Spatial summation of perimetric stimuli has been used to derive conclusions about the spatial extent of retinal-cortical convergence, mostly from the size of the critical area of summation (Ricco's area, RA) and critical number of retinal ganglion cells (RGCs). However, spatial summation is known to change dynamically with stimulus duration. Conversely, temporal summation and critical duration also vary with stimulus size. Such an important and often neglected spatiotemporal interaction has important implications for modeling perimetric sensitivity in healthy observers and for formulating hypotheses for changes measured in disease. In this work, we performed experiments on visually heathy observers confirming the interaction of stimulus size and duration in determining summation responses in photopic conditions. We then propose a simplified computational model that captures these aspects of perimetric sensitivity by modelling the *total retinal input*, the combined effect of stimulus size, duration, and retinal cones-to-RGC ratio. We additionally show that, in the macula, the enlargement of RA with eccentricity might not correspond to a constant critical number of RGCs, as often reported, but to a constant critical *total retinal input*. We finally compare our results with previous literature and show possible implications for modeling disease, especially glaucoma.

*spatial summation*) and duration (

*temporal summation*) has been shown to be altered following retinal ganglion cell (RGC) loss from glaucoma (Mulholland et al., 2015; Redmond, Garway-Heath, Zlatkova, & Anderson, 2010; Rountree et al., 2018; Swanson, Felius, & Pan, 2004; Yoshioka, Zangerl, & Phu, 2018). Both spatial and temporal summation are characterized by a biphasic response, with a steeper reciprocal relationship between stimulus area/duration and contrast at threshold for smaller/shorter stimuli (total summation) and a shallower change for larger/longer stimuli (partial summation). The response is often characterized in terms of the point of transition between these two phases (

*critical size/duration*) (Mulholland et al., 2014). The physiological basis of spatial and temporal summation has been extensively studied. Although models solely based on RGCs exist (Glezer, 1965), spatial summation has been linked to cortical magnification and to the convergence of RGCs onto cells of the visual cortex (Kwon & Liu, 2019). This phenomenon is often referred to as

*cortical pooling*, and it is the favored model for explaining spatial summation (Kwon & Liu, 2019; Pan & Swanson, 2006). Cortical pooling can be modeled through a linear combination of filter elements tuned to different spatial frequencies (Pan & Swanson, 2006).

*spatiotemporal summation*). Models exist to describe temporal summation in isolation (Gorea & Tyler, 1986; Swanson, Pan, & Lee, 2008; Watson, 1979). Many of these authors acknowledge the effect of stimulus configuration (Gorea & Tyler, 1986; Watson, 1979) and adaptation state (Swanson, Ueno, Smith, & Pokorny, 1987) on critical duration. Direct experimental evidence of the interaction between size and duration for simple circular stimuli (Barlow, 1958; Mulholland et al., 2015; Owen, 1972) suggests a combined integration of the total input by the visual system. Some attempts have been made to describe such an interaction, mainly in the field of motion detection (Anderson & Burr, 1991; Fredericksen, Verstraten, & van de Grind, 1994), but this phenomenon has been little explored for perimetry (Owen, 1972). Another aspect that has been overlooked is the effect of retinal convergence. One common assumption is that spatial summation at different eccentricities can be exclusively explained by the change in density of RGCs (Kwon & Liu, 2019). However, similarly to cortical convergence, individual RGCs might carry a different weight in terms of retinal input at different eccentricities because they receive input from a different number of photoreceptors (larger in the periphery), with significant changes in the composition and density of their mosaic.

^{2}. Calibration was performed in a dark room before every experiment through an automated procedure implemented by the manufacturer. As it is convention in perimetry, the intensity of the stimulus in dB is expressed as attenuation of the maximum possible stimulus intensity (3,185 cd/m

^{2}), so that higher contrast equates to lower dB values. This quantity can be converted to Weber contrast (

*W*) using Equation 1. However, for simplicity in our calculations, we report the values as differential light sensitivity (DLS), which is simply the sensitivity value in dB/10.

_{c}*SD*) of 1 dB and a guess/lapse rate of 3%. The prior distribution was updated at each response to generate a posterior distribution. The posterior distribution was used as the prior distribution for next step in the strategy. The stimulus was chosen as the mean of the prior distribution at each step, rounded to the closest integer dB value. This has been shown to provide unbiased estimates of the 50% detection threshold for a yes/no task (King-Smith et al., 1994). The determination of each threshold terminated when the posterior distribution reached a standard deviation < 1.5 dB (dynamic termination criterion).

