*− 1). The reason is that the temporal filter of Equations 6 and 7 is carefully crafted (Brown, 2000) to give the correct phase characteristics of the (causal) low-pass filter, given a finite time step Δ

*t*. Causality is of no concern when dealing with spatial filters; hence, Equations 8 and 9 provide the optimal filter, with an impulse response that is simply a sampled exponential irrespective of the step size Δ

*s*.

*q*(

*n*) ≔

*g*

_{1}

*q*(

*n*+ 1) +

*g*

_{2}

*q*(

*n*) for decreasing

*n,*turns out to produce a perfect two-sided exponential pulse spread function. In the equation above, ≔ denotes assignment, to indicate that the

*q*(

*n*) to the left is different from the

*q*(

*n*) to the right. For finite arrays [1:

*N*], boundary effects have to be taken into account. It can be shown that by choosing the first output element for the filtering into the forward direction as

*q*(1) =

*g*

_{1}

*b*+

*g*

_{2}

*p*(1), and the first output element for the filtering into the reverse direction as

*q*(

*N*) ≔ (

*q*(

*N*) +

*g*

_{1}

*b*)/(1 +

*g*

_{1}), the filter output is identical to that of an infinite input array with the values beyond the boundary given by

*b*(thus,

*b*is the intensity of the surround if the input is an array of intensities).

*λ*

_{S}and with

*λ*

_{L}and by combining the results according to the weighting of Equation 4.

*s*= 0.3° and 10° field size (i.e., a circular hexagonal grid with about 1,000 cones) takes 8 s on a single processor.

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

τ _{ R} | Time constant of R* inactivation | ms | 3.4 | 1.4 to 3.8 |

τ _{ E} | Time constant of E* inactivation | ms | 8.7 | 4.5 to 19 |

c _{ β} | Rate constant of cGMP hydrolysis in darkness | (ms) ^{−1} | 2.8×10 ^{−3} | Fixed |

k _{ β} | Rate constant of cGMP hydrolysis | (ms) ^{−1}/td | 1.0×10 ^{−4} | 4.7×10 ^{−5} to 3×10 ^{−4} |

n _{ X} | Apparent Hill coefficient of CNG activation | – | 1 | Fixed |

τ _{ C} | Time constant of Ca ^{2+} extrusion | ms | 3 | Fixed |

a _{ C} | Scaling constant of GC activation | au | 0.2 | 0.09 to 0.32 |

n _{ C} | Hill coefficient of GC activation | – | 4 | Fixed |

τ _{m} | Capacitive membrane time constant | ms | 2.3 | Fixed |

γ | Parameter of membrane nonlinearity | – | 0.7 | Fixed |

a _{is} | Scaling constant of membrane nonlinearity | au | 0.1 | 3.5×10 ^{−2} to 3.7×10 ^{−1} |

τ _{is} | Time constant of membrane nonlinearity | ms | 90 | Fixed |

t _{delay} | Delay time | ms | 3 | 1.3 to 3.5 |

g _{t} | Parameter of transmitter activation curve | au | 15 | 8 to 21 |

V _{n} | Parameter of transmitter activation curve | mV | 5 | 3.3 to 8.1 |

s | Parameter of transmitter activation curve | – | 0.4 | 0.3 to 0.4 |

τ _{1}, τ _{2} | Time constants of cone–horizontal cell loop | ms | 5.7 | 4 to 8 |

τ _{h} | Time constant of cone–horizontal cell loop | ms | 7.0 | 4 to 20 |

λ _{S} | Short length constant of spatial filter | μm | 20 | 5 to 30 |

λ _{L} | Long length constant of spatial filter | μm | 300 | 100 to 440 |

w _{S} | Relative weight of short spatial filter | – | 0.15 | 0 to 0.27 |

τ _{p,max} | Parameter of adaptive τ _{p} loop | ms | 25 | 18 to 34 |

c _{p} | Parameter of adaptive τ _{p} loop | au | 0.25 | 0.22 to 0.46 |

I _{p} | Parameter of adaptive τ _{p} loop | au | −25 | −18 to −26 |

τ _{itp} | Parameter of adaptive τ _{p} loop | s | 10 | Fixed |

c _{h} | Parameter of adaptive g _{h} loop | au | 0.25 | 0.10 to 0.44 |

I _{h} | Parameter of adaptive g _{h} loop | au | −20 | −17 to −27 |

τ _{itd} | Parameter of adaptive g _{h} loop | s | 10 | Fixed |

*RC*time) of the cell is expected to be shorter for small fields than for large fields. This is expected because current can also flow laterally in the case of small fields (making

*R*and, thus, the time constant effectively smaller) but less so in the case of large fields when the cell is more isopotential. It is possible that including this into the model would resolve the discrepancy in Figure 6D.

