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1Department of Anatomy and Neurobiology and 2Department of Biomedical Engineering, Washington University School of Medicine, St. Louis, Missouri
Submitted 29 December 2004; accepted in final form 17 March 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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; Cohen and Andersen 2002), their spatial location is no longer known whenever gaze, the position of the eyes in space, is either actively or passively redirected. To compensate for potential gaze changes and maintain a stable neural representation of the outside world, the brain must continuously adjust its internal representation of the environment using extraretinal signals. This ability, often referred to as "updating," has been demonstrated often for intervening voluntary eye movements (Hallett and Lightstone 1976; Herter and Guitton 1998; McKenzie and Lisberger 1986; Ohtsuka 1994; Schlag et al. 1990; Zivotofsky et al. 1996).
In everyday life, gaze changes relative to objects of interest also occur when our head and body rotate or translate in space. Medendorp and colleagues (2003) recently showed that updating is accurate when humans actively moved their head sideways by bending at the lower back and neck. Importantly, the amount of body displacement during translation must be weighted differently in the updating of far versus near targets, as shown in Fig. 1A, which illustrates the geometrical need for a larger eye movement for near compared with far targets during translation. This occurs because updating the stored representation of object location must anticipate the consequences of motion parallax, a geometrical property whereby near objects slip on our retina more than those on the horizon (Howard and Rogers 1995). Medendorp et al. (2003) showed that this geometrical property is taken into account by adjusting the amplitude of memory saccades during active translations and rotations. For the latter, incorporating the geometry of motion parallax is also necessary (although much smaller) when the head rotates about an axis that is different from the rotational axis of the eye, thus involving an eye translation relative to the target, as illustrated in Fig. 1B.
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study can arise from multiple modalities, including vestibular, proprioceptive, and efference copy signals. Are we equally capable of performing these tasks in response to gaze-in-space changes associated with passive displacements of the body, such as those occurring when riding in a car or train? Under these conditions, neither efference copies of the motor command nor neck proprioceptive signals can provide veridical information about the intervening movement, leaving the vestibular system as the main extraretinal source of motion-related information. Indeed, there is growing evidence that vestibular signals are important for path integration (Bremmer and Lappe 1999; Cohen 2000; Israel et al. 1996, 1997; Stackman and Herbert 2002) and can be used to drive voluntary eye movements (Israel and Berthoz 1989).
It is presently unknown whether vestibular signals can be used to update memory-guided eye movements during translation, similarly as previously shown for head and body rotation (Baker et al. 2003; Israel et al. 1999; Klier et al. 2005). The main goal of the present study was to investigate whether memory-guided saccades are scaled by both motion amplitude and viewing distance during passive displacements. Our results support this hypothesis, suggesting that vestibular information can be used centrally to anticipate the visual consequences of motion parallax for spatial memory, as shown previously for low-level, reflexive eye movements (Paige and Tomko 1991; Schwarz and Miles 1991; Schwarz et al. 1989). In addition to characterizing the accuracy of the updating process for the first time during passive translations, we also quantify the properties of the version and vergence eye movements elicited during both the stationary and motion memory depth tasks. Preliminary results have been presented in abstract form (Li et al. 2004).
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Two rhesus monkeys were chronically implanted with a head ring anchored to the skull with stainless steel or titanium screws (Angelaki 1998). To record eye movements, traditional scleral eye coils were surgically implanted under the conjunctiva of both eyes to measure horizontal and vertical eye movements. Animals were trained to perform memory-guided saccadic eye movements (Baker et al. 2003; Vogelstein et al. 2003). All surgical procedures and animal handling were in accordance with institutional and National Institutes of Health guidelines.