*SD*(which model the 50% threshold and the slope of the FOS curve, respectively) were simultaneously fitted as free parameters. The test was terminated when the entropy of the combined posterior distribution was ≤ 4.5. For the purpose of this preliminary step, the four spatial locations were considered interchangeable. Therefore, only four FOS curves were determined, one for each stimulus combination. The prior distribution for the mean was itself a Gaussian distribution with a

*SD*of 4 dB, centered on the average of the sensitivity estimates obtained from the ZEST procedure for the tested locations (eight estimates for each stimulus combination, i.e., four locations each tested twice) and limited over a domain of ±5 dB around its mean. The prior distribution for the

*SD*of the FOS curve was a uniform between 1 and 10 dB, with steps of 0.5 dB.

*SD*for the Gaussian FOS curves was used to determine the contrast levels to be tested for each stimulus combinations in the actual MOCS. We tested seven steps for each location and each condition. The steps were placed at the following quantiles of the Gaussian FOS (neglecting lapse/guess rate): {0.0001, 0.1, 0.3, 0.5, 0.7, 0.9, 0.9999}. We, however, ensured that all the steps were at least 1 dB apart (the minimum interval allowed by the device) and that the two most extreme contrast levels were at least 10 dB above and below the estimated 50% threshold. The 50% threshold was calculated as the average of the two test results obtained from the ZEST strategy for each location. Each contrast level was presented 25 times, and each spatial location was tested fully and independently, for a total of 2,800 presentations. A break of at least 10 min was introduced every 350 presentations, and the whole test was split into two sessions performed on two separate days. The individual presentations were fully randomized across test locations, stimulus area/duration combinations, and contrast levels. Pauses between presentations and false responses were determined as described above for the main experiment.

*SD*(σ), lapse rate (λ), and guess rate (γ) were free parameters (see Equation 2). Mean (µ) and σ were hierarchical parameters that varied for each of the four tested locations. Information, however, was propagated across different locations to improve the robustness of the fit of the parameters for each testing condition. Lapses and guesses were instead modeled as global parameters for the whole test. Details of the implementation of the Bayesian model are reported in the Appendix.

*spatiotemporal input*. We integrated the spatiotemporal input into a computational model of the response of RGC mosaics, partially based on the work by Pan and Swanson (2006) and Bradley et al. (Bradley, Abrams, & Geisler, 2014). The key novel aspect of our modeling was that the linear response from the RGC mosaic was pooled and integrated over time so that changes in duration and size of the stimulus would both simultaneously affect the temporal and spatial response of the system. We further modeled the retina as a two-stage mosaic, where the response from individual photoreceptors active in photopic adaptation conditions (cones) was integrated by the RGC mosaic to explore the effect of retinal convergence in the central visual field. The density of the two mosaics was varied to reproduce the effect of eccentricity. We refer to the combined effect of the spatiotemporal input and changes in retinal structure (i.e., density of the photoreceptor and RGC mosaics) as

*total retinal input*. The model was implemented in MATLAB (The MathWorks, Natick, MA, USA) and is described in detail below.

^{2}), we modeled only the cone mosaic. In this retinal model, individual RGCs pool the response from the photoreceptors according to their receptive fields (RFs). To improve the efficiency of computation, each hexagonal lattice was rearranged in a regular lattice with anisotropic spacing (see Figure A.1). This simplifies the pooling operation, which can be computed via simple convolution of the regularized lattice with the RGC-RF filter (see next section), also rearranged accordingly on the same regular lattice. The response of the photoreceptor mosaic was simply computed by multiplying the mosaic by the stimulus. In its simplest form, this is equivalent to assigning a value of 1 to all the photoreceptors that fall within the stimulus area, leaving the others to 0. However, in its final implementation, this was modified to include the effect of optical blur (see later). Only the Parasol OFF RGC mosaic was used for the calculations (P-OFF-RGC), assuming that the ON and OFF mosaics operate on parallel redundant channels for the detection of simple round stimuli. Parasol cells were chosen because there is experimental evidence that these cells preferentially mediate sensitivity to briefly flashed stimuli, such as those used in perimetry. The calculations were repeated with the midget OFF RGC mosaic (mOFF-RGC) and reported as supplementary material for comparison with some previous literature (Kwon & Liu, 2019).

