Human Visual Search Does Not Maximize the Post-Saccadic Probability of Identifying Targets
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Auteurs : Camille Morvan [États-Unis] ; Laurence T. Maloney [États-Unis]Source :
- PLoS Computational Biology [ 1553-734X ] ; 2012.
Abstract
Researchers have conjectured that eye movements during visual search are selected to minimize the number of saccades. The optimal Bayesian eye movement strategy minimizing saccades does not simply direct the eye to whichever location is judged most likely to contain the target but makes use of the entire retina as an information gathering device during each fixation. Here we show that human observers do not minimize the expected number of saccades in planning saccades in a simple visual search task composed of three tokens. In this task, the optimal eye movement strategy varied, depending on the spacing between tokens (in the first experiment) or the size of tokens (in the second experiment), and changed abruptly once the separation or size surpassed a critical value. None of our observers changed strategy as a function of separation or size. Human performance fell far short of ideal, both qualitatively and quantitatively.
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DOI: 10.1371/journal.pcbi.1002342
PubMed: 22319428
PubMed Central: 3271024
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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Human Visual Search Does Not Maximize the Post-Saccadic Probability of Identifying Targets</title><author><name sortKey="Morvan, Camille" sort="Morvan, Camille" uniqKey="Morvan C" first="Camille" last="Morvan">Camille Morvan</name><affiliation wicri:level="1"><nlm:aff id="aff1"><addr-line>Department of Psychology, New York University, New York, New York, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Psychology, New York University, New York, New York</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="aff2"><addr-line>Center for Neural Science, New York University, New York, New York, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Center for Neural Science, New York University, New York, New York</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="aff3"><addr-line>Department of Psychology, Harvard University, Cambridge, Massachusetts, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Psychology, Harvard University, Cambridge, Massachusetts</wicri:regionArea></affiliation></author><author><name sortKey="Maloney, Laurence T" sort="Maloney, Laurence T" uniqKey="Maloney L" first="Laurence T." last="Maloney">Laurence T. Maloney</name><affiliation wicri:level="1"><nlm:aff id="aff1"><addr-line>Department of Psychology, New York University, New York, New York, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Psychology, New York University, New York, New York</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="aff2"><addr-line>Center for Neural Science, New York University, New York, New York, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Center for Neural Science, New York University, New York, New York</wicri:regionArea></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">22319428</idno><idno type="pmc">3271024</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3271024</idno><idno type="RBID">PMC:3271024</idno><idno type="doi">10.1371/journal.pcbi.1002342</idno><date when="2012">2012</date><idno type="wicri:Area/Pmc/Corpus">002184</idno><idno type="wicri:Area/Pmc/Curation">002184</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Human Visual Search Does Not Maximize the Post-Saccadic Probability of Identifying Targets</title><author><name sortKey="Morvan, Camille" sort="Morvan, Camille" uniqKey="Morvan C" first="Camille" last="Morvan">Camille Morvan</name><affiliation wicri:level="1"><nlm:aff id="aff1"><addr-line>Department of Psychology, New York University, New York, New York, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Psychology, New York University, New York, New York</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="aff2"><addr-line>Center for Neural Science, New York University, New York, New York, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Center for Neural Science, New York University, New York, New York</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="aff3"><addr-line>Department of Psychology, Harvard University, Cambridge, Massachusetts, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Psychology, Harvard University, Cambridge, Massachusetts</wicri:regionArea></affiliation></author><author><name sortKey="Maloney, Laurence T" sort="Maloney, Laurence T" uniqKey="Maloney L" first="Laurence T." last="Maloney">Laurence T. Maloney</name><affiliation wicri:level="1"><nlm:aff id="aff1"><addr-line>Department of Psychology, New York University, New York, New York, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Psychology, New York University, New York, New York</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="aff2"><addr-line>Center for Neural Science, New York University, New York, New York, United States of America</addr-line></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>Center for Neural Science, New York University, New York, New York</wicri:regionArea></affiliation></author></analytic><series><title level="j">PLoS Computational Biology</title><idno type="ISSN">1553-734X</idno><idno type="eISSN">1553-7358</idno><imprint><date when="2012">2012</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>Researchers have conjectured that eye movements during visual search are selected to minimize the number of saccades. The optimal Bayesian eye movement strategy minimizing saccades does not simply direct the eye to whichever location is judged most likely to contain the target but makes use of the entire retina as an information gathering device during each fixation. Here we show that human observers do not minimize the expected number of saccades in planning saccades in a simple visual search task composed of three tokens. In this task, the optimal eye movement strategy varied, depending on the spacing between tokens (in the first experiment) or the size of tokens (in the second experiment), and changed abruptly once the separation or size surpassed a critical value. None of our observers changed strategy as a function of separation or size. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">PLoS Comput Biol</journal-id><journal-id journal-id-type="publisher-id">plos</journal-id><journal-id journal-id-type="pmc">ploscomp</journal-id><journal-title-group><journal-title>PLoS Computational Biology</journal-title></journal-title-group><issn pub-type="ppub">1553-734X</issn><issn pub-type="epub">1553-7358</issn><publisher><publisher-name>Public Library of Science</publisher-name><publisher-loc>San Francisco, USA</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">22319428</article-id><article-id pub-id-type="pmc">3271024</article-id><article-id pub-id-type="publisher-id">PCOMPBIOL-D-11-00705</article-id><article-id pub-id-type="doi">10.1371/journal.pcbi.1002342</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="Discipline-v2"><subject>Biology</subject><subj-group><subject>Neuroscience</subject><subj-group><subject>Cognitive Neuroscience</subject></subj-group></subj-group></subj-group><subj-group subj-group-type="Discipline-v2"><subject>Social and Behavioral Sciences</subject><subj-group><subject>Psychology</subject></subj-group></subj-group></article-categories><title-group><article-title>Human Visual Search Does Not Maximize the Post-Saccadic Probability of Identifying Targets</article-title><alt-title alt-title-type="running-head">Suboptimal Visual Search</alt-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Morvan</surname><given-names>Camille</given-names></name><xref ref-type="aff" rid="aff1"><sup>1</sup></xref><xref ref-type="aff" rid="aff2"><sup>2</sup></xref><xref ref-type="aff" rid="aff3"><sup>3</sup></xref><xref ref-type="corresp" rid="cor1"><sup>*</sup></xref></contrib><contrib contrib-type="author"><name><surname>Maloney</surname><given-names>Laurence T.</given-names></name><xref ref-type="aff" rid="aff1"><sup>1</sup></xref><xref ref-type="aff" rid="aff2"><sup>2</sup></xref></contrib></contrib-group><aff id="aff1"><label>1</label><addr-line>Department of Psychology, New York University, New York, New York, United States of America</addr-line></aff><aff id="aff2"><label>2</label><addr-line>Center for Neural Science, New York University, New York, New York, United States of America</addr-line></aff><aff id="aff3"><label>3</label><addr-line>Department of Psychology, Harvard University, Cambridge, Massachusetts, United States of America</addr-line></aff><contrib-group><contrib contrib-type="editor"><name><surname>Kording</surname><given-names>Konrad P.</given-names></name><role>Editor</role><xref ref-type="aff" rid="edit1"></xref></contrib></contrib-group><aff id="edit1">Northwestern University, United States of America</aff><author-notes><corresp id="cor1">* E-mail: <email>morvan@fas.harvard.edu</email></corresp><fn fn-type="con"><p>Conceived and designed the experiments: LTM CM. Performed the experiments: CM. Analyzed the data: CM. Contributed reagents/materials/analysis tools: CM. Wrote the paper: LTM CM.</p></fn></author-notes><pub-date pub-type="collection"><month>2</month><year>2012</year></pub-date><pmc-comment> Fake ppub added to accomodate plos workflow change from 03/2008 and 03/2009 </pmc-comment>
