poletti - optimal multimodal integration in spatial localization

7/27/2019 Poletti - Optimal Multimodal Integration in Spatial Localization

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Behavioral/Cognitive

Optimal Multimodal Integration in Spatial Localization

Martina Poletti,1 David C. Burr,3,4 and Michele Rucci1,2

1Department of Psychology and 2Program in Neuroscience, Boston University, Boston, Massachusetts 02215, 3University of Florence, Florence 50125, Italy,

and 4Institute of Neuroscience, National Research Council, Pisa 56124, Italy

Saccadic eye movements facilitaterapid and efficient exploration of visual scenes, but alsopose serious challengesto establishing reliablespatial representations. This process presumably depends on extraretinal information about eye position, but it is still unclear whether

afferent or efferent signals areimplicatedand howthesesignalsare combined with thevisual input.Usinga novel gaze-contingentsearchparadigm with highly controlled retinal stimulation, we examined the performance of human observers in locating a previously fixated

target aftera variable numberof saccades,a taskthatgeneratescontrastingpredictions for different updatingmechanisms. We showthat

while localization accuracy is unaffected by saccades, localization precision deteriorates nonlinearly, revealing a statistically optimalcombination of retinal and extraretinal signals.Theseresults provide direct evidence for optimal multimodal integration in the updatingof spatial representations and elucidate the contributions of corollary discharge signals and eye proprioception.

IntroductionHow do we keep track of the locations of objects in a scene?Imagine looking for a specific book in a bookcase. While goingthrough the shelves, you see another potentially interesting book,but keep looking for the desired one until realizing that it is notthere. Then, you confidently look back to reach the book youpreviously spotted. Although for most people this is an effortlessoperation, it requires complex computations. The projections ofstationary objects shift on the retina every time a saccade occurs,and many visual functions progressively decline with eccentric-ity, making localization of non-fixated objects difficult and oftenimpossible. Thus, spatial localization cannot rely solely on theimage currently present on the retina, but needs to be based on astable internal representation of the scene.

Theretinal image is not theonly sourceof spatial information.Important contributions to the updating of spatial representa-tions come from efferent oculomotor signals (von Helmholtz,1925) (known as corollary discharges; Sperry, 1950), as revealedbyexperiments in which stimuli are displayed immediately be-fore or during saccades (Hallett and Ligthstone, 1976; Wurtz,2008). It has long been argued that another extraretinal signal,

extraocular muscle proprioception, may also play a key role(Sherrington, 1918). However, experimental evidence on thefunction of eyeproprioception has remained controversial (Don-aldson, 2000), leading to the idea that this signal is primarily usedfor oculomotor calibration and learning rather than for spatialrepresentation (Lewis et al., 2001). Furthermore,it is unclear how

retinal and extraretinal signals are combined to yield the stableand unified representation of space that is necessary for accu-rately localizing objects across saccades.

Theoretically, the updating of spatial representations couldrely on a single source, such as the corollary discharge. However,a more robust and effective strategy would be to use informationfrom all the available sources. In tasks in which multiple sensorycues are available, human observers often follow a strategy of cueintegration similar to maximum likelihood (Ernst and Bulthoff,2004; Knill and Pouget, 2004), an approach that minimizes thevariance of the resulting perceptual estimate (Clark and Yuille,1990). As exemplified in Figure 1, a similar strategy could inprinciple be used in spatial localization to optimally combinelocation estimates obtained in different modalities.

To distinguish among different possible mechanisms ofspatial updating, we developed a new procedure of gaze-contingent display, which allowed us to test spatial localiza-tion after a variable number of spontaneous, unconstrainedsaccades. In this task, different mechanisms yield differentpredictions. If spatial representations are updated solely onthe basis of the efferent corollary discharge (effectively a formof dead reckoning), each saccade would contribute its ownerror, so the variance of the localization error should increasewith the number of saccades performed after fixation on agiven target. On the other hand, if the updating process reliesonly on the retinal image and/or extraocular muscle proprio-ception, localization precision should not depend on the num-ber of previous saccades. The statistically optimal integrationof all the available sources gives yet a different scenario: itpredicts the variance of the localization error to increase withthe first few saccades following target fixation and then satu-rate, as the contribution of the corollary discharge progres-sively looses reliability and is eventually no longer considered.

We show that localization of a previously fixated target is wellpredicted by the statistically optimal integration of retinal,efferent, and afferent signals.

Received Feb. 1, 2013; revised June 6, 2013; accepted July 16, 2013.

Author contributions:M.P.andM.R.designedresearch;M.P.performedresearch;M.P.,D.C.B.,andM.R. analyzed

data; M.P., D.C.B., and M.R. wrote the paper.

Thiswork wassupported byNationalInstitutesof Health GrantEY18363andNationalScienceFoundationGrant

BCS1127216(M.R.)and EuropeanResearchCouncilGrant STANIB (D.C.B.).We thankHaroldE. Bedell,Dan Bullock,

and Jonathan D. Victor for helpful comments.

Correspondence should be addressed to Michele Rucci, Department of Psychology, 2 Cummington Mall, Boston

University, Boston, MA 02215. E-mail: [email protected]:10.1523/JNEUROSCI.0523-13.2013

Copyright 2013 the authors 0270-6474/13/3314259-10$15.00/0

The Journal of Neuroscience, August 28, 2013 33(35):1425914268 14259


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Materials and MethodsSubjects. Four emmetropic subjects (two malesand two females), all nave about the purposeof the study, participated in the experiments.All fourtookpartin Experiment1, and three ofthem participated in Experiments 2 and 3. In-formed consent was obtained from all partici-pants following the procedures approved bythe Boston University Charles River CampusInstitutional Review Board.

Apparatus. Horizontal and vertical eye posi-tions were sampled at 1 kHz with a Generation6 DPI eye tracker (Fourward Technologies)and recorded for subsequent analysis. A dental-imprint bite barand a head-rest kept subjects at afixed distance from the monitor (126 cm) andprevented head movements. Stimuli wereobserved monocularly with the right eye,while the left eye was patched. They were dis-played on a fast-phosphor CRT monitor(Iiyama HM204DT) with a vertical refresh rateof 200 Hz and spatial resolution of 800 600

pixels. Stimuli were rendered and modified inreal time by means of EyeRIS (Santini et al.,2007), a system for gaze-contingent display thatenables precise synchronization between eyemovements data and the refresh of the image onthe monitor, as well as accurate spatial localiza-tion of the line of sight. Gaze-contingent proce-

dures were implemented by custom-developed

software based on OpenGLand real-timeC routines running in parallel

on the host CPU and on EyeRIS digital signal processor, respectively.

Stimuli and procedure. Data were collected in separate experimental

sessions. Every session started with preliminary setup operations that

lasted a few minutes and included positioning the subject optimally and

comfortably in the apparatus, tuning the eye tracker, and calibratingEyeRIS to accurately convert the eyeposition measurements given by the

eyetracker into screen coordinates. Subjects were never constrained in

the experimental setup for 15 min consecutively.

Accurate localization of theline of sight is importantin this study.To this

end,before eachrecordingsession, we conducted a gaze-contingent calibra-

tion,which providesmore accuratelocalizationthan thestandardprocedure

used in oculomotor research (Poletti et al., 2010). This calibration consisted

of two phases. In the first phase, average eye positions were measured as

subjects sequentially fixated on each of thedots of a three-by-three grid, as iscustomary in eye-tracking experiments. The initial mapping from eye posi-

tion coordinates to degrees of visualanglewas determined by bilinear inter-

polation over the mean eye positions measured at these nine points. In the

second phaseof the calibration procedure, subjects refined this gaze-to-pixel

mapping. They fixated again on each of the points of the grid and, at each

point, corrected the estimated location of the center of gaze, which was

displayed in real time on themonitor as a retinally-stabilized redcross. Sub-jects used dedicated controls on the EyeRIS joypad to move the cross on the

horizontal andverticalaxesuntilit matched theirperceivedfixationlocation.