*K*= 0.01 *

_{s}*K*, where

_{c}*K*is the peak sensitivity of the excitatory center. The

_{c}*SD*of the surround was 6.7 times larger than the

*SD*for the center (average reported by Croner & Kaplan, 1995). The

*SD*for the center was scaled so that the

*radius*of the center component was equal to the intercell spacing of the mosaic (defined by its density). The radius was defined by Croner and Kaplan as the distance from the center at which the excitatory Gaussian component has value

*K*/

_{c}*e*. The corresponding

*SD*was approximated as

*SD*= Cell spacing/1.414. Note that, while the center-surround proportions are based on Croner and Kaplan (1995), the actual extent of the RGC-RFs in our model depends only on the intercell spacing of the RGC mosaic.

*k*is the summation exponent (4 in this study) (Meese & Williams, 2000; Pan & Swanson, 2006; Quick, 1974; Robson & Graham, 1981; Swanson, Felius, & Pan, 2004; Tyler & Chen, 2000), and

*S*is the total spatial input defined as

*R*is the response of an individual ganglion cell to the stimulus. Note that the contribution of individual RGCs (

_{i}*R*) can change because of the location of the RGC with respect to the stimulus (edge as opposed to center) and the effect of retinal convergence (RGCs in the periphery will have a bigger contribution when fully stimulated because of their larger pooling from the photoreceptors). The temporal profile of the stimulus is represented by

_{i}*f*(

*t*), which is a step function with value 1 when the stimulus is on and 0 otherwise. As previously mentioned, the combined effect of stimulus size, stimulus duration, RGC density, and retinal convergence defines the

*total retinal input*. Much like other temporal filters, this operation can also be implemented through temporal convolution. Note that such an approach to spatiotemporal summation is very similar to what was described in Frederiksen et al. (Fredericksen, Verstraten, & van de Grind, 1994) and Anderson and Burr (1991) for motion detection. Since only the P-OFF-RGC mosaic was considered for our calculations, the RGCs that were assigned a negative input were considered inhibited by the stimulus. Their negative contribution to the sum can be interpreted as an inhibition of their background activity. Obviously, such a simple approach would not account for other filter choices with a strong biphasic response, where a simple summation would always result in a zero net sum. From the examples in Figure 3, we can see that this pooler has the desired properties when the response is computed for different stimulus sizes and durations (i.e., a shorter duration determines a larger critical area and vice versa). One additional convenient property of this pooler is that the critical size and duration depend on the integration constant τ. The integration constant τ is therefore the scaling factor of the pooler and can be used to test the hypothesis of constant input integration across the VF. If the hypothesis of constant integration response for the same amount of total retinal input is correct, we do not expect important changes in the integration constant across different testing conditions and eccentricities. An alternative approach would be to model individual RGCs (or higher-order visual detectors) as separate spatiotemporal integrators and to pool their response by vector summation (Pan & Swanson, 2006; Quick, 1974). Such an approach has the advantage of allowing the modeling of the response from specific classes of RGCs and produces sensible spatial and temporal summation responses. However, it fails to reproduce the interaction between spatial and temporal input that would be expected. For example, Ricco's areas in spatial summation curves would be unaffected by changes in stimulus duration. This is in contrast with evidence from the literature (Barlow, 1958; Baumgardt, 1959; Mulholland et al., 2015; Owen, 1972). It is worth noting that the current model could be extended to include the temporal response of individual classes of RGCs prior to pooling. However, this would increase the number of tunable parameters and would be beyond the objectives of the current study and what could be determined with our experiments.

*fminsearch*function in MATLAB; Lagarias, Reeds, Wright, & Wright, 1998) to minimize the root mean squared error (RMSE). The summation exponent was set to

*k*= 4 (Meese & Williams, 2000; Pan & Swanson, 2006; Quick, 1974; Robson & Graham, 1981; Swanson, Felius, & Pan, 2004; Tyler & Chen, 2000), and the RGC mosaic density was varied according to the eccentricity following the model by Drasdo et al. (Drasdo, Millican, Katholi, & Curcio, 2007; Montesano et al., 2020). These estimates were corrected with individual imaging data obtained from the OCT scans, as previously reported (Montesano et al., 2020; Raza & Hood, 2015). The model was fitted by tuning the parameter τ, which represents the integration constant of the spatiotemporal input. An additional parameter (additive in log-scale) allowed translation along the vertical axis (log-DLS, Offset term).

