What should a quantitative model of masking look like and why would we want it?

Quantitative models of backward masking appeared almost as soon as computing technology was available to simulate them; and continued interest in masking has lead to the development of new models. Despite this long history, the impact of the models on the field has been limited because they have fundamental shortcomings. This paper discusses these shortcomings and outlines what future quantitative models should look like. It also discusses several issues about modeling and how a model could be used by researchers to better explore masking and other aspects of cognition.

Backward masking refers to reduced visibility of a target stimulus when it is followed by a mask stimulus. The conditions under which masking occurs, and some special properties and uses of backward masking, are well summarized in other papers in this issue (Breitmeyer this volume, Enns, & Oriet, this volume).
This paper looks at the status of quantitative models, considers some issues and limitations about such models, and then explores how to proceed in a way that will improve the study and use of backward masking.
Studies of masking often vary the timing between the target and mask stimulus. A measure of target visibility plotted against the stimulus onset asynchrony (SOA) between the target and mask is called a masking function. Empirical work typically finds two types of masking functions, referred to as Type A and Type B.
A Type A masking function is shown in Figure 1a. The visibility of the target is minimized for common onset of the target and mask (SOA = 0). As the SOA increases, the target becomes more visible. A Type B masking function is shown in Figure 1b. The target is easily visible for common onset of the target and mask stimuli, but becomes less visible as the SOA increases. After reaching a minimum of visibility (maximum of masking) at some intermediate SOA, target visibility increases.
Whether Type A or Type B masking is produced depends on the target, mask, experimental task, and conditions of the experiment, as is discussed in other papers in this issue (Breitmeyer this volume, Bridgeman this volume, Herzog, this volume).
Scholarly papers on backward masking often describe it as mysterious, paradoxical, or surprising.
These claims about backward masking are of two types. First, it is surprising to some researchers that a trailing mask can affect the visibility of the leading target. Indeed, the phenomenological appearance of the target-mask sequence is sometimes that only the mask is presented. This result is surprising for some views of neural processing that supposes information proceeds in a feed forward manner. In some such views, the earlier target information would always be at a neural location where the mask information was not. In such a view, masking requires the mask information to lead ahead in space (or backward in time) to interfere with the target percept.  http://www.ac-psych.org (1993), Francis (1997), Herzog et al. (2003, and Bugmann and Taylor (2005). Such models demonstrate that many properties of backward masking are a natural part of visual processing.

AbstrAct
Why are there so many different models of backward masking? Considering this question reveals some important issues about modelling and backward masking. The first answer is that there are so many models of masking because there is no general theory of visual perception that might place constraints on the structure and properties of models. Without a general theory, it is fairly easy to introduce a new model and argue against other models.
Second, some aspects of masking, such as the existence of Type B masking (Breitmeyer & Öğmen, 2000) or common onset masking (Di Lollo et al., 2000) have been described as difficult to explain.
Modellers are drawn to challenges and so explore whether their model can account for the empirical results. Success is often reported, but it is often less because of the details of the model and more because many of the models explain aspects of masking with similar basic principles. For example, Francis and Cho (2005) show how a small system with four equations can produce a Type B masking function. Bugmann and Taylor (2005) used a system with 341 equations to also produce a Type B masking function. There are many important differences between the models and there are differences in the quantitative values of their masking functions.
Nevertheless, both models produce a Type B masking function for essentially the same reasons. There are many different models of masking, in part, because researchers end up repeating the same basic principles in a variety of models.
Such repetition is worthwhile. The model proposed by Francis and Cho (2005) demonstrates one of the simplest systems that can produce a Type B masking function. In contrast, the model of Bugmann and Taylor (2005) demonstrates that the same basic principle robustly applies even when it is embedded in a much more complicated system.
There is value to both kinds of implementations of the principle.
On the other hand, this kind of repetition is not often recognized as repetition. The models of Weisstein (1972) and Bridgeman (1971) have often been considered as very different models, but Francis (2000) showed that both models operate with a common ba- and this effect is a property of many different models of backward masking (Francis & Herzog, 2004). Thus, all of these models predict that if the target and task are held fixed, then variations in the mask (intensity, duration, or shape) could vary the shape of the masking function from Type A to Type B, but only such that the masking function curves do not intersect.
We have now identified several circumstances where this prediction does not hold (Francis & Cho, 2007;Francis & Herzog, 2004). Figure 2b combines data from two experiments in Francis and Cho (2007), where the target and task were always the same (identify the orientation of a half disk target among three full disk distracters), but the spatial shape of the mask varied.
The main finding is that variations in the spatial shape of the mask lead to Type A or Type B masking functions, but that these masking function shapes were not related to the overall strength of masking.
This data presents a significant problem for all of the current models. There is no variation of parameters that will allow the models to match this experimental finding. There needs to be entirely new kinds of models with properties quite different from the current models.
One of the key problems with the current models is that they do not have a sufficiently rich representation of the spatial properties of the target and mask stimuli (Herzog, this volume). For many of the models, the representation of the mask is simply a numerical value that changes over time. This is explicitly the case for the models by Weisstein (1972), Anbar and Anbar (1982), Bachmann (1994), Di Lollo et al. (2000), Francis (2003a), and Francis and Cho (2005). Even for models that include a spatial representation of stimuli, the calculations of masking often reduce the mask's effect on the target to a single numerical value. Francis (2000) showed that this was the case for the recurrent lateral inhibition model of Bridgeman (1971Bridgeman ( , 1978, and a similar conclusion appears to be true for the models of Francis (1997), Purushothaman et al. (2000), Herzog et al. (2003) and Bugmann and Taylor (2005).
The significance of this property is that a variation in the spatial shape of the mask, as in Figure 2b can only lead to a differing magnitude (or duration) of the corresponding mask's effect in the model. Thus, advancement of the models requires a substantial elaboration of the spatial aspects of the models. Interestingly, Weisstein (1972)  with current models sometimes take days or weeks (Francis, 1997;Purushothaman et al., 2000) to car- (a) Simulation results from the model of Francis and Cho (in press) show that the shape of the masking function is related to masking strength. Type A functions occur for strong masks and Type B functions for weaker masks, and the curves never cross. (b) An experimental study in Francis and Cho (in press) varied the spatial shape of the mask. The shape of the masking function is not related to masking strength and the curves cross.
http://www.ac-psych.org ry out key simulations. Models that include a richer spatial representation (e.g., Cao & Grossberg, 2005;Grossberg, 1997;Itti, Koch, & Niebur, 1998) will take many times longer on similar computer equipment. It is not clear whether modern computing power is sufficient to build the kind of model that appears to be needed. We return to this issue in a later section.

