![]() ![]() Different summation rules will make different predictions for the A+B threshold, and so the results from the experiment can be used to distinguish between them. These can then be used to predict the threshold for the compound stimulus “A+B” (e.g., Baldwin, Husk, Meese, & Hess, 2014 Graham & Nachmias, 1971 Graham & Robson, 1987 reviewed by Graham, 1989 Meese, 2010 Pirenne, 1943 Quick, Mullins, & Reichert, 1978 Sachs, Nachmias, & Robson, 1971). In the first method, detection thresholds for two component stimuli, here termed “A” and “B,” are measured. There are two ways in which experiments are typically conducted to investigate summation. It can be posed at any level of the brain from the basic summation of contrast signals (e.g., Hoekstra, Van der Goot, Van den Brink, & Bilsen, 1974) to the summation of semantic information between different modalities (e.g., visual and auditory stimuli in To, Baddeley, Troscianko, & Tolhurst, 2011), and there is no reason to expect the answer to be the same at every level and in every modality (although some degree of commonality would be parsimonious). This leads to one of the most basic questions in vision science: How are these outputs combined? Numerous studies have addressed this “summation” question. More complex stimuli are the norm, and to represent these the visual system must combine the outputs of the simpler mechanisms. Behaviorally, however, the stimuli that are ecologically relevant are not the simple features that these mechanisms are sensitive to. For example, simple cells are sensitive to luminance modulations of some specific frequency, orientation, and phase. The early visual system contains tuned mechanisms that respond to particular stimulus features at particular locations in the visual field. The methods described here can be readily applied using software functions newly added to the Palamedes toolbox. ![]() We also show how one can fit the formulas directly to real psychometric functions using data from a binocular summation experiment, and show how one can obtain estimates of τ and test whether binocular summation conforms more to PS or AS. We show how the probability (and additive) summation formulas can be used to simulate psychometric functions, which when fitted with Weibull functions make signature predictions for how thresholds and psychometric function slopes vary as a function of τ, n, and Q. Both formulas are general purpose, calculating performance for forced-choice tasks with M alternatives, n stimuli, in Q monitored mechanisms, each subject to a non-linear transducer with exponent τ. We derive formulas that employ numerical integration to predict the proportion correct for detecting multiple stimuli assuming PS under SDT, for the situations in which stimuli are either equal or unequal in strength. Modeling the equivalent of PS under SDT is, however, relatively complicated, leading many investigators to use Monte Carlo simulations for the predictions. PS is traditionally modeled under high threshold theory (HTT) however, tests have shown that HTT is incorrect and that signal detection theory (SDT) is the better framework for modeling summation. There are broadly speaking two ways this can occur: additive summation (AS) where inputs from the different stimuli add together in a single mechanism, or probability summation (PS) where different stimuli are detected independently by separate mechanisms. Many studies have investigated how multiple stimuli combine to reach threshold. ![]()
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