0042-6989/92 $5.00 + 0.00 Copyright 0 1992 Pergamon Press plc

Vision Res. Vol. 32, No. 3, pp. 513-526, 1992

Printed in Great Britain. All rights reserved

Spatial Localization Without Visual References* JANIS M. WHITE,?

DENNIS M. LEVI,? A. PHILIP AITSEBAOMOt

Received 27 December 1990; in revised form 24 July 1991

To explain the veridical percept of the spatial ordering of objects and the generation of eye movements to peripheral targets, Lotze (1885 Microcosmos. Edinburgh: T. & T. Clark) proposed that there is a position label (local sign) for each retinal element. To estimate the precision of local sign information, we measured absolute localization thresholds at various eccentricities in the nasal visual jield, in the complete absence of visual references. To eliminate perception of the visual surround, observers viewed a large display screen through a neutral density filter (2.0 log unit) in a dark room. TheJixation target was extinguished at various times (interstimulus intervals or ISIs)prior to the onset of the test stimulus. In general, our results show that localization thresholds are proportional to the target eccentricity at all ISIS. At each eccentricity, localization thresholds are elevated after the extinction of the visual reference compared to thresholds when the reference is present. However, relative to the referenced threshold, unreferenced thresholds are elevated by a greater proportion at smaller eccentricities than at larger eccentricities. Our threshold vs ISI data can be adequately modeled on the basis of an intrinsic positional uncertainty, which increases with eccentricity, and additive and multiplicative sources of noise. The additive noise appears to reflect primarily the increasing scatter in eye position when the fixation target is extinguished. Our model’s estimate of intrinsic positional uncertainty in the isoeccentric direction appears to reflect primarily the intrinsic positional uncertainty of the peripheral retina (the local sign), being very similar to cumulative cone position uncertainty and to the spacing between ON-P,,, ganglion cells. In the isoeccentric direction, the estimated precision of the local sign mechanism across eccentricities is slightly better than the precision of saccadic endpoints, suggesting that noise in the motor system must also contribute to the scatter of saccadic endpoints in the isoeccentric direction. Interestingly, in the radial direction, we$nd a surprising similarity in our observers’ positional uncertainty and the precision of saccadic endpoints. Positional uncertainty Psychophysics Eccentricity discrimination Vernier alignment Local sign

INTRODUCTION The “space relations of impressions pass as such into the soul, . . . each impression is conveyed to it by a distinct fibre, and the fibres reach the seat of the soul with their relative situation wholly undisturbed . . . Now, according to its particular position, for each one of these side points of the retina is required a peculiar amount and direction of movement of the eye, in order that the spot of most distinct vision may be exposed as a receptive surface to the rays. . . The fulfillment of this requirement presupposes that each of the several fibres . . . can transfer its stimulations, in a manner and degree peculiar to itself, to the various motor filaments, on whose variously graduated *or The Sound of One Hand Clapping. tuniversity of Houston, College of Optometry, 4901 Calhoun Boulevard, Houston, TX 772046052, U.S.A.

Periphery spatial sampling

Spatial interval

co-operation the extent and direction of the ocular movements depends” (Lotze, 1885). In order to explain the veridical percept of the spatial ordering of objects and the generation of precise eye movements to peripheral targets, Lotze (1885) proposed that there is a position label (local sign) for each retinal element. Although this theory has been criticized because Lotze apparently rejected spatial descriptions of brain events as an adequate explanation (for an erudite critique of Lotze’s theory, see Morgan, 1977), the notion of local signs has a great deal of appeal and a mechanism for local signs could be provided by the precise topographic mapping of the retino-cortical pathways (Morgan, 1977; Klein & Levi, 1987; Levi & Klein, 1990). As would be expected of a local sign mechanism based upon topographical mapping with central magnification, peripheral localization thresholds increase with increasing eccentricity (Westheimer, 1982; Levi, Klein & Aitsebaomo, 1985; Klein & Levi, 1987; Yap, Levi & Klein, 513

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1987a, 1989; Levi, Klein & Yap, 1988; Levi & Klein, 1990) however, techniques used to measure these thresholds are complicated by the presence of visual references (such as the fixation target and the surrounding visual environment). Therefore, the measured thresholds may actually represent relative localization or position thresholds. Under certain circumstances relative localization thresholds appear to depend more on the separation between the test and reference targets than upon eccentricity (Beck & Schwartz, 1979; Beck & Halloran, 1985; Morgan & Watt, 1989; Levi & Klein, 1989, 1990; Burbeck & Yap, 1990a; J. Palmer, personal communication). Furthermore, since thresholds for spatial interval and alignment judgments are unequal at the same separation (Klein & Levi, 1987; Levi & Klein, 1990; Morgan, Ward & Hole, 1990), it has been suggested that a local sign theory is not sufficient to account for pattern acuity. Rather, Morgan et al. (1990) have suggested that an additional stage of processing, where the relation between features is computed, is necessary. The purpose of the present study was to examine the effects of visual references on peripheral localization, by measuring localization thresholds at various eccentricities both with a visual reference and at various times following the extinction of the visual reference. Localization thresholds were measured in both the radial direction (i.e. threshold for offsets along an imaginary line connecting the eccentric point with the fixation point) and in the isoeccentric direction (threshold for offsets above and below the same imaginary line). Our results (threshold vs ISI) can be adequately modeled on the basis of an intrinsic positional uncertainty, which increases with eccentricity, and additive and multiplicative sources of noise. Our analysis suggests that the intrinsic positional uncertainty in the isoeccentric direction is similar to both the cumulative uncertainty in cone position and to the spacing between ON-P,, ganglion cells (WHssle, Grunert, Rohrenbeck & Boycott, 1989). A preliminary report of the findings has been presented (White, Levi & Aitsebaomo, 1990). METHODS

Localization thresholds were measured in both radial and isoeccentric directions. The stimuli for the two tasks are illustrated schematically in Fig. 1. Radial localization thresholds

Radial localization thresholds (i.e. thresholds for offsets along an imaginary line connecting the eccentric point with the fixation point) were measured at four eccentricities (2.2, 4.5, 10.0, and 17.4 deg) in the nasal visual field. The stimulus was a bright (approx. 50 times absolute detection threshold) vertical line (60 x 6 min arc in size) generated on the dark screen (50 x 38 deg) of a Commodore Pet computer. To eliminate perception of the visual surround, including the screen edges, the room was totally dark and the observer viewed through a 2.0 log unit neutral density filter. This filter ensured

STIMULI

Eccentrlc’ty

; iii;

