Contrast gain reflects the rapidity of response amplitude increase with increase in stimulus contrast. In physiology, contrast gain can be measured directly as the initial slope of cell contrast response function. In psychophysics, contrast gain estimation is not straightforward. Further, rod and cone contrast gains have not been measured psychophysically at mesopic light levels where both rods and cones are active, due to the difficulty in producing stimuli that excite rods and cones separately at the same adaptation level. Here, we estimated rod and contrast gains by fitting reaction time distributions measured at a light level in which rods alone (scotopic), rods and cones (mesopic), or cones alone (photopic) mediate vision. The reaction time distributions were modeled by two different strategies, a simplified diffusion model that assumes a stochastic accumulation process and a model we developed that begins with sensory input based on early visual processing impulse response functions and assumes the reaction time variability originates in the response criterion. Estimates of contrast gain from both models were comparable and consistent with primate physiology measurements.

*t, X*(

*t*), is determined as

*X*(0) is the starting value,

*W*(

*t*) is a Wiener process that has a normal distribution with mean of 0 and variance

*t, μ*is the drift rate, and

*σ*is a positive constant such that the variance of

*X*(

*t*) is equal to

*σ*

^{2}

*t*. Therefore,

*X*(

*t*) is normally distributed with mean of

*X*(0) +

*μt*and variance

*σ*

^{2}

*t*. The drift rate reflects how quickly the information will be accumulated on average and can be interpreted as the strength of sensory input, which is determined by the stimulus. Under the framework of sequential sampling, a decision is made once the value of

*X*(

*t*) passes a response boundary or criterion

*g*. Reaction time is then modeled as the sum of the decision time and an irreducible minimum reaction time (

*RT*

_{0}). In other words, reaction time is determined by five parameters in the diffusion model, including

*X*(0),

*μ, σ, g,*and

*RT*

_{0}. The variation in reaction time data can result from the Wiener process, variability in the parameters defining the Wiener process, or the irreducible minimum reaction time. Many forms of the Wiener diffusion model have been proposed, depending on whether the parameters are assumed to be deterministic or stochastic. For instance, the Ratcliff Wiener diffusion model (Ratcliff, 1978; Ratcliff & Smith, 2004) assumes that

*X*(0) and

*RT*

_{0}have uniform distributions and

*μ*has a normal distribution, while the EZ diffusion model (Wagenmakers, van de Maas, & Grasman, 2007) assumes no between-trial variation in the parameters, i.e.,

*X*(0),

*μ, g,*and

*RT*

_{0}are fixed and the only source of variation comes from the Wiener process itself.

*same*distribution of decision criteria was applied for

*all*the stimulus conditions. Our rationale for having a common distribution of decision criteria was based on neurophysiological studies that indicated highly stereotypical neural responses in primates from ganglion cells (Croner, Purpura, & Kaplan, 1993; Sun, Rüttiger, & Lee, 2004) to V1 in alert monkeys (Gur, Beylin, & Snodderly, 1997; Gur & Snodderly, 2006). Neural responses in the lower order neurons are too reliable to account for variability in decision time (Schall, 2002). We therefore inferred that the principal source of variance in RT need be in post-sensory processing. Since decision processing is cortically mediated, it is reasonable to hypothesize that decision processing could use similar rules no matter what type of photoreceptors mediate sensory processing in the visual system. We use a gamma distribution for criterion for two reasons. First, a gamma distribution was successfully used to describe reaction time distributions obtained with a wide range of conditions (McGill, 1963). Second, a gamma distribution has two parameters, a scale parameter and a shape parameter. Varying the scale parameter corresponds to changing the units or scales of the decision variable, therefore, the scale parameter could be easily linked to contrast gain in early visual processing. The shape parameter described the shape of the density function of reaction time or decision criterion; therefore, it could be thought as a parameter for variability in post-sensory processing. In a similar vein, a Weibull distribution that has three parameters for scale, shape, and shift (initial value) has been used for modeling reaction time (Rouder, Sun, Speckman, Lu, & Zhou, 2003). Under the current framework, the irreducible minimum reaction time

*RT*

_{0}is comparable to the shift parameter in a Weibull distribution. We restricted a shape parameter of the gamma distribution to be the

*same*for the measured reaction time over all 80 conditions. A variable criterion approach could be thought as a convenient way to describe the combined contributions of all of the sources of variation in post-sensory processing.