These refinements were then incorporated into the offsets and gains of the

bilinearinterpolation. Thismethod improvedlocalizationof the lineof sight

by approximately a factor of three on each axis.

Experiments were conducted in complete darkness, and special carewas taken in ensuring the absence of spurious light sources. The monitor

was set to minimum contrast and brightness settings, and stimuli were

displayed for brief periods (50 ms) at low luminance (0.6 cd m 2), con-

ditions that excluded influences from phosphor persistence, as demon-

strated in previous experiments (Poletti et al., 2010). Furthermore, to

prevent dark adaptation,each trial ended with the presentation of a whitenoise mask at maximum intensity, and subjects took frequent breaks

during which the room was illuminated normally.Observers were instructed to search for two small (20 radius) red

circles, which, they were told, were hiddenbehind the black background,

but would briefly appear when fixated. Their task was to report the loca-tion of the first object (the target) upon finding the second one (theresponse cue). In reality, each circle was displayed during a selected fix-ation, with the two fixations separated by a predetermined number ofsaccades (15, 9, or 10), which variedpseudorandomlyacrosstrials. Bothcircles were displayed at the center of gaze at fixation onset. In Experi-ments 1 (see Fig. 4) and 3 (see Fig. 7), subjects used a joystick to position

a 5 cross initially displayed at the response cue location. These twoexperiments differed only for the presence or absence of a visual refer-ence, a stationary 5 white dot at maximum contrast. In Experiment 2(see Fig. 6), subjects pressed a button when fixating on the rememberedtarget location.

Data analysis. Recorded eye-movement traces were segmented intoseparate periods of fixations and saccades on the basis of the velocity ofthe trajectory. Eye movements with minimal amplitude of 3 and peakspeed higher than 3/s were selected as possible saccades. Consecutiveevents closer than 15 ms were then merged together, a method thatautomatically excluded possible postsaccadic artifacts. Saccade ampli-tude was defined as the modulus of the vector connecting the two loca-tions at which eye speed became greater (saccade onset) and lower(saccade offset) than 3/s. Trials in which eye tracking was not continu-ous were discarded. For each trial, we measured the localization error,i.e., the vector difference between the estimated and real positions of thetarget (see Fig. 4a). For each subject, trials with the same numbers ofintervening saccades were pooled together, and the variability of local-izationerror wasquantified by means of itsdispersion area, defined as thearea of the 68th percentile confidence ellipse (Steinman, 1965). Thisquantity has been used extensively in the literature; since it estimates theareawithin1 SDfromthemean, it can beregardedas theextension oftheconcept of variance to two dimensions. For every subject, dispersionareas were estimated over an average of 64 trials for each considerednumber of intervening saccades. Since for each individual subject all thetrials with the same saccades number were collapsed into one singlemeasurement of the dispersion area, the variability of this estimate andwithin subject statistical comparisons were determined by means ofbootstrap (see Figs. 4 7, error bars).

Bootstrap was also used to determine whether measurements after 9and 10 saccades differed from the predictions of the standard corollarydischarge model of error accumulation. At each bootstrap iteration k, we

Figure1. Anexample of biologicallyplausibleintegrationof efferent,afferent,and retinal signals.a,WhilelookingatobjectA(red triangle, right), an observer plans a saccade toward object B (blue square), which was at the center of gaze at fixation 0, nsaccades before (left). b, Likelihoods of independent estimates of object Bs location in retinotopic coordinates. The afferent

estimate, LA,isproportionaltothedifferencebetweenthecurrenteyeposition, eN,andthepositione0 assumedbytheeyeduringfixation on B. The efferent estimate, L

E, is proportional to the sum of all the saccades, sk, which intervened between the two

fixations. The retinal estimate, LR, is determined by the position of object B on the retinal image. These three estimates can be

combined to maximize the likelihood of localization. FA

and FE

represent mappings into retinotopic coordinates. This scheme ismeant to provide an intuitive example of how independent estimates can be obtained and integrated in spatial localization.Several otherplausibleimplementations of thisstrategythat do notnecessaryrely on retinotopicrepresentations areconceivable.

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first obtained an estimate of thedispersion area of each subject, Dk(n),byrandom sampling with replacement from all of the n saccade trials (n 13) availablefor the considered subject. We then usedthe bootstrappedareas from all subjects to estimate, by means of least-squares interpola-tion, the areas Ek(n) predicted by a purely efferent mechanism of spatialupdating. Under the assumption of independence of the localizationerror introduced by each saccade, the corollary discharge model predictsthat each saccade enlarges the dispersion area by a fixed amount:

E n O2 nE

2, (1)

where O2 is an error associated with the reporting method (in Experi-

ment 1,placing a cursor on the display),andEis thecorollary dischargeerror. This method enabled estimation of the distributions of the corol-lary discharge predictions at n 9 and 10 saccades, which were com-pared with the dispersion areas measured experimentally by means oftwo-sample two-tailed ttests. The method was first tested by means ofsimulations with random data extracted from various distributions,which confirmed the validity of the test and the correctness of the result-ing levels of significance.

Since, as shown by our data, the dispersion area increases at a progres-sively lower rate with saccades, estimation of the corollary dischargemodelby linearregression of thedispersion areas measured after n 13

saccades actually underestimates the error predicted by a mechanism ofspatial updating entirely based on corollary discharge. Results did notchange if the corollary discharge model was estimated over the first foursaccades, rather than the just the first three. These two numbers of sac-cades, three and four, have been used previously in the literature tomeasure the influence of extraretinal signals in multiple-step saccadetasks (Bock et al., 1995; Collins, 2010).

The ideal observer curves shown in Figures 4 7 were obtained byleast-squares fitting the optimal integration model to the data. Thismodel assumes that the localization error is isotropic and normally dis-tributed. It has three parameters: the SDs of the efferent (E, after onesaccade), afferent (A), and retinal signals (R). After n saccades, themodel predicts that the localization error will have variance M

2 (n):

M

2

n

E

2

n nE2

A

2

n A2

R

2

n R2

, (2)

where

E 1/ nE

2

1/ nE2 1/A

2 1/R

2 , (3)

A

1/A2

1/ nE2 1/A

2 1/R

2 , (4)

R 1/R

2

1/ nE2 1/A

2 1/R

2 . (5)

The predicted dispersion area, the area of the 68th percentile confidencecircle,is given byA(n) 2log[(1 0.68)1]

M

2 (n). Theassumption ofradial symmetry is supported by the data in Figure 4, but the model canbe easily extended to eliminate this assumption. In complete darkness(Experiment 1 and 2), R was set to infinity, and onlyA and E wereestimated. Identification of these two parameters is possible because theafferent signal gives a constant localization error independent of thenumber of saccades, whereas the localization error given by the efferentsignal increases with the number of saccades. In the presence of a visualreference, R was also estimated. In this case, the model cannot distin-guish betweenRandA, as both terms give a constant localization erroracross saccades,and we thereforesetA to thevalue estimatedpreviouslyin Experiment 1.