Experimental setup
During experiments, head-restrained monkeys sat in a primate chair that was mounted on top of a motion platform (Neurokinetics, Pittsburgh, PA), which was used to passively translate or rotate the animals. The initial fixation target was a light-emitting diode (LED) located about 1 m in front of the monkey at eye level. This "fixation" target, which was mounted on the motion platform, always remained at a fixed distance from the animal (head-fixed). This ensured a similar starting position of the eyes for all runs. Visual targets for the memory eye movement tasks (referred to here as "flashed" or "memory" targets) were provided by one of 5 LEDs lined up directly in front of the monkey at different distances: 12, 17, 22, 27, and 60 cm. These LEDs were mounted on a horizontal panel that was placed slightly below the monkey’s eyes and secured to the floor. Thus flashed/memory targets always remained fixed to the world in these experiments and required primarily horizontal eye movements to be foveated (see Data analyses). Because these LEDs were placed slightly below the head-fixed far target at 1 m, small vertical eye position changes (about 4–10°) were also typically called for. The position of the monkey from the target array at the beginning and end of each run could be manipulated using the motion platform (see Behavioral paradigms).
Binocular horizontal and vertical eye movements were measured using a 3-field magnetic coil system (CNC Engineering, Seattle, WA). The animal’s performance during these memory tasks was monitored on-line using behavioral windows for both version and vergence. Instantaneous version and vergence positions were calculated from the left (L) and right (R) eye positions, as (R + L)/2 and L – R, respectively. Eye-movement signals were low-pass filtered using 6-pole Bessel filters at 200 Hz. The data were digitized at 833.3 Hz with 16-bit resolution and stored for off-line analysis. A custom-written script in the Spike2 software controlled stimulus presentation and data-acquisition hardware (CED Power 1401; Cambridge Electronic Design, Cambridge, UK).
Behavioral paradigms
Animals were trained to perform memory saccades under 3 conditions, which will be referred to as "stationary," "translational motion," or "rotational motion" tasks. These 3 types of memory tasks were typically randomly interleaved within blocks of trials that were collected on multiple experimental days. The general outline of the tasks is illustrated in Fig. 2. All trials began with the onset of the fixation target in a dimly illuminated room (interval 1). During continued fixation of the 1-m fixation target (±2° version and ±0.75° vergence windows) for >500 ms, one of the closer, world-fixed LEDs flashed for 200 ms (interval 2). Then, during a variable delay period lasting 750–1,750 ms (interval 3), animals either remained stationary ("stationary" task) or were moved ("motion" tasks). This movement, which started within 50 ms after the flash, lasted 500 ms, was completed before the fixation target was turned off, and included either a 5-cm rightward/leftward translation or a 10° leftward/rightward rotation (with the axis of rotation 7.5 cm behind the eyes). All motions had a trapezoidal velocity profile with peak velocities of 10 cm/s and 20°/s for translation and rotation, respectively. For translational trials, peak linear acceleration was 1.25 m/s2. During the motion, animals were trained to suppress their VOR with eye position being maintained within the specified behavioral fixation windows.
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; Zee et al. 1992). The duration of individual trials varied from trial to trial (about 4–7 s) as a result of the variable delay period and variable reaction times for memory-guided and corrective eye movements (including the required 1-s memory and postmemory fixation periods once the eyes fell inside the respective behavioral windows). If either the version or vergence eye position fell outside the specified behavioral windows at any time during the task, the trial was aborted and the data were discarded.
Motion and stationary trials differed in the motor error needed to generate accurate memory-guided eye movements. For stationary trials, the required memory eye movement was toward the location of the flash, and thus motor error was approximately equal to retinal error (Klier and Crawford 1998) and no updating was required. In contrast, because at the end of the rotational/translational motion trials, animals ended up at a different angle (and slightly different distance) relative to the flashed target, the retinal location of the flash was no longer appropriate as a motor error for the memory eye movement. Instead, for the monkeys to perform this task accurately, extraretinal sensory cues related to the intervened movement were required to compute the motor error for the memory eye movement.
To directly compare memory eye movements for motion trials (where updating was necessary) with those for stationary trials (where no updating was needed), data for the stationary task were gathered at all initial and final positions of the motion trials. This included a total of 25 combinations, corresponding to the 5 flashed targets at any one of 5 positions relative to the target array: original orientation (with memory targets directly in front), rotated 10° rightward/leftward, or translated 5 cm rightward/leftward of the target array. With animals in the original orientation and flashed targets directly in front, the horizontal eye movement called for during the memory period was a purely vergence response (depth change). In contrast, when the eye movements were made from the rotated or translated stationary positions, changes in both the version (direction change) and vergence (depth change) were necessary.