<pub-date pub-type="ppub"><month>2</month><year>2012</year></pub-date><pub-date pub-type="epub"><day>2</day><month>2</month><year>2012</year></pub-date><pub-date pub-type="ecorrected"><day>03
</day><month>2</month><year>2012</year></pub-date><volume>8</volume><issue>2</issue><elocation-id>e1002342</elocation-id><history><date date-type="received"><day>19</day><month>5</month><year>2011</year></date><date date-type="accepted"><day>21</day><month>11</month><year>2011</year></date></history><permissions><copyright-statement>Morvan, Maloney. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.</copyright-statement><copyright-year>2012</copyright-year></permissions><abstract><p>Researchers have conjectured that eye movements during visual search are selected to minimize the number of saccades. The optimal Bayesian eye movement strategy minimizing saccades does not simply direct the eye to whichever location is judged most likely to contain the target but makes use of the entire retina as an information gathering device during each fixation. Here we show that human observers do not minimize the expected number of saccades in planning saccades in a simple visual search task composed of three tokens. In this task, the optimal eye movement strategy varied, depending on the spacing between tokens (in the first experiment) or the size of tokens (in the second experiment), and changed abruptly once the separation or size surpassed a critical value. None of our observers changed strategy as a function of separation or size. Human performance fell far short of ideal, both qualitatively and quantitatively.</p></abstract><abstract abstract-type="summary"><title>Author Summary</title><p>Vision is most sensitive to fine detail at the center of gaze (the fovea). We typically move our eyes several times a second to build up an accurate picture of the world around us and find objects of interest. Very recently, researchers have developed models of how a visual system like ours could search a scene for a specific target with the smallest possible number of eye fixations. In two experiments, we tested the assumptions underlying such models. We set up visual “games” in which observers were rewarded for their performance in moving their eyes once to recognize simple targets. To do well (earn the maximum possible reward), observers had to move their eyes according to the predictions of recent models of eye movement. We found that our observers failed to choose optimal eye movement strategies and failed to maximize their potential winnings. Our results suggest a simpler picture of eye movement selection, driven by a few simple heuristic rules that lead to good but not optimal performance in everyday tasks.</p></abstract><counts><page-count count="11"></page-count></counts></article-meta></front><body><sec id="s1"><title>Introduction</title><p>For many detection and discrimination tasks, performance decreases rapidly with increasing distance from the center of vision. Observers overcome this limitation by making discrete eye movements (saccades) as often as three times per second, in effect scanning the environment. Such serial scanning is not limited to humans or to the visual modality. It is commonly found whenever the sensory range is limited spatially but the sensors can be displaced. Examples include exploratory whisking by rats <xref ref-type="bibr" rid="pcbi.1002342-Hartmann1">[1]</xref> echo-location by bats <xref ref-type="bibr" rid="pcbi.1002342-Chiu1">[2]</xref>, and haptic exploration by humans <xref ref-type="bibr" rid="pcbi.1002342-Lederman1">[3]</xref>.</p><p>The pattern of eye movements depends on the observer's goals <xref ref-type="bibr" rid="pcbi.1002342-Yarbus1">[4]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Hayhoe1">[5]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Hayhoe2">[6]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Land1">[7]</xref>. In visual search, for example, the observer is searching for a specified target within the visual field. Following each eye movement the visual system gains access to new information as a result of the most recent eye movement and must decide whether to terminate the search because the target has been located, to continue the search by planning a further eye movement, or to abandon the search. If the search is continued then a key question is, how does the visual system plan the next saccade given the visual information gathered so far?</p><p>Models of eye movement planning fall roughly into two categories. The first class, salience models, uses the current retinal image to assign a numerical measure called salience to each location in the retina <xref ref-type="bibr" rid="pcbi.1002342-Itti1">[8]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Parkhurst1">[9]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Peters1">[10]</xref>. Salience is often linked to physical measures such as luminance or local contrast. Salience models differ in how salience is computed and in how the visual system uses the salience map to plan the next saccade.</p><p>Models of the second class, optimal statistical models, are designed to optimize a specified criterion <xref ref-type="bibr" rid="pcbi.1002342-Geisler1">[11]</xref>. These models take into account the visual sensitivity of the eye across the retina and make use of all of the information gained in past searches to plan a sequence of saccades that, for example, minimizes the expected number of saccades needed to locate the target <xref ref-type="bibr" rid="pcbi.1002342-Eckstein1">[12]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Najemnik1">[13]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Najemnik2">[14]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Palmer1">[15]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Renninger1">[16]</xref>. The information gathered during initial fixation and with each successive saccade is a measure of the likelihood that the target is at each possible retinal location, a likelihood map (<xref ref-type="fig" rid="pcbi-1002342-g001">Figure 1</xref>).</p><fig id="pcbi-1002342-g001" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1002342.g001</object-id><label>Figure 1</label><caption><title>A likelihood map.</title><p>The map is a plot of the likelihood that a specified target is present at each retinal location. The task is to select the retinal destination for the next saccade. Two strategies are illustrated. The first computes a saccade to the retinal location with highest likelihood (Λ<sub>max</sub>). The second computes a saccade that maximizes the probability of identifying the target location after the first saccade. We refer to the destination of this saccade as the Π<sub>max</sub>. Najemnik & Geisler (2005) emphasized that Λ<sub>max</sub> and Π<sub>max</sub> can be different when one takes into account the information gathering capabilities of the retina away from fovea and considers adaptive strategies that plan future saccades based on information gathered in previous saccades.</p></caption><graphic xlink:href="pcbi.1002342.g001"></graphic></fig><p>While the ability of such statistical models to predict eye movements behavior in natural scenes has been challenged <xref ref-type="bibr" rid="pcbi.1002342-Henderson1">[17]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Henderson2">[18]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Tatler1">[19]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Einhauser1">[20]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Nystrom1">[21]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Tatler2">[22]</xref> and alternative models have been proposed, in particular that of Tatler and colleagues <xref ref-type="bibr" rid="pcbi.1002342-Tatler3">[23]</xref> incorporating high level features, statistical models allow to model ideal (optimal) behavior and compare human performance to ideal <xref ref-type="bibr" rid="pcbi.1002342-Geisler1">[11]</xref>.</p><p>The differences between salience models and statistical models are less than they first appear to be. Likelihood, for example, is arguably a candidate measure of salience. However, a major difference between the two classes of model is the rules for planning the next saccade. With salience models these rules are typically ad hoc, chosen to capture known features of human visual search. They usually propose that the next saccade go to the currently “most salient” location but with some mechanism inhibiting return to those that have already been searched (inhibition of return <xref ref-type="bibr" rid="pcbi.1002342-Klein1">[24]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Wang1">[25]</xref>). The planning algorithms for statistical models, on the other hand, are dictated by the requirement that search be optimal by a previously specified criterion. The modeler typically has no further choices once visual sensitivity across the retina is measured and the criterion to be optimized is selected.