The optimal integration model was also compared to the corollarydischarge linear model that best interpolated all of the available datapoints (n 15, 9, and10 saccades).Differentmodels were compared by

means of a corrected version of the Akaike information criterion (AIC;Akaike, 1974; Hurvich and Tsai, 1989). This criterion determines whichmodel among those compared is more likely to be closer to the true

model in the sense of minimizing KullbackLeibler discrepancy. AICCvalues reported in the text are averages across subjects.

Our corollary discharge model in Equation 1 assumes that the local-ization error increases linearly with the number of saccades. However,the predictions of a purely efferent mechanism of spatial updating maydeviate fromlinearity because of two factors: (1) lack of independence inthe errors introduced by separate saccades and (2) the finite size of thedisplay. If correlations between successive saccades exist, inaccuracies in

the corollary discharge could yield correlations in the localization errorsgiven by separate saccades. In this case, the dispersion area would in-crease nonlinearly with rate determined by the extent of the correlation.Furthermore, in Experiment 1, subjects could report the position of thetarget by moving the cursor only up to the edges of the monitor. Thislimitation in the working area implies that the reported dispersion areacannot grow larger than the surface of the monitor. For this reason, evena perfect corollary discharge mechanism of spatial updatingone inwhich the internal copy of the saccade is veridicalwould eventuallydeviate from linearity and saturate.

To examinethe impact of both factors, for each subject, we conductedMonte Carlo simulations of the individual experimental trials. Thesesimulations estimatedthe error distribution that maybe expected fromalinear model of the corollary discharge, given the sequences of saccadesperformed by the subject in the experiments. We assumed that, on eachCartesian axis k, the internal representation of the saccade displacementSk was linearly related to the actualsaccade shift, Sk: Sk kSk k k,where k is a random error with normal distribution N(0, k). For eachindividual subject, k and k were estimated on the basis of the localiza-tionerrors measured in theone-saccade trials,em (e1,m, e2,m), by linearregression of the points (ek,m, Sk,m), where Sk,m represents the kth com-ponent (k 1, 2) of the saccade performed in the mth trial. The remain-ing parameter k and a possible offset at zero saccades, ok, representingthe variability associated with the reporting method (the process of plac-ing the cursor on the display) were estimated by optimally fitting, in theleast-squares sense, the SDs of the localization errors measured on thecorresponding axis after one to three saccades.

In the Monte Carlo simulations, for each experimental trial (i.e., thecombination of targetposition andrecordedsaccade sequence), we com-

puted thedistribution of theestimatedtarget location resulting from thismodel and compared it to the experimental data. To take into accountthe effect of the monitor size, estimated positions outside of the displayboundaries were adjusted by taking the closest point on the monitoredges. The results of these simulations are given in Figure 5. They showthat correlations across saccades had little effect on the predictions of acorollary discharge model, even when combined with the bounding ef-fect of the limited working area of the display. The data in Figure 5 wereobtained by fitting the corollary discharge model in a coordinate systemaligned withthe saccade (asin Fig. 4b). Very similar results were obtainedby fitting the model in the original reference system (as in Fig. 4a).

ResultsObservers were instructed to search for two small circles, which,

they were told, were hidden behind the black background butwould briefly appear when fixated. Their task was to report thelocation of the first object (the target) upon finding the secondone, either by placing a cursor (Experiment 1) or by looking backat its remembered location (Experiment 2). In reality, the twocircles were displayed sequentially at the center of gaze, each dur-ing a separate fixation (Fig. 2a). By presenting visual stimuli atdifferent locations in space but at the same position on the ob-servers retina, this procedure always resulted in the same retinalstimulation independent of the eyemovements performedby theobserver.

We first examined performance in total darkness, a conditionin which spatial localization can only occur on the basis of effer-

ent and/or afferent signals. These two mechanisms yield differentpredictions: if spatial representations are updated from the effer-ent corollary discharge, each saccade will contribute its own in-

Poletti et al. Optimal Integration in Spatial Localization J. Neurosci., August 28, 2013 33(35):1425914268 14261


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dependent error, so the variance of the localization error shouldincrease linearly with the number of saccades performed afterexposure to the target (Fig. 2b). On the other hand, an afferentsignal from eye proprioception should be independent of thenumber of intervening saccades, so, apart from a possible mem-ory decay with time, the resulting variance should remain con-stant (Fig. 2c).

Both strategies are, however, suboptimal. Use of the corollarydischarge may work well for a recently fixated target, but aftermany saccades, the error will eventually become unacceptablylarge. In contrast, an estimate based on extraocular propriocep-tion may be relatively imprecise (Donaldson, 2000), but will out-perform the corollary discharge after a sufficient number of

saccades. The probabilistically optimal strategy is to combineboth signals based on their precision (Fig. 2d), an approach thatyields consistent estimates with minimum variance (Clark andYuille, 1990; Ernst and Bulthoff, 2004; Knill and Pouget, 2004).This strategy predicts that the variance of the localization errorwill increase almost linearly with the first few saccades, as it ini-tially relies heavily on the more precise efferent signal, and thenprogressively saturate as more weight is allocated to the afferentsignal (Fig. 2e).

Even though observers were asked to search in complete dark-ness, sequences of saccades were very similar to those measuredwhen the same task was conducted in the presence of a visualreference and, as shown in Figure 3, also similar to those occur-

ring during free viewing of pictures of natural scenes.Interestingly, localization accuracy was little affected by sac-cades (Fig. 4a,b). For all subjects, on both the horizontal and

vertical axes, the mean localization error after 10 saccades wasvery close to zero (averages across observers,x-axis, 0.14 0.97;

p 0.79; y-axis, 0.03 0.39) and statistically indistinguish-able from the error measured after just one saccade (p 0.43,paired two-tailed ttest). However, the variability of the estimatedlocation increased with saccades so that, for all subjects and onboth Cartesian axes, variances after 10 saccades were significantlylarger than those measured after just one saccade (p 0.002,two-tailed Ftest of equality of variances). Thus, spatial localiza-tion remained accurate on average, but lost precision as the num-ber of saccades increased.

We quantified the precision of localization after a given num-ber of intervening saccades by the dispersion area, defined as the

area of the68th percentile confidence ellipse (the directextensionof the variance to two dimensions; Steinman, 1965). With only afew saccades, the dispersion area appeared to increase almostlinearly, a finding that,without additionaldata points, couldhavebeen easily mistaken as evidence for a corollary discharge mech-anism of spatial updating. However, it deviated from linearitywith additional saccades (Fig. 4c; see Fig. 5 for individual subjectdata). In all subjects, the increment in variability occurring withthe first five saccades (13.8 1.7 deg2) was significantly largerthan the change caused by the following five saccades (6.8 2.3deg2; p 0.05, paired two-tailed t test). As a consequence, thedispersion areas after 9 and 10 saccades were significantly smallerthan the predictions of the standard corollary discharge model of

error accumulation based on the rate of error increment with thefirst three saccades (p 0.001, two-tailed ttest). This deviationfrom the prediction of constantly increasing variance cannot be

Figure2. Experimentalprocedureand theoreticalpredictions.a,Two20 radiuscircles,thetargetandtheresponsecue,weresequentiallydisplayedatthecenterofgazeafter n saccades(s1

sn). Observers were asked to search for the two cues and report the remembered location of the target upon appearance of the response cue, either by placing a cursor (Experiment 1, visuallocalization) or bylooking back (Experiment2, oculomotor localization).The spatial positionsof both thetarget andtheresponsecue (X

TandX

R) variedacross trialsdepending onthe subjectseye

movements.bd, Predicted precision of different localization strategies.The variance of the localization error is expected to increase proportionally to the number of saccades between the targetandthe cuewitha purelyefferentmechanismof spatial updating (b),and toremainconstant with a purely afferentone (c). d, e, Theoptimalintegrationmethod is tocombine both sources,eachweightedinverselytoitsvariance(2

k).d, This strategypredictsthatthe localizationerrorwillfirstincrease almostlinearlyandthensaturateas saccadesoccur,since (e) weights are progressively

reallocated from the corollary discharge (E) to eye proprioception (

A). L

E, L

A, and L

Orepresent location estimates in a gaze-centered frame of reference.