Animals were extensively trained with this task for longer than 3 months. To verify that a trial-by-trial sensorimotor transformation was performed during these spatial memory tasks, trained animals were also tested with additional experimental blocks that included novel motion amplitudes (but no stationary trials). For these new protocols, 3 translation (3, 4, or 5 cm) and 3 rotation (5, 8, or 10°) amplitudes were used (for a total of 12 x 5 = 60 conditions). Because we wanted to investigate how accurate these eye movements were, and to avoid excluding runs where memory performance was poor, no behavioral window was imposed for the memory period during these variable motion amplitude blocks. Thus all runs in which the monkeys successfully ignored the flash and refixated the memory target after it was relighted were rewarded and saved for off-line analysis.
Finally, data were also collected during interleaved rotation and translation motion trials to the 17-, 22-, 27-, and 60-cm targets for 2 different locations of the initial fixation, head-fixed LED: one at 1 m (similar as in all other experimental protocols), the other at 12 cm. This additional protocol aimed at comparing the accuracy of updating and the scaling of the memory saccade amplitude by distance, separately for convergent and divergent eye movements.
Data analyses
Only successful trials (with reward delivery) were analyzed off-line using Matlab (The MathWorks, Natick, MA). Eye movements were calibrated using a daily fixation task with positive eye position corresponding to upward and rightward directions. A custom-written script allowed the experimenter to examine each trial manually by plotting the horizontal and vertical positions of each eye, as well as horizontal version and vergence. A semiautomatic procedure was used to identify saccades when eye velocity (resultant of horizontal and vertical components) exceeded (or fell below) 25°/s. Memory responses were aligned at saccade onset and averaged across trials.
For each experimental run, 4 sets of eye position values were computed by averaging eye position over 20-ms time intervals: 1) The initial fixation was computed 50 ms before the head-fixed target was turned off. 2) The endpoint of the memory saccade was computed 50 ms after the end of the memory saccade. 3) Because of an often slowly changing vergence angle, the end of the memory period (also referred to here as memory eye position) was computed 50 ms before the reillumination of the memory target (i.e., 950 ms after the eyes were within the specified memory windows; see above). 4) For a comparison of the memory-guided with the visually guided movement, the postfixation eye position was also computed after the memory target was relighted at the end of the trial. These postfixation data were used as the "ideal" eye movement necessary to foveate the target.
The changes in horizontal version (direction) or vergence (depth) eye position after the memory-guided eye movement were then calculated as the difference between the endpoint of the saccade or the memory period and the initial fixation position. These values were then compared with the required change in eye position, computed as the difference between the postfixation and the initial fixation values. These variables defined the "accuracy" of memory-guided eye movements and correspond to a vergence/version generalization of what has been referred to as "systematic error" in previous studies (White et al. 1994). In addition, to estimate the variability of memory-guided eye movements, we computed "variable errors" defined as the SD of version and vergence endpoints about their mean. Because the vertical saccadic component was small (see Experimental setup), analyses have focused on horizontal eye movements.