_{10}– log

_{10}scale. The parameters for the curves were fitted accounting for the optical blur (based on each participant's average pupil size and iris pigmentation). Densely sampled curves were numerically calculated using these parameters to estimate Ricco's area. These curves were calculated without the effect of optical blur. This simulates removing the estimated effect of optics on the size of Ricco's area. Note that accounting for convergence in the fitting process will not change Ricco's area, as parameters are optimized to fit the same data.

*p*values were corrected using a Bonferroni–Holm correction. All calculations were performed in R (R Foundation for Statistical Computing, Vienna, Austria) using the

*lme4*package (Bates, Mächler, Bolker, & Walker, 2015). All comparisons were performed on log

_{10}-transformed values of Ricco's area, integration constant, and number of P-OFF-RGCs, unless otherwise specified. Eccentricity was treated as a discrete factor.

_{10}(stimulus area), the raw log

_{10}(number of RGCs), or the convergence weighted log

_{10}(number of RGCs) as predictors in a mixed-effect model. The unexplained residual variance (including random effects) was 1.93 dB

^{2}for the log

_{10}(stimulus area), 1.79 dB

^{2}for the unweighted log

_{10}(number of RGCs) (7.2% reduction), and 1.77 dB

^{2}for the convergence weighted log

_{10}(number of RGCs) (8.1% reduction).

_{10}(stimulus area) or the log

_{10}(spatiotemporal input) as predictors in a mixed-effect model. The unexplained residual variance (including random effects) was 11.4 dB

^{2}for the log

_{10}(stimulus area) and 3.7 dB

^{2}for the log

_{10}(spatiotemporal input), a 67.5% reduction.

*SD*). As expected, the estimated Ricco's area increased toward the periphery (Figure 7C and Table 2), with no significant differences between the areas calculated with and without accounting for convergence. However, such a change did not correspond to a constant number of P-OFF-RGCs being stimulated. Instead, the estimated number of P-OFF-RGCs at Ricco's area was consistently larger toward the fovea (Figure 6D). This was mirrored by a change in the integration constant τ with eccentricity. However, this trend in τ was completely eliminated by accounting for the change in cone/RGC convergence (Figure 6A and Table 2). This effect of convergence was larger when modeling the mOFF-RGC mosaic (supplementary material). This result can alternatively be visualized by multiplying the number of P-OFF-RGCs at Ricco's area by the corresponding convergence factor (Figure 6D and Table 2). Note that this is a post hoc calculation and not an output from the model (accounting of convergence is expected to have an effect on the model's parameters but not on Ricco's area and the shape of the fitted response profile). There was a small significant increase in the vertical Offset with eccentricity, which was reduced by accounting for convergence (Figure 6B and Table 2).

*SD*) and 1.40 ± 0.41 dB for the 200-ms stimuli. This can be compared to the 0.96 ± 0.35 dB average RMSE obtained from fitting the 200-ms data alone at the same eccentricity. For context, the root mean squared difference in sensitivity between the two repetitions of the retested combinations was 2.44 dB, and the root mean squared deviation from the average of the two repetitions was 1.22 dB. An example of the calculation for one location in one subject is also shown (Figures 7A, B). There was a strong correlation between the parameter estimates obtained by fitting data from all stimulus durations and 200 ms alone (previous section), at the same eccentricity (correlation coefficient: 0.83 for log

_{10}(τ) and 0.89 for the sensitivity offset; Figures 7C, D). However, the two estimates appeared to have a consistent significant difference (

*p*< 0.0001), approximately constant in log

_{10}-scale. The median [interquartile range] was 34.65 [25.31, 56.04] × 10

^{2}for the τ constant and 2.36 [2.31, 2.42] dB/10 for the offset. These values were both significantly smaller than those reported in Table 2 for the same eccentricity (

*p*< 0.0001 and

*p*= 0.00298, respectively). Significant differences were also present for all the other parameters, including Ricco's area and the number of P-OFF-RGCs at Ricco's area (all

*p*< 0.0001). Numeric values of Ricco's area and corresponding P-OFF-RGC counts are reported in Table 3 for all durations. Differences in Ricco's areas between different durations were not tested as such differences are assumed by the model.