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ascribed to differences in scan path for different numbers of sac-cades. The screen area covered by three saccades was, on average,56 5 deg2, compared with 52 8 deg 2 for nine saccades.

Similarly, there was no difference in the average distance betweenthe target and the response cue, on average 4.9 0.8 and 4.1 0.6 after three and nine saccades, respectively. These results arenot compatible with the widely held assumption that spatial rep-resentations are updated exclusively on the basis of the corollarydischarge.

As shown in Figure 4b, the data closely followed the predic-tions of a strategy of optimal integration of both afferent andefferent signals. The relationship of dispersion area to saccadenumber was best fit (least-squares) by a maximum likelihoodmodel in which the SD of the corollary discharge (E, the vari-ability after one saccade) was approximately half the SDA of theproprioceptive signal (mean across subjects, E 1.31 0.13,

A 2.44 0.79). This model explained almost all the varianceof the data (adjusted R2 0.95 0.03) and provided a muchbetter fit than the best (least-squares) discharge model of erroraccumulation estimated on the dispersion areas measured withthe first three saccades, a model that yielded a negative coefficientof determination. The optimal integration model was also, onaverage,801 times more likely tobe close to the true model, inthesense of minimizing the KullbackLeibler distance, than the stan-dard corollary discharge model used in the literature, the best-fitting first-order linear model of all measured dispersion areas(mean AICC, 11.36 3.58). This latter model explained asmaller amount of variance (adjusted R2 0.73 0.18) andpredicted an error with no saccades (8.53 1.57 deg2) that was

implausibly as large as the error after one saccade.According to the optimal integration model, eye propriocep-tion contributed 20% after one singlesaccade,a value similar to

the estimates obtained at fixationby previous studies (Gauthier etal., 1990; Bridgemanand Stark,1991). However, the contributionof proprioception increased with saccades so that this signalquickly became the predominant source of information (Fig. 4d).

These findings cannot be explained by possible memory de-cays, which would only further increase localization errors in thetrials with larger numbers of saccades: regardless of possiblememory contributions, subjects were far more precise thanexpected in these trials. Results were also not influenced byfactors that could make the predictions of a purely efferentmodel of spatial updating deviate from linearity, such as pos-sible correlations in saccade directions and the limited area ofthe display (Fig. 5).

If saccades within a trial are correlated, systematic inaccura-cies in the efferent signals would cause statistical dependencies inthe localization errors resulting from separate saccades. In thiscase, the model of Equation 1 no longer holds, and the predicteddispersion area would increase nonlinearly with saccades. Fur-thermore, limitations in the working area of the displayi.e., thefact that subjects could report the position of the target by mov-ing the cursor only up to the edges of the monitorimply thatthe dispersion area cannot grow larger than the surface of themonitor, so that even a perfect corollary discharge mechanism ofspatial updating will eventually deviate from linearity and satu-rate. These effects, however, did not alter our conclusions, asdemonstrated by the results of dedicated Monte Carlo simula-tions in Figure 5. The estimated dispersion areas at 9 and 10saccades differed significantly from the predictions of a corollary

discharge model of spatial localization even when (a) this modelwas individually fit for each subject, (b) it was applied to thesequences of eye movements recorded in the experiments, and

Figure3. Characteristicsof eyemovements. Probabilitydistributionsof fixation durations(top) andsaccade amplitudes(bottom)in theabsence(left; datafrom Experiments1 and2 combined)andpresence(center)of a visual reference.For comparison,fixationdurationsand saccade amplitudesmeasuredduring normal examinationof a sceneare alsoshown (right).In thiscondition, thesame observers freely viewed pictures of natural scenes, each presented for 10 s. Data from all subjects were pooled together. The red line and number in each panel represent the mean of thedistribution.



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(c) localization errors were corrected to take into account thefinite size of the display (see Materials and Methods).

Furthermore, very similar results were also obtained whenobservers reported the target location by looking back at the re-membered position rather than by placing a cursor (Experiment2; Fig. 6), a condition in which the finite size of the monitor was

not an issue. These results reveal that the information providedby extraocular muscle proprioception is taken into account inevaluating the location of a previously fixated target and that the

contributions of afferent and efferent signals are weighted in away that conforms to the rules of optimal cue integration.

The results ofFigures 4 6 were obtained in complete dark-ness. To determine whether a scheme of optimal cue integrationcontinues to hold in the presence of visual landmarks, we re-peated Experiment 1 while maintaining a single visual reference

(a 5 dot) fixed at the center of the display for the entire durationof the trial (Experiment 3). In this way, the target and the refer-ence were simultaneously visible, and the task could in principle

Figure4. Visuallocalization(Experiment1).a,Summaryofalltrials.Eachdotrepresentsthelocalizationerrorinanindividualtrial.Differentpanelsshowtrialswithdifferentnumbersofsaccadesbetween the target and the response cue. The mean error (red dot) and the 95% confidence ellipse are shown in each panel together with the marginal probability distributions and their best

Gaussian fits,(,) (red curves). Data from all subjects (N 4) were pooled together. b, Same data as in a after rotating the axes to align the abscissa with the cue-target direction. c, Meandispersionareaacrosssubjectsasafunctionofthenumberofsaccades.Asterisksmarksignificantdeviations( p 0.001,two-tailedpairedttests),fromthepredictionsofapurelyefferentestimate,as given by the linear regression of the measurements obtained with the first three saccades (blue line). The black curve represents the least-squares fit of the ideal observer model. d, Optimalweighting of afferent and efferent estimates.As thenumber of saccades increases,proprioceptionis weighted morestrongly andeventuallybecomesthe predominantsource of information. Errorbars and shaded regions in cand drepresent SEM.



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be accomplished just on the basis of the visual input, by remem-bering the targets location relative to the reference. Like propri-oception, this retinal cue does not depend on the number ofsaccades occurring after presentation of the target.

The visual reference greatly improved the precision of local-ization (Fig. 7). For each individual subject, the dispersion area

was now smaller by a factor of 2 after 1 saccade (p 0.05, pairedtwo-tailed ttest) and by a factor of 4 after 10 saccades (p 0.001;confront with Fig. 4c). Yet, apart from this scale change, theshapeofthe error function remained strikingly similar to that observed incomplete darkness. Thelocalizationerror sharplyincreased with thefirst three saccades and settled on a constant value after four sac-cades, so that the dispersion areas measured after 9 and 10 saccadesweresignificantlydifferent fromthose predicted by a linearmodel oferror accumulation based on the data points measuredwith the firstthree (or four) saccades (p 0.04; two-tailed ttest).

As in complete darkness, experimental data followed closelythe predictions of an optimal cue integration strategy (Fig. 7a).When efferent and afferent parameters were constrained to be

equal to those obtained in Experiment 1, the maximum likeli-hood model that best fittedthe data was 3316 times more likelytobe correct than the best-fitting corollary discharge model of error

accumulation (mean AICC, 15.44 2.55). Furthermore, whenboth E and R were left free to vary, the optimal integrationmodel yielded an SD of the corollary discharge signal that wasvery similar to that measured in the previous experiments (E 1.08 0.13).

