INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Vergence eye movements are critical for good binocular vision, serving to align both eyes on the same object: the nearer the object of regard, the greater the required convergence. The visual control of vergence depends heavily on the slight difference in the positions of the images on the two retinas, which is referred to as binocular disparity and effectively defines the vergence error. Although there are a number of complex visual cues to viewing distance that can influence vergence, here we shall be concerned solely with disparity, which is the most potent (for recent review, see Collewijn and Erkelens 1990). The ability of pure disparity errors to drive vergence was first demonstrated by Rashbass and Westheimer (1961), who used a Wheatstone stereoscope to present identical targets independently to the two eyes. Targets with crossed disparity errors (equivalent to objects nearer than the plane of fixation) elicited increased convergence, and targets with uncrossed disparity errors (equivalent to objects farther than the plane of fixation) elicited decreased convergence, exactly as expected of a negative feedback system working to achieve and maintain appropriate binocular alignment. Most studies have been concerned with horizontal vergence, perhaps in large measure because it is the means by which binocular alignment is shifted between objects in different depth planes. However, good binocular vision requires that the two lines of sight be aligned vertically as well as horizontally, and vertical disparities have been shown to elicit appropriate vertical vergence, although the effective range of disparities is much smaller, the responses have much more sluggish dynamics, show more extensive spatial integration, and are less sensitive to instruction, than the horizontal vergence responses associated with horizontal disparities (Howard et al. 1997, 2000; Kertesz 1983; Stevenson et al. 1997).

In the present study on humans, we have been concerned solely with the initiation of vergence by disparity steps applied to large random-dot stereograms. Previous studies on humans have generally, although not always, used small targets and reported latencies ranging from 150 to 200 ms (Erkelens and Collewijn 1991; Houtman et al. 1977, 1981; Jones 1980; Mitchell 1970; Rashbass and Westheimer 1961; Westheimer and Mitchell 1956). Using monkeys, we have recently reported that, when large textured patterns are used, horizontal disparity steps of suitable magnitude consistently elicit horizontal vergence eye movements with latencies of <55 ms (Busettini et al. 1994a, 1996b). These short-latency vergence responses were in the appropriate direction only when steps were small (<2-3°), and large steps (up to 12.8°) yielded responses with both isogonal and orthogonal components that were largely independent of whether the steps were crossed or uncrossed, horizontal or vertical (so-called, "default" responses). We argued that the responses to small disparity steps reflected the operation of a servomechanism that normally functions to correct residual vergence errors, and the (default) responses to large steps were due to residual local correlations. We also reported that small disparity steps applied in the immediate wake of a saccadic eye movement yielded appropriately directed vergence eye movements with much higher initial accelerations than did the same steps applied some time later. Further, this transient postsaccadic enhancement seemed to be dependent, at least in part, on the visual stimulation associated with the prior saccade because similar, although sometimes weaker, enhancement was observed when the disparity steps were applied in the wake of saccadelike shifts of the textured pattern (so-called, "simulated saccades"). In the present paper on humans, we report similar findings: latencies are short (although 25 ms or so longer than in monkeys), there are default responses with large steps, and there is transient enhancement in the wake of real and simulated saccades. The only significant methodological difference between the present study and our previous one on monkeys was the visual stimulus: the collage of arbitrary geometrical shapes used previously was replaced by a random-dot pattern, permitting a more formal description of the stimulus. We also report here that initial vergence responses have orthogonal components that reach an asymptote with steps of a few degrees and persist (as default responses) with large steps. Further, we show that the initial vergence responses to small disparity steps (<2-3°) are relatively insensitive to eightfold changes in the size of the texture elements (random dots), although responses to steps of intermediate size (3-6°) are larger with larger elements. In addition, we report that the vertical vergence responses to vertical disparity steps have only slightly longer latencies but are appreciably smaller in magnitude than the horizontal responses to horizontal steps, although the general form of the disparity tuning curves is very similar. Finally, we use the same new random-dot stimuli to extend our earlier report on monkeys and show that the initial vergence responses of monkeys share most of the features of the human, the differences between the two species being quantitative rather than qualitative. Some of these findings on humans have been published in abstract form (Busettini et al. 1994b).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

The eye movements evoked by brief exposure to disparity steps applied to large projected random-dot patterns were recorded from seven adult human subjects. All procedures were approved by the Institutional Review Committee concerned with the use of human subjects, and all subjects provided informed consent. Three of the subjects (FM, RK, and GM) each participated in numerous recording sessions, and four more (MB, KP, HA, and JG) each participated in only a few sessions on selected abbreviated paradigms. All of these subjects had stereoacuities better than 40 s of arc (Titmus test) and no known oculomotor or visual problems other than refractive errors that were corrected with spectacles when necessary. Only two subjects (FM and RK) had prior experience (as subjects) in eye-movement studies. We also recorded the vergence responses of three rhesus monkeys using the exact same stimuli as for the human recordings to permit a quantitative comparison between the two species. Except for the details of the stimuli (described in Recording and stimulation procedures), the methodology used on monkeys was exactly the same as in our earlier study and will not be described here (Busettini et al. 1996b).