_I

RAD,AL

FIXATION

TEST Eccentricity

====z _-__TEST

I soECCENTRIC

-7 FIXATION

FIGURE 1. Schematic representation of the stimuli used for measurement of localization thresholds, in the nasal visual field. The fixation line was presented centrally for 750msec. At varied intervals after offset of the fixation line (ISIS from 0 to 16OOmsec), the test line was briefly presented at one of the five positions shown. Orientation of the lines differed for the radial and isoeccentric tasks.

that the observer was unable to see the screen edges and also eliminated visible persistence of the fixation and test lines following their offset. Each trial began with the presentation of a central fixation line (of the same dimensions as the test line) for 750 msec. The interstimulus interval (ISI) began with the extinction of this fixation line and varied in duration from 100 to 1600 msec, during which the observer attempted to maintain fixation at the location of the extinguished fixation line in the now totally dark room. The observers reported that the task was more difficult when eye movements were purposefully made during the ISI. At the end of the ISI, the test line was presented for 150 msec. For each trial at each eccentricity, the test line was presented in one randomly-chosen position of a set of five (at the nominal eccentricity and displaced 1 or 2 units to the right or left of it; see Fig. 1). Following the extinction of the test line, the observer’s task was to judge the direction and magnitude of the horizontal displacement of the test line relative to its average position (i.e. the nominal eccentricity) by giving integer numbers from -2 to 2. For example, on each trial in the radial task, the test line was presented (in random order) at an eccentricity of 9, 9.5, 10, 10.5, or 11 deg and the observer’s task was to judge the target eccentricity relative to the nominal eccentricity of 10 deg. After each trial, an auditory signal provided feedback regarding the actual position of the test line. The positions of both the fixation and test lines were jittered from trial to trial. Data were collected in blocks of 125 trials at each eccentricity and IS1 and each block was preceded by about 20 practice trials. We refer to this task as “radial” because the displacement (offset) cue is along a radial line between the fixation and the peripheral targets. For comparison, localization thresholds with a simultaneous visual reference were also measured. In these trials, the central fixation line was presented for 750 msec and after 600 msec, the test line was also presented for 150 msec. The two lines were extinguished simultaneously. Otherwise, the paradigm was as above. Note that in the presence of a simultaneous visual reference,

SPATIAL

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WITHOUT

the radial task is essentially a 2-line spatial interval (width) discrimination. Isoeccentric localization thresholds

Isoeccentric localization thresholds (i.e. thresholds for offsets above and below an imaginary line connecting the eccentric point with the fixation point) were also measured at the four eccentricities in the nasal field. The procedure was similar to that for radial thresholds except the target lines (both fixation and test) were horizontal and the test line was either aligned with an imaginary horizontal line connecting the fixation line and the test line, or displaced 1 or 2 units above or below that imaginary line (Fig. 1). We call this task “isoeccentric” because the displacement (offset) cue is along an isoeccentric arc. Note that when a simultaneous visual reference is present, this isoeccentric task is essentially a 2-line Vernier alignment task. Additional measurements (i) Fovea1 Vernier thresholds. In addition to measuring localization thresholds at the four eccentricities, Vernier thresholds were measured for abutting lines located at the fixation position, across ISIS of 200-1600 msec. Thresholds were measured with the lines oriented both horizontally and vertically. The measurements with vertically-oriented lines are similar to those of Matin, Pola, Matin and Picoult (1981) who measured Vernier thresholds with sequentially flashed lines in a dark room. As discussed below, these measures provide a psychophysical estimate of the noise added by eye movements when the fixation target is extinguished. (ii) Localization threshold

along direrent meridians.

Both radial and isoeccentric thresholds of one observer (DL) were also measured at 10 deg in the inferior visual field and at 10 deg in the inferior nasal field (along an oblique meridian). Thresholds in both directions were also measured at 17.4 deg in the inferior and at 17.4 deg in the inferior nasal field of another observer (TN). To measure thresholds along the inferior meridian, the screen was turned 90 deg so that the fixation and test lines were oriented horizontally to measure radial thresholds and vertically for isoeccentric thresholds. For thresholds along the oblique meridian, the screen was turned by 45 deg causing the fixation and test lines to be oriented obliquely.

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Observers

Four experienced psychophysical observers with normal or corrected to normal vision participated in the study. The observers included the three authors and one other who was not aware of the purpose of the experiment. Viewing was monocular (the left eye was occluded with a patch), with the natural pupil and a prescription lens to correct for the distance from the eye to the display screen. Head position and alignment were fixed by a chincup and headrest.

RESULTS Radial thresholds

Figure 2 (left) shows peripheral radial localization thresholds plotted as a function of IS1 for observers DL and JW. In this and in subsequent figures, the symbol size is proportional to the target eccentricity. The effect of eliminating visual references varied across subjects, the extremes of which are represented by the data of DL and JW. At each eccentricity, radial thresholds of DL increased with increasing dark ISI, whereas radial thresholds of JW increased, but by much less. When plotted with a linear ordinate (as in Fig. 2), the absolute elevation of thresholds is somewhat larger at larger eccentricities (e.g. DL) or is similar at all eccentricities (e.g. JW). However, when compared to the referenced threshold at each eccentricity, the relative (or percentage) threshold elevation is actually greater at small eccentricities. This relative elevation of thresholds is seen more clearly in Fig. 5, in which the data of Fig. 2 have been replotted with a logarithmic ordinate. With the logarithmic ordinate, one can see that at small eccentricities, radial thresholds with an IS1 as short as 200 msec are elevated compared to thresholds with a visual reference, with little further increase in thresholds with increasing ISI. In contrast, at large eccentricities, thresholds are only slightly elevated (relative to the referenced threshold) by the presence of any dark interval between presentation of the fixation line and the peripheral test line. Radial thresholds with a visual reference increased with increasing eccentricity of the test line and were consistent with previous results (Klein & Levi, 1987; Levi & Klein, 1990). Similarly, at each ISI, thresholds without visual references increased with increasing eccentricity.

Analysis

Data were analyzed using a maximum-likelihood multicriterion probit analysis (Levi, Klein & Aitsebaomo, 1984). Threshold was defined as the offset giving a d’ value of 0.675, corresponding to 75% correct. Blocks of trials which were too difficult or too easy (i.e. in which the d’ values for the 1 unit offset were ~0.35 or > 1.9) were excluded from the analysis. All other blocks were included in the analysis and the plotted thresholds represent the means of from 2 to 6 runs per eccentricity and ISI, weighted by the inverse variance. The error bars indicate f 1 SE and include both within and between run variance.

