10.1.1 The visual field inhomogeneity in log contrast sensitivity is bilinear
Previously, the decline in log contrast sensitivity for grating patches was understood to be a linear function of eccentricity. For a wide range of spatial frequencies (1.6 - 12.8 c/deg) the slope of the decline was constant when eccentricity was expressed in stimulus carrier cycles (Pointer & Hess, 1989). The study presented in Chapter 4 establishes that the decline in log contrast sensitivity in the central visual field (within 18 cycles of fixation) is better charac- terised as bilinear, initially steep before then shallowing to around half that gradient. As a 3D surface that is a function of both horizontal and vertical eccentricity, this bilinear sensitivity decline takes the form of a witch’s hat (Baldwin et al., 2012).
Like the Pointer and Hess (1989) result, the bilinear sensitivity functions reported in this thesis are scale invariant across a wide range of spatial frequencies (0.7 - 4.0 c/deg) and show a shal- lower decline across the horizontal meridian than across the vertical meridian. The superior and inferior declines of the vertical meridian were asymmetric, with the second slope of the in- ferior decline being more shallow than that of the superior decline. This is in agreement with the vertical meridian asymmetry reported previously (for a review, see Abrams et al., 2012). The study conducted here did not find any strong or consistent effect of stimulus orientation within the central visual field, indicating that these results can be generalised across different orientations.
The attenuation surface developed in Chapter 4 provides a more accurate account of the con- trast sensitivity decline within the central visual field than has been reported previously. It is used elsewhere in this thesis both in models of the visual system (Chapter 5) and to transform stimuli in order to counteract the effects of the inhomogeneity in sensitivity (see below). It is not clear what physiological feature or features of the visual system might cause the scale invariant bilinear decline reported here. Ahumada and Watson (2011) suggested that sensi- tivity could be predicted from the retinal cone density functions (see Section 4.7.6). Although this correctly predicts the concave shape of the log contrast sensitivity declines, a model based on cone density would fail to predict the scale invariance found here. In order for the effect to be scale invariant, the strongest part of the inhomogeneity must occur where neurones are spatial frequency selective. The earliest point at which this is seen is the ganglion cell layer in the retina. Recent work by Bradley and Geisler (2012) has attempted to build a bottom-up model of visual processing starting from the ganglion cell stage. I have been in contact with the authors to try to establish whether this model can predict the witch’s hat attenuation surface.
10.1.2 Area summation is spatially extensive and occurs according to a single rule
By applying the inverse of the attenuation surface to the stimuli Chapter 6 demonstrates that the summation of threshold contrast occurs under a single (fourth-root) rule over at least 33 cycles once the visual field inhomogeneity in contrast sensitivity is factored out. These data were fit by a variety of summation models (developed in Chapter 5). The summation slopes found were consistent with either a probability summation model or noisy energy model. The results of previous studies (e.g. Meese & Summers, 2012) lead me to favour the noisy energy model interpretation of the results. As the final stage of the noisy energy model is a linear sum performed over the local filtered, transduced, and template-weighted responses to the stim- ulus image, this suggests that either i) a linear summing mechanism exists in the visual system that combines the outputs from simple cells over at least 33 cycles, or ii) several such mech- anisms of smaller sizes (e.g. complex cells) exist, whose outputs are then also combined by a linear sum over at least 33 cycles (a series of two linear summation operations being indistin- guishable from a single linear sum).
The findings in Chapter 7 show that area summation occurs according to a single (fourth-root) summation rule in the fovea, parafovea, and periphery. As above, the noisy energy model is favoured over the probability summation model based on the results from other studies (e.g. Meese & Summers, 2012). This finding suggests that the most of the differences in the results from summation experiments conducted at different locations in the visual field would be ac- counted for by the visual field inhomogeneity in contrast sensitivity. After the initial attenua- tion and filtering stages, the responses from the detectors appear to have been combined by a common set of processes at the three different locations tested in this study. One observer did show some additional summation in the parafovea which may have been caused by elongation of the filter elements (simple cell receptive field analogues) with eccentricity, and another ob- server showed some unusually high thresholds for very small stimuli in the periphery, however in general the model provided an excellent account of the data with a single fitted parameter (global sensitivity) for each observer.