Recording and stimulation procedures

The presentation of stimuli and the acquisition, display, and storage of data were controlled by a PC (Hewlett Packard Vectra, 486) using a Real-time EXperimentation software package (REX) developed by Hays et al. (1982). The horizontal and vertical positions of both eyes were recorded with an electromagnetic induction technique (Robinson 1963) using scleral search coils embedded in silastin rings (Collewijn et al. 1975). Coils were placed in each eye following application of 1-2 drops of anesthetic (proparacaine HCl), and wearing time ranged up to 73 min for the three main subjects and 30 min for all others. The AC voltages induced in the scleral search coils were led off to a phase-locked amplifier that provided separate DC voltage outputs proportional to the horizontal and vertical positions of the two eyes with corner frequencies (3 dB) at 1 kHz (CNC Engineering). The outputs from the coils were calibrated at the beginning of each recording session by having the subject fixate small target lights located at known eccentricities along the horizontal and vertical meridia. Peak-to-peak voltage noise levels were equivalent to an eye movement of 1-2 min of arc. Interocular distance was measured to the nearest millimeter.

The subject was seated in a fiberglass chair with his/her head stabilized by means of a chin support and forehead rest combined with a head strap and faced a translucent tangent screen (distance, 33.3 cm; subtense, 85 × 85°) onto which two identical, overlapping patterns were back-projected. Orthogonal polarizing filters in the two projection paths and matching filters in front of each eye ensured that each pattern was visible to only one eye: dichoptic stimulation. The screen was constructed of material specially designed to retain the polarization (Yamaboshi, Tokyo, Japan). The patterns consisted of black circular dots with randomly distributed centers on a white background; overlaps were freely allowed, resulting in irregular clustering, and the screen image had equal amounts of black and white (50% coverage). Dot diameters were 2° except for one special study in which this parameter was varied systematically (when all dots had one of the following diameters in a given experiment: 0.5, 1.1, 2.2, or 4.4°). The luminance of the images on the screen was measured with a photometer (Spectra Pritchard), sampling the screen through the polarizing filters so as to mimic the subjects' view. With this arrangement, the luminance measured through the matching polarizing filters was 0.13 cd/m² in the light areas of the patterns and 0.0026 cd/m² in the dark areas. The equivalent measures through the nonmatching (orthogonal) polarizing filters were 0.0011 cd/m² in the light areas and 0.00060 cd/m² in the dark areas. Subjects were unaware of the "ghost" images seen through the orthogonal filters. Pairs of mirror galvanometers (General Scanning, M3-S with vector tuning) positioned in each of the two light paths in an X/Y configuration were used to control the horizontal and vertical positions (and thereby the horizontal and vertical disparities) of the two images. These galvanometers were driven by the DAC outputs of the PC at a rate of 1 kHz with a resolution of 12 bits (optical range, ±50°). Voltage signals separately encoding the horizontal and vertical positions of both eyes together with the positions of the four mirror galvanometers were low-pass filtered (Bessel, 6-pole, 180 Hz) and digitized to a resolution of 16 bits, sampling at 1 kHz. All data were stored on a hard disk and, after completion of each recording session, were transferred to a workstation (Silicon Graphics) for subsequent analysis. The rise time of the mirror galvanometers was <2 ms, and to determine their true dynamics and exact timing they were monitored with a Tektronix digital storage oscilloscope linked to a PC (386).

Paradigms

In a given experiment, the stimulus parameter(s) under study were varied from trial to trial in a pseudorandom sequence, in part to discourage prediction/anticipation, and in part to distribute any effects due to short-term changes in nonvisual factors such as arousal, attention, fatigue, and so forth. It was usual to collect data over several sessions until each condition had been repeated a sufficient number of times to permit good resolution of the responses to be achieved (through averaging) even when we were exploring the limit of the responsive range with stimuli of marginal efficacy.