Isoeccentric thresholds

Isoeccentric localization thresholds as a function of IS1 are shown for observers DL and JW in the right panel of Fig. 2. At each eccentricity, isoeccentric thresholds increased with increasing dark IS1 for both observers. The absolute elevation of thresholds was similar at all eccentricities. However, again when compared to the referenced

threshold at each eccentricity,

the relative (or percentage) elevation of isoeccentric thresholds was greater at small eccentricities than at large (see Fig. 5). At small eccentricities, compared to referenced thresholds, isoeccentric thresholds with an IS1

JANIS M. WHITE et al.

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“OV

0:o INTE~STIMU~US

INTERVAL

0:n

014

oi

o:e

1.0

1.2

1.4

1.6

(seconds)

FIGURE 2. Peripheral localization thresholds (radial on the left and isoeccentric on the right) as a function of dark ISI, for observers DL (top) and JW (bottom). Note that the axes are linear. The size of the symbols indicates eccentricity, with larger symbols representing greater eccentricities. The lines connect the data points at each eccentricity. Thresholds measured in the presence of a visual reference are the leftmost points.

as short as 200 msec were already markedly elevated (by approx. 2-5 times) and increased only slightly more with increasing ISI. In contrast, at large eccentricities (10 deg and greater), the effect of IS1 on isoeccentric thresholds, relative to referenced thresholds, was much less so that thresholds without visual references were only slightly elevated. For example, at 17.4deg, the isoeccentric threshold for JW increased from 0.14 deg with a visual reference to 0.20deg at an IS1 of 0.4sec (about a factor of 1.4) and to 0.24 deg at an IS1 of 0.8 set (about a factor of 1.7). All observers showed a similar pattern of results. Isoeccentric thresholds with a visual reference increased with the eccentricity of the test line, consistent with previous results (Klein & Levi, 1987; Levi & Klein, 1990). Unreferenced thresholds at a specific IS1 increased with increasing eccentricity only at larger eccentricities and overlapped at smailer eccentricities. The number of eccentricities at which thresholds overlapped varies across subjects and is probably limited by eye movements, as discussed below. At each eccentricity, isoeccentric thresholds (both with and without visual references) were 2-6 times lower than radial thresholds. The relative superiority in the isoeccentric task, which has been reported previously (Klein & Levi, 1987; Yap, Levi & Klein, 1987b), may be partly due to an anisotropy in the underlying neutral substrate, partly due to perceptual interactions (such as variability in judgments of size between the reference and test lines)

present only in the radial or width task and/or partly due to differences in the internal reference used for the two tasks. The difference in thresholds will be discussed further below. Fovea1 localization thresholds Each observer’s fovea1 localization thresholds were measured using abutting lines with dark ISIS from 200 to 1600 msec. Similar to previous results (Matin et al., 1981; Foley, 1976; Findlay, 1974; Fahle, 1991) the fovea1 localization thresholds increased with increasing ISI, The solid symbols in Fig. 5 show the fovea1 localization thresholds for three observers, for vertical lines (left panels) and horizontal lines (right panels). There is close agreement between our fovea1 data for vertically oriented lines and the psychophysical data of Matin et al. (1981), measured with a similar paradigm. All but one of our observers showed quantitatively similar thresholds for offsets in the horizontal and vertical directions across ISIS. DL, the exception, showed substantially higher thresholds for horizontal offsets across all ISIS. Localization threshold along inferior and oblique meridians To further investigate the difference between radial and isoeccentric thresholds at each eccentricity, both localization thresholds were determined in the inferior field and in the inferior nasal field of two subjects. For both tasks, the pattern of thresholds across ISIS was

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and/or cortical inhomogeneity. In our attempt to measure uncertainty of the position signal (without visual references), we introduced a dark interval. At large eccentricities, l~~tion of a single target in the dark is almost as precise as when a visual reference is simultaneously present; however, at small eccentricities, localization is poorer when there is no visual reference present. Elimination of visual references evidently creates noise which elevates localization thresholds proportionally more (relative to the baseline thresholds at an IS1 of 0) at small than at large eccentricities, and more .. 60 75 90 30 0 15 at long than at short ISIS. The noise could be additive or multiplicative. Meridian (degrees) In a model with purely additive noise, the variance of FIGURE 3. Localization thresholds (at an eocentricity of 17.4deg) as a function of the meridian of the test line (0 deg is in the nasal fieid, the noise would add to the intrinsic ~sitional uncer45 deg in the inferior nasal and 90 deg in the inferior field) for subject tainty (expressed as variance), resulting in elevation of TN. Circles represent radial thresholds and squares, &eccentric measured thresholds. The effect of the additive noise on thresholds. Solid symbols are thresholds at an IS1 of 400 msec and localization thresholds would depend upon the magniopen symbols, thresholds in the presence of a visual reference. Note the tude of the noise relative to the magnitude of the intrinsic relative superiority of isoeccentric thresholds in the major meridians. positional uncertainty, as shown schematically in Fig. 4(a). ~~sholds, shown as a function of IS1 in Fig. 4(a), similar to that found in the nasal field (presented would depend upon: (i) intrinsic positional uncertainty, above). That is, at each eccentricity (i.e. 10.0 and which increases with eccentricity (as shown by the thin 17.4 deg), thresholds without visual references were only solid line at an eccentricity of 0 deg and by the thick solid slightly elevated relative to the threshold with a visual line at 10 deg); and (ii) an additive source of noise reference. (variance), which grows with increasing ISI (as shown by The relative superiority of isoeccentric vs radial the dotted~dash~ line). As seen in the figure, at a small thresholds was present in each meridian however, the eccentricity where the noise exceeds the intrinsic posmagnitude of the superiority was greatly minimized itional uncertainty (i.e. 0 deg), additive noise would along the oblique meridian. Figure 3 shows referenced cause localization thresholds (solid circles) to increase and unreferenced (at an IS1 of 400 msec) thresholds for markedly with increasing ISI. The thresholds would be observer TN, at an eccentricity of 17.4 deg, as a function essentially equal to the additive noise (the threshold is of retinal meridian (note that 0 deg is in the nasal field, determined by Pythagorean summation of the two noise 45 deg in the inferior nasal and 90 deg in the inferior sources). However, at a larger eccentricity where additive field). At this large eccentricity, there is a striking similarity between thresholds with a reference and noise is small relative to the intrinsic positional uncertainty (i.e. lOdeg), the localization threshold (open thresholds 400 msec after the fixation line was extincircles) would be high at an IS1 of 0 (due to the high guished. In the major meridians (0 and 9Odeg), both referenced and unreferenced isoeccentric thr~holds are intrinsic position un~rtainty) and with increasing ISI, would be only minimally elevated (relative to the about 6 times lower (better) than radial thresholds, threshold at an IS1 of 0) by additive noise. whereas in the oblique meridian (45 deg), the difference In contrast, in a model with purely multiplicative is less than a factor of 2. The much reduced difference in thresholds along the oblique meridian is due to an noise, the effect of the noise on localization thresholds would be an elevation by a constant percentage of the increase in isoeccentric thresholds along the oblique meridian compared to isoeccentric thresholds along intrinsic positional un~rtainty, as shown ~hemati~lly in Fig. 4(b). Thresholds, shown as a function of IS1 in the major meridians (Heeley & Tinney, 1988). Radial thresholds are about the same along the oblique and Fig. 4(b), would depend upon: (i) intrinsic positional uncertainty, which increases with eccentricity (as shown major meridians. by the thin solid line at an eccentricity of 0 deg and by The relationship between unreferenced localization the thick solid line at 10 deg); and (ii) multiplicative threshold and intrepid positional ~cert~~ty noise, which grows with increasing ISI. The effect of the The notion of local signs (Lotze, 1885) implies that multiplicative noise would be a multiple of the intrinsic each visual mechanism (i.e. “retinal element”) has its positional uncertainty at each eccentricity, resulting in own unique position label, without reference to the an elevation of thresholds by the same percentage at each fovea. To put this another way, localization judgments eccentricity. Therefore, in absolute terms, thresholds must be made at a level of representation at which the would be elevated more at large eccentricities where spatial position of the target is made explicit, with a the intrinsic positional unce~ainty is large (relative to precision sufficient to support the observed thresholds small eccentricities). However, when compared to the (Morgan et al., 1990). There will be uncertainty associ- threshold at an IS1 of 0, multiplicative noise would result ated with the position signal of each element, which in an equal percentage elevation of thresholds at the two increases with increasing eccentricity due to retinal eccentricities. When plotted on a logarithmic ordinate, 1