</p><p>Recently, Najemnik & Geisler <xref ref-type="bibr" rid="pcbi.1002342-Najemnik1">[13]</xref> analyzed the performance that could be expected of a statistical model designed to minimize the number of saccades needed to locate a target. Given the current likelihood map, it is intuitively appealing to plan the next saccade to the location most likely to be the target, the maximum likelihood point denoted <inline-formula><inline-graphic xlink:href="pcbi.1002342.e001.jpg" mimetype="image"></inline-graphic></inline-formula> in <xref ref-type="fig" rid="pcbi-1002342-g001">Figure 1</xref>. Najemnik & Geisler <xref ref-type="bibr" rid="pcbi.1002342-Najemnik1">[13]</xref> demonstrate that the correct optimal strategy, minimizing the expected number of saccades to locate a target, typically aims at a location <inline-formula><inline-graphic xlink:href="pcbi.1002342.e002.jpg" mimetype="image"></inline-graphic></inline-formula> that need not coincide with <inline-formula><inline-graphic xlink:href="pcbi.1002342.e003.jpg" mimetype="image"></inline-graphic></inline-formula>. <inline-formula><inline-graphic xlink:href="pcbi.1002342.e004.jpg" mimetype="image"></inline-graphic></inline-formula> is the location that allows the visual system to best use its extra-foveal retinal sensitivity to evaluate multiple locations simultaneously as illustrated in <xref ref-type="fig" rid="pcbi-1002342-g001">Figure 1</xref> and in the accompanying inset. The likelihood of a target at the location <inline-formula><inline-graphic xlink:href="pcbi.1002342.e005.jpg" mimetype="image"></inline-graphic></inline-formula> may be very low, as indicated in <xref ref-type="fig" rid="pcbi-1002342-g001">Figure 1</xref>. Moreover, the optimal strategy also considers possible information gathered from future searches contingent on information gained from the current, much as a strong chess player thinks beyond the immediate consequences of his current move.</p><p>Najemnik & Geisler <xref ref-type="bibr" rid="pcbi.1002342-Najemnik1">[13]</xref> compared human performance in searching for a small Gabor patch in <inline-formula><inline-graphic xlink:href="pcbi.1002342.e006.jpg" mimetype="image"></inline-graphic></inline-formula> noise background to the predictions of an optimal statistical model and found qualitative agreement between the model and performance, at least in overall performance.</p><p>One difficulty in comparing performance between human and model is that the stimuli are complex and it is difficult, to predict trial by trial, where the ideal observer should fixate. Here we present a simplified visual search task which allows us to test whether the visual system uses its extra-foveal sensitivity (as Najemnik & Geisler <xref ref-type="bibr" rid="pcbi.1002342-Najemnik1">[13]</xref> propose) to minimize the number of saccades required to identify the target.</p><p>In this task, the observer makes only one saccade per trial and we restrict the observer's possible choices of saccadic destinations to three. The observer must saccade to one of these three possible locations, marked by gray squares arranged horizontally, above or below his initial fixation (<xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2A</xref>). When the observer has completed the saccade, the target appears at either the left or right location, but never in the center. The trial is aborted if the observer tries to execute a second saccade.</p><fig id="pcbi-1002342-g002" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1002342.g002</object-id><label>Figure 2</label><caption><title>Experiment 1: The configural task and saccadic strategies.</title><p>A. There were three tokens (grey squares) present in the visual search array. They are equally spaced horizontally with separation <italic>δ</italic> (deg). The observer initially fixated the fixation cross that is either above the central token or below (shown as above). The observer chooses one of three saccadic strategies, side-left, center, or side–right, saccading to one of the three tokens. If the observer saccaded to any other location, the trial was terminated. Once the saccade was completed, the configural target appeared with equal probability in either the left or right token but never in the center token. The target was one of two configurations shown in the inset, labeled dot-up or dot-down. In the figure the target appears on the right. The other side token and the center token remained grey squares. The observer's task was to judge whether the target is dot-up or dot-down. B. The observer's ability to correctly judge whether the target is dot-up or dot-down depended on the retinal eccentricity of the target E. It was captured by a retinal sensitivity function <italic>ψ(E)</italic> that varied from nearly 1 (fovea) to 0.5 (far periphery). The retinal sensitivity function for one observer (S01) is shown in the figure. The retinal sensitivity function and separation <italic>δ</italic> determined the choice of strategy that maximized the observer's probability of a correct response in the task. C. For observer S01, the expected probability correct corresponding to the center strategy (always saccade to the center token) is plotted in blue and the expected probability correct corresponding to the side strategy (always saccade to one of the side tokens) is plotted in red. <italic>P[Correct|Center]</italic> is greater than or equal to <italic>P[Correct|Side]</italic> from 0° to a separation <italic>E<sup>opt</sup></italic>, the optimal switch point. Beyond the optimal switch point, <italic>P[Correct|Side]</italic> is greater than <italic>P[Correct|Center]</italic>. For separations between 0° and 3° about the difference between the two strategies is slight. However, the observer seeking to maximize expected probability correct for separations between 6° and 20° (the range of Experiment 1) should adopt the center strategy for separations less than <italic>E<sup>opt</sup></italic> and then switch to the side strategy. The value of <italic>E<sup>opt</sup></italic> depends on the observer's retinal sensitivity function <italic>ψ(E)</italic> and may differ for different observers.</p></caption><graphic xlink:href="pcbi.1002342.g002"></graphic></fig><p>The target consists of a grey square and a small white dot that is either near the top of the square (dot-up configuration) or near the bottom (dot-down configuration). The two configurations are shown in an inset in <xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2A</xref>. The observer's task is to discriminate whether the target is dot-up or dot-down. He receives a small amount of money for each correct discrimination.</p><p>The observer's probability of correct discrimination is determined by his retinal sensitivity function <inline-formula><inline-graphic xlink:href="pcbi.1002342.e007.jpg" mimetype="image"></inline-graphic></inline-formula> where <inline-formula><inline-graphic xlink:href="pcbi.1002342.e008.jpg" mimetype="image"></inline-graphic></inline-formula> (eccentricity) is the distance from the fovea to the target. For the discrimination task we employed, <inline-formula><inline-graphic xlink:href="pcbi.1002342.e009.jpg" mimetype="image"></inline-graphic></inline-formula> is a decreasing function of <inline-formula><inline-graphic xlink:href="pcbi.1002342.e010.jpg" mimetype="image"></inline-graphic></inline-formula>. We plot an example of <inline-formula><inline-graphic xlink:href="pcbi.1002342.e011.jpg" mimetype="image"></inline-graphic></inline-formula> versus <inline-formula><inline-graphic xlink:href="pcbi.1002342.e012.jpg" mimetype="image"></inline-graphic></inline-formula> for one observer in <xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2B</xref>.</p><p>The observer has only three possible choices of strategy. He may saccade to the leftmost token, the center token, or the rightmost. If the observer adopts the center strategy, then the separation between where the observer is fixated and the target on left or right is just the spacing between the locations (denoted <inline-formula><inline-graphic xlink:href="pcbi.1002342.e013.jpg" mimetype="image"></inline-graphic></inline-formula> in the <xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2A</xref>). The probability of correct discrimination is<disp-formula><graphic xlink:href="pcbi.1002342.e014"></graphic><label>(1)</label></disp-formula></p><p>If the observer adopts a side strategy then the separation will be either 0 (if he has chosen the location where the target appears) or <inline-formula><inline-graphic xlink:href="pcbi.1002342.e015.jpg" mimetype="image"></inline-graphic></inline-formula> (if he has chosen the location on the side opposite to the location where the target appears). Since the target appears at the left or right location with equal probability, the observer's probability of correct discrimination is<disp-formula><graphic xlink:href="pcbi.1002342.e016"></graphic><label>(2)</label></disp-formula></p><p>In <xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2C</xref> we plot <inline-formula><inline-graphic xlink:href="pcbi.1002342.e017.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e018.jpg" mimetype="image"></inline-graphic></inline-formula> for the observer whose retinal sensitivity map is shown in <xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2B</xref>. If the tokens are close together (<inline-formula><inline-graphic xlink:href="pcbi.1002342.e019.jpg" mimetype="image"></inline-graphic></inline-formula> is small) then the probability correct is close to 1 for both strategies. When the separation is between <inline-formula><inline-graphic xlink:href="pcbi.1002342.e020.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e021.jpg" mimetype="image"></inline-graphic></inline-formula>, use of the center strategy would lead to higher probability correct. Beyond the point marked <inline-formula><inline-graphic xlink:href="pcbi.1002342.e022.jpg" mimetype="image"></inline-graphic></inline-formula> use of the side strategy would maximize expected probability correct. This critical value is determined by the observer's retinal sensitivity function. If the human observer is using his peripheral sensitivity to maximize the probability of correct discrimination, we would expect an abrupt change in strategy when the separation of center and side tokens exceeds <inline-formula><inline-graphic xlink:href="pcbi.1002342.e023.