*λ*

_{S},

*λ*

_{L}, and

*w*

_{S}), and these can be quite different for different H1 cells even at a similar eccentricity (Packer & Dacey, 2002, 2005).

*V*

_{h}(and consequently

*V*

_{d}) becomes more positive,

*V*

_{s}becomes more negative, thus shifting the working point to shallower parts of the synaptic activation function. As a result, the modulation of

*V*

_{s}attributable to the test stimulus produces smaller modulations in

*I*

_{t}than when

*V*

_{h}is negative. The effect is rather modest in the fitted curve in Figure 7A but larger in the generic curve (dashed blue line). For different H1 cells, the effect varies from small to considerable (B. B. Lee, personal communication), which may correspond in the present model to slight variations in the shape of the synaptic activation function and in the working point. Kraaij, Spekreijse, and Kamermans (2000) have investigated the influence of the nonlinear activation function on the feedback in fish horizontal cells, and the consequences for color vision are discussed in Kamermans, Kraaij, and Spekreijse (1998).

*β*nonlinearity in combination with the calcium feedback loop (Figure 1A).

*τ*

_{itd}is not as long as assumed (10 s). Therefore, I checked if a short

*τ*

_{itd}would be consistent with the measurements in Figure 7. I found that the fits deteriorated when

*τ*

_{itd}was made small enough to influence the dynamics on the time scale of the experiment (∼1.6 s). This supports the notion that the alleged adjustment of the feedback gain is indeed a rather slow process that is not easily detected with stimuli confined to relatively short time scales.

^{2+}(Fahrenfort, Sjoerdsma, Ripps, & Kamermans, 2004; Packer & Dacey, 2005). The main effect is that the slow sagging in the response to an on-step is strongly reduced and that the response amplitude to such steps gradually decreases in the course of the application of the drug and eventually vanishes altogether.

^{2+}), as shown in Figure 9. The black traces at Δ

*V*

_{s}= 0 mV and at Δ

*V*

_{s}= −6.5 mV are measured responses (Packer & Dacey, 2005) to a 100% modulated square wave (2.44 Hz, 10° field diameter, 1,000 td mean intensity). The curve at Δ

*V*

_{s}= 0 mV is the response as measured before application of carbenoxolone, and the response at Δ

*V*

_{s}= −6.5 was recorded a few minutes after the start of carbenoxolone application, when the response dynamics were clearly affected, but before the response had vanished (Packer & Dacey, 2005). The red traces are model calculations of the H1 response,

*V*

_{h}, made on the assumption that the main effect of carbenoxolone (or Co

^{2+}) is to gradually shift

*V*

_{s}(see Figure 1C) to more negative values (denoted by the numbers, Δ

*V*

_{s}in millivolts, below the curves). Two equivalent ways to put this are, first, that carbenoxolone gradually shifts the synaptic activation function to more positive values or, second, that carbenoxolone gradually shifts the feedback signal

*V*

_{d}to more positive values. The model predictions based on this assumption are shown by the red traces, which fit well with the measured traces at Δ

*V*

_{s}= 0 mV and at Δ

*V*

_{s}= −6.5 mV. The slow sagging after an on-step gradually disappears, whereas the slow sagging after an off-step survives longer. Moreover, the response eventually vanishes altogether.

*V*

_{s}in response to the square wave. When Δ

*V*

_{s}becomes more negative,

*V*

_{s}shifts to shallower parts of the activation curve. As a result, the gain decreases, and thereby, the effectiveness of the feedback decreases as well. Therefore, the voltage swing of

*V*

_{s}gradually becomes larger (i.e., closer to that of

*V*

_{p}, the signal at the cone pedicle). The negative part of

*V*

_{s}leads to a strongly compressed negative part of

*V*

_{h}. This reduces or even abolishes the slow sagging, which is, in the model, attributed to local processes (in particular a nonlinear membrane) in the cone inner segment, axon, and pedicle ( Figure 1A; van Hateren, 2005). The positive part of

*V*

_{s}is still in a steeper part of the activation curve, and therefore, the positive part of

*V*

_{h}is left mostly intact, including the slow sagging occurring at off-steps. Only at strongly negative Δ

*V*

_{s}are both negative and positive parts of the response nearly vanished in

*V*

_{h}.

^{+}concentration (Vessey et al., 2005). Any change in the maximum gain of

*g*

_{h}can be formally compensated in the model by a reciprocal change in the synaptic gain

*g*

_{t}. This only scales

*V*

_{p},

*V*

_{d}, and

*V*

_{s}(which have not been measured and are, therefore, undetermined to some extent) while keeping the measured

*V*

_{h}the same.