T-

7

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JANIS M. WHITE ef af.

Our threshold vs IS1 data were fit best by a model incorporating three independent sources of noise or variance: (i) intrinsic positional uncertainty, which grows with eccentricity; (ii) additive noise; and (iii) multiplicative noise [Fig. 4(c)]. To estimate the intrinsic positional uncertainty at each eccentricity (a,)* from our experimental data, we fit the data with a model of the form:

ADDITIVE NOISE

Th = ([o:*(l 0.0

0.4

0.8

1.2

1.6

3 ?! F

(b)

lo

MULTIPLICATIVE NOISE

s

+ Km*ISI)] -t”K,*ISI)‘/2

where Th is the localization threshold and a, is the intrinsic positional uncertainty at each eccentricity, as described by equation (2). a, = 7”*(1 + E/E,)

ADDITIVE P MULTIPLICATIVE NOISE

.wi ! 0.0

.

I

I

0.4

.

0.8 IS1

I

1

1.2

1.6

(seconds)

FIGURE 4. Schematic representation of three models showing the effect of noise on localization thresholds. In each, thresholds as a function of dark IS1 are shown at two eccentricities: at the fovea (solid circles and dashed lines) and at 10 deg (open circles and dashed lines). (a) The effect of additive noise. Thresholds depend upon: (i) intrinsic positional uncertainty (shown by the thin solid line at eccentricity 0 deg and by the thick solid line at eccentricity 10 deg); and (ii) an additive source of noise, which grows with increasing IS1 (shown by the dotted-dashed line). The threshold (squared) is equal to the sum of the squares of the two factors. (b) The effect of ~uItiplicuiive noise. Thresholds depend upon: (i) intrinsic positional uncertainty (shown by the thin solid line at eccentricity Odeg and by the thick solid line at eccentricity lode&; and (ii) multiplicative noise, which grows with increasing ISI. The threshold is equal to the product of the two factors. (c) The effect of both additive and multiplicative noise on localization thresholds.

this is represented by parallel curves for fovea1 and peripheral thresholds [Fig. 4(b)]. The effect of bath additive and multiplicative noise on localization thresholds is illustrated in Fig. 4(c). A quantitative model, incorporating intrinsic positional un~rtainty and both additive and multiplicative sources of noise, is presented below. As discussed below, the model shown in Fig. 4(c) provides the best fit to our data set. *Since our thresholds are estimated at d’ = 0.675, o, is actually equal to 0.67% the SD of the intrinsic positional uncertainty.

(1)

(2)

where T, is the fovea1 intrinsic positional uncertainty at time 0 (i.e. no ISI), E is the eccentricity of the target, and E2 is a factor which describes the variation in positional uncertainty with eccentricity. E2 represents the eccentricity at which the uncertainty is twice the fovea1 value (Levi et al, 1985; Yap et al., 1989). K,,, and K,, in equation (1), are multiplicative and additive factors, respectively. These factors represent multiplicative and additive sources of noise. In equation (l), both factors are multiplied by IS1 so that both have a larger effect on the l~a~zation threshold with increasing ISI. The multiplicative factor is also multiplied by the intrinsic positional uncertainty at each eccentricity (expressed as variance). Note that the additive factor is in deg2. IgorTM was used to fit the entire isoeccentric (and then, radial) data set of each observer according to the model. Igor uses a Levenber&Marquardt iterative algorithm to minimize the error between the experimental data and the model fit. Specifically, Igor fit four parameters to about 24-28 data points (per observer), providing estimates of the parameters: T,, E2, Km, and K, . Threshold vs ISXdata and the model fits (shown by the dotted lines) for three observers are shown in Fig. 5. The model provides a reasonable fit to much of the data, with reduced x2 values ranging from about 3 to 5. Table 1 shows the model’s estimates of the four parameters for each observer. The model [equation (I)] provides estimates of the intrinsic positional uncertainty at each eccentricity, which are shown for each observer in Fig. 6. Peripheral positional ~certainty increases with increasing eccentricity however, radial uncertainty (open symbols) is higher (by a factor of approx. 4-6) than isoeccentric uncertainty (solid symbols) at all ~centri~ties. Note that at the two largest eccentricities (10 and 17.4 deg), the isoeccentric uncertainties are approx. 0.1 and 0.17 deg respectively, equal to about 1% of the target eccentricity. Table 2 shows the peripheral intrinsic uncertainties estimated from the model fits, specified as a fraction of eccentricity + &. This is the “eccentricity Weber fraction”, kECC (Klein & Levi, 1987). For the isoeccentric task, kECC is approx. 0.01 and for the radial task, it averages about 0.045. For comparison, estimates of kECC from two previous studies, using quite different approaches, are given (these are discussed further in the Discussion section).