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><p>In Experiment 1 we first measured observers' retinal sensitivity functions. In the main part of the experiment, observers chose between center and side strategies as we varied separation <inline-formula><inline-graphic xlink:href="pcbi.1002342.e024.jpg" mimetype="image"></inline-graphic></inline-formula> over the range 8 to 24 degrees. Observers received a small monetary reward for each correct discrimination. We compared observers' choices of strategy (center or side) to the choice of strategy maximizing expected gain. The observer maximizing expected gain would pick the strategy, center or side, offering the larger probability correct in Equations 1 and 2, switching strategy at the optimal switch point <inline-formula><inline-graphic xlink:href="pcbi.1002342.e025.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><p>In Experiment 2 we varied the size of the targets rather than distance to manipulate <inline-formula><inline-graphic xlink:href="pcbi.1002342.e026.jpg" mimetype="image"></inline-graphic></inline-formula>. Observers chose between the same array of tokens in <xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2A</xref> but now the tokens varied in size. There is still an optimal point in <inline-formula><inline-graphic xlink:href="pcbi.1002342.e027.jpg" mimetype="image"></inline-graphic></inline-formula> where the observer should switch from a center strategy to a side strategy but now it is expressed in size.</p><p>Each experiment consisted of three phases, sensitivity mapping, decision, and verification, illustrated in <xref ref-type="fig" rid="pcbi-1002342-g003">Figure 3</xref> and described in the <xref ref-type="sec" rid="s4">Methods</xref> section. In the sensitivity mapping phase, we measured sensitivity for the visual task for different eccentricities of targets (Experiment 1) and for different sizes of targets at different eccentricities (Experiment 2). In the second (decision) phase, we tested human ability to select eye movements that maximize expected gain. In the last phase (verification), we repeated the decision phase but forced the observer to make the saccade that our model (see <xref ref-type="sec" rid="s4">Methods</xref> section) predicted would maximize expected gain. By doing so, we verified that, had they followed this strategy, they would have increased their expected gain to the maximum possible expected gain predicted by the model.</p><fig id="pcbi-1002342-g003" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1002342.g003</object-id><label>Figure 3</label><caption><title>Experiment 1: A. Retinal mapping session.</title><p>The observer first fixated a fixation cross for 700 msec. Then a target appeared on the right or left of the fixation point. The target was displayed for 500 ms and then disappeared. A response screen was displayed until observers indicated their response. B. Decision session. The observer first fixated a fixation cross for 700 msec. Then square markers appeared at the three locations to which the observer was permitted to saccade. Once the observer completed the saccade, the target configuration appeared at one of the side locations chosen at random. The observer then judged whether the target was dot-up or dot-down. We compared human performance to performance maximizing probability of correct discrimination. C. Verification session. The protocol was the same as the Decision session except for the addition of a 400 ms phase, before the saccade, where one of the 3 tokens disappeared. This token indicated the position that the observer should saccade to. If the observers made his saccade to another location (fixation position not within a 1° radius of the indicated object) the trial was aborted and replayed later. Actual eye movement data for one trial for one observer is superimposed in red on the stimulus arrays.</p></caption><graphic xlink:href="pcbi.1002342.g003"></graphic></fig><p>In the separation and size experiments, we assumed that the target is always presented and we considered only the first saccade. If we were to modify the task slightly so that, although the grey squares appeared, the target configuration (dot-up or dot-down) was not presented on one half of the trials (that is, all of the grey squares were uniform, without a marked configuration), then on trials where the observer fails to detect the location of the configuration after one saccade, he must make one or two additional saccades to determine if the target configuration is present at all and in what configuration. The strategy in our task which maximizes probability correct also minimizes the number of saccades needed to be correct in this modified task.</p><p>In the experiments reported, we interleaved separations (Experiment 1) and sizes (Experiment 2) rather than presenting the same separation or size repeatedly in in a single experimental block. Observers could potentially learn the blocked task by simply trying different saccadic strategies and seeing which is more rewarding as a function of separation (Experiment 1) or size (Experiment 2). In effect they learn to pair strategy and separation/size by reinforcement learning. But the prediction of the class of statistical models we consider is that observers will take into account their own retinal sensitivity in planning saccades without an extensive history of reinforcement <xref ref-type="bibr" rid="pcbi.1002342-Maloney1">[26]</xref>. Reinforcement learning plays no role in these theories.</p></sec><sec id="s2"><title>Results</title><sec id="s2a"><title>Sensitivity mapping phase</title><p>The sensitivity mapping plots are presented in <xref ref-type="fig" rid="pcbi-1002342-g004">Figure 4A</xref> for Experiment 1 and in <xref ref-type="fig" rid="pcbi-1002342-g004">Figure 4B</xref> for Experiment 2. The maps show the percentage of correct responses as a function of Eccentricity for Experiment 1 and Size for Experiment 2. The results for each observer are used to predict the ideal observer performance in the decision phase.</p><fig id="pcbi-1002342-g004" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1002342.g004</object-id><label>Figure 4</label><caption><title>Retinal sensitivity mapping.</title><p>A. Experiment 1. The observer's probability of correctly identify the stimulus configuration (dot-up or dot-down) is plotted as a function of the eccentricity at which the configuration was presented. Data is presented for all five observers. The smooth curve is a maximum likelihood fit of a psychometric function. The observer's probability of correctly identify the stimulus configuration (dot-up or dot-down) is plotted as a function of the stimulus size for stimuli presented at each of three retinal eccentricities. B. Experiment 2. Data is presented for both observers. Observer S04 in Experiment 2 was the same observer as Observer S04 in Experiment 1. See text.</p></caption><graphic xlink:href="pcbi.1002342.g004"></graphic></fig></sec><sec id="s2b"><title>Decision phase</title><sec id="s2b1"><title>Saccadic choices</title><p>We analyzed the experimental data by first identifying observers' decisions. For each trial, the saccade was categorized as being directed to the closest token, computed in Euclidean distance (see Figure S1 in <xref ref-type="supplementary-material" rid="pcbi.1002342.s001">Text S1</xref> and <xref ref-type="sec" rid="s3">discussion</xref>).</p></sec><sec id="s2b2"><title>Strategy</title><p><xref ref-type="fig" rid="pcbi-1002342-g005">Figure 5</xref> shows the probability to saccade to a side object as a function of the tokens separation for Experiment 1 (<xref ref-type="fig" rid="pcbi-1002342-g005">Figure 5A</xref>) or size for Experiment 2 (<xref ref-type="fig" rid="pcbi-1002342-g005">Figure 5B</xref>). For each observer and experiment, we obtained <inline-formula><inline-graphic xlink:href="pcbi.1002342.e028.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e029.jpg" mimetype="image"></inline-graphic></inline-formula>, the optimal switch points, using the psychometric function estimated from the mapping data. The optimal strategy is plotted as a solid line. The optimal strategies for our choices of <inline-formula><inline-graphic xlink:href="pcbi.1002342.e030.jpg" mimetype="image"></inline-graphic></inline-formula> are to saccade to the middle for short separations or big sizes, and vice versa for the big separations and the small sizes. The optimal strategy switches (vertical line) at a specific value <inline-formula><inline-graphic xlink:href="pcbi.1002342.e031.jpg" mimetype="image"></inline-graphic></inline-formula> or <inline-formula><inline-graphic xlink:href="pcbi.1002342.e032.jpg" mimetype="image"></inline-graphic></inline-formula>of the independent variable from a 0% probability of a side saccade (i.e. 100% probability of a center saccade) to a 100% probability of a side saccade (either left or right).</p><fig id="pcbi-1002342-g005" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1002342.g005</object-id><label>Figure 5</label><caption><title>Strategy choice.