Relationships between variables were quantified using linear regressions, obtained by minimizing either the vertical offset (Regress function of Matlab) or the perpendicular offset of the data to the line (using a nonlinear least-square algorithm based on the interior-reflective Newton method; Coleman and Li 1994, 1996). The latter analysis was used for independent variables (e.g., Fig. 9), with 95% confidence intervals computed using bootstrapping with replacement thus, the confidence intervals were typically asymmetric. Other comparisons between variables were made using ANOVA.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Monkeys were trained to perform memory-guided eye movements to targets flashed at different distances, as illustrated in Fig. 3. For example, the trial shown in Fig. 3A started with fixation of a far (1 m), head-fixed, central LED, while a world-fixed target at a distance of 12 cm was briefly flashed (Fig. 3; intervals 1 and 2). Then, as soon as the fixation target was turned off (Fig. 3; end of interval 3) and while stationary in complete darkness, the monkey made an eye movement toward the memorized world-fixed target location (Fig. 3; interval 4). Because the central fixation and flashed targets were located at different depths, the two eyes did not move equally. In fact, because the flashed target in this example was in the midsagittal plane, there was an opposite change in the horizontal position of the two eyes, resulting in a large vergence angle change (Fig. 3A, last row). This change in vergence for near targets was typically less than expected from a visually guided eye movement, as illustrated by the corrective movement that brought the eyes closer to the target when it was turned back on (Fig. 3A, interval 5). For targets flashed further away from the animal, the observed change in vergence was smaller (e.g., for a 60-cm target; Fig. 3B).
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Memory eye movements for the motion tasks
For the stationary trials, results of which have been summarized above, the required memory eye movement was toward the location of the flash and no updating was required. In contrast, for motion trials, animals were either translated 5 cm or rotated 10° to the right/left during the delay period, such that the spatial location of the memory eye movement goal differed from the location of the flash (Fig. 2). Examples of two such motion trials, with the flash at 12 and 60 cm and a 5-cm leftward displacement during the delay period, are illustrated in Fig. 7. If the animal did not update the goal of the memory eye movement, a purely vergence response would be expected, as illustrated under identical stationary conditions in Fig. 3. In contrast, for the memory eye movement to be appropriate for the now eccentric location of the memory target, a conjugate rightward horizontal eye movement, whose amplitude should depend on flashed target distance, would be expected to accompany the change in vergence. Indeed, despite an identical displacement, the memory saccade depended on the distance of the flash: its amplitude was larger for near than far targets (Fig. 7, A and B).
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; Hine and Thorn 1987; Snyder and King 1992; Wei and Angelaki 2004).
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Similar saccade sizes were evoked during convergence and divergence trials [ANOVA, F(1,651,1) = 1.56, P >> 0.05]. When motion data were plotted versus the respective stationary data (as in Fig. 9), regression line slopes were larger for convergence than for divergence trials (mean ± 95% confidence intervals were 0.84 ± 0.60 and 0.50 ± 0.40, respectively) (Fig. 11, black and gray lines). Although slopes were different, confidence intervals were overlapping, suggesting that updating performance was not statistically different between convergence and divergence trials. Therefore the memory update appears to be invariant with the depth location of the first target, although the exact saccade scaling by distance might depend on the direction and intensity of the ongoing vergence.
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Because stationary condition trials were not tested in these blocks, mean values of the memory saccades for translational and rotational motion trials were plotted as a function of the inverse distance of the flash, as shown in Fig. 12, A and B. Monkeys compensated for both the actual distance of the flash and the amplitude of the imposed motion [factorial ANOVA, F(1,790,4) = 300 for translation, F(1,801,4) = 90 for rotation, P < 0.001 for both]. These relationships were quantified using linear regression (Fig. 12, solid lines). Superimposed in the figure is the ideal performance that would correspond to the different motion amplitudes (Fig. 12, dashed lines). It is obvious by comparing the solid and dashed lines of the same color that, following translational updating, the evoked memory saccades: 1) scaled less with inverse distance than expected according to the geometry and 2) did not asymptote to zero saccade amplitude at an inverse viewing distance of zero (i.e., viewing at infinity; Fig. 12A).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Neural estimate of target distance: origin of vergence errors during the stationary spatial memory task
We observed systematic and consistent errors in memory-guided vergence during the stationary spatial memory task. These results in monkeys are quantitatively similar to what was previously shown in human subjects (Medendorp et al. 2003). In the human experiments, targets were flashed in complete darkness, unlike the present experiments where the room was dimly illuminated during the time of the flash. Yet, under both conditions, memory vergence was less for near targets, compared with the visually guided vergence position acquired when the target was turned back on at the end of the trial. Because memory-guided vergence was accurate for far targets (Fig. 6), we consider unlikely that these errors arise from a range effect (Kapoula 1985; Kumar et al. 2003).
