Binocular information acquisition

and visual memory

Thomas A. Busey

Indiana University

Geoffrey R. Loftus

University of Washington

Please send correspondence to: Thomas A. Busey

Department of Psychology

Indiana University

Bloomington, IN 47405

email: busey@indiana.edu

We investigate mechanisms underlying binocular combination of visual information within the context of a visual information acquisition theory proposed by G. Loftus, T. Busey, and their colleagues (e.g, as described by Busey & Loftus, 1994). A central assumption of the theory is that of a sensory threshold. By the theory, this threshold engenders an information loss in that information processing subsequent to the threshold is assumed to occur only when the magnitude of a sensory representation triggered by the stimulus exceeds the threshold. In this article, we describe research designed to investigate whether the presumed threshold is situated prior to or subsequent to the point at which the information from the two eyes combines. We also consider more general mechanisms of binocular combination and conclude that one must take the location of the sensory threshold into account when trying to interpret the nature of the binocular combination mechanisms. Evidence from three experiments provides strong conclusions both about the location of the sensory thresholds and about the mechanisms of binocular combination and binocular information acquisition. We conclude that a linear summation mechanism, an independent sampling information acquisition model and both pre- and post-combinatorial sources of information loss are required to account for the data. The model quantitatively accounts for the finding that binocular superiority decreases relative to monocular performance as the presentations to the two eyes overlap less in time.

Visual information acquisition begins at the perceptual level with photons arriving at each of an observer's two eyes. The information contained in the pattern of arriving photons may represent something as simple as a monochromatic patch of light or as complicated as a naturalistic scene. Regardless of the stimulus, however, the geometry of binocular vision and the separation between the two eyes generally dictate that slightly different versions of this information arrive at each eye.

Despite initially acquiring two separate representations of the visual scene, it is only under rare circumstances that observers report seeing more than one view of the world. This plus other evidence from the stereo depth perception literature (e.g. Julesz, 1971), and from the binocular combination literature (e.g. Blake & Fox, 1973; Blake, Sloane, & Fox, 1981), implies that the visual system must combine the outputs of the two eyes into a single unitary sensory representation and then based on this representation, proceed with further information processing mechanisms to acquire task-relevant information.

In this article, we report research to investigate binocular coding within the context of a visual information acquisition theory described by Loftus, Busey and their colleagues (Loftus, Busey & Senders, 1993; Loftus & Ruthruff, 1994; Busey & Loftus, 1994). The theory consists of a sensory front end followed by information acquisition components. The sensory front end is responsible for generating the sensory representation indicated above, and subsequent information acquisition mechanisms are responsible for acquiring task-relevant information. This theory successfully predicts performance in a number of different information processing paradigms, including digit recall (Loftus, Busey & Senders, 1993; Loftus, Duncan & Gehrig, 1992; Loftus & Ruthruff, 1994; Busey & Loftus, 1994); picture recognition (Loftus & McLean, in preparation); and synchrony judgment and partial report (Loftus & Irwin, under revision). Despite these successes, however, the current conceptualization of the theory does not specify how information from the two eyes combines to create a unitary sensory stimulus representation.

We have two goals in this article. Our first goal is to account for the mechanisms that combine information from the two eyes in information processing tasks. Our second goal is to examine the sources of information loss that result from a sensory non-linearity that proved essential to account for past data. We will demonstrate that these two goals cannot be addressed separately, but are intimately related. Below we propose an extension of the Loftus and Busey theory, and focus on a crucial component of it termed a sensory threshold, which entails a specific source of information loss in the information processing pathway. We address two central questions: First, is the sensory threshold prior or subsequent to this point at which information from the two eyes combines? Second, what mechanisms are responsible for combining the two sources of information into a single unitary representation?

The major empirical question guiding binocular summation experiments has been: How much better are two eyes than one eye? The answer has not been simple. As detailed in excellent reviews by Blake and Fox (Blake & Fox, 1973; Blake, Slone & Fox, 1981), binocular performance may exceed monocular performance by various magnitudes. Binocular performance above monocular performance can be expected on statistical grounds, since a binocular stimulus provides two opportunities to detect a near-threshold stimulus. Blake and Fox (1973) formalized this into the Probability Summation prediction, which is implied by the assumption that the two opportunities were made independently. This model assumes an or rule for combination of the information from both eyes, such that the stimulus is detected if either the left or the right eye detect it. Other models of binocular combination can be produced through consideration of the neural mechanisms that combine the signals from two eyes, which include super-additive summation (interaction), additive (linear) summation, and sub-additive summation (inhibition). Interaction implies that the output of a system combining the inputs from the two eyes exceeds the sum of the inputs. Linear summation implies that the output is the sum of the inputs, and sub-additive summation implies that the output is less than the sum of the inputs. The probability summation prediction has become somewhat of a measuring stick from which conclusions about the combinatorial mechanisms can be derived. Blake and Fox (1973) conclude that if binocular performance exceeds a probability summation prediction based on monocular performance, this implies genuine neural interaction (that is, an additive or super-additive summation mechanism). They argue that the probability summation model accordingly provides a baseline to detect combinatorial mechanisms that imply more interaction than the or rule described above.

Although the focus of the binocular combination experiments has been on the nature of the combination mechanisms (i.e. neural interaction, linear summation or inhibition), we will demonstrate that such questions cannot be addressed without consideration of both pre- and post-combinatorial sources of information loss. In particular, we will argue that a post-combinatorial threshold mechanism must play a major role when binocular combination data is used to infer the mechanisms of combination.

The question of the combinatorial mechanisms subserving binocular combination is best addressed by varying the onsets of the presentations to each eye. For example, consider the probability summation model, that assumes that each eye provides an independent chance to detect a near-threshold stimulus. If this model holds, performance will remain constant if the two eyes are stimulated simultaneously, or if first one eye and then the other eye is stimulated. Conversely, improved binocular performance with simultaneous presentation compared to successive presentation would constitute evidence against the probability summation model. Thus the temporal overlap of the two presentation in a binocular condition provides a strong test of binocular summation models, and below we review relevant temporal overlap experiments.

Three studies of binocular combination demonstrate the utility varying the temporal onsets of a binocular stimulus to each eye, and all three sets of authors have argued for neural interactions (predictions above probability summation) between the two eyes to produce binocular performance. Westendorf, Blake & Fox (1972) presented contrast increments to both eyes in a detection task. The stimulus onset asynchrony (SOA) between the two presentations was either 0 ms or 100 ms. They found binocular performance above a probability summation prediction for the simultaneous presentations, but binocular performance consistent with probability summation for the 100 ms SOA. They concluded that binocular performance cannot be predicted solely from probabilistic considerations, but that the superiority at 0 ms SOA resulted from neural interaction between the two eyes.

An earlier study by Matin (1962) demonstrated the utility of varying the temporal overlap of the two presentations. He varied the interval separating the two presentations over a wide range, and found that SOAs less than 100 ms produced binocular performance that is superior to the probability summation prediction. Matin concluded that the two eyes were not independent detectors, but detection instead resulted from neural summation between the two eyes.

Although much of the work on binocular summation has relied on detection tasks, Eriksen and his colleagues (Eriksen, Greenspon, Lappin & Carlson, 1966; Eriksen & Greenspon, 1968) examined letter identification in a three-alternative forced-choice procedure. As with previous findings, they found that for SOA’s shorter than 50 ms, binocular performance exceeds the probability summation prediction.

In short, the available evidence disconfirms a probability summation mechanism, and has led investigators to posit neural interaction as the combination mechanism mediating binocular combination. However, these data also clearly indicate that no single combinatorial mechanism can account for binocular summation at all SOA’s, since binocular performance changes with SOA. Thus a complete description must account for the amount of temporal overlap that is produced by a non-zero SOA binocular stimulus. The present article represents such an approach, although it is motivated, in part, by extant binocular summation models. Two of the most influential are the binocular energy detector model of Legge (1984a, 1984b) and the interocular suppression model of Cogan et. al. (Cogan, 1987; Cogan, Clarke, Chan & Rossi, 1991).