</title><p>A. Experiment 1. For each observer we plot the proportion of time they picked either the side strategy as a function of eccentricity of the side locations markers. The proportions predicted by an ideal observer maximizing probability of correct response are shown as solid blue curves. These are step functions, going from 0 to 1 at the optimal switch point for the observer which is computed from each observer's retinal sensitivity function. Observers failed to shift strategy with changes in eccentricity. B. Experiment 2. For each observer we plot the proportion of time they picked either side strategy as a function of size of the targets.</p></caption><graphic xlink:href="pcbi.1002342.g005"></graphic></fig><p>If observers were optimal their probability of choosing a side strategy would follow this step function, and any deviation from this step function reduced their expected gain and probability correct. The results, shown as blue dots (<xref ref-type="fig" rid="pcbi-1002342-g005">Figure 5</xref>) are unequivocal: observers were markedly suboptimal. Not only did they fail to switch strategy at <inline-formula><inline-graphic xlink:href="pcbi.1002342.e033.jpg" mimetype="image"></inline-graphic></inline-formula> or <inline-formula><inline-graphic xlink:href="pcbi.1002342.e034.jpg" mimetype="image"></inline-graphic></inline-formula>, but more strikingly, they did not change their strategy at all as a function of the separation or size of the tokens.</p></sec><sec id="s2b3"><title>Saccadic latencies</title><p>Some observers took their decisions very quickly while others were slow. Also, within a given observer there was some variability in the speed of their decisions. We examined whether the speed with which observers made their decision correlated with how “good” the decision was by plotting the likelihood of the saccades to be correct as a function of their latency. As shown in Figure S2 in <xref ref-type="supplementary-material" rid="pcbi.1002342.s001">Text S1</xref>, the quickness of the decision does not seem to affect whether the decision was closer to or further from optimal or not.</p></sec><sec id="s2b4"><title>Saccade length</title><p>Recall that the distances from initial fixation to the two side objects were not the same. Whenever the observer chose to saccade to one of the side objects, they preferred the nearer. The proportion of side strategy saccades to the nearer of the two objects were 0.85, 0.64, 0.95, and 0.86 for the four observers in Experiment 1, 0.99 and 0.77 for the two observers in Experiment 2.</p></sec><sec id="s2b5"><title>Inter-trial dependencies</title><p>One possible explanation of observers' sub-optimal performance is that switching strategies (side to center or vice versa) might entail a “cognitive cost”. We tested whether observers tended to repeat strategies by computing the conditional frequency with which a choice of a given strategy on one trial is followed by a choice of the same strategy on the following across all trials. We compared this conditional frequency to the (non-conditional) frequency with which observers used a given strategy estimated across all trials. The conditional frequency of using a specific strategy was significantly greater than the non-conditional frequency: 0.76 versus 0.5 for the side strategy and 0.6 versus 0.5 for the center strategy. One-tailed t-tests indicated that the difference is statistically significant in both cases with p = 0.04 and p = 0.03 respectively for side and center strategies. We conclude that some observers tended to repeat strategies above chance.</p><p>The observed inter-trial dependencies provide some characterization of human performance in our task beyond the simple claim that it is sub-optimal. Of course, given that the trials were randomized, any evidence of inter-trial (sequential) dependencies in choice of saccadic strategy is further evidence for sub-optimal choice of strategy. We cannot conclude that these dependencies account for failures in human judgment in our task since any causal connection may well be in the reverse direction: the observer, ignorant of the correct strategy, may tend to default to whatever he chose on recent trials.</p><p>Similarly, the experimental conditions (separation or size) were interleaved and randomized in both experiments. On occasion, a particular separation or size was followed by the same separation or size. We considered the possibility that observers chose the optimal strategy for the repeated condition with greater frequency. It was not the case: the probability to pick an optimal saccade was 0.57 on average, and 0.59 for repeated conditions, the difference was non-statistically significant on one-tailed t-test (p = 0.19).</p></sec><sec id="s2b6"><title>Performance (Gain)</title><p>In the verification phase we directly confirmed that observers could have increased their gain by employing the optimal strategies predicted by our model. The trials in the verification phase were identical to those in the decision phase excepted that on each trial the observer was instructed to saccade to the location that we predicted would maximize his expected gain (see <xref ref-type="sec" rid="s4">Methods</xref>). We plot mean gain for each observer in the decision phase (red circles) and in the verification phase (blue circles) in <xref ref-type="fig" rid="pcbi-1002342-g006">Figure 6</xref>. In the decision phase, observers did not maximize expected gain and their choices of strategies reduced their expected winnings by 9% on average (max = 17%, min = 6.1%, SD = 0.0398). When forced to choose optimal strategies on each trial, observers' gain increased and is indistinguishable from maximum expected gain as predicted by the model.</p><fig id="pcbi-1002342-g006" position="float"><object-id pub-id-type="doi">10.1371/journal.pcbi.1002342.g006</object-id><label>Figure 6</label><caption><title>Comparison to Maximum Expected Gain.</title><p>Each observer's mean gain in the decision phase (in red) and in the verification phase (in blue) is plotted versus the maximum expected gain possible for that observer. The red points are consistently below the 45 degree line indicating that observers failed to maximize expected gain as predicted by the model in the decision phase. The blue points on the other hand are scattered around the 45 degree line. Observer's gain approached the maximum possible gain when the observer executes the eye movement strategy that maximizes expected gain as predicted by the model.</p></caption><graphic xlink:href="pcbi.1002342.g006"></graphic></fig><p>In the analyses just reported here, we used estimates of retinal sensitivity function <inline-formula><inline-graphic xlink:href="pcbi.1002342.e035.jpg" mimetype="image"></inline-graphic></inline-formula> from the Sensitivity Session to predict the visual strategies that would maximize expected gain in the Decision Session. However, in the Decision Session, each visual judgment was preceded by a saccade while in the Sensitivity Session it was not. There is considerable evidence that a preceding saccade can briefly alter perception <xref ref-type="bibr" rid="pcbi.1002342-Kowler1">[27]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Deubel1">[28]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Castet1">[29]</xref> and we can conjecture that it may also alter sensitivity.</p><p>However, we can readily estimate <inline-formula><inline-graphic xlink:href="pcbi.1002342.e036.jpg" mimetype="image"></inline-graphic></inline-formula> based on data in any of the three sessions and use these estimates to predict the visual strategy maximizing expected gain in the Decision Session. In the Figure S3 of the <xref ref-type="supplementary-material" rid="pcbi.1002342.s001">Text S1</xref> we compare observers' data to these predictions. Inspection of this plot shows that each observer's data is inconsistent with the optimal strategy based on <inline-formula><inline-graphic xlink:href="pcbi.1002342.e037.jpg" mimetype="image"></inline-graphic></inline-formula> in any session. Whichever session we choose to estimate <inline-formula><inline-graphic xlink:href="pcbi.1002342.e038.jpg" mimetype="image"></inline-graphic></inline-formula>, we reach the same conclusion: observers did not adapt their strategies to the separation of the tokens (Experiment 1) or to changes in size (Experiment 2).</p><p>To test whether observers maximized expected gain, we performed a nested-hypothesis test <xref ref-type="bibr" rid="pcbi.1002342-Mood1">[30]</xref> comparing each observer's winning to those of an observer maximizing expected gain. The results for Experiment 1 with all observers combined indicated that observers overall did not maximize expected gain (<inline-formula><inline-graphic xlink:href="pcbi.1002342.e039.jpg" mimetype="image"></inline-graphic></inline-formula>) and that three of the four observers considered in isolation did not maximize expected gain: S01 (<inline-formula><inline-graphic xlink:href="pcbi.1002342.e040.jpg" mimetype="image"></inline-graphic></inline-formula>), S03 (<inline-formula><inline-graphic xlink:href="pcbi.1002342.e041.jpg" mimetype="image"></inline-graphic></inline-formula>) and S04 (<inline-formula><inline-graphic xlink:href="pcbi.1002342.e042.jpg" mimetype="image"></inline-graphic></inline-formula>). The winnings of the remaining observer S02 were not significantly different from maximum expected gain (<inline-formula><inline-graphic xlink:href="pcbi.1002342.e043.jpg" mimetype="image"></inline-graphic></inline-formula>). For Experiment 2, the overall test lead to rejection of the hypothesis that the observers maximized expected gain (<inline-formula><inline-graphic xlink:href="pcbi.1002342.e044.jpg" mimetype="image"></inline-graphic></inline-formula>) and to rejection of the hypotheses that each observer maximized expected gain (S04: <inline-formula><inline-graphic xlink:href="pcbi.1002342.e045.jpg" mimetype="image"></inline-graphic></inline-formula>, S05:<inline-formula><inline-graphic xlink:href="pcbi.1002342.e046.jpg" mimetype="image"></inline-graphic></inline-formula>).