10.1.3 Summation of threshold contrast over area is normal in amblyopia
Spatial summation in amblyopia had previously been explained by a probability summation model (Hess & Howell, 1978; Hess & Campbell, 1980), however the rejection of this model as an account of summation in the normal visual system (in favour of a noisy energy model) prompted me to investigate whether summation in amblyopia also needs to be reevaluated. The “Battenberg” study on area summation in amblyopia presented in Chapter 8 shows that spatial summation is normal in amblyopia in at least some cases and is best explained by a model which features linear filtering followed by a linear sum of the squared filter outputs. This is consistent with the noisy energy model if it cannot match a template to the contrast modulation in the Battenberg stimulus and instead pools contrast over the stimulus extent (as was reported in the original study conducted in normal observers by Meese, 2010).
It is suspected that the results for some of the observers in this study were confounded by artefacts introduced by the checkerboard modulation of the Battenberg stimuli, leading to anomalous short-range summation results for the normal observer at medium spatial frequen- cies (4 - 8 c/deg) and for some of the amblyope observers. This effect could only reduce the amount of measured summation however, and so could not account for the cases where greater summation was found than that predicted by the probability summation model (as is the case for half of the amblyopes tested in Chapter 8). From this finding it appears that amblyopes show the same spatial summation behaviour as normal observers. This suggests that the higher- level mechanisms that sum the responses from detectors positioned across the visual field are intact in amblyopia.
10.1.4 The summation of orientation signals is a noisy two-stage process
By extending the Battenberg stimulus paradigm to the orientation domain (Chapter 9), I have found that the spatial configuration of the signal areas in a stimulus has a significant effect on its coherence threshold. This is surprising because previous studies have indicated that the ar- rangement of the stimulus elements does not affect performance (Dakin, 2001). For the stim- uli that contained a checkerboard pattern of potential signal and noise regions (“noise check” condition), thresholds initially doubled compared to the “full” stimulus for the smallest check sizes (consistent with the observer combining information from all elements in the stimulus) before decreasing to a factor of approximately√2above the full stimulus threshold for the medium check sizes (consistent with the observer only monitoring potential signal elements). The data from the noise check experiment were well-described by a two-stage “hybrid model” that first performed a mandatory local integration of elements within a fixed radius before then combining the outputs of that first stage ideally. The only data points not fit by this model were those from the largest check size, where performance diverged dependent on whether most of the signal was presented to the fovea or in the periphery. Limiting the observer to only use information from the elements presented within a square 19 by 19 degree (76 by 76 stim- ulus carrier cycle) aperture in the centre of the display provided a much better fit to the data. This is similar to the result from a previous experiment on the summation of threshold con- trast over large areas that found a maximum integration region (Baker & Meese, 2011). Note that the aperture size proposed here is much larger than the region tested in the area summa- tion experiment presented in Chapter 6, which found no limit on the summation of contrast over 33 stimulus carrier cycles.
Coherence thresholds for Battenberg stimuli were similar when the noise regions were re- placed by blank space (“signal only” condition). This was an unexpected finding, as based on the information from the stimulus alone a√2threshold elevation would be expected for all of the Battenberg conditions (as there are no noise elements in the non-signal regions to impair performance). This indicates that the noise introduced by the randomly oriented elements in those regions was not the limiting factor on performance. This result would be consistent with performance for the task being limited by internal noise that is proportional to the area being monitored, regardless of the element density in that area. This can be built into the hy- brid two-stage model by adding internal noise at each location in the input stage, however the mandatory summation in the first stage of the model means that the limiting internal noise could also be placed between the first and second stages. Further work is needed to explore the nature of the limiting internal noise, and to determine whether this model can be adjusted to account for the differences in performance for the noise check and signal only conditions when signal is presented to the periphery.