The transition from progressive loss of precision with the first

few intervening saccades to saturation with the following sac-cades is incompatible with a mechanism of spatial updating thatrelies on one single signal, but emerges naturally from the opti-mal integration of multiple signals. Again, this transition cannotbe explained by memory influences. The number of saccades isinevitably correlated with the duration of the trial, which maylead to loss of precision due to memory restraints. However, anymemory-driven effects should result in increased variability forlargernumbers of saccades,whereas ourresults show that the rateof increase in dispersion errors decreases with saccade number,contrary to predictions from decay of memory traces. The opti-malintegrationmodelattributes theinitial loss in precisionto thedecline in reliability of the corollary discharge estimate, which is

as precise as the retinal signal after one single saccade (R 1.27 0.20), but becomes less reliable as more saccades occur.Interestingly, proprioception continued to be considered, and its

1 5 100

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Figure 5. Individual subject data. The dispersion areas measured in the experiments are compared to the predictions of a corollary discharge model of spatial localization adjusted to take intoaccount the finite size of the display and possible biases in the internal representation of saccades (blue line). The adjustment was obtained by means of Monte Carlo simulations of the individualexperimentaltrials,inwhichthedistributionofpossibletargetpositions,boundedbythemonitoredges,wasestimatedbyapplyinganindividuallyfittedcorollarydischargemodeltotherecorded

sequenceofeyemovements.Asterisksmarkstatisticallysignificantdifferencesbetweenpredictedandmeasureddispersionareas(p 0.02,two-tailedttest).Forcomparison,thepredictionofthecorollarydischargemodelestimatedonarandomconcatenationofsaccadesfromdifferenttrialsandwithoutconsiderationofthemonitorboundariesisalsoshown(dashedline).Theleast-squaresparameters of the optimal integration model are shown in each panel (

Eand

A). Error bars indicate SEM.



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contribution grew from 10% after one saccade to 20% after10 saccades (Fig. 7b). These data strongly support a scheme ofspatial updating based on the optimal integration of differentcues, each weighted by its current reliability.

DiscussionStable spatial representation is essential for visual exploration,motor control, and ultimately for survival. Therefore, the visualsystem must have developed strategies to optimize the mainte-

nance of stable representations under evolutionary pressure. Theresults of this study show that the accuracy and precision bywhich human observers locate a previously fixated target closely

follow the predictions of the optimal integration of efferent, af-ferent, and retinal signals. These findings are notcompatible withan updating mechanism based on a single signal, either the cor-ollary discharge or the retinal input.

In this study, stimuli were modified in real time according tothe observers eye movements. This gaze-contingent procedureprovides a departurefrom thestandard methods used to examinethe influence of saccades on spatial localization (Hallett andLigthstone, 1976). Traditional methods only allowanalysis of the

effects of a few saccades, as their extensions to longer saccadicsequences yield confounding factors in the interpretation of ex-perimental data. For example, the extension of the dual-step sac-

Figure6. Saccadiclocalization(Experiment2).a,Meandispersionareaacrosssubjectsasafunctionofthenumberofsaccades.Asterisksmarksignificantdeviations( p 0.05,two-tailedpairedttests) from the predictions of a purely efferent estimate, as given by the linear regression of the measurements obtained with the first three saccades (blue line). The black curve represents theleast-squares fit of the ideal observer model. b, Optimal weighting of afferent and efferent estimates. Symbols and graphic conventions are the same as in Figure 4.

Figure7. Influenceofavisualreference. a, Mean dispersion area acrosssubjectsas a functionof thenumberof interveningsaccades. Theblack curve represents theleast-squaresfit ofthe idealobservermodelthatintegratesafferent,efferent,andretinalsignalswiththeoptimalweightcombination(b).Conditionswereidenticalto thoseof Experiment1,exceptforthepresenceofa 5 dotat the center of the display throughout each trial. Symbols and graphic conventions are the same as in Figure 4.



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cade task to multiple saccades (Bock et al., 1995; Collins, 2010)implies that both memory requirements (the number of loca-tions the subject needs to remember) and visual input character-istics (the stimulated retinal locations) change with the numberof saccades. Our approach circumvents these problems. In ourexperiments, the memory load was equalized, as subjects onlyneeded to remember one spatial location, no matter how many

saccades performed. Furthermore, the spatiotemporal stimuluson the retina was always the same, because all stimuli were dis-played at the center of gaze. Since observers performed normalexploratory saccades, our method also eliminated the possibilityof planning more than one saccade at the same time (McPeeketal., 2000). To our knowledge, only one previous pioneering studyused a somewhat similar approach (Karn et al., 1997), but thisstudy only examined performance after two and five saccades,making it impossible to determine whether localization precisiondeteriorated in a linear or nonlinear fashion.

The mechanismsresponsible for establishing and maintainingstable representations during eye movements have been the sub-ject of fierce unsolved controversy (Sherrington, 1918; von

Helmholtz, 1925; for reviews, see Medendorp, 2011; Tatler andLand, 2011; Hamker et al., 2011). Our findings contribute toreconciling two vast bodies of conflicting experimental observa-tions. The first set of data consists of results often taken as evi-dence that spatial representations are exclusively updated on thebasis of the corollary discharge. These observations include (a)the retained capability of compensating for presaccadic oculardisplacements after deafferentiation of the extraocular muscles(Guthrie et al., 1983), a procedure that is supposed to eliminateeye proprioception; (b) the results of double- and multiple-stepsaccades (Bock et al., 1995; Collins, 2010) and the correspondingimpairments measured after inactivation of neural pathways in-volved in signaling corollary discharges (Sommer and Wurtz,2002); and (c) the illusory motion experienced during passive eye

rotations (von Helmholtz, 1925) and ocular paralysis (Matin etal., 1982). Allof these observations are, however, fullycompatiblewith the optimal integration scheme proposed here, as they wereobtained by means of experimental paradigms that only required afew saccades. As shown by our data, the optimal combination strat-egy weights heavily the corollary discharge under these conditions.

In contrast with this literature, a second set of experimentalobservations has provided support to a role for proprioception inthe updating of spatial representations. These results include ac-curate localization of targets after very large numbers of saccades(Skavenski and Steinman, 1970), correction for passive displace-ment of the line of sight in darkness (Skavenski, 1972), as well asthe localization errors occurring after passive eye rotations (Gau-

thier et al., 1990; Bridgeman and Stark, 1991) or following alter-ations of proprioceptive signals (Allin et al., 1996; Lennerstrandet al., 1997; Balslev and Miall, 2008). These results are also wellexplained by a model of optimal integration that uses eye propri-oception, in addition to other signals, to localize objects in space.In this regard, it is worth observing that the typical durations ofnatural fixationsare sufficiently long for proprioceptivesignals toreach the cortex (Xu et al., 2011). Even though stimuli were dis-played very briefly in our experiments, the fixations in whichtargets appeared were significantly longer than average, allowingsufficient time for proprioception to contribute to the process ofspatial updating (Fig. 8).