The binocular energy detector model extends the signal detection model of Green and Swets (1966, 1974) to binocular summation. The model consists of two monocular channels that are subjected to input noise before undergoing a squaring function. The results are linearly summed and the final binocular output results from a compressive nonlinearity and the addition of central noise. This model predicts a quadratic summation relationship between binocular and monocular thresholds, which results from the fact that when two noisy signals are added, the standard deviation increases by the square root of the number of signals. While this model can account for a diverse range of low-contrast phenomenon, all of the binocular stimuli used to test the model consist of simultaneous presentations to both eyes. The model does not include a temporal representation of the physical stimulus, and thus it cannot be used to account for non-zero SOA binocular presentations. As described above, these stimuli provide strong tests of binocular summation models since they can be used to determine how the range of binocular summation varies with the onsets of the two presentations of a binocular stimulus.

Alexander Cogan (Cogan, 1987; Cogan et. al. 1991) presents a model that can qualitatively account for the effects of varying the SOA between the two presentations of a binocular stimulus. We describe this model and fit it to our data in the General Discussion section; briefly it is as follows. The combination mechanism is relatively complex: information registering in either eye both provides excitatory input to the monocular channel and inhibits the other monocular channel. In a separate mechanism, the information from the two eyes combines into a binocular fused channel, and the final binocular output is the sum of the combined monocular channels and the binocular fused channel. This model accounts for increments and decrements in binocular and monocular presentations and will also qualitatively account for the effects of non-zero SOA’s in binocular presentations. However, Cogan does not provide quantitative fits to variable-SOA data. The importance of quantitative fits will become apparent in the discussion section of the current article, but to anticipate, two models that share similar theoretical structures and thus provide qualitatively identical predictions do not give the same quantitative predictions. Indeed, we will test and disconfirm Cogan's model based on poor quantitative fits.

One conclusion is clear from the existing binocular summation data and theories: A complete account of binocular summation must include both descriptions of how information from two eyes combines, as well as how this relationship changes as the SOA between the two presentations is varied. This, along with a description of the sources of information loss that contribute to performance, is the major theoretical contribution of the current work.

The remainder of this article is organized as follows. First, we explicitly describe the task that we are trying to model. Second, we briefly outline the Busey/Loftus theory, and propose two (not mutually exclusive) candidate extensions to it that may account for performance in tasks designed expressly to investigate binocular combination. Third, we describe data from three experiments designed to evaluate the two extensions and, more generally, to reveal the nature of the information combination mechanisms. Fourth, we conclude that neither extension alone will completely account for all of the observed characteristics of the data, although a combination of the two does provide a good account of the data. Fifth, we make specific conclusions about the nature of the information combination mechanisms and, based on these conclusions, describe theoretical implications for researchers studying the nature of the combinatorial mechanisms. Finally, we apply Cogan’s model to our data to examine how the choice of combination mechanisms affects our conclusions. We will argue that the question of the nature of the combinatorial mechanism is intimately tied to the question of the location of the sensory threshold and that above-probability summation binocular performance need not imply neural interactions between the two monocular channels.

The task that we will attempt to model is complex enough to provide generalizability to everyday tasks, but simple enough to be plausibly described with a relatively simple and concise theory. In this task, four digits are presented to an observer whose job is to report as many of them as possible, in their correct order, guessing if necessary. The basic performance measure, p, is the proportion of correctly-reported digits, in their correct locations, adjusted for the guessing probability of 0.10.

Busey & Loftus (1994) have described a theory, which they termed the LST (Linear Systems Theory) model, that has successfully accounted for performance in this digit-recall task. In this section we summarize the theory, while in the next section, we develop two extensions of it designed to account for the combinatorial mechanisms involved in binocularly viewed stimuli. This theory conjoins a linear-filter front end, common in the vision literature, with an independent information sampling model that operates on the output of the linear-filter front end to acquire task-relevant information. Figure 1 provides three functions that will be relevant throughout our discussion of the theory. Note also that compete equations for all models are found in Appendix A, while the units and dependencies of all model parameters and functions are found in a Glossary in Appendix B.

The initial response in the visual system to a briefly-presented stimulus can be modeled as the output of a band-pass or low-pass linear filter (e.g. Sperling and Melchner, 1968; Watson, 1986). Rapid temporal changes in the stimulus are temporally blurred by a visual system that cannot keep up with the stimulus; slower temporal changes are better represented. The exact nature of the response can be derived by specifying an impulse-response function for the linear filter that describes the severity of the temporal blurring, or, thought of in another way the amount of response that persists following the termination of the physical stimulus. Under low-luminance conditions such as the ones used in our experiments, the filter is usually assumed to be low-pass, meaning that the visual system reproduces progressively higher temporal frequencies progressively less faithfully.

The shape of the impulse response function, g(t), dictates the type of temporal blurring. In keeping with previous sensory literature (e.g. Watson, 1979, 1986) we have chosen this function to be the difference of a two gamma functions, each with a different time constant,

where t represents the time-constant of the gamma function and provides an estimate of the temporal response properties of the mechanisms mediating a given task. The sensory response function component of the LST theory is based on work from Andrew Watson (Watson, 1986), George Sperling (Sperling and Sondhi, 1968) and others working in the temporal domain of perception. Figure 2, left panel, shows example impulse response functions.

The first term in Eq. 1 represents an excitatory component, while the second term represents an inhibitory component of the response, which tends to sharpen the response and allows it to respond to higher temporal frequencies. The parameter r represents the ratio of the time-constant of the inhibitory component of the response to the excitatory component of the response, and s represents the magnitude of the temporal inhibition component. The parameter n represents the number of exponentially-decaying stages in each gamma process, and for the current modeling we fix it at 9, although the fit of linear-filter models to data is relatively unaffected by different values of n (Burr & Morrone, submitted).

Identification data for letter stimuli such as those used in the present study are often modeled by setting s to 0 (Busey, 1994), which gives a monotonic impulse response function g(t) as shown as a solid curve in the left panel of Figure 2. An alternative way of representing the same information is by taking the Fourier transform of the impulse response function g(t), which results in a temporal contrast sensitivity function (TCSF). The TCSF plot show the sensitivity of a system to different temporal frequencies. The TCSF corresponding to the solid line in the left panel of Figure 2 is given by the solid line in the right panel. These curves represent a purely sustained response, and give a monotonically-decreasing TCSF, as shown in Figure 2, right panel.

Detection data for stimuli containing mainly low spatial frequencies, or stimuli presented on bright backgrounds, often are modeled by s > 0 (Watson, 1986). In this case the impulse response inhibits processing after an initial excitatory response, which results in an inhibitory lobe in the impulse response function g(t) (dashed line in Figure 2, left panel) and a characteristic TCSF with a decrease in sensitivity at low temporal frequencies (dashed curve in Figure 2, right panel).

The temporal characteristics of the physical stimulus are summarized by a function describing contrast over time since stimulus onset, called the stimulus input function. This represents the physical stimulus, not the visual systems response to this stimulus. For the stimuli discussed here, the digits simply appeared at some contrast, F, for some controlled stimulus duration, and then disappeared. Thus the stimulus input function f(t) can be described as:

where F represents stimulus contrast and t is time since stimulus onset.

We term the visual system's response to such a stimulus a sensory response function, denoted a(t), which is computed by convolving the stimulus input function with the impulse response function:

where "*" represents convolution. The sensory response function represents the output of the linear-filter stage of the theory, and describes the initial representation of a stimulus in the visual system.

Further information acquisition mechanisms act on this representation to acquire task-relevant information. However, prior to becoming available for information acquisition, the sensory response function is assumed to pass through a threshold device, such that only the portion of the sensory response function that exceeds a sensory threshold is available for information acquisition. Thus we define a new function aQ(t):

where Q is the magnitude of the sensory threshold. Information acquisition cannot begin until the sensory response, a(t), exceeds the sensory threshold.

As described below, once the information acquisition process begins it proceeds at a rate that is proportional to aQ(t). Before turning to this issue, however, some notes on the nature of the sensory non-linearity are in order. The behavior of the sensory threshold described in Eq. 4 can be mimicked by other non-linearities, and in particular by a power function of the form:

where b represents the magnitude of non-linear compression. This power function has been used by Watson (1979) in his probability summation in time model and by Legge (1984a, b) in a binocular summation model with b = 2. Raising the sensory response function a(t) to an exponent greater than 1 has similar properties to a hard threshold. It tends to produce more overall area when a sensory response function is tall and thin, verses short and wide (See Figure 2 in Watson 1979 for an example of this effect, where |g(ti)| represents the height of the sensory response function, a(t). As b increases, the psychometric function becomes steeper and approximates a hard threshold). The hard-threshold model has the same behavior, and thus the two models can mimic each other. We have fit models with both hard-threshold and power-function non-linearities, and in general we find the two formulations equivalent. However, the threshold model gives slightly better fits to the data and is also slightly easier to demonstrate graphically. In addition, there is neurological evidence in favor of this formulation from Ohzawa & Freeman (1986), as discussed in the General Discussion. For these reasons, as well as for consistency with previous formulations (Busey & Loftus, 1994), we present the model predictions in terms of the threshold model. However, we provide model parameters and RMSE’s for the Power Function model in Table 3.