^{2}), threshold behavior should be close to Weber's law at least for a G-III stimulus (Bierings, de Boer, & Jansonius, 2018; Swanson et al., 2014). Retinal illuminance can be reduced by media opacity (such as cataract), but this is likely to be negligible in a young healthy cohort. Pupil size can also affect retinal illuminance, especially if below 3 mm (Swanson et al., 2014), but the average pupil size in our cohort was 5.9 ± 0.8 mm.

_{10}units, which is very similar to the change measured by Tuten et al. (Tuten, Cooper, & Tiruveedhula, 2018) with adaptive optics (AO). Taking the summation over the absolute value instead produced an average change of 0.37 log

_{10}units, which is closer to what was reported by Dalimier and Dainty (2010) for similar experiments. Ultimately, a definitive answer to these questions could only be obtained by performing these same experiments with coupled AO-corrected stimuli and imaging, so that accurate estimates of individual RGCs can be obtained and the effect of optical aberrations eliminated (Liu et al., 2017).

*z*score transformation and would require numerous catch trials to determine the individual response bias (Klein, 2001). In our data, the response bias and lapse rate were estimated from the response to stimuli that were likely to be much above or below the 50% threshold (as determined using a pilot using QUEST+ to estimate threshold and psychometric function slope), and all participants were encouraged to maintain a low false-alarm rate during the experiments. Both the guess and lapse rates were very close to 0 and are therefore unlikely to have greatly affected the estimates of the psychometric function.

*p*= 0.005). This comparison was performed for the log

_{10}-RGC number with a linear mixed model using the hemifield as a fixed effect and the eccentricity as a random effect, nested within the subject, to perform a paired same-eccentricity comparison.

**G. Montesano**, CenterVue (C); Relayer (O);

**P.J. Mulholland**, Heidelberg Engineering (R), Visual Field Sensitivity Testing (P), LKC Inc. (R);

**G. Ometto**, Relayer (O);

**D.F. Garway-Heath**, Carl Zeiss Meditec (C), CenterVue (C), Heidelberg Engineering (F), Moorfields MDT (P), ANSWERS (P), T4 (P), Visual Field Sensitivity Testing (P);

**J. Evans**, None;

**D.P. Crabb**, CenterVue (C), ANSWERS (P), T4 (P).

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_{c}and σ

_{c.}The four locations were modeled as hierarchical random effects on µ

_{c}and σ

_{c}, with no correlations between the two parameters. The lapse rate (λ) and guess rate (γ) were modeled as global parameters for the whole test. The response (yes/no) was modeled as binomial process with 25 trials. Following Prins (2019), the prior distribution for the mean of the parameters µ

_{c}and σ

_{c}for each stimulus combination was a noninformative normal distribution with a standard deviation of 30 dB. The prior distribution for the variance of the parameters µ

_{c}and σ

_{c}for each stimulus combination was a noninformative uniform distribution between 0 and 1,000 dB. The random effects for each location were modeled as a normal distribution with mean µ

_{c}and standard deviation σ

_{c}. The prior distribution was linked to the parameter σ

_{c}via a logarithmic function. The parameters γ and λ were nonhierarchical and had a Beta prior distribution with shape parameters 2 and 50.

_{10}– log

_{10}coordinates) of the same template response by an amount equivalent to the log

_{10}change in duration. Note that the selection of the filter scale does not need to depend solely on the stimulus duration but more generically on the retinal input, to include the effect of cone/RGC convergence, duration, background illumination, or, for example, RGC loss in disease. For the sake of simplicity, everything except duration was held constant for these calculations. The combined effect is best represented by a summation surface, shown in Figure A.3. In the figure, three summation curves are isolated by cutting through the surface at different stimulus durations and correspond to using a different filter scale. Importantly, temporal summation responses can be obtained by cutting through the surface along the orthogonal (duration) axis. Because the surface is obtained by proportionally translating the same spatial summation curve, temporal summation responses also follow the same template curve, proportionally shifted with different stimulus sizes. This would produce the same results obtained with our more generic input summation model. With this interpretation, although a strict retina–V1 convergence cannot be defined, testing in partial summation condition (i.e., long stimulus durations and high background illumination) would allow the calculation of the smallest possible spatial scale for a given retinal location.