</p></sec></sec></sec><sec id="s3"><title>Discussion</title><sec id="s3a"><title>Summary</title><p>We reported two experiments intended to determine whether observers correctly employed their extrafoveal retinal sensitivity to optimize visual search. Each experiment consisted of three phases, sensitivity mapping, decision, and verification. In the sensitivity mapping phase, we measured each observer's retinal sensitivity as a function of target eccentricity (Experiment 1) and/or target size (Experiment 2).</p><p>On each trial of the decision phase, observers first executed a saccade to one of three retinal locations, left, center or right (<xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2A</xref>). Following the saccade, a target would appear at either the left or right location but never in the center location. The observers then attempted to discriminate whether a small white dot within the target was near the top or bottom of the target.</p><p>Their probability of success in discriminating depended on the location to which they had saccaded and the location at which the target appeared. Observers received a monetary reward for each correct discrimination and their challenge was to decide which location to saccade to so as to maximize their expected gain. We refer to their choices as a saccadic strategy. There were only three possible strategies, left, center, or right, and two of these strategies, left and right, were effectively equivalent (see <xref ref-type="sec" rid="s4">Methods</xref>). We refer to them collectively as the side strategy.</p><p>A center strategy led to better discrimination for smaller eccentricities (Experiment 1) or larger sizes (Experiment 2). Whether the target appeared on the left or right side, the small eccentricity (large size) meant that the observer could discriminate above chance while fixated at the center location. For large enough eccentricities in Experiment 1, the center strategy resulted in performance near chance. In contrast, either side strategy resulted in better performance since, if the target appeared on the same side as the observer chose to saccade to, then he could readily discriminate it. This would occur on half the trials and on the remaining trials, when the target appeared on the side not chosen, the observer would be near chance in responding. Overall, the side strategy would lead to performance better than that expected with the center strategy. See <xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2C</xref>. The same conclusion holds in Experiment 2 where we varied size.</p><p>Consequently, as the experimenter increased the eccentricity of the side locations or decreased the size of the target, the observer optimizing expected gain or, equivalently, probability correct, should switch from a center strategy to either one of the side strategies at a specific optimal switch point.</p><p>We used the data from the sensitivity mapping phase to predict the optimal switch point as a function of eccentricity or in size for each observer. We compared observers' choices of strategy to the choices predicted to maximize their probability correct in the discrimination task.</p><p>None of our 6 observers switched strategy at the optimal point. All had evident, idiosyncratic biases, toward either the side or center strategies, but, most strikingly, they chose the center and side strategies equally often for all eccentricities and sizes of target. They did not adapt their strategy to the stimulus configuration at all.</p><p>In a separate verification phase we reran the main part of the experiment but now indicating to the observer where to saccade on each trial, “forcing” the observer to adopt the saccadic strategy that our model predicted would maximize probability correct and expected gain. We found that observers' mean gain increased when they executed the strategy predicted to maximize expected gain and that their mean gain was in good agreement with the maximum expected gain predicted by the model.</p><p>In summary, observers did not respond to variations in token separation and size at all. They apparently ignored the independent variable in each of the two experiments. We had expected that, for example, their performance might be qualitatively consistent with that of the ideal observer. That is, they might abruptly switch strategies at some separation (size) but not at the separation (size) that maximized expected gain. Or they might have been inconsistent in choosing strategies but only near the switch point so that probability of picking the side strategy might smoothly decrease from 1 to 0 instead of following the ideal step function. Neither of these outcomes occurred. We see no evidence that the visual system is sensitive to the factors we varied in the experiment.</p><p>Humans sometimes do not make single saccades even when it is possible instead producing two or three saccades <xref ref-type="bibr" rid="pcbi.1002342-Zelinsky1">[31]</xref>. We considered the possibility that this particular aspect of our protocol is responsible for observed sub-optimality. An argument against this possibility is the ease with which our subjects adapted to the task. The ratio of excluded trials due to blinks or second saccade combined was between 8% and 24% of the trials (15% on average).</p><p>If the observer's saccade did not arrive within one degree of a token, we terminated the trial. This occurred on between 2.4% and 34% of trials (mean 17%), across observers. However, the horizontal and vertical standard deviations of saccades in our experiment (see Figure S1 in the <xref ref-type="supplementary-material" rid="pcbi.1002342.s001">Text S1</xref>) are large compared to the one degree cutoff we imposed and, even if the observer attempted to saccade to the center of a token on every trial, many of the resulting saccades would fall outside the one degree limit. Hence, we cannot infer that a failure to saccade to within one degree of a token (the criterion for success) indicates that the observer intended to saccade anywhere other than to the token. In particular, there is no basis to conclude that normal eye movement planning and execution has been altered by the constraints we impose.</p><p>Had we used a less stringent criterion for termination of a trial (or imposed no criterion), observers might have chosen to saccade to a location away from any of the three tokens. Our analysis depends on knowing what strategies are available to the observer and which of these they chose. We also verified that the observer's distributions of saccades to each token were approximately centered on the token and not off to one side. The distributions of saccade endpoints for all observers are show in Figure S1 of the <xref ref-type="supplementary-material" rid="pcbi.1002342.s001">Text S1</xref> as well as descriptive statistics for all the observers.</p><p>If planning consumes cognitive resources then the choice of optimal plan should reflect the “cost” of these resources to the organism <xref ref-type="bibr" rid="pcbi.1002342-Simon1">[32]</xref>. The key problem, though, is to develop experimental methods that allow us to demonstrate that these hidden costs are real and that they explain the observer's behavior.</p><p>In conclusion, we find little evidence that observers correctly use their visual sensitivity outside the fovea to optimize visual search.</p></sec><sec id="s3b"><title>Heuristic based planning</title><p>Our results are in apparent conflict with the predictions of optimal statistical models discussed in the introduction <xref ref-type="bibr" rid="pcbi.1002342-Eckstein1">[12]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Najemnik1">[13]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Najemnik2">[14]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Palmer1">[15]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Renninger1">[16]</xref>. Najemnik & Geisler <xref ref-type="bibr" rid="pcbi.1002342-Najemnik1">[13]</xref>, for example, asked observers to locate a Gabor patch in a 1/f field of noise. They compared human performance to ideal performance minimizing the expected number of saccades to find the target. As we explained in the introduction, the strategy that maximizes expected gain and probability correct in our task also would minimize the number of saccades needed to correctly discriminate the target configuration.</p><p>Our task is designed so that the visual system must have access to estimates of retinal sensitivity as a function of size or eccentricity in order to plan saccades that maximize expected gain. We, in effect, compared choice of saccade on each trial to the choice of saccade that would maximize expected gain, something we could do because of the simplicity of our design.</p><p>The key predictions of Najemnik & Geisler's model are more difficult to match to human performance in their experiments. They, for example, predict the length of the first saccade and find that the distributions of lengths of first saccades are matched to that of the ideal. However, this does not imply that any particular saccade, triggered by a particular combination of signal and noise, is in itself optimal or even close.