Once the sensory response function exceeds the sensory threshold, information acquisition can begin. At time t following stimulus onset information acquisition proceeds at an instantaneous rate, termed the information acquisition rate, denoted r(t), which is proportional to the product of (1) the magnitude of above-threshold sensory response and (2) some decreasing function of already-acquired information. The particular form of the information acquisition formulation is similar to independent sampling models proposed by Townsend (1981), Shibuya & Bundesen (1988), Rumelhart (1970), and Massaro (1970); viz., information is presumed to be acquired at random and with replacement. The independent-sampling assumption implies that the rate of acquiring information is constant, and governed by a scaling parameter 1/cs. However, because information is sampled from the stimulus with replacement, the rate of acquiring new information is proportional to the amount of not-yet-acquired information.

More specifically, let I(t) be the proportion of acquired information at time t. The information acquisition rate, r(t), is by definition the derivative of I(t) with respect to time, and is given by:

Eq. 5 states that the rate of acquiring new stimulus information is proportional to the product of the above-threshold sensory response and the remaining stimulus information, with a constant of proportionality 1/cs. The units of cs and r(t) are 1/ms. Appendix B contains a Glossary of all model parameters and their units.

It is simple to show (e.g., Busey & Loftus 1994) that with this rate function, the function relating total proportion of acquired information, denoted I(°), to the above-threshold area under aQ(t), denoted AQ(°), is given by,

The units of AQ(°) are milliseconds, and I(°) has units of proportion of total information.

One additional linking hypothesis is required to make quantitative predictions, which is that p, the proportion of correctly-recalled digits, equals the total proportion of acquired information I(°). Given this and Eq. 6, our fundamental prediction emerges:

An important consequence of Equation 7 is that performance, p, is monotonically related to the above-threshold area under the sensory-response function. Thus if some condition i results in more above-threshold area, AQ(°) than another condition j, the theory predicts that performance in condition i will exceed that in condition j. In addition, any two conditions that produce the same above-threshold area AQ(°), are predicted to have the same performance. Although Equation 7 provides a stronger description of the area/performance relation than monotonicity, for purposes of exposition it is convenient to think in terms of the consequences of the weaker monotonicity relation, which is that more above-threshold-area implies greater performance.

At the risk of redundancy, we emphasize that these relations are a consequence of Eqs. 1, 3, 4, 5 and 6 and are not assumptions of the theory. It is also worthwhile to note that the prediction of Eq. 7 holds for any stimulus wave form, not just the rectangular wave form given in Eq. 2, and used in the present experiments.

Testing Eq. 7 requires an experimental design that varies AQ(•), the above-threshold area under the sensory response function. We typically vary the stimulus duration within several different types of conditions, including those involving binocular, monocular and dichoptic stimuli, in order to produce empirical data that provides a strong test of the theory. When plotted against stimulus duration, digit-recall performance in a given condition is called a performance curve.

The above model formulations do not explicitly assume the presence of noise, although, as discussed below, the sensory threshold may be thought of as representing the amount of signal lost to noise. The binocular energy model of Legge (1984a, 1984b) assumes both input and central sources of noise, although formulations by Cogan (1987) do no explicitly assume noise. Noise could be explicitly incorporated into the model without changing the formulations by assuming that the height of the sensory response function gives the probability that the signal exceeds the noise at that particular instance. This is the interpretation adopted by Watson’s probability summation in time (Watson, 1979), and this makes the power-function formulation of the model similar to Legge’s (1984a, b) model.

Complete equations for all model formulations are provided in Appendix A.

The sensory threshold described in the previous section is assumed to exist prior to the information acquisition components of the information processing pathway. As a result, we consider it a sensory component. Because no further information processing takes place unless the sensory response exceeds threshold, the threshold can be considered a source of information loss. We now ask: Does this information loss occur before or after the information from each eye is combined into a central unitary representation?

Below we describe two models that address this question. Each model extends the previously-described theory by assuming that information enters each eye and engenders in each eye an initial peripheral sensory response function. These two peripheral sensory response functions then combine centrally to provide a single central sensory response function. Information is acquired based on this central sensory response function and further processing, or a task-relevant response, is made based on the amount of acquired information. We describe these two models within the context of Experiment 1, which is designed to evaluate them.

In Experiment 1 we used monocular, binocular, and dichoptic stimuli. A monocular stimulus is presented to one eye only. A binocular stimulus is presented to both eyes simultaneously. A dichoptic stimulus consists of two temporal stimulus halves, presented successively, with zero ISI: the stimulus is presented first to one eye and then immediately to the other eye.

As we shall see, comparison of performance curves for these three presentation types will allow us to isolate the location of the presumed sensory threshold. To do so, we must extend our theory such that it takes these three presentation types into account. We first assume that information from each eye is carried by a monocular channel, and that each monocular channel generates a peripheral sensory response curve. The information from the monocular channels combines to produce a central sensory response curve, upon which memory performance is based. As an extension of linearity, we further assume that the two peripheral sensory response curves sum to produce the central sensory response curve.

Support for our assumption of a linear combination mechanism comes from single-cell recording done with simple cells in cat by Ohzawa and Freeman (1986). Using sine-wave gratings tuned to the cell’s optimal orientation, they found that the light-evoked neural responses from each eye are summed linearly to produce the output of a binocular cell. Of some relevance to our theory is that they also found a minority of cells that deviate from this linearity prediction, which they attribute to a threshold mechanism that occurs after the linear binocular summation. We will return to this in our Discussion section.

The linearity assumption also allows quantitative evaluation of the sources of information loss, both before and after the site of binocular combination. Such a summation mechanism was adopted by Legge (1984a, b), although his additional assumptions about noise may provide for different predictions of binocular performance. A more complex combination mechanism developed by Cogan (1987) is discussed in a subsequent section.

Given the assumptions described above, we are able to ask: is there a sensory threshold for the peripheral sensory response functions, for the central sensory response curve, or for both?

Figure 3 illustrates these two possibilities. To illustrate them, we show the responses generated by an 80-ms dichoptic presentation (which, of course, involves a 40-ms presentation to one eye followed by a 40-ms presentation to the other eye) are shown. The top half of Figure 3 shows the peripheral sensory threshold model, by which a peripheral threshold, QP, is assumed on the sensory response functions generated by each monocular channel. Only the above-peripheral-threshold area from the peripheral sensory response curves combines to form the central sensory response curve. All of the central sensory response curve area contributes to performance.

The bottom half of Figure 3 shows the central sensory threshold model, by which a central threshold, Qc, is assumed on the central sensory response curve. All of the peripheral-curve area contributes to the central sensory response curve, but only the above-central-threshold area contributes to performance.

Both thresholds are implemented through the Eq. 4 thresholding operation.

Evaluating these two candidate models entails the following logic. Consider an 80 ms monocular presentation and its corresponding sensory response curve, shown in the left panel of Figure 4. This monocular presentation implies the same above-central-threshold area (1.4 in this example) for both peripheral and central sensory threshold models. The location of a sensory threshold is irrelevant for this condition because information is presented to only one eye, and there is no information to combine from the other eye. This makes the monocular condition a good comparison stimulus because monocular performance is predicted to be the same by both models.

Now consider a dichoptic presentation in which the 80-ms duration is broken into two 40 ms presentations, the first to one eye, and the second to the other eye with a zero ISI. The peripheral and central sensory threshold models yield different predictions for this condition. The middle panel of Figure 4 shows the central sensory response curve prediction for the peripheral-threshold model. Only the above-peripheral-threshold area contributes to the central sensory response curve, but all of the central-sensory-response-curve area contributes to performance. Because the area reaching the central sensory response curve is smaller than the above-threshold area in the monocular condition, the peripheral-threshold model predicts dichoptic performance to be lower than monocular performance. This can be seen clearly in the middle panel of Figure 4 as a notch in the central sensory response curve of the dichoptic presentation. This loss of area results from two peripheral thresholds being assessed separately on two different peripheral sensory response curves, as compared with the single peripheral threshold assessed on a single peripheral sensory response curve in the monocular case.