STANDARD PARADIGM. At the beginning of each trial, the two patterns on the screen were positioned in register and overlapped exactly (zero disparity) for a minimum period in excess of 1 s to allow adequate time for the subject to acquire a convergent state appropriate for the near viewing (33.3 cm). Disparity steps were initiated in the wake of a saccade into the center of the screen because preliminary experiments had shown that the vergence responses were subject to transient postsaccadic enhancement. This was accomplished by having the subject transfer fixation between suitably positioned target spots projected onto the scene through polarizing filters so as to be seen by the right eye only (to avoid any possible disparity conflict with the background patterns). The initial target spot appeared 10° right of center 1 s after the beginning of the trial. When the subject's right eye had been positioned within 1-2° of the spot for a period of time that was randomly varied (1-1.5 s), the spot was extinguished and a new one appeared at the center of the screen. This new target was extinguished as soon as the computer detected a saccadic eye movement, using as a criterion an eye speed >54°/s. If this saccade achieved a speed in excess of 180°/s and arrived within 4° of the position of the new target (which was now no longer visible), then it was deemed appropriate and the disparity step was initiated with a postsaccadic delay of 50 ms (measured from the time when eye speed fell below 36°/s). We used the convention that rightward and upward movements were positive, and stimulus disparity was computed by subtracting the horizontal (or vertical) position of the right image from the horizontal (or vertical) position of the left image. This meant that horizontal disparity steps were positive when the left stimulus stepped to the right and/or the right stimulus stepped to the left (crossed-disparity steps), and negative when the left stimulus stepped to the left and/or the right stimulus stepped to the right (uncrossed-disparity steps); vertical disparity steps were positive when the left stimulus stepped upward and/or the right stimulus stepped downward (left-hyper disparity steps), and negative when the left stimulus stepped downwards and/or the right stimulus stepped upward (right-hyper disparity steps). The disparity steps had a rise time of <2 ms and were applied symmetrically to the patterns seen by each of the two eyes (equal amplitudes, opposite directions), except when stated otherwise. The amplitude of the disparity steps varied from trial to trial (absolute values: 0.2, 0.4, 0.8, 1.2, 1.6, 2.0, 2.4, 3.2, 4.0, 4.8, 5.6, 6.4, 9.6, and 12.8°), with crossed, uncrossed, left-hyper, and right-hyper steps being randomly interleaved. Because we were interested only in the initial vergence responses, exposure to the disparity steps lasted only 200 ms, and, if there were no saccades during this time, then the data were stored on a hard disk; otherwise, the trial was aborted and subsequently repeated. At this point in the paradigm, electro-magnetic shutters in the light paths were used to blank both images for 500 ms, marking the end of the trial; when the images reappeared, they were once more in register for the start of the next trial. The projected patterns always filled the entire screen, and their initial positions were pseudorandomized (9 positions, with horizontal and/or vertical offsets of 0 or 10°) to reduce the impact of local anisotropies in the patterns on the mean vergence responses. Subjects were instructed to make saccades into the center of the pattern by following the projected target spots and then to refrain from making any further saccades until the screen was blanked, signaling the end of the trial. Subjects were given no instructions in regard to the disparity step stimuli but were asked to restrict their blinks to the inter-trial period. Direct observation, in addition to the eye movement profiles (blinks being associated with transient horizontal convergence and downward version), indicated that subjects had no problem following this instruction. By applying the disparity steps in the immediate wake of centering saccades we ensured that 1) the subject was alert during the steps, 2) the stimulus pattern was always centered on the retina at the onset of the steps, and 3) the vergence responses were enhanced (postsaccadic enhancement). Note that all experiments included control trials in which no steps were applied (saccade-only trials).

DEPENDENCE ON A PRIOR SACCADE OR SIMULATED SACCADE. Preliminary experiments (Busettini et al. 1994b) had revealed that, like ocular following (Gellman et al. 1990), disparity vergence was subject to transient postsaccadic enhancement, and we now conducted a series of experiments in which the postsaccadic delay was varied systematically to characterize the time course of the enhancement. Stimuli were 1.6° crossed- and uncrossed-disparity steps, and the postsaccadic delay intervals were 50, 100, 200, 400, and 800 ms (randomly interleaved).

OCULAR FOLLOWING: DEPENDENCE ON A PRIOR SACCADE OR SIMULATED SACCADE. To allow a direct comparison of the effects of a prior saccade or simulated saccade on disparity vergence with those on ocular following (Gellman et al. 1990), additional experiments were undertaken that were identical to those just described except that the disparity steps were replaced with conjugate steps in which the patterns seen by both eyes stepped together 0.8° (leftward or rightward): ocular following stimulus. Note that the previous study of the dependence of human ocular following on a prior saccade used velocity step stimuli (Gellman et al. 1990).

Data collection and analysis

The horizontal and vertical eye position data obtained during the calibration procedure were each fitted with a third-order polynomial that was then used to linearize the horizontal and vertical eye position data recorded during the experiment proper. The latter were then smoothed with a cubic spline of weight 10⁷, selected by means of a cross-validation procedure (Eubank 1988), and all subsequent analyses utilized these splined data. To be consistent with our conventions for defining the polarity of the disparity stimuli, rightward and upward eye movements were defined as positive, and vergence position was computed by subtracting the horizontal (or vertical) position of the right eye from the horizontal (or vertical) position of the left eye. This meant that horizontal vergence was positive (denoting increased convergence) when the left eye moved rightward with respect to the right eye, or the right eye moved leftward with respect to the left eye. Likewise, vertical vergence was positive when the left eye moved upward with respect to the right eye, or the right eye moved downward with respect to the left eye (so-called, left sursumvergence or right deorsumvergence). Horizontal (or vertical) version position, equivalent to cyclopean gaze position, was computed by averaging the horizontal (or vertical) positions of the two eyes. Vergence (or version) velocity was obtained by two-point backward differentiation of the vergence (or version) position data.

The vergence position and vergence velocity temporal profiles recorded in all of the trials using a given stimulus condition were displayed together (synchronized to the disparity step) with an interactive graphics program that allowed the deletion of the occasional trials with saccades or blinks.