</p><p>An alternative explanation for the results of Najemnik & Geisler <xref ref-type="bibr" rid="pcbi.1002342-Najemnik1">[13]</xref> is that human visual search is based on simple heuristics analogous to those postulated in salience models. Tatler & Vincent <xref ref-type="bibr" rid="pcbi.1002342-Tatler4">[33]</xref> for example, presented compelling evidence that saccade selection could be better predicted by oculo-motor preferences than by visual information or task (although they did not provide evidence of predictive power of these biases relative to chance) Under this account, the visual system has heuristic preferences for saccades of certain lengths or possibly a tendency to saccade to the center of mass of clusters of objects in the periphery <xref ref-type="bibr" rid="pcbi.1002342-He1">[34]</xref>, <xref ref-type="bibr" rid="pcbi.1002342-Ottes1">[35]</xref>. The second heuristic, under specific circumstances, might mimic selection of the optimal point <inline-formula><inline-graphic xlink:href="pcbi.1002342.e047.jpg" mimetype="image"></inline-graphic></inline-formula> in <xref ref-type="fig" rid="pcbi-1002342-g001">Figure 1</xref> not because it is the saccade that minimizes the expected number of saccades but because it lies near the centroid of a cluster of items in the visual field.</p><p>Such a heuristic-based approach may approximate ideal performance in some tasks while failing utterly in others. The experimenter who considers performance in a limited range of scenes may record behavior that approximates optimal but is in fact no more than a lucky coincidence of a heuristic rule and experimental conditions. Such “apparent optimality” is not rare in behavioral studies of animals <xref ref-type="bibr" rid="pcbi.1002342-Janetos1">[36]</xref> or humans <xref ref-type="bibr" rid="pcbi.1002342-Gigerenzer1">[37]</xref>. And, since the stimuli of Najemnik & Geisler <xref ref-type="bibr" rid="pcbi.1002342-Najemnik1">[13]</xref> were chosen to mimic the statistical properties of natural scenes, it is not surprising that application of visual heuristics lead to good performance in such scenes.</p><p>If human saccade decisions are based on such heuristics rather than on a computation that requires knowledge of visual sensitivity maps, we would expect a failure of adjustment when one's visual sensitivity map is changed. In fact, when observers' foveae were artificially shifted with gaze-contingent techniques, their performances in visual search were significantly worse than predicted by the ideal-observer model <xref ref-type="bibr" rid="pcbi.1002342-Michel1">[38]</xref>.</p><p>In contrast, we designed our simple task so that the visual system can only succeed if it has access to estimates of visual sensitivity for the range of sizes and eccentricities we considered. We compared human performance to optimal on a trial by trial basis. We conjecture that observers failed in our task because it is not well matched to the collection of visual heuristics that guide saccadic selection.</p></sec></sec><sec sec-type="methods" id="s4"><title>Methods</title><sec id="s4a"><title>Apparatus</title><p>Experiments were programmed in C++ using Microsoft DirectX APIs on a Pentium 3 computer running Windows XP. Stimuli were displayed on a 19-inch Sony Trinitron Multiscan G500 monitor run at a frame rate of 100 Hz with 1280×1024 resolution in pixels. A forehead bar and chinrest were used to help the observer maintain a viewing distance of 57 cm. At that distance, the full display subtended 40.4°×30.3°. The observer viewed the display binocularly. Eye movements were recorded using an Eye Link II (SR Research, Toronto, Canada) sampling eye position at 500 Hz.</p></sec><sec id="s4b"><title>Subjects</title><p>The subjects were NYU undergraduate students. Four subjects participated in the Experiment 1 (3 female) and two in Experiment 2 (1 female). They were unaware of the purpose of the experiment and all had normal or corrected-to-normal vision.</p></sec><sec id="s4c"><title>Stimuli and task</title><p>Stimuli were presented against a uniform gray background (50% white). The target configuration, represented in the inset in <xref ref-type="fig" rid="pcbi-1002342-g002">Figure 2</xref>, consisted of a light gray square with a superimposed light gray dot at either the top (dot-up configuration) or at the bottom (dot-down configuration). The tokens subtended 1° of visual angle in Experiment 1 and between 0.6° to 1.8° of visual angle in Experiment 2. The observer's task was to report whether the target was dot-up or dot-down. Observers responded by rotating the mouse wheel in one direction corresponding to dot-up, the other corresponding to dot-down. Observers were rewarded for correct responses and they were aware that they would be rewarded. Observers were instructed to reply as accurately as possible and no time was imposed on their response. They were not given any feedback regarding their response. The maximum reward was $20.</p></sec><sec id="s4d"><title>Experimental design</title><p>Each experiment comprised three phases, sensitivity mapping, decision and verification. We ran two experiments, in Experiment 1 we varied only the separation and in Experiment 2, only the size. The different separations in Experiment 1 were randomly interleaved, the different sizes in Experiment 2 were also randomly interleaved.</p><sec id="s4d1"><title>Sensitivity mapping phase (<xref ref-type="fig" rid="pcbi-1002342-g003">Figure 3A</xref>)</title><p>In the sensitivity mapping phase of each experiment a red fixation cross was first displayed at the center of the screen. After the observer's fixation was stable (eye velocity less than 10°/s and eye position within a 1° radius circle around the fixation cross) for 700 ms the cross turned white and the target appeared unpredictably to the left or right of the cross at any of five target eccentricities ranged from 0° to 12° by steps of 3°, in Experiment 1, and from 0° to 12° by steps of 6° in Experiment 2. After 500 ms, the stimulus was replaced by a response screen. The observer responded whether the target was dot-up or dot-down and had unlimited time to do so. Observer's fixation had to be stable during the entire trial or the trial was discarded otherwise. Also, blinking was not allowed. The target subtended 1° of visual angle in Experiment 1 and from 0.6° to 1.8° by steps of 0.3° in Experiment 2. Experiment 1 had 20 ( = 5×2×2) conditions repeated 20 times each (400 trials total). In Experiment 2 there were 60 ( = 3×2×2×5) conditions each repeated 25 times (1500 trials total).</p></sec><sec id="s4d2"><title>Decision phase (<xref ref-type="fig" rid="pcbi-1002342-g003">Figure 3B</xref>)</title><p>In the decision phase the observers started by fixating a red cross, positioned vertically ±4° relative to the middle of the screen. The horizontal position of the cross relative to the tokens was observer-specific in Experiment 1 (see details below) and ±3° in Experiment 2. After 700 ms of stable fixation, we displayed three aligned tokens 4° above or below the fixation plane. The tokens remained visible until the end of the trial. The central token was always centered horizontally on the screen and the side tokens were equidistant on either side.</p><p>As mentioned above the fixation cross was not centered horizontally but slightly offset to the right or to the left. If it were centered horizontally it would be equidistant to each side tokens (the potential targets) but closer to the center token. But in fact, we are not testing the choice between the right and left targets but between either target or the center token. Therefore the fixation cross was displayed in a point that was equidistant to the center token and either side token. Even if the observer preferred shorter saccades, he would always have a choice between a side saccade and a central that were of equal length. In addition, given that the separations between the side token were observer specific, so was the position of the cross.</p><p>Observers were instructed to make one and only one saccade towards one of the tokens. Once the saccade landed and the eyes were stable for 50 ms, a dot appeared in one of the side tokens. The display remained visible for 500 ms if the observer's fixation remained stable and if the eyes remained within 1° of the position where the eyes landed. If the observer blinked or tried to make another saccade the trial would be aborted and replayed later in the experiment.</p><p>We chose different separations for different observers based on the observers' sensitivity mapping data. We first fit the data by a least square criterion using a four parameter psychometric function based on the logistic cumulative distribution function <xref ref-type="bibr" rid="pcbi.1002342-Johnson1">[39]</xref>:<disp-formula><graphic xlink:href="pcbi.1002342.e048"></graphic><label>(3)</label></disp-formula>where <inline-formula><inline-graphic xlink:href="pcbi.1002342.e049.jpg" mimetype="image"></inline-graphic></inline-formula> is used to denote the eccentricity of the stimulus in degrees, the value <inline-formula><inline-graphic xlink:href="pcbi.1002342.e050.jpg" mimetype="image"></inline-graphic></inline-formula> is the probability of correct classification at the fovea (typically close to 1 for our stimuli) and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e051.jpg" mimetype="image"></inline-graphic></inline-formula> is the asymptotic probability of correct classification for large eccentricities, typically close to 0.5 which is chance performance. The parameter <inline-formula><inline-graphic xlink:href="pcbi.1002342.e052.jpg" mimetype="image"></inline-graphic></inline-formula> is the point where the probability of correct detection is <inline-formula><inline-graphic xlink:href="pcbi.1002342.e053.jpg" mimetype="image"></inline-graphic></inline-formula>. The resulting fits are shown in <xref ref-type="fig" rid="pcbi-1002342-g004">Figure 4</xref>. The values of <inline-formula><inline-graphic xlink:href="pcbi.1002342.e054.