Our finding that spatial localization follows the predictions of

an optimal integration strategy is consistent with previous results(Niemeier et al., 2003; Munuera et al., 2009; Ziesche and Hamker,2011) and adds to a considerable body of evidence showing that

humans tend to perform in a statistically optimal manner in taskswhere multiple cues are available (Ernst and Banks, 2002; Stockerand Simoncelli, 2006; Freeman et al., 2010; Geisler, 2011). Severalindependent signals convey information about the location of apreviously fixated object. These signals can be easily transformedinto a common coordinate system, such as saccadic vectors orpositions in retinotopic or head-centered coordinates. Unlikemany tasks, however, the relative precision of these estimatesdepends not only on the characteristics of each individual signal,but also on the observers recent behavior, i.e., on the sequence of

saccades performed. This important feature distinguishes a max-imum likelihood model from previously proposed multimodalmodels in spatial localization. Previous experiments that at-tempted to quantify the relative contribution of afferent and ef-ferent signals have yielded different outcomes (Gauthier et al.,1990; Bridgemanand Stark,1991; Li and Matin,1992).Ourstudyshows that variable contributions are to be expected dependingon the cues available in each specific task and on the observersoculomotor activity.

Interestingly, in our experiments with a visual reference, ob-servers weighted the efferent signal by approximately the sameamount as the retinal signal, a finding that appears to contrastwith the general notion that the retinal input prevails over ex-

traretinal contributions (Matin et al., 1982). In these experi-ments, subjects were presented with an isolated stationary dot forthe entire duration of the trial. A stationary dot provides, in prin-ciple, an optimal reference for establishing spatial relationships,but also differs greatly from the visually rich scenes that humansnormally encounter. Future studies will need to examine the con-tribution of the retinal input with more natural stimulation andtest the specific prediction raised by our model that also thissignal is weighted according to its reliability.

Under natural viewing conditions, the need emerges to keeptrack of multiple locations, each observed with one or more ded-icated fixations. Our findings suggest that the visual system up-dates the representation of each location on the basis of how

many saccades have occurred since the corresponding fixation.There are various plausible ways in which this spatial updatingscheme may be implemented in neuronal populations. Several

Reference

No reference

Fixa

tion

dura

tion

(ms

)

Target Others OthersTarget

Figure 8. Fixation duration. Comparison between the mean duration of the fixations inwhich the target was displayed (Target) and the mean duration of the other fixations in the

searchtask (Others).Values represent averages acrosssubjects.Data fromExperiments1 and 2were combined. *p 0.047 (two-tailed paired ttest).



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mechanisms have been proposed for statistically optimal combi-nation of neural signals, with supporting evidence (Ma et al.,2008; Fetsch et al., 2011). An interesting possibility, compatiblewith current theories, is that the same populations of neuronsthat convey each individual estimate also signal its reliability. Forexample, the precision of the location estimated by the corollarydischarge could be inferred by the spread of activity in a popula-

tion of neurons with shifting receptive fields. If each remappingoperation causes an enlargement of the focus of activity, thespread in the pattern of activity representing each target will beproportional to number of elapsed saccades, therefore effectivelysignaling the reliability of the estimate. A similar mechanism mayalso occur in theother modalities. For example, in thepresenceofa crowded scene,the activity of a visualmap signalingthe locationof a targetmay be more distributed, therefore implicitly signalinga lower reliability of the cue. Further work is needed to elucidatethe neural mechanisms by which the visual system optimallycombines afferent, efferent, and retinal signals to efficiently local-ize objects in space.

ReferencesAkaike H (1974) A new look at the statistical model identification. IEEE

Trans Automat Control 19:716723. CrossRefAllin F, Velay JL, Bouquerel A (1996) Shift in saccadic direction induced in

humans by proprioceptive manipulation: a comparison between memory-guidedand visuallyguided saccades. ExpBrainRes 110:473481. Medline

Balslev D, Miall RC (2008) Eye position representation in human anteriorparietal cortex. J Neurosci 28:89688972. CrossRef Medline

Bock O, Goltz H, Belanger S, SteinbachM (1995) On the role of extraretinalsignals for saccade generation. Exp Brain Res 104:349350. CrossRefMedline

Bridgeman B, Stark L (1991) Ocular proprioception and efference copy inregistering visual direction. Vision Res 31:19031913. CrossRef Medline

Clark JJ, Yuille AL (1990) Data fusion for sensory information processingsystems. Boston: Kluwer.

Collins T (2010) Extraretinal signal metrics in multiple-saccade sequences.

J Vis 10(14):7 114. CrossRef MedlineDonaldson I (2000) The functions of the proprioceptors of the eye muscles.

Phil Trans R Soc Lond B Biol Sci 335:16851754. MedlineErnst MO, Banks MS (2002) Humans integrate visual and haptic informa-

tion in a statistically optimal fashion. Nature 415:429433. CrossRefMedline

Ernst MO, Bulthoff HH (2004) Merging the senses into a robust percept.Trends Cogn Sci 8:162169. CrossRef Medline

Fetsch CR, Pouget A, DeAngelis GC, Angelaki DE (2011) Neural correlatesof reliability-based cue weighting during multisensory integration. NatNeurosci 15:146154. CrossRef Medline

Freeman TC, Champion RA, Warren PA (2010) A Bayesian model of per-ceived head-centered velocity during smooth pursuit eye movement.Curr Biol 20:757762. CrossRef Medline

Gauthier GM, Nommay D, Vercher JL (1990) The role of ocular muscleproprioception in visual localization of targets. Science 249:58 61.CrossRef Medline

Geisler WS (2011) Contributionsof idealobservertheory to vision research.Vision Res 51:771781. CrossRef Medline

Guthrie BL, Porter JD, Sparks DL (1983) Corollary discharge provides ac-curate eye position information to the oculomotor system. Science 221:11931195. CrossRef Medline

Hallett PE, Lightstone AD (1976) Saccadic eye movements towards stimulitriggered by prior saccades. Vision Res 16:99106. CrossRef Medline

HamkerFH, Zirnsak M, Ziesche A, Lappe M (2011) Computational modelsof spatial updating in peri-saccadic perception. Phil Trans R Soc Lond BBiol Sci 366:554571. CrossRef

Hurvich CM, Tsai CL (1989) Regression and time series model selection insmall samples. Biometrika 76:297307. CrossRef

Karn KS,Mller P, HayhoeMM (1997) Reference framesin saccadic target-ing. Exp Brain Res 115:267282. CrossRef Medline

Knill DC, Pouget A (2004) The Bayesian brain: the role of uncertainty inneural coding and computation. Trends Neurosci 27:712719. CrossRefMedline

Lennerstrand G, Tian S, Han Y (1997) Effects of eye muscle proprioceptive

activation on eyepositionin normaland exotropicsubjects. Graefes ArchClin Exp Ophthalmol 235:6369. CrossRef MedlineLewis RF, Zee DS, Hayman MR, Tamargo RJ (2001) Oculomotor function

in the rhesus monkey after deafferentation of the extraocular muscles.Exp Brain Res 141:349358. CrossRef Medline

Li W, Matin L (1992) Visual direction is corrected by a hybrid extraretinaleye position signal. Ann N Y Acad Sci 656:865867. CrossRef

Ma WJ, Beck JM, Pouget A (2008) Spiking networks for Bayesian inferenceand choice. Curr Opin Neurobiol 18:217222. CrossRef Medline

Matin L, Picoult E, Stevens JK, Edwards MW Jr, Young D, MacArthur R(1982) Oculoparalytic illusion: Visual-field dependent spatial mislocal-izations by humans partially paralyzed with curare. Science 216:198201.CrossRef Medline

McPeek RM, Skavenski AA, Nakayama K (2000) Concurrent processing ofsaccades in visual search. Vision Res 40:24992516. CrossRef Medline

Medendorp WP (2011) Spatial constancy mechanisms in motor control.