*k*is the scaling factor,

*C*is the Michelson contrast, and

*C*

_{e}is the effective contrast of the sensory processing signal used by the visual system. In this case, the contrast gain is equal to

*k*. The stimulus Weber contrast was not converted to Michelson contrast, because for sinusoidal stimuli as used in physiological study (Purpura et al., 1988), Weber contrast equals Michelson contrast (Sun, Mitchell, & Swanson, 2006). For cone stimuli at 20 and 200 Td, a saturating response is evident in physiological measurements (Purpura et al., 1988). The contrast–response function is therefore described by the Michaelis–Menten saturation function:

*C*

_{sat}is the semi-saturation contrast. When

*C*

_{sat}is much larger than

*C,*Equation 3 becomes a linear function approximately, therefore, Equation 2 can be thought as a simplified form of Equation 3. We used a linear function in Equation 2 for the purpose of reducing the number of free parameters in model fits. The contrast gain (the initial slope) can be obtained as the first derivative of Equation 3 at zero contrast (Shapley & Enroth-Cugell, 1984), that is,

*k*/

*C*

_{sat}. Purpura et al. (1988) did not use

*k*/

*C*

_{sat}as the contrast gain at high light levels. They used the first few points to estimate the initial slope using linear regression without forcing a zero intercept, which yields lower values than

*k*/

*C*

_{sat}. These differences, however, are very small.

*t, X*(

*t*), is determined by Equation 1. A decision is made once the value of

*X*(

*t*) passes a response boundary or criterion

*g*. This is a first-passage time problem in stochastic processes. With the EZ diffusion model that assumes a fixed starting value, drift rate, and criterion, a closed form solution of probability density function (

*τ*) can be obtained from the normal density function of

*X*(

*τ*) that has a mean of

*X*(0) +

*μτ*and variance

*σ*

^{2}

*τ*(Sigman & Dehaene, 2005):

*RT*

_{0}). The probability density function of reaction time

*RT*was

*g*can be set as 1. To estimate contrast gain from the EZ diffusion model, we assumed that the drift rate,

*μ,*is a function of contrast at each light level. We used a linear function (see Equation 2) for rod stimuli at all light levels (0.002–20 Td) and cone stimuli at 2 Td, or the Michaelis–Menten saturation function (see Equation 3) for cone stimuli at 20 and 200 Td to describe the relation between the drift rate (

*μ*) and stimulus contrast.

*α*) for all stimulus conditions. We set the scale parameter

*β*to be 1. The model free parameters included one shape parameter (

*α*) for the criterion distribution for all stimulus conditions, one scaling factor (

*k*) for rod incremental and cone stimuli at each light level, and one

*C*

_{ sat}for cone stimuli at each light level. We initially set one rod irreducible minimum reaction time (

*RT*

_{0}) for all 5 light levels as we did for modeling mean RT, but the RT distribution fits were not satisfactory, further confirming the importance of modeling RT distributions to estimate contrast gain. We required two different values of

*RT*

_{0}, one for the light levels ≤0.2 Td, presumably mediated by the

*slow*rod–rod bipolar-Amacrine II pathway (Sharpe & Stockman, 1999), and the other for the light levels ≥2 Td, presumably mediated by the

*fast*rod–rod cone gap junction pathway. For rod decremental stimuli, the scaling factor (

*k*) at each light level was set to be proportional to that for rod incremental stimuli with

*p*as a free parameter. In total, there were 16 free parameters (both rod and cone stimuli: 1

*α*; rod stimuli: 5

*k,*1

*p,*2

*RT*

_{0}, one for the slow rod pathway, one for the fast rod pathway; cone stimuli: 3

*k,*3

*C*

_{sat}, 1

*RT*

_{0}). For each condition, we generated 1000 random numbers for decision criteria based on a gamma distribution. The RT distribution was simulated from the IRF-based model. We adopted the weighted least square (WLS) approach to fit the model by minimizing the weighted sum of the squared errors (SSEs) between the measured and fitted quantile RTs (in seconds) for all conditions (Ratcliff & Smith, 2004). The quantiles used were 0.1, 0.3, 0.5, 0.7, and 0.9. The weights were set as 2 for 0.1 and 0.3 quantiles, 1 for the 0.5 and 0.7 quantiles, and 0.5 for 0.9 quantile, as recommend by Ratcliff and Smith (2004). The parameters were searched until the total weighted SSE between the measured and simulated RT quantiles was minimized.