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e055.jpg" mimetype="image"></inline-graphic></inline-formula> were typically close to 0.5 and 1 for all observers with <inline-formula><inline-graphic xlink:href="pcbi.1002342.e056.jpg" mimetype="image"></inline-graphic></inline-formula> corresponding to the 75% correct point. The parameter <inline-formula><inline-graphic xlink:href="pcbi.1002342.e057.jpg" mimetype="image"></inline-graphic></inline-formula> controls the slope of the psychometric function as <inline-formula><inline-graphic xlink:href="pcbi.1002342.e058.jpg" mimetype="image"></inline-graphic></inline-formula>. The estimated values of <inline-formula><inline-graphic xlink:href="pcbi.1002342.e059.jpg" mimetype="image"></inline-graphic></inline-formula> for the four observers were S01: 5.93; S02: 4.16; S03: 4.38 and S04: 4.83 degrees with mean 4.82 degrees. The separations between the side tokens for each observer included the seven values<disp-formula><graphic xlink:href="pcbi.1002342.e060"></graphic><label>(4)</label></disp-formula>and the additional values [8°, 24°]. There were therefore nine separations in total, seven chosen based on the observers' sensitivity mapping data and two common to all observers. For S01 these were, for example, [8°, 8.85°, 9.85°, 10.85°, 11.85°, 12.85°, 13.85°, 14.85°, 24°]. There were 96 trials per separation which summed to 864 trials per observer in Experiment 1. The trials were performed in 4 different sessions preceded by training trials (25 for the first session and 5 thereafter) that were not included in the analysis.</p><p>In Experiment 2 the separation between the side tokens was a constant 12°. The token sizes ranged from [0.6°, 0.9°, 1.2°, 1.5°, 1.8°] with 112 trials per size. Thus there were 560 trials performed in 7 separate sessions preceded by training trials, as in Experiment 1.</p></sec><sec id="s4d3"><title>Verification phase (<xref ref-type="fig" rid="pcbi-1002342-g003">Figure 3C</xref>)</title><p>The purpose of this phase was to verify that our predictions of performance in the decision phase, based on measurements in the sensitivity mapping phase, were accurate. Instead of allowing the observer free choice of saccade locations, we instructed them which location to saccade to. Here we will show data for the cases where they were instructed to saccade to the optimal location. The design, represented in <xref ref-type="fig" rid="pcbi-1002342-g003">Figure 3B</xref>, was similar to the decision phase except that at the start of each trial the observer was instructed which of the three locations to saccade to: after 700 ms of initial stable fixation, one token disappeared for 400 ms and the observer could start his saccade to this indicated token as soon as it reappeared. Once the observer completed the saccade, we verified that the observer's fixation was within 1° of the indicated token.</p></sec></sec><sec id="s4e"><title>Model</title><p>As explained in the Introduction, there is a given separation (Experiment 1) or size (Experiment 2) at which observers should switch strategy. We call the optimal switch point <inline-formula><inline-graphic xlink:href="pcbi.1002342.e061.jpg" mimetype="image"></inline-graphic></inline-formula> and, in Experiment 1, it is defined as the separation between the side tokens for which<disp-formula><graphic xlink:href="pcbi.1002342.e062"></graphic><label>(5)</label></disp-formula></p><p>The right hand side of the equation describes the observer's performance when he has chosen to saccade to one of the side tokens. On half the trials, the target will appear at that side location and he will discriminate correctly with probability <inline-formula><inline-graphic xlink:href="pcbi.1002342.e063.jpg" mimetype="image"></inline-graphic></inline-formula> (he is fixating the target). On the other half the trials, the target will appear on the other side, a distance <inline-formula><inline-graphic xlink:href="pcbi.1002342.e064.jpg" mimetype="image"></inline-graphic></inline-formula> from fixation. He will discriminate correctly with probability <inline-formula><inline-graphic xlink:href="pcbi.1002342.e065.jpg" mimetype="image"></inline-graphic></inline-formula>. The overall probability of correct discrimination is the right hand side of Equation 3. The left hand side is the performance expected with a center strategy. Whether the target appears on left or right, it is a distance <inline-formula><inline-graphic xlink:href="pcbi.1002342.e066.jpg" mimetype="image"></inline-graphic></inline-formula> from fixation and the observer discriminates correctly with probability <inline-formula><inline-graphic xlink:href="pcbi.1002342.e067.jpg" mimetype="image"></inline-graphic></inline-formula>. The switch point is the point at which the two strategies lead to equal discrimination performance. For eccentricities <inline-formula><inline-graphic xlink:href="pcbi.1002342.e068.jpg" mimetype="image"></inline-graphic></inline-formula> with <inline-formula><inline-graphic xlink:href="pcbi.1002342.e069.jpg" mimetype="image"></inline-graphic></inline-formula>, saccading to the center square results in a higher probability of correct classification. For <inline-formula><inline-graphic xlink:href="pcbi.1002342.e070.jpg" mimetype="image"></inline-graphic></inline-formula> with <inline-formula><inline-graphic xlink:href="pcbi.1002342.e071.jpg" mimetype="image"></inline-graphic></inline-formula>, saccading to either of the side tokens leads to better performance.</p><p>We derived a similar equation for Experiment 2 but now in terms of target size. The optimal switch point <inline-formula><inline-graphic xlink:href="pcbi.1002342.e072.jpg" mimetype="image"></inline-graphic></inline-formula> is defined by<disp-formula><graphic xlink:href="pcbi.1002342.e073"></graphic><label>(6)</label></disp-formula>with <inline-formula><inline-graphic xlink:href="pcbi.1002342.e074.jpg" mimetype="image"></inline-graphic></inline-formula> denoting the sensitivity mapping function for each observer and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e075.jpg" mimetype="image"></inline-graphic></inline-formula>denotes the sensitivity function as a function of size for eccentricity <inline-formula><inline-graphic xlink:href="pcbi.1002342.e076.jpg" mimetype="image"></inline-graphic></inline-formula>. At this point, both strategies have the same probability of success. The ideal observer that maximizes expected probability correct will switch strategy precisely at <inline-formula><inline-graphic xlink:href="pcbi.1002342.e077.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e078.jpg" mimetype="image"></inline-graphic></inline-formula>.</p><p>We estimated <inline-formula><inline-graphic xlink:href="pcbi.1002342.e079.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e080.jpg" mimetype="image"></inline-graphic></inline-formula> for each observer in each experiment using Equations 5 and 6 and numerical optimization. We also verified that the optimal point is unique. The <inline-formula><inline-graphic xlink:href="pcbi.1002342.e081.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e082.jpg" mimetype="image"></inline-graphic></inline-formula> for each observer are shown together with the results in the next section.</p><p>The sensitivity functions <inline-formula><inline-graphic xlink:href="pcbi.1002342.e083.jpg" mimetype="image"></inline-graphic></inline-formula> and <inline-formula><inline-graphic xlink:href="pcbi.1002342.e084.jpg" mimetype="image"></inline-graphic></inline-formula> could also be estimated using the data from the decision or verification phase. We used the sensitivity function derived from the sensitivity mapping phase in the analysis reported in the main text. Using the data from either of the other two phases only led to small changes in estimated optimal switch point that do not affect our conclusions. We report those in Figure S3 in the <xref ref-type="supplementary-material" rid="pcbi.1002342.s001">Text S1</xref>.</p></sec></sec><sec sec-type="supplementary-material" id="s5"><title>Supporting Information</title><supplementary-material content-type="local-data" id="pcbi.1002342.s001"><label>Text S1</label><caption><p><bold>Supplementary figures.</bold></p><p>(DOC)</p></caption><media xlink:href="pcbi.1002342.s001.doc" mimetype="application" mime-subtype="msword"><caption><p>Click here for additional data file.</p></caption></media></supplementary-material></sec></body><back><fn-group><fn fn-type="conflict"><p>The authors have declared that no competing interests exist.</p></fn><fn fn-type="financial-disclosure"><p>This work was funded by NIH/NEI EY019889 (<ext-link ext-link-type="uri" xlink:href="http://www.nih.gov/">http://www.nih.gov/</ext-link>) and the Humboldt Foundation (<ext-link ext-link-type="uri" xlink:href="http://www.humboldt-foundation.de/web/home.html">www.humboldt-foundation.de/web/home.html</ext-link>). 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