To best illustrate the temporal structure of the responses, mean vergence velocity profiles were computed for each stimulus condition. To eliminate any effects due to postsaccadic vergence drift, the mean vergence velocity profile recorded during the control saccade-only trials was subtracted from the mean vergence velocity profiles obtained for each stimulus condition. All of the vergence velocity traces in the figures have been so adjusted, and upward deflections of these traces represent convergent or left sursumvergent velocities.

LATENCY MEASURES. An objective algorithm was used to estimate the latency of onset of vergence using data obtained with disparity steps that gave close-to-maximal responses. Visual inspection of the mean vergence velocity profiles had indicated that the average latencies were generally 75-85 ms. Accordingly, the individual vergence velocity profiles over the time window, 52-118 ms (measured with respect to the onset of the disparity step), were fitted with a function that assumed that, up to a time, T, the response profiles were flat (preresponse period), and then incremented linearly (response period). The fitting was done using the nonlinear regression method implemented in BMDP 3R (Dixon et al. 1990), based on a modified Gauss-Newton algorithm (Jennrich and Sampson 1968). For times <T, the function had a constant value (P₁), and for times T, the function had a value, at time t, of P₁ + [P₂(t  T)]. The value of the baseline (P₁), the starting point of the linear segment (T, which was our estimate of the response latency), and the slope of the linear segment (P₂), were all free parameters. Starting values were 75 ms (T), 0°/s (P₁), and 0°/s² (P₂).

AMPLITUDE MEASURES. Estimates of the amplitude of the initial disparity vergence response were obtained by measuring the change in vergence position over a 67-ms time interval starting 90 ms after the onset of the disparity step. It will be seen that the mean latency of onset is about 80 ms so that this amplitude measure is restricted to the period prior to the closure of the feedback loop, when eye movements begin to influence the disparity: initial open-loop response. The measures from all trials were then used to calculate the mean change in vergence, together with the SD, for each stimulus condition. Of course, sampling the response at a fixed time with respect to the onset of the stimulus meant that the sample would be sensitive to changes in the latency of onset of the response. The latency of vergence showed little dependence on any of the stimulus parameters examined in the present study, although the initial vergence responses could become vanishingly small as the limits of the response range were explored. (At such times, response-locked measures would default to later components of the response.) To eliminate any effects due to postsaccadic vergence drift, the mean change in vergence during the saccade-only (control) trials was subtracted from the mean change in vergence for each stimulus condition, and these adjusted measures are the ones given in the text and plotted in the figures (where they are referred to as "change in vergence position").

OCULAR FOLLOWING. The analysis of the ocular following data was very similar to that of the vergence data except that it was carried out on the splined position measures obtained from the right eye. Quantitative estimates of the amplitudes of the initial responses were obtained by measuring the change in eye position during the 67-ms time interval starting 90 ms after the onset of the conjugate position step, and two-point backward differentiation was used to obtain eye velocity profiles. The position measures and velocity temporal profiles obtained for all trials were then averaged for each stimulus condition, and any effects due to postsaccadic drift were eliminated by subtracting the data obtained from saccade-only (control) trials. These adjusted eye-position measures (referred to as "change in eye position") and eye-velocity temporal profiles are used for all figures and textual references.

Phoria measurement

Dissimilar target images were presented to the two eyes dichoptically, and the subjects' verbal reports of their relative positions were used to bring the targets into subjective alignment using a staircase procedure, thereby providing an estimate of each subject's phoria. Subjects viewed the tangent screen in the usual way, and separate projectors with orthogonal polarizers were used to present an upright cross (+) continuously to one eye and an oblique cross (×) transiently to the other. Each cross was black on a white background and spanned 2.3° with arms 10 min of arc thick. The subject fixated the upright cross, and, following a warning tone, the oblique cross appeared for 50 ms. The subject was required to report the horizontal and vertical position of the oblique cross with respect to the upright cross, i.e., right/left/aligned, above/below/aligned. If the subject reported misalignment, then, on the next trial, the oblique cross was repositioned, by a discrete amount, so as to reduce its apparent horizontal and vertical separation from the upright cross. If the subject reported alignment along one or the other axis, no adjustment was made along that axis for the next trial. At the start of each block of trials, the oblique cross could have horizontal and vertical offsets (relative to the upright cross) of 0, +5, or 5° (varied randomly). The repositioning steps were initially 2°, and then reduced with each successive report of alignment or reversal in the apparent misalignment, to 1°, then to 0.5°, and finally to 0.2°. When the horizontal and vertical steps had both decremented to 0.2°, the oblique cross had invariably reached an asymptotic position, and the block was continued for 20 more trials before starting a new block with a new misalignment. Each subject completed 9-20 blocks of trials, and his/her phoria was estimated from the average misalignment of the 2 crosses for the last 20 trials in each block.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Vergence responses to horizontal disparity steps