As we will show next, a finding of poorer dichoptic relative to monocular performance is not only consistent with a peripheral-threshold model, but also disconfirms any model in which the sole threshold is central.

The right panel of Figure 4 shows the central sensory response curve for a dichoptic presentation predicted from the central sensory threshold model. The dashed curves are the peripheral sensory response curves before they combine to produce the central sensory response curve. Because the two peripheral sensory response curves combine before the central sensory threshold, the dichoptic central sensory response curve is identical to the monocular central sensory response curve, and they have the same above-central-threshold area. Thus the central-threshold model predicts monocular and dichoptic performance to be identical. It is for this reason that a finding of poorer dichoptic compared to monocular performance disconfirms a purely central-threshold model.

A second set of predictions involves comparison of the monocular and binocular conditions. In general the means by which information from two eyes is combined to produce binocular information–and, correspondingly, the relation between performance based on a monocular versus a binocular presentation–is the source of some debate (Legge, 1984b; Blake & Fox, 1973; Blake, Sloane & Fox, 1981). Two reasonable propositions have been put forth: first that the combination occurs via probability summation, and second that the combination occurs via quadratic summation. In our General Discussion we evaluate a third proposition that assumes neural interaction between the two ocular channels in a binocular presentation.

The supposition that the information combines via probability summation is easily incorporated by our theory. In what follows we develop a version of the theory that gives a prediction consistent with probability summation, and to do so we begin with probability summation and work backwards. We are assuming that the two opportunities to view a binocular stimulus are equally affected by a sensory non-linearity, and thus we are considering a version of the model that consists only of a peripheral threshold (although the same arguments below would hold assuming no thresholds).

Consider any two information sources (e.g., first and second presentations; left eye and right eye; etc.) and denote the proportions correct based on these two sources as p1 and p2. If pB denotes proportion correct given both information sources, then probability summation implies that,

or,

Substituting (Eq. 7) for each p and simplifying we find:

where AQ,L(°) and AQ,R(°) represent the peripheral sensory response curves from the left and right eyes, respectively, after a peripheral threshold has been assessed. A Q,B(°) represents the sensory-response function for both eyes. Taking the natural log of each side of the equation,

Thus Equation 10 may be interpreted to mean that two separate areas (e.g., two areas corresponding to performance from the two eyes separately) sum to produce a total area corresponding to performance in the combined (e.g., binocular) condition. Thus probability summation entails simply summing the two peripheral sensory response curves over t to generate the central sensory response curve. Assuming equal contributions from both eyes, i.e. AQ,L(°) = AQ,R(°) = AQ,M(°), then AQ,B(°) = 2AQ,M(°) and

where pB and pM correspond to binocular and monocular performance respectively. We stress that this assumes that the sensory threshold Q is assessed before the site of combination, and that the information-processing rates cs are the same for both eyes. Thus we are considering the peripheral threshold model for the moment.

We began this section by using the probability summation equation (Eq. 9), and worked backwards to produce a version of the theory that is consistent with Eq. 9. The goal was to detail those assumptions necessary to produce the probability summation prediction. To summarize, if, in addition to the assumptions embodied in Eqs. 4, 5 and 6, we also assume a linear combination mechanism and no post-combination information loss, then the theory will make a prediction that is consistent with probability summation (Eq. 11). This is because the above-threshold area engendered by a binocular presentation is twice that engendered by a monocular presentation under a peripheral threshold model.

We have developed a model that makes a prediction consistent with probability summation, but uses a linear summation combination rule. This may come as a surprise to some readers, who associate an or combination rule with the probability summation model (the stimulus is detected if the left eye detects it or if the right eye detects it, or both). A linear summation rule is typically associated with neural summation, which gives a prediction for binocular performance above probability summation. How then can we develop a model with a linear summation rule that still predicts probability summation? The answer to this apparent paradox lies in the equation for converting above-threshold area to performance (Eq 7). The exponential in this equation allows us to re-express the traditional probability summation equation (Eq 9) in terms of the sums of areas, which implies a linear summation mechanism. However, this formulation is formally equivalent to the or rule of probability summation. Probability summation contains the intrinsic notion of diminishing returns: if two observers or eyes each have a 50% chance of detecting the stimulus, the stimulus will be detected 75% of the time. Our formulation moves this notion of diminishing returns to the exponential in the information processing stages. If we instead computed individual probabilities for the left and right eyes via Eq 7, we would have to adopt the or rule expressed in Eq 9 as our combination mechanism. Thus our adoption of a linear summation mechanism depends upon the information extraction portion of the theory (Eq 7). We should note that the component of the model is essential, in that it accounts for performance as a function of exposure duration.

Note that the probability summation prediction is a consequence of the assumptions of (1) a linear summation mechanism, (2) no post-combinatorial information loss and (3) our exponential information extraction mechanism. Other models have adopted a linear summation mechanism and other assumptions and derive a different prediction of binocular performance based on monocular data.

At this point we wish to clearly distinguish between a probability summation prediction, which is a prediction of binocular performance based on monocular data, and a theory that assumes a linear-summation combination mechanism. The latter is a model of how the information from two eyes combines. When a linear-summation mechanism is conjoined with additional assumptions of no post-combinatorial information loss and an independent sampling information-acquisition mechanism, then a linear-summation mechanisms will make a prediction that is consistent with probability summation. However, as we will show below, a theory that assumes a linear-summation combination mechanism will make predictions that differ from probability summation if a different set of assumptions about thresholds are made. Thus the question of combination mechanisms is intimately tied to the question of threshold locations; one cannot simply compare monocular and binocular data and infer the combinatorial mechanisms without also considering pre- and post-combination information losses. This is a major theme of the current work. We shall refer to a model that sums the information from two eyes as a linear-summation combination mechanism, which under certain assumptions will predict probability summation, but which under other assumptions will not.

When we refer to predictions of binocular data based on monocular performance, we will call this a probability summation prediction. When we refer to a specific model of binocular information combination that explicitly assumes a linear summation combination mechanism and may, under certain assumptions, make a prediction of binocular performance that is consistent with probability summation, we shall refer to this as a linear-summation model.

We are now in a position to discuss the model’s predictions about the relation between the monocular and the binocular conditions. First consider the predictions for a peripheral threshold model, as shown in the top panel of Figure 5. In a monocular presentation, the digits appear in only one eye and generate only a single peripheral sensory response curve. The area that survives the peripheral threshold constitutes the central sensory response curve.

In a binocular presentation, the digits are presented to both eyes, and generate two peripheral sensory response curves. These two curves will both pass through peripheral sensory thresholds, and because the contrast of the two presentations is always the same, the two peripheral curves will be identical. As a result, the binocular conditions central sensory response curve will always be twice as high as the monocular condition's central sensory response curve, and thus the binocular area will always be twice as great. This implies that under a peripheral threshold model binocular performance will fulfill a probability summation relation with monocular performance: . Note that this is in accordance with the probability summation prediction of Eq. 11, although we have now described a specific model that makes a probability summation prediction. This model makes two key assumptions: no post-combinatorial source of information loss and a linear-summation combinatorial mechanism.

By the central-threshold model, however, binocular performance is predicted to be greater than is predicted by probability summation. The logic underlying this assertion is illustrated in the bottom panel of Figure 5. In the absence of peripheral thresholds, the two curves are the original monocular and binocular sensory response curves. As in the top panel the binocular curve is overall twice as tall as the monocular curve. However, there is now a central sensory threshold, and only the above-central-threshold area determines performance.

Note that much of the monocular area falls below the central threshold, leaving only an area of 1.4 above threshold in this example. In contrast the binocular curve, by virtue of being higher than its monocular counterpart, is relatively less affected by the same threshold; hence the binocular above-threshold area (4.1 in this example) is greater than twice the monocular above-threshold area (of 1.4). This implies binocular performance above the probability-summation prediction (i.e. ). This above-probability-summation prediction holds for any non-zero central sensory threshold (except for the trivial case when neither curve exceeds threshold and performance is zero in both conditions). This above-probability summation prediction is a direct result of the central sensory threshold.