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FIGURE 5. Localization thresholds (radial on the left and isoeccentric on the right) as a function of dark ISI, for three observers. Note that the ordinate is logarithmic. Open symbols are peripheral localization thresholds (with the size of point indicating eccentricity) and solid symbols, fovea1 localization thresholds. Thresholds measured in the presence of a visual reference are the leftmost points. The dotted lines were fit to the data, according to our model which considers intrinsic, additive and multiplicative noise [equation (l), see text]. Note that the dotted lines fit to the fovea1 data extend toward the lower left and that they estimate To where they intersect an IS1 of 0.

TABLE 1. Model fit parameters

Task Isoeccentric

TO C&d

E2 (dcg)

Radial

Observer

Model parameters

JW 0.008

f

0.001*

0.43 f 0.1

DL

TN

0.004 f 0.002

0.007 f 0.002 0.67 f 0.19

0.35 f 0.16

PA

zn

0.018 2.64 +f 0.0006 0.47

0.066 2.16kO.54 f 0.003

0.068 1.02 f 0.003 0.42

To(de3 E2 (deg)

0.013 f 0.003 0.39 rfr0.09

0.007 f 0.003 0.25 *0.11

0.006 f 0.0904 0.42 f 0.02

0.007 f 0.002 0.3 f 0.04

2

0.016 1.03 + f 0.001 0.14

0.17 9.2kO.11 + 0.01

0.07 1.02 + f 0.006 0.02

0.067 1.79 f 0.003 0.3

*Correction for heterogeneity would increase the error bars by about a factor of 1.5-2.

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JANIS M. WHITE

ECCENTRICITY (degrees) FIGURE 6. Estimated intrinsic positional uncertainty (radial, open symbols; isoeccentric, solid symbols) for each observer as a function of eccentricity. Positional uncertainty was estimated from our model [equation (I), see text]. The labeled lines show several retinal factors which could limit positional uncertainty. See text for discussion.

DISCUSSION

The most surprising result of this study is that at large eccentricities (i.e. 10 deg or greater), elimination of visual references has little effect on localization thresholds, when compared to the referenced localization thresholds. This result is surprising because it is often assumed that the visual system makes an explicit computation based upon the position and nature of the visual reference (Morgan et al., 1990; Beck & Halloran, 1985). In the absence of visual references, our task is essentially the localization of a single line in the dark. Our unreferenced isoeccentric thresholds, averaging about 1% of eccentricity at the large eccentricities in the horizontal and vertical meridians, are only slightly higher than our referenced thresholds (2-line localization) and previously-measured referenced thresholds (2- and 3-line localization; Klein & Levi, 1987; Levi TABLE 2. Eccentricity “Webber fraction” (kECC) No. of

Task Isoeccentric

lines

Observer

kECC

1

JW TN DL SK ws DL JT KH JW TN DL PA JT CN DL SK ws DL JT CN DL KH

0.008 0.010 0.012 0.005 0.010 0.010 0.007 0.009 0.034 0.078 0.027 0.048 0.027 0.035 0.031~.04 0.014 0.016 0.011 0.011 0.017 0.016 0.015

3

3 Radial

1

2

3

3

Study Present

Klein and Levi (1987)

Levi and Klein (1990) Present

Levi and Klein (1990)

Klein and Levi (1987)

Levi and Klein (1990)

et al.

& Klein, 1990). Similarly, in the radial direction, our unreferenced thresholds are about 5% of the target eccentricity, only slightly higher than our referenced thresholds and previously-measured referenced thresholds (Klein & Levi, 1987; Levi & Klein, 1990-see Table 2). We shall argue that at large eccentricities, isoeccentric localization thresholds with or without visual references represent a reasonable estimate of the intrinsic positional uncertainty of the peripheral retina, i.e. of the peripheral local sign. In contrast, at small eccentricities, elimination of visual references causes a relative increase in localization thresholds. Although unreferenced thresholds are higher than visually-referenced thresholds in both the radial and the isoeccentric tasks, the difference is more robust in the isoeccentric task. Below, we consider potential sources of additive, multiplicative and other noise which may contribute to the elevation of thresholds as IS1 increases. Additive noise

In this section, we argue that the primary source of additive noise is eye movements. The local sign notion (Lotze, 1885) implies that each “retinal element” has a position signal, without reference to the fovea. Our paradigm to measure uncertainty of the position signal included introduction of a dark interval between fixation and test lines. One important consequence of the dark interval is that scatter in eye position increases (Matin, Matin & Pearce, 1970: Skavenski & Steinman, 1970). Assuming these eye movements are not registered, increasing scatter in eye position will add noise to the position signal and thereby elevate localization thresholds. Indeed, Matin et al. (1981) have shown that Vernier thresholds for horizontal offsets at the fovea increase with increasing dark IS1 and that the increases can be accounted for primarily by the increasing scatter of horizontal eye position with increasing dark IS1 (see also Fahle, 1991). From our model [equation (l)], the additive noise in the radial and isoeccentric tasks for each observer has been estimated. Figure 7 shows the estimated additive noise in the radial task as a function of dark ISI. For comparison, the solid squares show the mean SD of horizontal eye position recorded by Matin et al. (1970) at various times after the fixation target was extinguished. Matin’s data has been multiplied by 0.675 to make them directly comparable to ours. The eye movement data fall in the middle of our estimates of additive noise, suggesting that scatter in eye position primarily accounts for our additive noise. The significance of added eye position noise will depend upon the magnitude of that noise relative to the magnitude of the intrinsic positional uncertainty, as illustrated in Fig. 4. The data of Matin et al. (1970) show that the scatter (SD) in horizontal eye position increases from about 0.1 deg 500 msec after extinction of the visual reference to about 0.2 deg 1.5 set after its extinction. Thus, at the fovea and small eccentricities, scatter in eye position may be equal to or greater in magnitude

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additive model and 3.88 + 0.49 for the complete model). It is of some interest that the multiplicative factor varies considerably between observers and tasks. For example, it is highest in observer DL’s radial data, accounting, in large part, for the increase in his thresholds with IS1 at large eccentricities (Fig. 2). Multiplicative noise

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FIGURE 7. Estimated additive noise of each observer as a function of dark ISI. Additive noise is estimated from our model [equation (l), see text] fit to the radial localization thresholds. For comparison, the solid squares show the mean scatter in horizontal eye position (n = 2) at different times following the extinction of a fixation target (from Matin et al., 1970).