Phil Trans R Soc Lond B Biol Sci 366:476491. CrossRefMunuera J, Morel P, Duhamel JR, Deneve S (2009) Optimal sensorimotor

control in eye movement sequences. J Neurosci 29:30263035. CrossRefMedline

Niemeier M, Crawford JD, Tweed DB (2003) Optimal transsaccadic inte-gration explains distorted spatial perception. Nature 422:76 80.CrossRef Medline

Poletti M, Listorti C, RucciM (2010) Stability of thevisual world duringeyedrift. J Neurosci 30:1114311150. CrossRef Medline

Santini F, Redner G, Iovin R, Rucci M (2007) EyeRIS: a general-purposesystem for eye movement contingent display control. BehavRes Methods39:350364. CrossRef Medline

Sherrington CS (1918) Observation on the sensual role of the propriocep-tive nerve supply of the extrinsic ocular muscles. Brain 41:332343.CrossRef

Skavenski AA (1972) Inflow as a source of extraretinal eye position infor-mation. Vision Res 12:221229. CrossRef Medline

Skavenski AA, Steinman RM (1970) Control of eye position in the dark.Vision Res 10:193203. CrossRef Medline

Sommer MA, Wurtz RH (2002) A pathway in primate brain for internalmonitoring of movements. Science 296:14801482. CrossRef Medline

Sperry RW (1950) Neural basis of the spontaneous optokinetic responseproduced by visual inversion. J Comp Physiol Psychol 43:482489.CrossRef Medline

Steinman RM (1965) Effect of target size, luminance, and color on monoc-ular fixation. J Opt Soc Am 55:11581165. CrossRef

Stocker AA, Simoncelli EP (2006) Noise characteristics and prior expecta-tions in human visual speed perception. Nat Neurosci 9:578585.CrossRef Medline

Tatler BW,LandMF (2011) Vision andthe representation of the surround-ings in spatial memory. Phil Trans R Soc Lond B Biol Sci 366:596610.CrossRef

von HelmholtzH (1925) Treatiseon physiologicaloptics. New York: Dover.Wurtz RH (2008) Neuronal mechanisms of visual stability. Vision Res 48:

20702089. CrossRef MedlineXu Y, Wang X, Peck C, Goldberg ME (2011) The time course of the tonic

oculomotor proprioceptive signal in area 3a of somatosensory cortex.J Neurophysiol 106:7177. CrossRef Medline

Ziesche A, Hamker FH (2011) A computational model for the influence ofcorollary discharge and proprioception on the perisaccadic mislocaliza-tion of briefly presented stimuli in complete darkness. J Neurosci 31:1739217405. CrossRef Medline