Let us summarize two critical predictions for Experiment 1. First, a finding that monocular performance exceeds dichoptic performance disconfirms a purely central-threshold model, and confirms a peripheral-threshold model. Second, a finding that binocular performance exceeds what would be predicted on the basis of probability summation from the monocular condition disconfirms a purely peripheral model and confirms a central-threshold model. Table 1 lists the assumptions and predictions for the Peripheral and Central Threshold models.

There were three main conditions in Experiment 1. In the first (monocular) condition a single d-ms stimulus was displayed to only one eye. In the second (dichoptic) condition there were two stimulus presentations, each d/2 ms in duration, separated by a zero-ms ISI. The first d/2 ms was displayed to one eye, while the second d/2 ms was displayed to the opposite eye. In the third (binocular) condition a d-ms stimulus was presented to both eyes simultaneously.

Three experienced vision researchers participated in Experiment 1. Observer TB is the first author. Observers GW and TAK were visiting faculty members.

Stimulus presentation (and response collection) in Experiments 1-3 were carried out on a Macintosh II computer. Stimuli were displayed on an Apple Monochrome monitor in conjunction with a Modified Wheatstone mirror stereoscope (see Blake & Fox, 1973). The refresh rate (67 Hz) of the display device was sufficiently fast to eliminate flicker, and poses little interpretational problem for our theory (See Busey & Loftus, 1994, Appendix C).

Subjects sat approximately 57 cm away from the screen in a dimly lit room, and used the computer keypad to respond. A chin rest positioned the observer's head in front of two mirror arrangements that projected each half of the screen to one of the observer's eyes. An enclosing box eliminated reflections from the screen and assisted with eye fusion.

Considerable effort was taken at the start of each block of trials to fuse the two halves of the monitor into one image without eye strain. Observers adjusted each of the four mirrors until the fixation point fused into a single image. They then adjusted the mirrors to create a cross out of two right angles, each of which was shown only to one eye. Figure 6 shows the experimental apparatus and the image used to adjust the mirrors. To further enhance eye fusion within a session, the background was a uniform gray field, and the fixation point never disappeared.

The digits were presented in 24 point Times-Roman font. The digits were each 0.50° high by 0.40° wide, separated by 0.75° vertically and 0.40° horizontally. Table 2 lists the luminances and contrasts. Darker letters on a lighter background were used to avoid phosphor-decay problems. The experiment was programmed using a timing and display package described by Ames & Palmer (1992).

There were 8 monocular and dichoptic stimulus durations, ranging from 30 - 240 ms in 30-ms increments. In the monocular condition the entire d-ms duration was shown to either the right or left eye. In a corresponding dichoptic condition, d/2 ms of a d-ms presentation was shown to either the right or left eye, followed immediately by the second (d/2)-ms presentation to the opposite eye, with zero ms ISI. Eye position was randomized across all conditions. In the binocular condition, there were 7 durations, ranging from 30 ms to 119 ms in 15-ms increments. During monocular and dichoptic presentations, the eye opposite the stimulated eye saw the blank background field.

Observers completed 18, 72-trial blocks, which provided 54 replications of each conditions per observer.

Figure 7 shows the data for Experiment 1, along with the predictions from the peripheral threshold model. Details of the parameter fitting procedures and equations are provided in Appendix A. No notable performance differences exist for the three observers, and thus the data shown are averaged across observers. The upper (binocular) solid line represents the model's prediction of binocular performance based on probability summation from the monocular condition. In this, as in all data figures, error bars represent standard errors, and the "RMSE" at the lower right is the root-mean-square error between the data points and the predicted values.

Two findings and associated conclusions are apparent. First, the finding that dichoptic performance is below monocular performance allows us to reject a pure central-threshold model. Second, the finding that binocular performance is above the probability summation prediction allows us to reject a pure peripheral threshold model.

The failure of both single-threshold models to account for performance in all three conditions leaves open the possibility that there is both a peripheral and a central threshold. Let us summarize the logic underlying this assertion.

The observed binocular performance above that predicted from probability summation indicates that a complete model requires a central threshold. This central threshold hinders monocular performance more than binocular performance, because the two monocular channels of a binocular presentation combine before the central threshold is assessed. The above-threshold binocular sensory response curve is therefore more than twice as high as an equal-duration monocular curve.

The observed inferiority of dichoptic relative to monocular performance indicates that a complete model must also include a peripheral threshold. The performance decrement for dichoptic presentations compared to monocular performance implies some form of information loss prior to the combination of the two ocular channels; such loss occurs with a peripheral threshold.

To summarize, we have rejected the pure forms of the peripheral and central sensory threshold models, and concluded that both the peripheral and central sensory thresholds are necessary. We now alter our theory to account for these findings. Essentially, we do this by dividing the formerly single sensory threshold into peripheral and central sensory thresholds.

Figure 8 illustrates this modified model, which we call the dual-threshold model. Each ocular channel produces an sensory response function, complete with peripheral threshold. The above-threshold areas from each channel sum to produce the central sensory response function, which includes a central threshold.

where aQ,L(t) is the sensory response from the left eye after a peripheral threshold  Qp is assessed. The response from the right eye, aQ,L(t), is computed in the same fashion. These values combine together to create a central sensory response function, ,

We then assess a central threshold,

where aQ,c(t) represents the height of the sensory response function after the central threshold has been assessed. Aqc(°) is the integral of aQ,c(t) from 0 to °, and performance is related to the above-central threshold area, AQc(°), by Eq. 7.

Figure 9 shows the modified theory's best fit to the Experiment 1 data. The fit is extremely good, with a RMSE of 0.0189. The assumption of a central threshold implies a model that contains a linear-summation mechanism but no longer makes a probability summation prediction of . Table 1 lists the assumptions and predictions for the Dual Threshold model.

Note that the site of information loss is relative to the combinatorial mechanisms. Information lost during the combination process, such as sub-additive summation, would be considered post-combinatorial information loss because it would affect both the left- and right-eye information. Pre-combinatorial information losses affect only the information from a single eye.

Table 3 lists the parameter values for each observer and for the fit in Figure 9. The parameter values for the Dual-Threshold Model come from the temporal variables in Eq. 1 and the threshold values from Eqs. 12 and 14. The information-processing rate parameter cs comes from Eq. 7. We also provide the parameter values for the power-function version of the model, with the analagous peripheral and central b values.

Given the necessity of two thresholds to adequately describe the Experiment 1 data, we might ask how previous experiments using similar stimuli, such as those described in Loftus, Busey & Senders (1993), Loftus & Ruthruff (1994) and Busey and Loftus (1994), could be fit by a single-threshold model. The answer is that, mathematically, the previously assumed single threshold subsumes both thresholds. The two thresholds simply combined and were represented by a single threshold. Only when the binocular and dichoptic conditions were included did the two thresholds reveal themselves. The dual-threshold model would, of course, also fit the data from Loftus, Busey & Senders (1993), Loftus & Ruthruff (1994) and Busey and Loftus (1994), albeit with 100% tradeoff between the two threshold parameters.

A further test of the dual-threshold model can be made as follows. Experiment 4 of Busey and Loftus (1994) included the present monocular condition, as well as a related condition called a gap-monocular condition, in which the total duration (d) of the monocular presentation was split into two temporal (d/2)-ms halves, separated by a 250-ms blank screen. Observers performed substantially better in the monocular compared to the gap-monocular condition. As Busey and Loftus showed, this is predicted by the theory, which implies more sensory response area above threshold in the monocular than in the gap-monocular condition.

The dichoptic condition of Experiment 1 is analogous to a gap-monocular condition, except for the zero ms ISI and the presentation of each pulse to different eyes. Dichoptic condition performance is, like gap-monocular condition performance, worse than monocular performance. This is implied by the theory in part because a dichoptic condition suffers the effects of two peripheral thresholds, while a monocular presentation suffers from only a single peripheral threshold.

The dichoptic and gap-monocular conditions are related in that in both, the stimulus is divided into two temporal halves, and performance in both conditions is worse than monocular performance. However, the dual-threshold model predicts a performance difference between these two conditions. The reason will be described in detail below, but briefly it is this. The dichoptic condition's zero ISI allows the two peripheral sensory response curves to overlap and sum, and thus provides more above-central-threshold area under the central sensory response curve compared to a gap-monocular condition. The gap-monocular condition's large ISI prevents the two peripheral sensory response curves from overlapping, leaving less above-central-threshold area. Thus, on the basis of the slightly more area above the dichoptic curve's central sensory threshold, the dual-threshold version of the theory predicts a slight performance advantage for the dichoptic condition, compared to the gap-monocular condition. Expressed differently, it appears that we cannot treat two separate presentations to separate eyes as simply independent, to-be-summed events, as would be dictated by a simple, peripheral-threshold-only model. The dichoptic peripheral sensory response curves overlap and push more area above the central sensory threshold, whereas the two curves from the two pulses in a gap-monocular presentation cannot overlap. The extra area pushed above the central threshold predicts that dichoptic performance is slightly above gap-monocular performance.