than the intrinsic positional uncertainty and will have a substantial effect on localization thresholds. In contrast, at large eccentricities (i.e. 17.4 deg), where intrinsic positional uncertainty is large relative to the eye position scatter, the effect of eye movements should be less. The implication for estimating intrinsic positional uncertainty is that at small eccentricities (and especially in the isoeccentric task), increasing scatter in eye position in the dark must be considered. We originally fit the data with an additive model, which considered intrinsic positional uncertainty and additive noise only. However, at some eccentricities (especially larger eccentricities), this additive model did not fit the data well. For example, PA’s radial localization thresholds, at the fovea and at 10.0 deg, as a function of dark IS1 are shown by the symbols in Fig. 8. The predicted localization thresholds based on a purely additive model are shown by the dashed lines, in the upper panel. fro obtain the additive model fit, we actually used equation (l), with the multiplicative factor (K,,,) set to 0.1 Also shown are the additive model estimates of PA’s intrinsic positional uncertainty at the fovea and at 10.0 deg (depicted by the horizontal solid lines). At the fovea, the additive model provides a good description of the data. In contrast, at 10.0 deg, the localization thresholds predicted by the additive model underestimate the measured thresholds at most ISIS. For comparison, the fits based on our complete model [equation (l)], considering intrinsic position uncertainty, additive and multiplicative noise, are shown by the dashed lines in the lower panel. Recall that the effect of multiplicative noise is proportional to the intrinsic positional uncertainty; since the intrinsic uncertainty at the fovea is small, the effect of multiplicative noise is small. Therefore, at the fovea, the predicted thresholds based on the complete model [equation (l)] are similar to those based on the additive model. In contrast, at 10.0 deg, intrinsic uncertainty is large, resulting in a large effect of multiplicative noise. Here, the predicted thresholds based on the two models differ with the complete model fitting the data more accurately (x2 = 8.8 + 0.72 for the

Increasing scatter in eye position is only one source of noise elevating localization thresholds when visual references are eliminated. Although we are unsure of the source(s) of the multiplicative noise, our model which considers multiplicative, in addition to additive, noise definitely provides a better fit to the data as a whole (see Fig. 8). Potential sources of the multiplicative noise might include inaccuracies in the internal reference used for the task (Klein & Levi, 1987; Levi & Klein, 1990, see Appendix), the computation involved in the comparison process, both of which will be discussed below, and decreasing memory of the fovea1 position in the dark. The contribution of memory loss to the elevation of localization thresholds with a dark IS1 has been suggested by Matin et al. (1981). Other sources of noise The visual system may achieve accurate localization thresholds with visual references by comparing the position signal of the peripheral locus (at which the target is imaged) to the position signal of the fovea. It is possible that in localizing a single line in the dark, the observer makes a comparison between the position signal of the peripheral locus and the remembered or judged position signal of the fovea. In such a comparison scheme, a different type of nonvisual internal reference, based upon a knowledge of horizontal and vertical due to gravity (Heeley & Buchanan-Smith, 1990), could also be used. We argue that such a nonvisual internal reference might account, in part, for both the superiority of isoeccentric vs radial thresholds along the major meridians and the elevation of isoeccentric thresholds along oblique meridians compared to along the major meridians. Both with and without visual references, isoeccentric thresholds were consistently lower than radial thresholds at the same eccentricity. This threshold difference has been reported previously for visually-referenced thresholds (Klein & Levi, 1987; Levi & Klein, 1990; Yap et al., 1987b). Potential explanations are that: (i) there is an anistropy in the underlying neural substrate and/or (ii) there are task-specific differences. (i) There are several lines of evidence for differences in performance for targets oriented radially vs isoeccentrically (Rovamo, Virsu, Laurinen & Hyvarinen, 1982; Temme, Malcus & Noell, 1985; Fahle, 1986; Yap et al., 1987b), and these have been variously accounted for on the basis of an anisotropy in retinal (Levick & Thibos, 1980) and cortical receptive fields (Leventhal, 1983; Schall, Perry & Leventhal, 1986), or in cortical magnification (Hubel & Wiesel, 1977; but see Tootell, Switkes, Silverman & Hamilton, 1988). However, the magnitude

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FIGURE 8. Radial localization thresholds of PA (at the fovea, solid symbols and at 10 deg, open symbols) as a function of dark ISI. The top panel shows the fit of a model with intrinsic positional uncertainty and additive noise (shown by the dashed lines) but no multiplicative noise. The model’s estimate of PA’s intrinsic uncertainty at each eccentricity is shown by the solid lines. The lower panel shows the fit (dashed line) based on our complete model, incorporating intrinsic uncertainty and both additive and multiplicative noise [equation (l)]. The solid lines show this model’s estimate of PA’s intrinsic uncertainty at each eccentricity.

of the difference in our psychophysical thresholds and in our estimates of intrinsic positional uncertainty suggests that the results cannot be accounted for on the basis of simple retinal or cortical mapping considerations (e.g. Schwartz, 1980). Moreover, we believe that the anatomical and physiological anisotropies described above are smaller than needed to account for the observed difference in our localization thresholds. (ii) The threshold difference could also result from differences in the two tasks. As referred to above, a

nonvisual internal reference may be used to achieve accurate localization threshold. For the isoeccentric task (see Fig. I), the observer may judge whether the test line is higher or lower than an imaginary horizontal line, so an internal reference for horizontal (or vertical in the inferior field) may be important. We speculate that an alignment comparison process is critical for minimum isoeccentric thresholds and that we have a very accurate internal reference for horizontal and vertical alignment, but not for oblique alignment. In the absence of visual