DevelopInG A neW moDel oF bAckWArD mAskInG
Since a new kind of model appears to be needed, this is a good opportunity to consider the desired proper- Significantly, the models have almost never been used to explain other aspects of cognition, perception, or consciousness. This is notable because masking techniques are often used to experimentally investigate these topics. Apparently, the properties and features of current models are not sufficient to contribute to the discussion of those topics. This lack of model use is not a healthy arrangement for the field. Ideally, nonmodellers would use the models to explore aspects of cognition and introduce new ideas that would drive model development.
So what would a new model of masking ideally look like? Given the problems with the current models described above, the new model must combine models of spatial vision and models of temporal vision. Some of these model parts may already exists, but putting them together may not be trivial. In particular, models of spatial vision simply may not work properly when temporal dynamics are considered.
There is a tendency for scientists to want simple models, but a system that mathematically deals with both spatial and temporal aspects of visual perception is unlikely to be simple. There may be simple parts of the model and there may be principles that guide the main computations of the model, but the most interesting parts of perception will involve interactions between the simple model parts. When such interactions involve feedback and non-linear relationships, the resulting behaviour is unlikely to be simple.
Indeed, past research indicates that there may be no way to predict the behaviour of such a system except by direct simulation. In this respect, the model will have to be studied in a way that is similar to psychophysical studies of human perception. Researchers will have to identify simulation experiments that test the behaviour of the system. This is a different view of modelling than most psychologists imagine. For most psychologists the definition of the model is essentially Here, we briefly describe the experiments because it helps to demonstrate how some masking effects exist across a variety of contexts and tasks. Breitmeyer (1978) had observers vary the luminance of a comparison stimulus to match the perceived brightness of a target disk that was masked by a surrounding annulus.
The experiment varied the SOA between target and mask and varied the duration of the mask. Figure 3 plots target visibility for varying mask durations averaged across the various SOAs. In this experiment there is a sharp drop in target visibility as mask duration increases. Di Lollo, Bischof, and Dixon (1993) had observers report the orientation of a gap that was placed on one side of a target outline square. The mask was an outline square with a gap on each side. They kept the SOA at zero, but varied the mask duration. Again, Figure 3 shows that there is a drop in percentage correct as mask duration increased. Francis, Rothmayer, and Hermens (2004) had observers report the orientation of a target half disk among three distracting full disks. The mask was a set of annuli that surrounded the target and distracter elements. SOA, target duration, and mask duration were all varied. Figure 3 shows the effect of mask duration averaged across all SOAs and two target durations. Although the slope is more shallow than for the other data sets, again percentage correct decreases as mask duration increases. Although it also used a variety of mask durations, the study by  and Bruner (1974) found that masking was stronger when observers were dark adapted. The data in Figure   4 are averaged across several SOAs. In contrast, Bischof and Di Lollo (1995) found that masking was absent when observers were dark adapted, but strong when observers were light adapted. The data in Figure   4 are from the faintest stimuli in each condition, averaged across many SOAs. Both studies appear to be conducted properly, so the conclusion is that the effect of dark adaptation is sensitive to many details of the task, stimuli, observers, and other experimental conditions. As a result, a model's explanation of the effect of dark adaptation needs to be similarly sensitive. In such a model, one would expect that changes in model parameters would lead to rather different model behaviours with regard to light adaptation.