http://dx.doi.org/10.1109/TAC.1974.1100705http://www.ncbi.nlm.nih.gov/pubmed/8871106http://www.ncbi.nlm.nih.gov/pubmed/8871106http://www.ncbi.nlm.nih.gov/pubmed/8871106http://dx.doi.org/10.1523/JNEUROSCI.1513-08.2008http://dx.doi.org/10.1523/JNEUROSCI.1513-08.2008http://www.ncbi.nlm.nih.gov/pubmed/18768690http://dx.doi.org/10.1523/JNEUROSCI.1513-08.2008http://www.ncbi.nlm.nih.gov/pubmed/18768690http://dx.doi.org/10.1016/S0079-6123(08)61800-2http://dx.doi.org/10.1016/S0079-6123(08)61800-2http://www.ncbi.nlm.nih.gov/pubmed/7672027http://dx.doi.org/10.1016/0042-6989(91)90185-8http://www.ncbi.nlm.nih.gov/pubmed/1771774http://dx.doi.org/10.1167/10.14.7http://www.ncbi.nlm.nih.gov/pubmed/21135254http://www.ncbi.nlm.nih.gov/pubmed/11205338http://www.ncbi.nlm.nih.gov/pubmed/11807554http://www.ncbi.nlm.nih.gov/pubmed/11807554http://dx.doi.org/10.1016/j.tics.2004.02.002http://dx.doi.org/10.1016/j.tics.2004.02.002http://www.ncbi.nlm.nih.gov/pubmed/15050512http://dx.doi.org/10.1016/j.tics.2004.02.002http://www.ncbi.nlm.nih.gov/pubmed/15050512http://dx.doi.org/10.1038/nn.2983http://www.ncbi.nlm.nih.gov/pubmed/22101645http://dx.doi.org/10.1016/j.cub.2010.02.059http://dx.doi.org/10.1016/j.cub.2010.02.059http://www.ncbi.nlm.nih.gov/pubmed/20399096http://dx.doi.org/10.1016/j.cub.2010.02.059http://www.ncbi.nlm.nih.gov/pubmed/20399096http://dx.doi.org/10.1126/science.2367852http://www.ncbi.nlm.nih.gov/pubmed/2367852http://dx.doi.org/10.1016/j.visres.2010.09.027http://www.ncbi.nlm.nih.gov/pubmed/20920517http://dx.doi.org/10.1126/science.6612334http://dx.doi.org/10.1126/science.6612334http://www.ncbi.nlm.nih.gov/pubmed/6612334http://dx.doi.org/10.1126/science.6612334http://www.ncbi.nlm.nih.gov/pubmed/6612334http://dx.doi.org/10.1016/0042-6989(76)90083-3http://dx.doi.org/10.1016/0042-6989(76)90083-3http://www.ncbi.nlm.nih.gov/pubmed/1258395http://dx.doi.org/10.1016/0042-6989(76)90083-3http://www.ncbi.nlm.nih.gov/pubmed/1258395http://dx.doi.org/10.1093/biomet/76.2.297http://dx.doi.org/10.1007/PL00005696http://www.ncbi.nlm.nih.gov/pubmed/9224855http://www.ncbi.nlm.nih.gov/pubmed/15541511http://www.ncbi.nlm.nih.gov/pubmed/15541511http://dx.doi.org/10.1007/BF00941731http://www.ncbi.nlm.nih.gov/pubmed/9147952http://dx.doi.org/10.1007/s002210100876http://dx.doi.org/10.1007/s002210100876http://www.ncbi.nlm.nih.gov/pubmed/11715079http://dx.doi.org/10.1007/s002210100876http://www.ncbi.nlm.nih.gov/pubmed/11715079http://dx.doi.org/10.1111/j.1749-6632.1992.tb25277.xhttp://dx.doi.org/10.1111/j.1749-6632.1992.tb25277.xhttp://dx.doi.org/10.1111/j.1749-6632.1992.tb25277.xhttp://dx.doi.org/10.1016/j.conb.2008.07.004http://dx.doi.org/10.1016/j.conb.2008.07.004http://www.ncbi.nlm.nih.gov/pubmed/18678253http://dx.doi.org/10.1016/j.conb.2008.07.004http://www.ncbi.nlm.nih.gov/pubmed/18678253http://dx.doi.org/10.1126/science.7063881http://www.ncbi.nlm.nih.gov/pubmed/7063881http://dx.doi.org/10.1126/science.7063881http://www.ncbi.nlm.nih.gov/pubmed/7063881http://dx.doi.org/10.1016/S0042-6989(00)00102-4http://dx.doi.org/10.1016/S0042-6989(00)00102-4http://www.ncbi.nlm.nih.gov/pubmed/10915889http://dx.doi.org/10.1016/S0042-6989(00)00102-4http://www.ncbi.nlm.nih.gov/pubmed/10915889http://dx.doi.org/10.1098/rstb.2010.0089http://dx.doi.org/10.1098/rstb.2010.0089http://dx.doi.org/10.1098/rstb.2010.0089http://dx.doi.org/10.1523/JNEUROSCI.1169-08.2009http://dx.doi.org/10.1523/JNEUROSCI.1169-08.2009http://www.ncbi.nlm.nih.gov/pubmed/19279239http://dx.doi.org/10.1038/nature01439http://www.ncbi.nlm.nih.gov/pubmed/12621435http://dx.doi.org/10.1038/nature01439http://www.ncbi.nlm.nih.gov/pubmed/12621435http://dx.doi.org/10.1523/JNEUROSCI.1925-10.2010http://dx.doi.org/10.1523/JNEUROSCI.1925-10.2010http://www.ncbi.nlm.nih.gov/pubmed/20720121http://dx.doi.org/10.1523/JNEUROSCI.1925-10.2010http://www.ncbi.nlm.nih.gov/pubmed/20720121http://dx.doi.org/10.3758/BF03193003http://www.ncbi.nlm.nih.gov/pubmed/17958145http://dx.doi.org/10.1093/brain/41.3-4.332http://dx.doi.org/10.1093/brain/41.3-4.332http://dx.doi.org/10.1016/0042-6989(72)90113-7http://dx.doi.org/10.1016/0042-6989(72)90113-7http://www.ncbi.nlm.nih.gov/pubmed/5033686http://dx.doi.org/10.1016/0042-6989(72)90113-7http://www.ncbi.nlm.nih.gov/pubmed/5033686http://dx.doi.org/10.1016/0042-6989(70)90115-Xhttp://dx.doi.org/10.1016/0042-6989(70)90115-Xhttp://www.ncbi.nlm.nih.gov/pubmed/5440782http://dx.doi.org/10.1016/0042-6989(70)90115-Xhttp://www.ncbi.nlm.nih.gov/pubmed/5440782http://dx.doi.org/10.1126/science.1069590http://dx.doi.org/10.1126/science.1069590http://www.ncbi.nlm.nih.gov/pubmed/12029137http://dx.doi.org/10.1126/science.1069590http://www.ncbi.nlm.nih.gov/pubmed/12029137http://dx.doi.org/10.1037/h0055479http://www.ncbi.nlm.nih.gov/pubmed/14794830http://dx.doi.org/10.1364/JOSA.55.001158http://dx.doi.org/10.1038/nn1669http://www.ncbi.nlm.nih.gov/pubmed/16547513http://dx.doi.org/10.1038/nn1669http://www.ncbi.nlm.nih.gov/pubmed/16547513http://dx.doi.org/10.1098/rstb.2010.0188http://dx.doi.org/10.1016/j.visres.2008.03.021http://dx.doi.org/10.1016/j.visres.2008.03.021http://www.ncbi.nlm.nih.gov/pubmed/18513781http://dx.doi.org/10.1016/j.visres.2008.03.021http://www.ncbi.nlm.nih.gov/pubmed/18513781http://dx.doi.org/10.1152/jn.00668.2010http://www.ncbi.nlm.nih.gov/pubmed/21346201http://www.ncbi.nlm.nih.gov/pubmed/22131401http://dx.doi.org/10.1523/JNEUROSCI.3407-11.2011http://www.ncbi.nlm.nih.gov/pubmed/21346201http://dx.doi.org/10.1152/jn.00668.2010http://www.ncbi.nlm.nih.gov/pubmed/18513781http://dx.doi.org/10.1016/j.visres.2008.03.021http://dx.doi.org/10.1098/rstb.2010.0188http://www.ncbi.nlm.nih.gov/pubmed/16547513http://dx.doi.org/10.1038/nn1669http://dx.doi.org/10.1364/JOSA.55.001158http://www.ncbi.nlm.nih.gov/pubmed/14794830http://dx.doi.org/10.1037/h0055479http://www.ncbi.nlm.nih.gov/pubmed/12029137http://dx.doi.org/10.1126/science.1069590http://www.ncbi.nlm.nih.gov/pubmed/5440782http://dx.doi.org/10.1016/0042-6989(70)90115-Xhttp://www.ncbi.nlm.nih.gov/pubmed/5033686http://dx.doi.org/10.1016/0042-6989(72)90113-7http://dx.doi.org/10.1093/brain/41.3-4.332http://www.ncbi.nlm.nih.gov/pubmed/17958145http://dx.doi.org/10.3758/BF03193003http://www.ncbi.nlm.nih.gov/pubmed/20720121http://dx.doi.org/10.1523/JNEUROSCI.1925-10.2010http://www.ncbi.nlm.nih.gov/pubmed/12621435http://dx.doi.org/10.1038/nature01439http://www.ncbi.nlm.nih.gov/pubmed/19279239http://dx.doi.org/10.1523/JNEUROSCI.1169-08.2009http://dx.doi.org/10.1098/rstb.2010.0089http://www.ncbi.nlm.nih.gov/pubmed/10915889http://dx.doi.org/10.1016/S0042-6989(00)00102-4http://www.ncbi.nlm.nih.gov/pubmed/7063881http://dx.doi.org/10.1126/science.7063881http://www.ncbi.nlm.nih.gov/pubmed/18678253http://dx.doi.org/10.1016/j.conb.2008.07.004http://dx.doi.org/10.1111/j.1749-6632.1992.tb25277.xhttp://www.ncbi.nlm.nih.gov/pubmed/11715079http://dx.doi.org/10.1007/s002210100876http://www.ncbi.nlm.nih.gov/pubmed/9147952http://dx.doi.org/10.1007/BF00941731http://www.ncbi.nlm.nih.gov/pubmed/15541511http://dx.doi.org/10.1016/j.tins.2004.10.007http://www.ncbi.nlm.nih.gov/pubmed/9224855http://dx.doi.org/10.1007/PL00005696http://dx.doi.org/10.1093/biomet/76.2.297http://dx.doi.org/10.1098/rstb.2010.0229http://www.ncbi.nlm.nih.gov/pubmed/1258395http://dx.doi.org/10.1016/0042-6989(76)90083-3http://www.ncbi.nlm.nih.gov/pubmed/6612334http://dx.doi.org/10.1126/science.6612334http://www.ncbi.nlm.nih.gov/pubmed/20920517http://dx.doi.org/10.1016/j.visres.2010.09.027http://www.ncbi.nlm.nih.gov/pubmed/2367852http://dx.doi.org/10.1126/science.2367852http://www.ncbi.nlm.nih.gov/pubmed/20399096http://dx.doi.org/10.1016/j.cub.2010.02.059http://www.ncbi.nlm.nih.gov/pubmed/22101645http://dx.doi.org/10.1038/nn.2983http://www.ncbi.nlm.nih.gov/pubmed/15050512http://dx.doi.org/10.1016/j.tics.2004.02.002http://www.ncbi.nlm.nih.gov/pubmed/11807554http://dx.doi.org/10.1038/415429ahttp://www.ncbi.nlm.nih.gov/pubmed/11205338http://www.ncbi.nlm.nih.gov/pubmed/21135254http://dx.doi.org/10.1167/10.14.7http://www.ncbi.nlm.nih.gov/pubmed/1771774http://dx.doi.org/10.1016/0042-6989(91)90185-8http://www.ncbi.nlm.nih.gov/pubmed/7672027http://dx.doi.org/10.1016/S0079-6123(08)61800-2http://www.ncbi.nlm.nih.gov/pubmed/18768690http://dx.doi.org/10.1523/JNEUROSCI.1513-08.2008http://www.ncbi.nlm.nih.gov/pubmed/8871106http://dx.doi.org/10.1109/TAC.1974.1100705

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