Horizontal disparity steps of suitable amplitude applied to large random-dot patterns elicited consistent horizontal vergence responses at short latencies. Figure 1 shows sample horizontal vergence velocity temporal profiles from 1 subject in response to 174 horizontal crossed-disparity steps of 2° applied 50 ms after 10° leftward centering saccades. Data are shown for all trials except the few (8) contaminated with saccades. Also shown in Fig. 1 is the mean horizontal vergence velocity profile (±SD), together with the mirror galvanometer feedback signals indicating the horizontal positions of the images seen by the left and right eyes. The estimated mean latency (objectively determined, see METHODS) for the data shown in Fig. 1 was 82.5 ms and is indicated by the arrows. For 2° crossed-disparity steps such as those used in Fig. 1, the mean latency for seven subjects was 80.9 ± 3.9 (SD) ms. Individual mean latencies (±SD), together with the number of measures contributing to the estimates (when the latency algorithm converged) and the total number of trials from which the data were drawn (after rejecting trials with saccades or blinks), were as follows: 82.5 ± 4.8 ms (subject RK, n = 174/174), 86.0 ± 8.7 ms (FM, 140/144), 84.6 ± 11.6 ms (GM, 147/177), 78.0 ± 6.2 ms (JG, 147/148), 77.7 ± 9.2 ms (KP, 165/168), 81.6 ± 7.0 ms (MB, 169/172), and 75.6 ± 9.2 ms (HA, 31/32). Because our major concern was with the initial open-loop vergence responses, the temporal profiles in all figures are discontinued 180 ms after the onset of the disparity steps.

View larger version (32K): [in this window] [in a new window]   Fig. 1. Sample disparity-vergence responses to multiple presentations of a 2° crossed-disparity step applied 50 ms after 10° leftward centering saccades (subject RK). Individual vergence velocity profiles (n = 174), obtained by differentiating splined vergence position traces and then subtracting the mean vergence velocity trace obtained in saccade-only control trials (to eliminate any effects due to postsaccadic drift). Data are collected together from several recording sessions. Arrow: estimated latency of onset. Calibration bar applies to ocular vergence traces only. Upward deflections represent convergent velocities.

BINOCULAR RESPONSE TO A BINOCULAR STIMULUS. These short latencies are similar to those reported for the ocular following elicited by conjugate ramps applied to large textured patterns (Gellman et al. 1990). Because visual motion detectors can integrate position steps over time to generate an apparent motion signal (Mikami et al. 1986a,b; Newsome et al. 1986), it was possible that the short-latency vergence responses resulted from independent monocular tracking, in which each eye tracked the apparent motion that it saw, rather than from the binocular misalignment per se. To test this idea, we restricted the step to one eye only, leaving the other eye to view a stationary pattern. The movements of the eye that saw the stationary pattern were of particular interest; any effects on this eye due to the monocular apparent-motion cues should have been either negligible (if motion stimuli affected only the eye that saw the motion) or in the same direction as the motion (if motion stimuli at either eye affected both eyes), whereas any effects on this eye due to the binocular misalignment (stereo) cues should have been in the direction opposite to the apparent motion stimulus.

View larger version (19K): [in this window] [in a new window]   Fig. 2. The effect on the horizontal vergence and version velocity responses of restricting the horizontal disparity step to one eye. Crossed-disparity steps of 2.4° were applied by 1) shifting the images seen by the left and right eyes equally (), 2) shifting the image seen by the right eye leftward while the other image remained stationary (- - - - -), 3) shifting the image seen by the left eye rightward while the other image remained stationary (· · ·). All traces are means (n = 183). Zero on the abscissa indicates the time of onset of the disparity steps. Subject RK.

DEPENDENCE ON THE MAGNITUDE AND DIRECTION OF THE STEPS. Vergence responses were often not exactly aligned with the direction of the disparity steps and could include appreciable orthogonal components ("directional errors"). That is, responses to horizontal steps included vertical vergence, and responses to vertical steps included horizontal vergence. Like the isogonal components, these orthogonal components showed a highly systematic dependency on the amplitude and direction of the disparity step, but the form of this dependency differed markedly for the two components and will be described separately.

View larger version (27K): [in this window] [in a new window]   Fig. 3. The vergence-velocity responses elicited by crossed-disparity steps: dependence on the amplitude of the step (sample mean traces for 3 subjects). Top row: vertical vergence velocity (v) over time (ordinate) is plotted against horizontal vergence velocity (h) over time (abscissa); dots on traces at 100, 120, 140, and 160 ms measured from the onset of the step; tick marks on the axes are at 1°/s intervals. Middle and bottom rows: temporal profiles for horizontal vergence velocity (h) and vertical vergence velocity (v), respectively; numbers at ends of traces, disparity steps (in deg); continuous lines, responses (to small disparity steps) that increment as stimulus amplitude increases; dashed lines, responses (to larger steps, indicated by numbers in parentheses) that decrement as stimulus amplitude increases; calibration bars, 2°/s; abscissa, time from onset of the disparity steps; upward deflections represent convergent (h) and left-sursumvergent (v) velocities. Column A, subject RK; column B, subject FM; column C, subject GM.

View larger version (24K): [in this window] [in a new window]   Fig. 4. The vergence-velocity responses elicited by uncrossed-disparity steps: dependence on the amplitude of the step (sample mean traces for 3 subjects). Layout and conventions as for Fig. 3.