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references, gravity could provide a very useful reference for horizontal and vertical (Heeley & Buchanan-Seth, 1990). This internal reference would cause isoeccentric thresholds to be better than radial thresholds and to be best along the major meridians compared to the oblique meridians, which is in accord with the present results. The important point is that since elimination of visual references has relatively little effect on isoeccentric thresholds at large eccentricities, the nonltisual internal reference for horizontal and vertical must be very precise. Another task-related factor which may elevate radial relative to isoeccentric thresholds is uncertainty in the perceived distance of the target from the observer. Under normal viewing, the perceived size of an object is related to the perceived distance of the object from the observer. When visual references are eliminated, uncertainty in the distance of the target from the observer will increase. Since our radial task requires a comparison to be made between the remembered position of the fovea and the peripheral target, the perceived size of the spatial interval between the two should vary with variations in the perceived distance of the target from the observer. Therefore, increasing uncert~nty in the perceived distance of the target could elevate radial thresholds. In contrast, since the isoeccentric task requires an alignment comparison, uncertainty in the perceived distance of the target from the observer should not affect these thresholds. Of course, there may be other effects of eliminating visual references that we have not considered however, at least at the larger eccentricities, it seems likely that the major source of noise influencing isoeccentric localization of a single line in the dark along the horizontal or vertical meridian is the intrinsic positional uncertainty since thresholds are not greatly improved by the addition of a visual reference. Furthermore, it is likely that for the radial task, our estimate of intrinsic positional uncertainty reflects the precision of the local sign and other task-related perceptual effects, Comparison with other studies Recently, Levi and Klein (1990) measured localization thresholds on isoeccentric arcs of different radii. The isoeccentric paradigm allows target separation to be varied while eccentricity is held constant. Their results were compatible with a two mechanism model: one mechanism was dependent on target separation but showed only a slight dependence on eccentricity and the other showed a strong eccentricity dependence but little effect of separation. Levi and Klein argued that the eccentricity dependent mechanism was essentially a “local sign” mechanism, like that proposed by Lotze (1885). In the radial direction, positional thresholds (Th) at large eccentricities are given by Th cs Ecc kEEC, where kEEC is a fraction of the target eccentricity. Interestingly, while Levi and Klein (1990) found values of kEEC of about 3-4% for 2-line spatial interval judgments, 3-line disc~mination (bisection) gave values of kEEC from 1.1 to 1.7%, much closer to the iso-

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eccentric thresholds (see Table 2). Burbeck and Yap (1990b) argue that thresholds for the 3-line spatial interval task should be halved relative to those of the 2-line spatial interval task due to the comparison between separations which is possible with the 3-line task. In comparison with the 2-line task, the 3-line also reduces the effects of memory and perceived viewing distance. Therefore, perceptual effects differentially interfering with the comparison in the 2-line radial task may contribute to the elevation of these thresholds. Intrinsic noise In this section, we argue that we can estimate the precision of the local sign from our unreferenced isoeccentric thresholds. To determine the precision of the local sign mechanism, we measured absolute localization thresholds. This required elimination of all visual references which introduces other unwanted sources of noise, such as the additive effect of scatter in eye position. With our model, we estimated the magnitude of additive and multiplicative noise introduced by our paradigm. Thus, the intrinsic noise in the localization mechanism of the visual system could be derived. Since there are perceptual effects (such as the lack of alignment compa~sons) which differentially elevate radial compared to isoeccentric thresholds, our estimates of intrinsic noise in the radial direction may be limited by both the precision of local sign and by these other effects. Furthermore, since our internal reference for horizontal and vertical may be superior to that for oblique meridians, the estimates of intrinsic noise in the isoeccentric direction along oblique meridians may also be limited by noise in the internal referencing vs the local sign mechanism. We consider the isoeccentric intrinsic noise estimates along the major meridians as most represen~tive of the precision of the local sign m~hanism and, at larger eccentricities, these estimates are very similar to the raw localization thresholds. To be clear, our hypothesis is that when a visual reference is present at the fovea, the visual system independently estimates the position of the fovea1 reference and that of the peripheral target. Each of these estimates will be limited by the precision of the local sign information. A comparison is made at a subsequent stage. In the absence of a fovea1 fixation target, in the dark, the comparison process uses nonvisual information, such as the ~member~ position of the fovea, or, for isoeccentric judgments in the horizontal and vertical meridians, the remembered horizontal or vertical. Based upon our results, we suggest that in the radial direction, the comparison process adds noise (less noise for 3-line bisection). In the isoeccentric direction, at least along the major meridians, the comparison process appears to add little additional noise. We argue that the limiting noise in making these isoeccentric judgments is the intrinsic positional uncertainty of the visual system because there is little effect of adding one or two reference lines. Although we did not actually measure fovea1 thresholds at an IS1 of 0, our model provides an estimate

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of the intrinsic positional un~rtainty at the fovea at time zero (T, in Table 1). Our estimates average about 25 set although the individual estimates have fairly large error bars. This average is considerably larger than visuallyreferenced localization thresholds at the fovea, which generally average about 5 or 6 sec. However, in the presence of a simultaneous nearby visual reference, fovea1 Vernier thresholds may be determined by the differential responses of orientation and size tuned spatial filters (where both features fall within the receptive field of a single filter, e.g. Wilson, 1986) rather than by a local sign m~hanism (Levi & Klein, 1990). Our model’s estimate of intrinsic uncer~inty at the fovea is compatible with other estimates using very different approaches. For example, Zeevi and Mangoubi (1984), using a perturbation analysis, arrived at a similar estimate and more recently. McKee, Welch, Taylor and Bowne (1990) also arrived at a similar estimate. Our estimate is compatible with an intrinsic uncertainty which is approx. 1% of the effective eccentricity (i.e. E f E,), where E2 is approx. 0.6 deg (i.e. 0.01 eO.6 = 0.006 deg or 22 arc set). The increase in intrinsic positional un~rtainty with eccentricity is shown in Fig. 6. For comparison, we have also plotted the spacing and positional uncertainty of retinal elements, The dotted line shows the spacing between a subset of ganglion cells, ON-P,, ganglion cells (Wassle et al., 1989) which is similar to our positional uncertainty estimates. The SD of cone separation increases more rapidly than actual cone separation (Hirsch & Miller, 1987; Hirsch & Curcio, 1989) and more recently, Wilson (1990) has suggested that the increase in uncertainty of the position of peripheral cones may limit peripheral positional acuity. Wilson’s estimates of c~ulative cone position unce~ainty in the human retina are replotted in Fig. 6 (dott~/dash~ line) and are also strikingly similar to our estimates of isoeccentric positional uncertainty. Wilson’s notion is only valid if the retina of each observer provides a statistical distribution of irregularities during the course of an experiment; however, Wilson has calculated that normal fixational eye movements would accomplish this. Therefore, we speculate that the precision of our intrinsic isoeccentric positional uncertainty may be accounted for on the basis of retinal elements, as suggested by Lotze (1885). It is interesting to note that c~ulative cone positional uncertainty and ganglion cell spacing agree so closely in the retinal periphery. The recent work of WIssle et al. (1989) also suggests that in the periphery, cortical magnification can be largely accounted for on the basis of the ganglion cell inputs. Intrinsic positional uncertainty and eye movements

Lotze originally proposed local signs as a basis for the generation of eye movements to peripheral targets, so we also compare our sensory position data with eye movement data. One of us (PA) had previously measured the horizontal scatter of endpoints of saccades made to briefly flashed targets in the nasal field (Aitsebaomo, 1988). The scatter (SD x 0.675 to allow direct com-