The sources of both the systematic and variable errors in memory-guided eye movements have been investigated for saccades on a frontoparallel plane (i.e., without associated changes in vergence). Based on the results from such studies, it is unlikely that significant errors may arise from the deterioration of spatial memory. For example, the temporal deterioration for memory-guided saccades in the frontoparallel plane occurs only for much longer delay times than those used in the present experiments (Gnadt et al. 1991; Ploner et al. 1998; White et al. 1994). Although it was originally proposed that the errors are introduced during the transformation from a perceptual memory of object location to a memory of the intended eye movement (Gnadt et al. 1991), there is increasing evidence that the systematic errors associated with the accuracy of conjugate saccades are introduced in the motor output downstream from the superior colliculus (Stanford and Sparks 1994) and downstream from the storage of remembered target location (Opris et al. 2003). Because the neural control of conjugate saccades and vergence eye movements are at least partly distinct (Gamlin 1999; Mays and Gamlin 1995; but see Zhou and King 1998), these conclusions on memory-guided saccades on a frontoparallel plane are not easily transferable to the vergence component of memory-guided eye movements.
Vergence errors could be the result of a perceptual mislocalization of the distance of the target. This issue has been studied extensively with pointing movements, where similar errors are found (Berkenblit et al. 1995; Foley 1985; Soecthing and Flanders 1989a). Such distance errors in pointing have been attributed either to nonlinear distortion of perceptual space (Wolpert et al. 1994) or to the underlying sensorimotor transformations (Soechting and Flanders 1989a,b). Similar errors of perceived distance have also been reported during walking (Philbeck and Loomis 1997), as well as in the scaling of the translational vestibuloocular reflex (TVOR) by a neural estimate of target distance (Angelaki et al. 2000; Schwarz and Miles 1991; Telford et al. 1997; Wei and Angelaki 2004; Wei et al. 2003). Because of such extensive evidence that, at least for visuomotor processing, near targets are typically mislocalized, Medendorp et al. (2003) hypothesized that the shortage of memory vergence for near targets in the absence of motion reflected the mislocalization of the distance of the flashed target.
However, as illustrated here (Fig. 4), memory vergence typically continues to increase throughout the memory period. In contrast, the dynamics of visually guided vergence are typically faster (Fig. 4B; see also Maxwell and King 1992; Zee et al. 1992). Thus at least some of the memory vergence errors for the stationary task might stem from the fact that vergence eye movements, typically requiring a closed-loop control system (Collewijn and Erkelens 1990; Horng et al. 1998; Leigh and Zee 1999; Mays and Gamlin 1995), were performed open- loop (i.e., in complete darkness, in the absence of visual feedback) during the spatial memory tasks.
Inherent property or context-specific adaptation?
An important issue to consider is whether the properties of memory-guided saccades during translation described here truly represent a natural state of the system or whether the ability for a distance-dependent compensation for translational/rotational movements arises as a result of extensive behavioral training. Specifically, unlike human subjects who can be asked to perform this task without visual feedback (i.e., memory target being turned on at trial end), monkeys must be extensively trained for several months to learn to perform memory saccades. However, the saccadic system is extremely plastic, with a great ability for "context-specific" adaptation, in which saccadic responses are trained to maintain different values according to some additional input, such as vertical gaze angle or gravitational information (Shelhamer and Clendaniel 2002).
Thus it might seem plausible that the saccadic system could be adaptively trained to exhibit exactly the functionality described here, even if the scaling of response magnitude with target distance were not visually appropriate, using target distance and the nature of provided motion as "contextual" inputs to modify the saccadic response. We have tried to address whether the animals performed an on-line sensorimotor transformation, rather than an arbitrary stimulus-response mapping, by interspersing multiple motion amplitudes within experimental sessions in which no behavioral window was imposed on the memory eye movement. Once animals were trained to perform these tasks, their ability to generalize to different amplitudes, target distances, and convergence/divergence movements suggests that this scaling by target distance might indeed be an inherent property, perhaps having evolved because of a global contextually specific adaptation that has come about simply because of the natural geometry of motion parallax.