View larger version (20K): [in this window] [in a new window]   Fig. 5. The vergence-velocity responses elicited by 12.8° disparity steps (sample mean traces for 3 subjects). Layout and conventions as for Fig. 3.

View larger version (36K): [in this window] [in a new window]   Fig. 6. The dependence of vergence responses on the amplitude of the horizontal disparity steps: tuning curves (3 subjects). Mean change-in-vergence measures (ordinate) are plotted (, , and ) against the amplitude of the disparity step (abscissa). A: horizontal (isogonal) vergence responses; curves indicate the least-squares best fits obtained using Eq. 2, for which r2 averaged 99.3%. B: vertical (orthogonal) vergence responses; curves indicate the least-squares best fits obtained using Eq. 1, for which r2 averaged >90% despite the poor fit for the crossed-step data of subject FM (r2 = 59.8%). Open symbols are the mean change-in-vergence measures obtained with 12.8° vertical disparity steps (data for right-hyper steps shown at left, data for left-hyper steps shown at right). Key identifies the data from each of the 3 subjects. Error bars, 1 SD.

Vergence responses to vertical disparity steps

Vertical disparity steps of suitable amplitude elicited consistent vertical vergence responses at short latencies. Using the vertical vergence velocity data obtained with 1.2° left-hyper steps, mean latency (±SD) determined by our objective method was 85.1 ± 3.4 ms for four subjects. Individual mean latencies (±SD), together with the number of measures contributing to the estimates and the total number of trials from which the data were drawn, were 86.9 ± 12.4 ms (subject RK, n = 159/179), 84.9 ± 12.3 ms (FM, 122/145), 88.2 ± 14.3 ms (GM, 152/177), and 80.5 ± 11.5 ms (HA, 26/31). These latencies are on average 3 ms longer than those listed earlier for the isogonal vergence responses (of these same 4 subjects) to horizontal disparity steps.

BINOCULAR RESPONSE TO A BINOCULAR STIMULUS. Once again, we tested the possibility that these short-latency (isogonal) vergence responses might have resulted from independent monocular tracking, rather than from disparity per se, by restricting the steps to one eye only, leaving the other eye to view a stationary pattern. The results of one such experiment on subject RK are shown in Fig. 7, in which vertical positive-disparity steps (that is, left-hyper steps) of 1.2° were applied by 1) shifting the pattern seen by the left eye 0.6° upward and the pattern seen by the right eye 0.6° downward (), 2) shifting the pattern seen by the left eye 1.2° upward (· · ·), or 3) shifting the pattern seen by the right eye 1.2° downward (- - - - -). The outcome was essentially the same as for horizontal vergence in that monocular steps were almost as effective as binocular ones in producing (isogonal) vergence, and the eye that saw the stationary pattern always moved in a direction appropriate for a stereo-driven response: when only the right eye saw a step (downward), the left eye moved upward, and when only the left eye saw a step (upward) then the right eye moved downward. Essentially identical data were obtained from two other subjects (FM and GM).

View larger version (28K): [in this window] [in a new window]   Fig. 7. The effect on the vertical vergence and version velocity responses of restricting the vertical disparity step to 1 eye. Left-hyper steps of 1.2° were applied by 1) shifting the images seen by the left and right eyes equally (), 2) shifting the image seen by the right eye downward while the other image remained stationary (- - - - -), 3) shifting the image seen by the left eye upward while the other image remained stationary (· · ·). All traces are means (n = 182). Zero on the abscissa indicates the time of onset of the disparity steps. Subject RK.

DEPENDENCE ON THE MAGNITUDE AND DIRECTION OF THE STEPS. The vergence data obtained with vertical steps showed a pattern of dependence on the amplitude and direction of the steps strongly resembling that reported above for horizontal steps. The relevant data for vertical steps are shown in Figs. 810, which are organized like Figs. 3-5 to permit a ready comparison with the data for horizontal steps. It is evident from these figures that the disparity tuning curves for the data obtained with vertical steps had the same general form as the curves obtained with horizontal steps: the curves for the isogonal component resembled the derivative of a Gaussian, and the curves for the orthogonal component were exponential (with the same polarity for positive and negative stimuli). However, there were consistent quantitative differences between the two sets of data.

View larger version (29K): [in this window] [in a new window]   Fig. 8. The vergence-velocity responses elicited by left-hyper disparity steps: dependence on the amplitude of the step (sample mean traces for 3 subjects). Top row: vertical vergence velocity (v) over time (ordinate) plotted against horizontal vergence velocity (h) over time (abscissa); dots on traces at 100, 120, 140, and 160 ms measured from the onset of the step. Middle and bottom rows: temporal profiles for horizontal vergence velocity (h) and vertical vergence velocity (v), respectively. Other conventions as for Fig. 3.

View larger version (31K): [in this window] [in a new window]   Fig. 9. The vergence-velocity responses elicited by right-hyper disparity steps: dependence on the amplitude of the step (sample mean traces for 3 subjects). Layout and conventions as for Fig. 8.