*k*) for rod incremental and cone stimuli, the

*C*

_{ sat}for cone stimuli at 2, 20, and 200 Td, and one

*σ*for each light level. For cone stimuli at all light levels (2, 20, and 200 Td), one

*RT*

_{0}proved sufficient to provide satisfactory data fits. For rod decremental stimuli, the scaling factor (

*k*) at each light level was set to be proportional to that for rod incremental stimuli with

*p*as a free parameter; and the

*σ*was set equal to that for rod incremental stimuli at the same light level. There were in total 23 free parameters in total (rod stimuli: 5

*k,*5

*σ,*1

*p,*2

*RT*

_{0}; cone stimuli: 3

*k,*3

*C*

_{ sat}, 3

*σ,*1

*RT*

_{0}). The model RT quantiles were simulated based on Equation 3. The parameters were fitted using the same WLS approach as the IRF-based model.

*DC*and 2.9% for

*AJZ*.

*DC*(left two columns) and

*AJZ*(right two columns) are shown in Figure 2. Each panel shows the RT quantiles of 5 Weber contrasts at one light level. For clarity, in each panel, the RT quantiles with the two highest contrasts and the two lowest contrasts were shifted horizontally (the highest contrast: −100 ms; the second highest contrast: −50; the lowest contrast: +100 ms; the second lowest contrast: +50 ms). The RT data for rod decremental stimuli had lower central tendency values compared to the rod incremental stimuli, but both distributions had similar shapes (RT quantiles to rod decremental stimuli are not shown). Although the minimization for model fits was based on RT quantiles, the RT distributions as well as model fits are also presented in the form of probability density function (

*DC*and Figure 4 for

*AJZ*). In Figures 3 and 4, the optimal bin widths of the RT density histograms for each stimulus condition were determined by a method derived by Scott (1979):

*w*is the optimal bin width,

*s*indicates the standard deviation, and

*n*is the number of trials. The top five rows show RT distributions (black bars) for rod incremental stimuli at 0.002 Td to 20 Td. The bottom three rows show RT distributions for cone stimuli at 2, 20, and 20 Td. Each panel shows the RT distribution at one contrast (labeled in panel) and one retinal illuminance level. Note that each column is not aligned with the contrast level because the contrast range varied with stimulus condition (from 5% to 160% Weber contrast).

*R*

^{2}= 0.77 for

*DC*and 0.98 for

*AJZ*), shown as a dashed line ( Figure 5, left panels). For rod stimuli, the relation between standard deviation and mean reaction time is described by a second-order polynomial function for

*DC*(

*R*

^{2}= 0.93) or a linear function for

*AJZ*(

*R*

^{2}= 0.78). The coefficient of variation (standard deviation/mean) of RT across conditions is shown in Figure 5 (right panels). Overall, the coefficient of variation (

*C.V.*) decreased with increases in contrast or retinal illuminance, reaching an asymptotic value of about 0.1, the same asymptotic value as reported by Chocholle (1940) in a study of simple reaction time to a 1000-Hz pure tone. The average coefficients of variation for observers

*DC*and

*AJZ*(

*AJZ*data in parentheses) were 0.12 (0.13) for cone RTs, 0.14 (0.17) for rod increment RTs, and 0.15 (0.17) for rod decrement RTs (shown as horizontal bars in Figure 5, right panels).