Macknik & Livingstone (1998) is not included in this
In general, robust experimental findings can be used to identify the main structure and properties of a model. Such findings are not so effective at identifying the particular parameters that define the model's behaviour. In contrast, sensitive experimental findings can be used to precisely parameterize a model, but tend to not be useful for characterizing the general structure and function of a model.

moDel structure AnD computAtIon
When constructing a model, one has to consider the units and mechanisms that make up the model components. Because backward masking is a tool that is used both by psychologists to explore aspects of human behaviour and by neuroscientists to explore properties of the brain, the ideal model will be defined

Sensitive effects of dark and light adaptation on masking.
In one study, masking is stronger with dark adaptation than with light adaptation. In the other study just the opposite was found. The small quantitative differences in the Purcell et al. (1974) data relative to that of Bischof & Di Lollo (1995) reflects differences in the experimental task rather than the strength of adaptation. Both findings were highly significant from a statistical point of view.
http://www.ac-psych.org unclear whether current computing power is sufficient to provide the spatial and temporal resolution that appears to be needed to emulate a backward masking experiment.
Some quick calculations explain why there may be a problem finding sufficient computing power. The temporal model of Weisstein (1972)

Feed forward and feedback models
There has been substantial discussion, both within the field of masking and elsewhere, about the importance of feedback within models. Some researchers have taken the stand that certain experimental findings rule out feed forward models (Di Lollo, Enns, & Rensink, 2000. This topic deserves some additional discussion because, contrary to common belief, Consider the two different anatomical systems in (1) Here, the capital letters indicate parameters and the terms -Ax(t) and -Cy(t) indicate passive decay. The activity from the higher level, y(t), feeds back in to the equation for activity at the lower level, x(t), through the term By(t). In this case, the mathematical layout of terms appears to match the anatomical structure.
For the feed forward system on the right there might be only one equation.
The term -Fx(t) again indicates passive decay and there is no feedback from higher areas.
Now let us add one further condition to the system.
This has a significant effect on how we can describe the rest of the feedback system. If we replace y(t) in equation (1) with the right hand side of equation (4), we get Now define the parameter If we combine the terms in equation (5) that multiply x(t), the equation becomes This is identical to equation (3)! In this case the behaviour of x(t) is mathematically identical in the feedback system and in the feed forward system. Thus, even if the anatomy of the visual system provides clear evidence of re-entrant or feedback signals, this does not guarantee that the system behaves any differently than a feed forward system. It is noteworthy too that, at first glance, equation 7 would seem like a very poor description of the feedback system in Figure   5. In fact, though, it fully captures the behaviour of the lower unit and the behaviour of the upper unit is just a multiple of the lower unit.
Of course, such isomorphism may not always be possible or practical, but one never knows for sure what the feedback signals actually do, and there are many other analogous situations that blur the distinction between feedback and feed forward systems. As Reeves (this volume) observes, mathematicians have noted that any feedback system can be approximated by a suitably complex feed forward system.
None of this is to say that re-entry, feedback, and non-linearities should be not investigated. To the contrary, their presence in the anatomy of the nervous system suggests that they need to be characterized and studied carefully. The problem with many of the current discussions of feedback in masking is that they fail to specify the exact nature of re-entry feedback (Di Lollo et al., 2000;Enns, 2004   http://www.ac-psych.org and expense. In an attempt to answer affirmatively we can consider some possible uses of such a model.

Create an ideal mask for a given target and task.
Backward masking is commonly used to study other aspects of cognition. At the moment the properties of the mask are found by experimental trial and error. Such work is frustratingly slow and inefficient. A good model might be able to speed up the process by identifying mask properties that would be able to mask the target properties most important to the experimenter.  (Braff & Saccuzzo, 1981;Green, Nuechterlein, & Mintz, 1994). A model may be able to help identify what mechanisms are different, which could lead to early detection and better understanding of how the disease operates.
Since backward masking is used as a tool to investigate many other neurophysiological and mental phenomenons, a good model would surely be useful in many other situations.

conclusIons
Backward masking is an important topic that is used throughout psychology both to investigate visual perception and as a tool to study other aspects of cognition. Unfortunately, there is currently no theory of how backward masking operates that can guide researchers on how to use masking. In particular, all of the quantitative models of backward masking have recently been shown to be invalid because they lack a sufficient representation of visual space.
These findings suggest that new types of models of backward masking are needed. It seems that a new model needs to deal with both space and time so that it can work with visual stimuli that are similar to those used in psychophysical experiments. The model needs to be flexible enough to operate in a variety of experimental situations and be connected to many different perceptual tasks. The model needs to be described in neurophysiological terms. The model needs to be structured in such a way that it can be used by nonmodelers. Finally, the model needs to be able to make particular predictions of neurophysiological and mental behaviour so that it can be tested and developed in a meaningful way.