View larger version (36K): [in this window] [in a new window]   Fig. 10. The dependence of vergence responses on the amplitude of the vertical disparity steps: tuning curves (3 subjects). Mean change-in-vergence measures (ordinate) are plotted (in filled symbols) against the amplitude of the disparity step (abscissa). A: vertical (isogonal) vergence responses; curves indicate the least-squares best fits obtained using Eq. 2, for which r2 averaged 99.4%. B: horizontal (orthogonal) vergence responses; curves indicate the least-squares best fits obtained using Eq. 1, for which r2 averaged >90% despite the poor fit for the left-hyper data of subject FM (r2 = 70.9%). Open symbols are the mean change-in-vergence measures obtained with 12.8° horizontal disparity steps (data for uncrossed steps shown at left, data for crossed steps shown at right). Key identifies the data from each of the 3 subjects. Error bars, 1 SD.

View larger version (22K): [in this window] [in a new window]   Fig. 11. Isogonal vergence responses to horizontal and vertical disparity steps. Ordinate, absolute mean change-in-vergence measures; abscissa, amplitude of disparity step (logarithmic scale). Filled symbols and continuous lines, horizontal responses (thick: crossed steps; thin, uncrossed steps); open symbols and dashed or dotted lines, vertical responses (dotted: left-hyper steps; dashed, right-hyper steps). Circles, subject RK; squares, subject FM; diamonds, subject GM. Error bars, 1 SD.

Quantitative comparison of the disparity tuning curves for horizontal and vertical steps

We fitted mathematical functions to the change-in-vergence measures for both the isogonal and orthogonal components of the vergence responses, and these functions are plotted as smooth curves in Figs. 6 and 10. The function parameters provide a succinct summary of the data and permit quantitative comparisons of the responses of the different subjects and, for a given subject, of the responses to horizontal versus vertical steps. Later, the function parameters will be used to compare the data for different stimulus patterns (large-dot patterns vs. small-dot patterns), as well as for human versus monkey.

The following exponential function (Eq. 1) was fitted to the mean orthogonal responses of each subject

(1)

where d is the disparity error created by the step, A_o is the asymptotic level of the exponential, C is an offset, and B_o is the space constant. It is clear from the curves in Figs. 6B and 10B that Eq. 1 generally provided a good fit to the orthogonal data with the exception of the responses of FM to crossed steps, which were unusual in showing a slight overshoot before settling at the asymptotic level. The best-fit parameters for all three subjects are listed in Table 1, which also includes an estimate of the goodness of fit, r², based on the percentage of the disparity-induced variation in vergence explained by the the sum of squares due to regression, with compensation for the mean: mean corrected r² (Cumming and Parker 1999; Engelman 1999). Most of the fits were clearly very good, r² averaging more than 90%, despite the poor fit for the data of FM. Table 1 indicates that the space constant, B_o, showed considerable inter-subject variation (range: 0.39-3.35°) and, for a given subject, could show substantial directional asymmetries for crossed versus uncrossed and right-hyper versus left-hyper stimuli. Excluding the crossed-step data for FM (because the fit was poor), B_o was always larger for the vertical vergence responses than for the horizontal, on average, by 130% (1.93 vs. 0.84°). It is also evident from Table 1 that A_o and C were generally negatively correlated. For example, on average, C is 21% of A_o (correlation coefficient 0.96) for the horizontal responses to vertical steps and 11% of A_o (0.74) for the vertical responses to horizontal steps. This means that the exponential fit generally showed a reversal in sign for the smallest disparities. Often, this reversal occurred below 0.2° (the smallest step that we used), and no reversal was evident in the recorded data. In such cases, it is possible that the y-offset in Eq. 1 (C) actually compensates for a small x-offset in the data, such as might occur, for example, if there were a small dead zone within which disparity is ineffective. This raises the possibility that an x-offset might be more appropriate in Eq. 1 than the y-offset that we have used. However, in other cases, the recorded responses to the smallest steps showed a clear reversal in sign: for examples, see the data of GM (uncrossed) and RK (crossed) in Fig. 6B. In such cases, where the fitted exponential function must show sign reversal, a separate y-offset term is critical.

The following function (Eq. 2) was fitted to the isogonal responses

(2)