A comparison of Busey and Loftus's (1994) Experiment 4, and the present Experiment 1 confirms this prediction: the difference between the dichoptic and monocular curves in the present Experiment 1 is not as great as the difference between the gap-monocular and monocular curves in Busey & Loftus's (1994) Experiment 4. Subtracting gap monoptic performance from monoptic performance reveals that 20-25 percentage points separates the two conditions in Experiment 4 of Busey & Loftus (1994). However, subtracting dichoptic performance from monocular performance reveals only a 10-15 percentage point separation between the two conditions of Experiment 1. Monoptic performance was similar in the two experiments, never differing by more than 10% across experiments.

Although these findings suggest that gap monoptic performance is inferior to dichoptic performance, this conclusion requires an across-experiment comparison. In Experiment 2 we replicate this finding by comparing the three conditions (monocular, dichoptic and gap-monocular) in a single experiment. Thus Experiment 2 provides a further test of the dual-threshold model, along with independent estimates of the model parameters.

From the perspective of the dual-threshold model, the fundamental difference between a dichoptic and a gap-monocular presentation is that the two peripheral sensory response curves of a dichoptic presentation are only processed independently up to the point where they combine to produce the central sensory response function. After this happens, the two peripheral sensory response curves overlap and sum to produce more area above the central sensory threshold in a dichoptic presentation, and thus the model predicts greater dichoptic performance compared to a corresponding gap-monocular condition.

To see why this is, consider Figure 10. The left panel shows the central sensory response curve for a 150 ms monocular presentation, with the central sensory threshold. The middle panel shows the central sensory response curve for a dichoptic presentation consisting of two 75 ms displays with a zero ISI. Even though the overall exposure duration is identical in the two conditions, the top of the dichoptic sensory response curve appears bowed when compared to the monocular curve. This happens because presenting all of the stimulus to the same eye as in a monocular presentation creates an sensory response curve that is wider above the sensory threshold than the two dichoptic sensory response curves (see left panel of Figure 10, solid vs. dashed curves). This results in more above-peripheral-threshold area for the monocular condition. The bow in the dichoptic central sensory response function (middle panel, Figure 10) results from the narrower dichoptic peripheral sensory response curves, that tend to lose more area to the peripheral sensory threshold.

The right panel of Figure 10 shows the central sensory response curve for a gap-monocular presentation consisting of two 75 ms displays separated by a 250 ms ISI. This panel demonstrates the sensory independence between the two curves: the ISI is long enough to prevent the first presentation from summing with the second. Accordingly, there is less area above the central sensory threshold, reducing performance for this condition compared to both the monocular and dichoptic conditions.

The two peripheral sensory thresholds ensure that dichoptic performance is below monocular performance, but the interaction of the two dichoptic peripheral sensory response curves at the central sensory response curve drives more area above the central sensory threshold. Thus dichoptic performance is predicted to be above the gap-monocular condition but below the monocular condition.

Let us summarize the predictions of Experiment 2. A finding of monocular performance above dichoptic performance implies a model with a peripheral threshold. A finding of dichoptic performance is above gap-monocular performance implies a central threshold.

Experiment 2 incorporates three basic conditions. In the monocular condition a single d-ms stimulus was presented to one eye. In the dichoptic condition there were two (d/2)-ms stimulus presentations, the first to one eye and the second to the other eye, separated by a zero ISI. In the gap-monocular condition there were two (d/2)-ms presentations, both to the same eye, separated by a 250 ms ISI.

Observers, stimuli and apparatus were identical to those of Experiment 1, except that Observer TAK was replaced by TK, a male graduate student.

The design and procedure were essentially identical to those of Experiment 1 except that the binocular condition was replaced by the gap-monocular condition. There were 8 durations with each of the 3 conditions. Observers completed 18, 72-trial blocks, which provided 54 observations per condition per observer.

Figure 11 shows the results of Experiment 2. Because there were qualitative differences between observer GW and the other two observers, we present individual data here. Observer TB's and TK's data (left and middle panels) confirm the foregoing prediction: dichoptic performance (filled squares) lies between monocular and gap-monocular performance. Observer TK's dichoptic data is close to the gap-monocular data, although dichoptic performance is significantly greater than the gap-monocular data, t(1520) = 3.87, p < 0.0001.

Observer GW's data are shown in the right panel. The most important difference is that dichoptic and gap-monocular performance are the same. The dual-threshold model, however, can fit each data pattern with different parameter values. Consider the comparison between observer TB and Observer GW. Figure 12 shows the central sensory response curves for observers TB and GW for each condition, as generated by the best-fitting model parameters. The top three panels demonstrate why a dual-threshold model predicts dichoptic performance between the monocular and gap-monocular performances: the low peripheral threshold and the temporal proximity of the two peripheral sensory response curves combine to produce more above-threshold area relative to the gap-monocular condition.

Observer GW's overall performance is approximately half that of the other two observers. From the model's perspective, this is a direct result of a high peripheral sensory threshold, causing a greater loss of area compared to observer TB. The bottom three panels of Figure 12 show the central sensory response curves given by the best-fitting model parameters for GW. A large peripheral threshold does not allow the two peripheral sensory response curves to overlap centrally as much. Without this large overlap, the two dichoptic curves will produce the about the same above-threshold area as the gap-monocular curve, and subsequently similar performance.

Because observer GW's dichoptic and gap-monocular performances were close together, a dual-threshold model with the central threshold set to zero (equivalent to the pure peripheral threshold model) will account for the data. However, the best fit to the data comes from peripheral and central sensory thresholds that are both positive. In addition, observer GW’s Experiment 1 data also required a central threshold to provide a good model fit.

Observer TK's overall performance lies somewhere in between the other observers and he has a peripheral threshold that is in between TB’s and GW’s. We see a concomitant reduction in the difference between dichoptic and gap-monocular performance. However, this data pattern is readily accounted for by the theory. The parameter values for the three observers in Experiment 2 are listed in Table 3.

Experiment 2 confirms the dual-threshold model, which is implied by Observer TB's and TK's data, and provides the best fit to Observer GW's data. In addition, the fit to Observer GW's data demonstrates how predictions that seem qualitatively different come from a single theory's different parameter values.

Experiments 1 and 2 have provided strong support for a peripheral threshold: monocular performance above dichoptic performance requires some form of pre-combination information loss. This information loss is represented in the dual-threshold theory by a peripheral threshold. All three observers in Experiments 1 and 2 demonstrated a clear superiority for monocular over dichoptic performance. However, Experiment 2 provides only weak support for a central threshold; only two of the three observers (Observers TB and TK) demonstrate a difference between dichoptic and gap-monocular performance. Perhaps the conditions used in Experiment 2 were not powerful enough to distinguish between a theory that requires a central threshold and a theory that does not. One goal of Experiment 3 was to test the central threshold component of the theory directly.

To test the central-threshold component of the dual-threshold theory, Experiment 3 incorporates dichoptically presented stimuli, and variation in the interstimulus interval (ISI) between the two pulses of a dichoptic presentation. To see how this manipulation identifies the existence of a central sensory threshold, consider a basic dichoptic presentation consisting of a 60-ms presentation to one eye followed by a 60-ms presentation of the same stimulus to the other eye. Consider two conditions that differ in the ISI between the two presentations. In the first condition, ISI is zero ms, while in the second condition it is 45 ms. Because the two conditions engender two identical peripheral sensory response curves, they both generate the same above-peripheral-threshold area passed to the combination mechanisms. As a result of these identical areas passed to the combination mechanism, any performance difference between the two conditions must be caused by post-combinatorial mechanisms; that is, a detrimental ISI effect must result from either a loss due to a central threshold or a loss in the combination of the two peripheral areas via nonlinear summation.

Figure 13 demonstrates the central sensory response curves resulting from monocular, 0-ms ISI dichoptic and 45-ms ISI dichoptic presentations, and demonstrates that a central-threshold assumption makes the prediction that as ISI increases, performance will decrease in the dichoptic condition.