*g*

_{h}, the actual

*g*

_{h}is varied by the model depending on how much transmitter is released in the triad as a result of the particular stimulus presented. Assuming ephaptic feedback, such a change in

*g*

_{h}might be produced by a modulation of the hemichannels (e.g., by protons; Sáez, Retamal, Basilio, Bukauskas, & Bennett, 2005), a shift in other conductances of the horizontal cell dendrites, or a shift in conductances of the surrounding neurons and structures involved in the triad (e.g., through a slow GABAergic system; Kamermans, Fahrenfort, & Sjoerdsma, 2002; reviews: Kamermans & Spekreijse, 1999; Schwartz, 2002; Wu, 1992; but see Verweij et al., 2003). With a more conventional chemical feedback mechanism, there is a range of possibilities for gain control. However, the model assumes that the feedback is primarily subtractive, and I find it difficult to see how a strongly nonsubtractive feedback system (i.e., multiplicative or divisive) could be equally consistent with the measurements considered here. This implies that a chemical feedback mechanism needs to be approximately subtractive over the voltage range normally occurring in the feedback. Although this is certainly feasible, it constrains the ways by which such a mechanism can be realized. Note that ephaptic feedback is inherently subtractive and, therefore, a particularly interesting possibility for the physiological substrate of the model proposed here.

*τ*

_{p}with such a presynaptic change in gain could not provide acceptable fits to the measurements. Therefore, either a more complicated process in the pedicle (possibly driven by the GABAergic system) or an electrically silent synaptic process restricted to each triad may be involved. A constraint is that it should be positioned before the subtraction (

*V*

_{p}−

*V*

_{d}) because I found that assuming an adaptive time constant as part of the main feedback loop did not produce adequate fits.

*V*

_{s}is too negative and the transmitter output of the triad is low, the increase in

*g*

_{h}will make

*V*

_{d}more negative and

*V*

_{s}more positive, mostly restoring normal transmitter output. Conversely, when the local stimulus is too dark, the high transmitter output will gradually decrease

*g*

_{h}and thereby make

*V*

_{d}more positive and

*V*

_{s}more negative, thus reducing transmitter output. Note that the (slow) change in

*g*

_{h}only affects the local triad and, therefore, assures that feedback remains effective despite large variations in loop gain that might otherwise result from the nonlinearity of the synaptic activation,

*I*

_{t}(

*V*

_{s}). Although such variations do occur on a fairly short time scale, steady variations are partly compensated by the change in

*g*

_{h}.

*g*

_{h}is slow, it forms a sort of spatial visual memory of the stimulus, and therefore, it will contribute to the occurrence of negative afterimages. When the eye fixates a bright spot for a while, the local gain,

*g*

_{h}, increases. When, subsequently, a homogeneous gray field is viewed, the large

*g*

_{h}at the spot position will drive

*V*

_{d}more negative and, therefore,

*V*

_{s}more positive. Thus,

*I*

_{t}is larger at that position than in the surround and, therefore, perceived as dark, that is, as the dark afterimage of a bright spot. Because the adaptation of

*g*

_{h}is assumed to take place locally at the cone triad synapse, the afterimage retains the full spatial resolution of the cone sampling grid. Note that the dynamics of afterimage formation and fading are determined by

*τ*

_{itd}in the model. In reality, this process might have more complicated dynamics than a first-order low-pass filter, but it may prove difficult to model this in more detail as long as the underlying physiological processes of this mechanism are not identified.

*I*

_{t}at the L cones will then be smaller than normal and

*I*

_{t}at the M cones will then be larger. This will gradually make

*g*

_{h}at the L cones larger (i.e., it will make

*V*

_{d}more negative and, thus,

*V*

_{s}more positive; hence,

*I*

_{t}will be larger and, thus, closer to normal) and

*g*

_{h}at the M cones smaller (hence,

*I*

_{t}will be smaller and, thus, closer to normal). The result is an

*I*

_{t}for both the L and M cones that is rather similar to the

*I*

_{t}that would arise from illuminating the scene with a neutral gray. In other words, the adaptation of

*g*

_{h}functions as chromatic adaptation, and the result is that perception (i.e.,

*I*

_{t}) is partly compensated for shifts in the chromaticity of the illuminant. Chromatic adaptation with a time constant in the order of 10–20 s has, indeed, been observed in both psychophysical measurements (Rinner & Gegenfurtner, 2000) and macaque retinal ganglion cells (Yeh, Lee, & Kremers, 1996).

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