The first term in Eq. 2 accounts for the nonzero asymptote, and, because this was independent of the direction of the step, we assumed that it shared the exponential development of the response to the orthogonal stimulus. Accordingly, the first term is an exponential function with an asymptotic level, A_i, and a space constant, B_i. The latter was fixed at the value of the space constant that was obtained when Eq. 1 was fitted to the pooled orthogonal data: for the horizontal isogonal fits, we pooled the horizontal responses to right-hyper and left-hyper steps, and for the vertical isogonal fits, we pooled the vertical responses to crossed and uncrossed steps. The second term in Eq. 2 is a Gabor function, in which  is the Gaussian width, f and  are the spatial frequency and phase of the cosine term, and G is a gain factor. Because the data were usually not symmetrical about zero, we incorporated a parameter, D, to allow the peak of the Gaussian to shift, and, because the rate at which responses incremented (with small disparities) was sometimes appreciably greater than the rate at which they decremented (with higher disparities), we scaled the disparity, d, by a compression factor, F, which could have values ranging downward from unity. A value for F of unity indicates no asymmetry, and lower values indicate that the rising phase of the tuning curve was steeper than the falling phase (increased skew). It is clear from the curves in Figs. 6A and 10A that Eq. 2 provided a good fit to the isogonal data for all three subjects, and the best-fit parameters for these curves are listed in Table 2, together with an estimate of the goodness of fit (again, r²), and P, the peak-to-peak amplitude of the best-fit functions. The mean r² indicates that, on average, the fit accounted for 99.4% of the disparity-induced variation in vergence. Table 2 indicates that the amplitude (P), Gaussian width (), and compression factor (F) were consistently greater for the responses to horizontal steps than for the responses to vertical steps: on average, P by 241%,  by 43%, and F by 31%. The spatial frequency (f) and phase () were quite variable (range for f, 0.028-0.164 cycles/deg; range for , 263.5-278.6°), but neither showed consistent differences between horizontal and vertical. The differences in F indicate that the asymmetry in the rising and falling phases of the tuning curves was greater for the vertical data.

Dependence on the size of the elements in the visual patterns

All of the above vergence responses were obtained with random-dot patterns in which the individual dots each had a diameter of 2°, and we now report the effects of changing the diameter of the dots up to eightfold while maintaining the same overall coverage (50%). These experiments were carried out on monkeys as well as humans to allow a comparison of the two species. We know from the study of Busettini et al. (1996b) that the disparity tuning curves for the isogonal horizontal responses of the monkey have the same general form as the curves of humans, resembling the derivative of a Gaussian with a nonzero asymptote. However, that earlier study did not document the orthogonal responses or the responses to vertical disparity steps and employed visual stimuli (irregular patterns) that cannot be formally described, so that direct quantitative comparisons with the present human data were not possible. We therefore undertook a comparison study on three monkeys and three humans using the same stimulus patterns.

The vergence responses of humans and monkeys showed a similar dependence on the size of the individual dots in the disparity stimuli, and Fig. 12 shows some representative data from one human subject. The most obvious effect on the isogonal disparity tuning curves was on the falling phase: for both horizontal and vertical steps, the transition from the peak to the asymptote tended to be more gradual as the dot size increased. The peak responses tended to occur at slightly higher disparities with the larger dots, but effects on the initial rapid rise to the peak were generally small and effects on the asymptote were somewhat variable. The latter was also evident in the orthogonal disparity tuning curves, and, in Fig. 12, B and D, this effect partially obscures a tendency for the space constant to increase with dot size.

View larger version (43K): [in this window] [in a new window]   Fig. 12. The dependence of vergence responses on the size of the individual random dots: tuning curves for a sample subject (RK). Mean change-in-vergence measures (ordinate) are plotted against the amplitude of the disparity step (abscissa). Top: isogonal vergence responses to horizontal (A) and vertical (C) disparity steps; curves indicate the least-squares best fits obtained using Eq. 2, for which r2 averaged 99.5%. Bottom: orthogonal vergence responses to horizontal (B) and vertical (D) disparity steps; curves indicate the least-squares best fits obtained using Eq. 1, for which r2 averaged 97.3 ± 1.7% (mean ± SD). Key identifies the diameters of the dots. Error bars, 1 SD.

Such quantitative effects were assessed by fitting Eq. 1 to the orthogonal data and Eq. 2 to the isogonal data, and the best-fit parameters are summarized in Tables 3 and 4, respectively, which show average data for the three humans and the three monkeys. The fits for the isogonal data were all extremely good for the human data (r², 99.4 ± 0.2% mean ± SD) and only slightly less so for the monkey data (r², 97.9 ± 1.5% mean ± SD).¹ The fits for some of the orthogonal data were poor, especially for the monkey vertical responses (to horizontal steps), mostly because the latter were vanishingly small and sometimes showed a very slight transient overshoot similar to that of subject FM seen in Figs. 6B and 10B. (Note that this was of little consequence for the fits of Eq. 2 to the monkey's vertical isogonal responses because the latter had correspondingly small asymptotes: see A_i in Table 4.) If the monkey vertical response data are excluded, however, r² for the orthogonal fits exceeded 0.8 in 32/36 data sets; again, the (4) data sets that were fitted poorly were of very small amplitude and showed slight overshoot.

Regarding the best-fit isogonal parameters, the only one that consistently showed a systematic dependence on dot size was the Gaussian width (), which increased with increasing dot size. For example, the value of  with the largest dots exceeded that with the smaller ones (on average) by 47% in humans (horizontal, 41%; vertical, 53%) and 152% in monkeys (133%, 172%). The compression factor, F, and the spatial frequency, f, sometimes showed a slight tendency to decrease with increasing dot size, and this, together with the changes in , largely accounted for the more gradual transition from the peak to the asymptote as the dot size increased. The only orthogonal par