Experiment 3 uses the monocular and dichoptic stimuli of Experiments 1 and 2. However, in the dichoptic condition the ISI between the first and second pulses was systematically varied from 0 to 90 ms. The duration of each of two pulses in each dichoptic presentation was either 30, 45, 60 or 75 ms.

Three observers participated in Experiment 3: the first author TB, a female graduate student LM, and a male graduate student MB.

The stimuli and apparatus were identical to those of Experiment 1. The contrast of the digits was set to 7.1%.

Both monocular and dichoptic stimuli were used. Within the dichoptic condition were 5 different ISI's ranging from 0 to 90 ms. The two pulses were the same duration, which was 30, 45, 60 or 75 ms. Within the monocular condition were 4 stimulus durations: 60, 90, 120 or 150 ms. Observers completed 24, 72-trial blocks, which provided 72 observations per condition per observer.

Experiment 3 results are summarized in Figure 14. All three observers produced data that demonstrate the same overall pattern, and thus averaged data are shown. The solid lines represent the dual-threshold model fit, whose best fitting parameters are provided in Table 3. Two findings in these data are noteworthy.

First, as ISI is increased in the dichoptic condition, performance decreases. This implies some form of post-combinatorial information loss, as modeled by a central sensory threshold in Figure 13. Prior to the combination of the information from each eye, dichoptic presentations with different ISI's all provide the same information; the ISI only becomes relevant once the responses from each eye are combined. However, under a linear-summation mechanism, increasing the ISI still cannot cause a decrease in performance without something like a central sensory threshold, because the various dichoptic conditions have the same overall area under their central sensory response functions. The central threshold causes a reduction in performance for longer ISI dichoptic conditions because it tends to eliminate more of the area under the wider, shorter central sensory response curves produced by a longer ISI dichoptic condition. Thus under the current dual-threshold model conceptualization, the Figure-14 data imply a central sensory threshold.

The second noteworthy finding in the Figure 14 data is that monocular performance is above the zero-ISI dichoptic condition's performance. This is most clearly evident for the short-duration presentations. Overall the magnitude of the effect is somewhat reduced when compared to the differences observed between these two conditions in Experiments 1 and 2. However, this pattern held for all three observers, with the exception of the longest-duration condition for Observer TB. The superiority of monocular performance in this as well as in Experiments 1 and 2, constitutes clear evidence for some form of pre-combinatorial information loss. Within the context of the dual-threshold model, this information loss implies a peripheral sensory threshold.

Taken together the findings shown in Figure 14 confirm the dual-threshold model and reconfirm the existence of a post-combinatorial source of information loss, modeled within the theory as a central sensory threshold.

To summarize the conclusions of Experiment 3, we have again disconfirmed a version of the linear-summation model that does not have some form of post-combinatorial information loss, and we have demonstrated that when this post-combinatorial information loss is modeled by a central sensory threshold we can provide good quantitative fits to the data.

Strong tests of binocular summation come from experiments that vary the temporal onsets of the two presentations of binocular and dichoptic stimuli. We have described a theory that quantitatively accounts for the temporal and ocular summation mechanisms that combine information over time and the two eyes. Central to this account are two sources of information loss, one prior to and one following the site of binocular combination. We have chosen sensory thresholds as the mechanism of the information loss. In what follows, we make several remarks about the thresholds in particular, along with general implications of our threshold notions for interpreting binocular superiority. Note that it is not our intent to provide a complete model of binocular combination that accounts for the wide variety of phenomenon that exist in the binocular combination literature. Rather, we have embedded a somewhat simplified binocular combination model within our general theory. This model is designed to (1) illustrate the implications of the sensory threshold, (2) examine the mechanisms of binocular combination in conjunction with this threshold, and (3) demonstrate the role of temporal interactions that result from overlapping sensory-response functions.

In their review of the binocular-summation literature, Blake and Fox (1973; see also Blake, Sloane & Fox, 1981) suggested that binocular performance above that predicted via probability summation from monocular performance implies "genuine neural interaction between the eyes, not just probability summation" (Blake et al., 1981, p. 266). They suggest that a modification of the combination rule is required to account for the superiority of binocular performance over the probability summation prediction. We conclude that this need not be the case. We have provided a model that has a linear summation mechanism and an exponential information acquisition stage. This model, with no post-combinatorial threshold, makes a prediction that is consistent with probability summation. However, this model will also account for the above-probability summation data from Experiment 1 without altering the nature of the combination mechanism. The existence of a post-combinatorial information loss (modeled by the central threshold) is completely responsible for this prediction.

The logic of this assertion is illustrated in the bottom panel of Figure 5. In the absence of a central threshold, the binocular curve is twice as high as a same-duration monocular curve (with or without a peripheral threshold); correspondingly, the binocular-curve area is twice as great as the monocular-curve area. In the absence of additional mechanisms, this would fulfill a probability-summation prediction of as a result of the exponential in the information processing components. However, when a central sensory threshold is introduced, binocular performance rises above what would be predicted from probability summation. This happens because the monocular curve's lower height renders it more vulnerable to a central sensory threshold than a binocular curve. From this perspective, binocular performance is observed to be above what is predicted by probability summation not because observers are doing better in the binocular task, but because they are doing worse in the monocular task. They are doing worse as a direct result of a post-combinatorial information loss, that differentially affects monocular performance more than binocular performance.

An explicit assumption of the current linear-summation model is that the above-threshold areas from the two monocular channels simply sum to produce the central sensory response function, upon which performance is based. While this combination mechanism has support from neuroscience (Ohzawa & Freeman, 1986) and has been adopted by Legge (1984 a, b) and others, alternative mechanisms have been proposed. We now consider in detail a model developed by Cogan (Cogan, 1987; Cogan et. al., 1991). A summary of this model is below; a more complete description is found in Figure 15. We have adapted this model to the current data, and we discuss the extensions to the model that are required to account for the present experimental data. We should point out that we are using Cogan’s model to fit proportion correctly-recalled digits rather than contrast thresholds, and thus we are not using his model as it was originally developed. In particular, Cogan develops an elegant quantitative formulation that allows him to calculate, rather than estimate, the parameters of the model. However, this formulation requires threshold data and thus cannot be applied the present proportion correct data. As a result, we are adopting only a portion of Cogan’s model, that which describes the binocular combination process. We are unable to calculate the values of the model parameters from our peformance curves, and thus we treat the model parameters as free parameters.

The Cogan model assumes that the two eyes perform transduction on incoming stimuli to produce ipsilateral excitation (eL, eR) and contralateral inhibition (iL, iR). The amount of inhibition in one eye is proportional to the excitation in the other eye, such that iR = c • eL and iL = c • eR. The combined either-eye output, bx is given by:

A separate, fused channel, bF, is given by:

and the net binocular effect is simply the sum of bx and bF.

Components of this model share similarities with the Dual-Threshold model presented here. Cogan et al. (1991) developed a linear-filter front end and used this model to fit two-pulse detection thresholds, in which the two pulses could either be the same contrast (++) or opposite contrast (+-). They systematically varied the stimulus onset asynchrony (SOA) and determined relative sensitivities to each type of stimulus. Although they did not obtain quantitative model fits, qualitative model predictions corresponded to the observed data.

Due to the qualitative success of the Cogan model to account for the two-pulse data, we fit the Cogan model to the present data. Before describing the results of this modeling, some comparisons between the two models are in order.

A major emphasis of the present work is the effects of post-combinatorial information loss on binocular and monocular stimulus representations. An example is the ability of the Dual-Threshold model with a central sensory threshold to account for above-probability-summation binocular performance, even with a linear summation combination mechanism that would otherwise provide a probability summation prediction. This results from the higher binocular central sensory response function that allows more area to survive a central sensory threshold. The binocular channel bF in the Cogan model can in principle have the same effect: if both eL and eR are active at once, bF will be large, giving a larger response and thus better performance. This is qualitatively similar to the two peripheral sensory response curves providing more above-central-threshold area when they co-occur. Thus the parameter k from Cogan’s binocular channel is somewhat analagous to the free parameter Qc in the Dual-Threshold model, and because we do not have threshold data we must estimate k from the data, making it a free parameter as well.

The interocular inhibition from the Cogan model serves to reduce the response in the opposite monocular channel. This pre-combinatorial information loss is somewhat analagous to the peripheral thresholds in the Dual-Threshold model. There are important differences, in that the peripheral thresholds act only on a single monocular channel, which the interocular inhibition crosses channels. However, within the model formulations and the application to our data, pre-combinatorial information loss is represented by a single free parameter, either c or QP.