*cdf,*Figure 2) or the fitted probability density function (

*pdf,*Figures 3 and 4) of RTs from the IRF-based model (red lines) and the EZ diffusion model (blue lines). The average weighted percentage variance (

*R*

^{2}) in the quantile RTs with all conditions explained by the model was ≥88% for both observers (see Table 1). The total SSE was between 0.088–0.152 s

^{2}. Since there were 325 RT quantiles assessed for each observer (25 quantile RTs in each panel in Figure 2, plus 125 quantile RTs for rod decremental stimuli), on average, the difference between the predicted quantile RTs and measured quantile RTs was in the order of 1.0 ms calculated from the total SSE for each observer. For the IRF-based model, the fitted shape parameter of the gamma distribution for the decision criterion was 7.1 for

*DC*and 4.8 for

*AJZ*( Figure 6). The

*RT*

_{0}for the slow and fast rod systems was 314 and 270 ms for the IRF-based model, or 309 and 282 ms for the EZ diffusion model (

*DC*), or 233 and 187 ms for the IRF-based model, or 234 and 188 ms for the EZ diffusion model (

*AJZ*). The difference in

*RT*

_{0}between the slow and fast rod systems was between 27 and 46 ms. The

*RT*

_{0}for the cone system was 243 ms from the IRF-based model and 225 ms from the EZ diffusion model (

*DC*), or 163 ms from the IRF-based model and 193 ms from the EZ diffusion model (

*AJZ*; see Table 1).

Parameters | IRF-based model | EZ diffusion model | |||
---|---|---|---|---|---|

DC | AJZ | DC | AJZ | ||

Rod increment | RT _{0} (slow), ms | 314 | 233 | 309 | 234 |

RT _{0} (fast), ms | 270 | 187 | 282 | 188 | |

Rod decrement | p | 1.43 | 1.48 | 1.47 | 1.43 |

Cone | RT _{0}, ms | 243 | 167 | 225 | 193 |

Rod and cone | α | 7.1 | 4.8 | – | – |

Mean weighted R ^{2} | 91% | 88% | 94% | 92% | |

Total weighted SSE (s ^{2}) | 0.145 | 0.088 | 0.152 | 0.144 |

*AJZ*). The ratio of cone to rod contrast gain was larger at 20 Td than that at 2 Td ( Figure 8, which also plotted log contrast gain ratio at 200 Td, assuming rod contrast gain at 200 Td is the same as 20 Td due to plateau effect, see Figure 7).

*s*) of matching cone luminance contrast (

*C*

_{cone}) as a function of rod contrast (

*C*

_{rod}) across the origin, i.e.,

*g*

_{cone}) to rod contrast gain (

*g*

_{rod}). Assuming at mesopic light levels, responses with rod or cone stimuli (

*R*

_{rod},

*R*

_{cone}) were linearly or nearly linearly related to rod or cone contrast, with a response at zero contrast as

*R*

_{0}:

*s*) of matching cone luminance contrast as a function of rod contrast across the origin. The slopes fitted based on the rod percept matching at 2, 10, and 100 Td were 0.35, 0.08, and 0.06 for

*DC,*or 0.45, 0.19, and 0.14 for

*IS*(Cao et al., 2008). Figure 8 showed that the log(1/

*s*) (averages of

*DC*and

*IS,*gray circles) as a function of log retinal illuminance were remarkably similar to the cone to rod contrast gain ratios estimated from RT distributions.

*slow*pathway hypothesized to mediate rod vision at low light levels. The second pathway transmits rod information via rod–cone gap junctions and ON and OFF cone bipolars. This is a

*fast*pathway hypothesized to mediate rod vision at high scotopic and mesopic light levels (Sharpe & Stockman, 1999). Consistent with the idea of two rod pathways, our initial attempts with one irreducible minimum reaction time for rod reaction times at all light levels did not yield satisfactory fits. Two irreducible minimum reaction times could characterize the rod reaction time data, one for the slow rod pathway (≤0.2 Td) and one for the fast rod pathway (≥2 Td). The estimated difference in two

*RT*

_{0}s between the slow and fast rod pathways was 27–46 ms, which is comparable with 33-ms delay between the two rod pathways estimated from self-nulling of rod flicker with 15-Hz modulation near 1.0 scotopic Td (in a range of 0.5–1.5 scotopic Td) or 0.13 photopic Td (in a range of 0.07–0.2 photopic Td; Sharpe, Stockman, & MacLeod, 1989; Stockman, Sharpe, Zrenner, & Nordby, 1991).

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