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FIGURE 9. (a) Comparison of sensory and saccadic data in the nasal field of observer PA. The dotted line represents our model’s estimate of PA’s radial intrinsic uncertainty across eccentricities. The triangles show radial (along the horizontal meridian) scatter of his saccadic endpoints (expressed as 0.675 times the SD of endpoints of saccades into the nasal field) vs saccade size, measured previously (Aitsebaomo, 1988). (b) Comparison of our sensory data with the saccadic data of van Gpstal and van Gisbergen (1989). The symbols show the averages of the individual estimates of positional uncertainty (circles, radial; square, isoeccentric). The lines show the scatter of saccadic endpoints (expressed as 0.675 times the SD) along the direction of the saccade (radial) and perpendicular to the direction of the saccade (isoeccentric) from van Gpstal and van Gisbergen (1989).

parison to our sensory data in which thresholds were specified at 75% correct) in his endpoints (triangles) is compared to our model’s estimate of his radial intrinsic noise (dotted line) in Fig, 9(a). There is surprising similarity between the sensory and saccadic data over the small range of overlap. van Opstal and van Gisbergen (1989) also measured the scatter of saccadic endpoints as a function of the size of the saccade but over a larger range of sizes. Saccades to targets along many meridians were elicited. To calculate scatter of saccadic endpoints, onfy first saccades were included and noise due to fixational eye movements was removed. Scatter, or SD, of endpoints was measured along the meridian of the saccades (i.e. in the radial direction) and perpendicular to the meridian of the saccades (i.e. in the isoeccentric direction) and, here, has been multiplied by 0.675 to allow direct comparison to our sensory data. Their saccadic scatter in the radial and isoeccentric directions along with our estimates of intrinsic positional uncertainty are shown in Fig. 9(b). Interestingly, the saccadic data show more scatter in the radial than in the isoeccentric direction, similar to our sensory data. The magnitude of the difference is less than in the sensory data. The saccadic data were averaged

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across saccades to all meridians since the authors saw no obvious change in scatter with changes in saccade meridian. However, their Fig. 5 shows somewhat more isoeccentric scatter in saceades to oblique meridians vs in those to major meridians, similar to our isoeccentric localization thresholds. This would cause the isoeccentric saccadic scatter, averaged across saccades to all meridians, to be somewhat elevated. In another study, van Opstal, van Gisbergen and Smit (1990) showed that an anistropy in scatter of saccadic endpoints (in the two directions) is also presetit in stimulation-induced saccades, implying that at least part of the anistropy is due to the neural organization of the superior colliculus (the structure stimulated to elicit saccades). Figure 9(b) also shows that, in the radial direction, saccadic scatter is very similar to our estimates of sensory uncertainty. The similarity is seen in the individual data of PA as well. However, as discussed above, our estimates of radial positional uncertainty probably reflect not only sensory limitations imposed by local signs but also perceptual limitations. It is possible that these perceptual limitations are also reflected in the accuracy of saccades along the meridian of the saccade (in the radial direction). Contrary to this possibility is the report of Wong and Mack (1981) of a situation in which the programming of saccades was based upon retinal error info~ation when that info~ation was m&de discrepant with perceptual information. In the isoeccentric direction, saccadic scatter parallels our estimated isoeccentric positional uncertainty but is greater in magnitude. As mentioned above, the averaging across saccades to all meridians (van Opstal & van Gisbergen, 1989) may have elevated isoeccentric saccadic scatter slightly but probably not enough to account for the difference between the sensory and saccadic data. Logically, the scatter in saccadic endpoints could result from noise in the sensory system and noise in the motor system. A simple additive model (amounting for the variance in saccadic endpoints by the sum of the variance in the sensory and motor systems) suggests that the motor system contributes more noise than the sensory system to the scatter endpoints in the isoeccentric direction. CONCLUSIONS

Our results and modeling suggest that in both the radial and isoeccentric directions intrinsic positional uncertainty increases with eccentricity. The large difference between the estimated radial and isoeccentric positional uncertainties at each eccentricity implies that a simple local sign model may not adequately explain localization thresholds. We speculate that when a single target is presented to the peripheral retina, its position is made explicit, with a precision that is limited by the precision of retino-cortical mapping. At a subsequent stage, there is a comparison process. In the isoeccentric direction, it appears that the comparison process adds little noise, so our isoeccentric thresholds provide, to a first approximation, a reasonable estimate of the

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intrinsic positional uncertainty of the peripheral retina. At large eccentricities, where neither eye movements nor perceptual factors (such as references or memory) appear to add si~fi~nt noise to our thresholds, intrinsic positional uncertainty is equal to about 1% of the target eccentricity, similar to the spacing of ON-PM ganglion cells (Wassle et al., 1989) and the cumulative jitter in cone positions (Wilson, 1990). Lotze (1885) originally proposed that each retinal element must have a position label, or local sign, to allow the accurate programming of saccadic eye movements to peripheral targets. Comparison of our sensory results with the saccadic results of van Opstal and van Gisbergen (1989) shows that in the isoeccentric direction, the scatter of saccadic endpoints is not as precise as the positional uncertainty (which we argue reflects retinal local sign information), suggesting that noise in the motor system must also contribute to the inaccuracy of saccades. Surprisingly, in the radial direction, the precision of saccadic endpoints closely matches our observers’ positional uncert~nty, raising the intoning possibility that the same source(s) of noise limit both the precision of spatial localization and of saccades to peripheral targets.

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Leventbal, A. G. (1983). Relationship between preferred orientation and receptive field position of neurons in cat striate cortex. Journal of Comparative Neurolqgy, 220, 416-483. Levi, D. M. & Klein, S. A. (1989). Both separation and eccentricity can limit precise position judgements. Vision Research, 29, 1463-1469. Levi, D. M. & Klein, S. A. (1990). The role of separation and eccentricity in encoding position. Vision Research, 30, 557-586. Levi, D. M., Klein, S. A. & Aitsebaomo, P. A. (1984). Detection and discrimination of the direction of motion in central and peripheral vision of normal and amblyopic observers. Vision Research, 24. 789-900.

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Acknowledgements-We are indebted to Stanley Klein for his critical comments on several earlier drafts of the manuscript and for his help in the data modeling. We are also grateful to Michael Morgan, Harold Bedell, Sarah Waugh and Hong Wang for their thoughtful comments and suggestions. Supported by a grant, ROIEY01728, from the National Eye Institute.