Qualitatively, the Dual-Threshold model and the Cogan model are similar and have the same number of free parameters when applied to proportion correct data. Given this, it is reasonable to ask whether the Cogan model can fit the data from Experiments 1-3. Such fits require an information-processing model component to deal with the change from a detection paradigm to a letter identification paradigm.

To assist in model comparison, we simply appended the information-processing mechanisms of the Dual-Threshold model to the output of the Cogan model (Eq 7). Consistent with the Dual-Threshold model, we assumed that the presentations to each eye engenders a peripheral sensory response function corresponding to Cogan’s eL or eR. These two response functions inhibit each other in proportion to their height at each moment in time, according to the free parameter c. A separate binocular channel was also created from the peripheral sensory response functions according to the free parameter k via Eq 14. The sum of these two channels gives a central sensory response function. The area of this central sensory response function is used to generate a performance prediction via Eq. 7. Table 1 lists the assumptions and predictions for the Modified Cogan model.

The Cogan model could not fit the data as well as the Dual-Threshold model. Parameter values are shown in Table 4. The left panels of Figure 16 show the fits to the averaged data from Experiments 1 and 3. The data from Experiment 2 could also not be fit, but are not shown, since the conclusions derived from the data mirror those from Experiment 1. The reason for these poor fits appears to come from a binocular excitation channel that cannot completely take the place of a central information loss that accounts for both the above-probability-summation binocular performance of Experiment 1 and the decrease in performance with increasing ISI from Experiment 3. When a central nonlinearity such as a central threshold is added to the Cogan model (Figure 16, right panels), the revised Cogan model gives very good fits in all cases. However, the addition of a central threshold gives the revised model one more free parameter that the Dual-Threshold model.

To summarize, while the Experiment 1-3 data disconfirm the simple version of the Cogan model, the model gives excellent quantitative fits if a central threshold or central power function is added. Such an addition might be required by the additional complexity of the task: detection and letter identification might require different spatial and temporal frequencies (Busey, 1994) and letter identification is best thought of as a classification task. The central information loss may result from inefficiencies or noise in the information processing mechanisms that are not required for simple detection.

The impulse response function used in the present work (Eq. 1) is monophasic, and therefore does not allow for temporal inhibition. Busey (1994) found weak evidence for temporal inhibition in digit identification tasks, and when the theory was modified to include temporal inhibition (via a biphasic impulse response function, setting s > 0 in Eq. 1) it successfully accounted for the data in these related tasks. However, application of this biphasic model to the present data did not markedly improve the fit over that provided by the monophasic model that we have described (despite two additional free parameters). This failure result from the peripheral thresholds, the effects of which tend to eliminate the inhibitory components of the biphasic peripheral sensory response curves. Similarly, a version of the theory that includes temporal inhibition but without peripheral sensory thresholds also did not provide a markedly better fit to any of the three data sets. A version of the Cogan model with temporal inhibition did not provide fits better than those shown in Figure 16, despite requiring two additional free parameters.

Using single-cell recording in cat, Ohzawa & Freeman (1986) found overwhelming evidence for linear summation in binocular cells. However, a minority of cells exhibited deviations from this linearity, in that the response from the cell was maximal when both eyes were equally stimulated, but dropped off at a rate faster than would be expected from a linear model when one eye was reduced. This is quite similar to the behavior of the central threshold observed in the current modeling (e.g. Figure 13). Indeed, Ohzawa & Freeman attribute this nonlinearity to a threshold mechanism that comes into play after the linear binocular summation. One possible mechanism for such a threshold is the threshold for firing on the post-synaptic cell receiving input from both eyes. Weak stimulation to either eye may not exceed this threshold and cause the cell to fire, although weak stimulation to both eyes may result in an action potential on the post-synaptic cell.

As noted above, the notion of peripheral and central sensory thresholds is similar to peripheral and central sources of noise. Denis Pelli (personal communication) has demonstrated that under some conditions, either the central or peripheral noise dominates the signal, and thus affects performance. He and Manoj Raghavan (Pelli, Raghavan & Ahuja, 1995; Raghavan & Pelli, 1995) demonstrate that the visibility of large, long-duration letters is limited by (central) cortical noise, and that visibility of small, brief-duration letters is limited by (peripheral) photon noise. The important implication for perception researchers is that a stimulus variable such as size can differentially affect two stages of processing (peripheral or central). Thus these two stages become important to model even in experiments that do not use monocular or dichoptic stimuli.

The thresholds as conceptualized above may be thought of as a constant drain on a system that is attempting to process a signal. Consider a metaphor in which the sensory response function represents the speed of some system, say the value of the speedometer in a car. The total amount of information, i.e. the area under the sensory response function, is analogous to the odometer reading after some period. Once the threshold is added, however, the metaphor changes to a motorboat traveling upriver. The speed of the current represents the height of the threshold; thus if motorboat speed does not exceed the current speed, then no upriver progress will be made.

We have extended a previously proposed theory of visual information processing to the domain of binocular combination. Application of this extended theory to data from three experiments helps identify the sources of information loss, both pre- and post-combination. Finally we have confirmed a version of the extended theory that combines both pre- and post-combinatorial sources of information loss with a linear-summation combination mechanism and an independent sampling information acquisition mechanism. This application demonstrates that the question of the combination mechanisms involved in binocular combination is intimately tied to the question of pre- and post-combination sources of information loss: one cannot simply look at monocular and binocular data and infer the combinatorial mechanisms without also considering the potential for something like peripheral and central sensory-response thresholds. In particular, the existence of a central threshold implies that binocular performance above a probability summation prediction based on monocular data need not implicate an above-linear-summation combination mechanism. A linear summation mechanism, an independent sampling information acquisition mechanism and a central threshold will provide an equivalent account.

The linear-filter model components also allows a good quantitative account of the degree to which the relationship between monocular and binocular performance changes as the asynchrony between the two pulses of the binocular stimulus changes. Such modeling reveals not only under what conditions the two monocular channels influence each other, but how they are affected by the post-combinatorial information loss. Finally, we have demonstrated that such quantitative fits are important when comparing two models that have similar structure and components and thus give qualitatively similar results.

Parameter fitting was accomplished using a simplex gradient descent method. To avoid the possibilities of local minima, we computed the best fitting parameters multiple times, each time from a different starting location in parameter space. The vast majority of parameter searches terminated at the same ending parameters for a given model fit.

All model fits derive the peripheral sensory response functions, ap(t), via the convolution of the stimulus wave form with the impulse-response function of the linear filter:

where g(t) is the gamma function from Eq 1. Note that for monocular and dichoptic stimuli, the left and right-eye f(t) functions will differ according to which eye is stimulated (or stimulated first in the dichoptic case).

We develop two forms of the non-linearity

A peripheral threshold, Qp, is assessed on the peripheral a(t) wave form:

for the hard threshold version of the theory, or a power function is applied to the peripheral a(t) wave form:

The two peripheral sensory response functions sum to create the central sensory response function, ac(t):

or

for the power function version of the theory.

In the Dual-Threshold model, a central threshold is assessed to create aQ,c(t):

for the hard threshold version of the theory, or a power function is applied to the central a(t) wave form:

The area under the aq,c(t) function is converted to performance via the following formula:

For the power-function version of the theory, performance is a function of the area under the ab,c(t) wave form:

Using the notation of the Busey & Loftus (1994) model, the Cogan model is formulated as follows:

and

where aR(t) and aL(t) represent eR and eL respectively. The central sensory response function ac(t) is computed via:

If a central threshold is applied after the point of combination, this formulation is added to create aQ,c(t):

for the hard threshold version, or a power function is applied to the central a(t) wave form:

In all three cases, performance is converted from the area under the central sensory response function via Eq 7:

were Ac(°) is replaced by Aq,c(°) when a central threshold is added or by Ab,c (°) when a central power function is added.

This research was supported by an NIMH predoctoral fellowship to Thomas Busey and an NIMH grant to Geoffrey Loftus. We would like to thank George Wolford for many helpful comments concerning the interpretation and modeling of these data. Helpful comments by Randy Blake and an anonymous reviewer were also appreciated. Please address all correspondece to Tom Busey, Department of Psychology, Indiana University, Bloomington IN 47405. busey@indiana.edu