Yet, it is important to point out that only experiments with human subjects who are asked to perform these tasks in the absence of visual feedback could provide an unequivocal answer to this question. Interestingly, humans can accurately update the goal of a saccadic eye movement during intervening passive roll head and body rotations only when the movement involves a change in spatial orientation relative to gravity, but not when the rotation occurs in the supine position (Klier et al. 2005). Israel et al. (1999) also reported that humans could perform accurate memory-guided saccades for yaw rotation only after a short period of training in the presence of visual feedback. These results could be interpreted to suggest that, at least for rotational updating, gravitational cues are critical for defining an allocentric (world-fixed) frame of reference. Whether an allocentric reference frame can be unequivocally defined during translation without visual feedback has yet to be tested with future human experiments.
Origin of extraretinal signal: a common target distance scaling of memory saccades and the vestibuloocular reflex?
Estimation of the memory-guided eye movement goal requires both retinal information about the spatial location of the flash and extraretinal information about the nature (rotation vs. translation), direction, and amplitude of the intervening head and body movement. What is the source of such extraretinal information? Unlike the active movements used in the study by Medendorp et al. (2003), extraretinal information in our experiments can arise neither from self-generated cues nor from an efference copy of the motor command. Thus the passive head and body displacements used here leave the vestibular system as the most likely sensory source for motion-related information.
A role of vestibular information in spatial perception has been demonstrated for path integration. Following a passive body displacement, human subjects are able to reproduce the amplitude of the displacement with good accuracy (Berthoz et al. 1995; Israel et al. 1995, 1997; Siegler et al. 2000). The traveled distance could be obtained through time integration of the velocity and acceleration information and stored in spatial memory (Berthoz et al. 1995; Israel and Berthoz 1989; Israel et al. 1997). The results of this study suggest that vestibular information can also interact with visual information to update the goal of memory-guided eye movements.
Scaling by viewing distance has been studied extensively during the TVOR. Similar to the findings of the present study (Fig. 13), the amplitude of compensatory eye movements during translation scales less than expected based on geometry (Angelaki et al. 2000; Schwarz and Miles 1991; Telford et al. 1997; Wei and Angelaki 2004; Wei et al. 2003). The fact that the memory-guided saccade amplitude errors described here are qualitatively similar to those characterizing the amplitude of compensatory eye movements during lateral motion raises the possibility that the two might have a common origin.
Viewing distance–dependent scaling in the TVOR arises primarily through a vergence scaling of the responses of certain premotor neuron groups in the prepositus hypoglossi and vestibular nuclei, known as the Burst-Tonic and Eye-Head cells (Chen-Huang and McCrea 1999; Meng and Angelaki 2003). Although the origin of the signals needed to update the goal of the memory-saccadic eye movements during motion tasks is unknown, it is likely that the necessary extraretinal signals originate from vestibular centers in the brain stem (and/or cerebellum). Under the assumption that visuospatial updating occurs in the visuomotor cortex (Andersen et al. 1997; Goldberg and Bruce 1990), there exist at least two possible pathways by which vestibular signals can reach these areas. The first pathway involves vestibular projections through the ventrolateral thalamus to the so-called parieto-insular vestibular cortex (PIVC) (Grusser et al. 1990a,b), an area that is bidirectionally interconnected with the frontal eye fields (Guldin et al. 1992; Huerta et al. 1987). Because no interconnection was found between frontal oculomotor areas and any of the other vestibular cortical areas (Guldin et al. 1992), it has been suggested that PIVC provides the necessary vestibular signals for vestibular memory-contingent saccades (Berthoz 1997).
Alternatively, the vestibular signals needed for visuospatial updating could follow a route similar to that of other extraretinal signals, i.e., through projections to the paralamellar mediodorsal and intralaminar nuclei of the thalamus (Sommer and Wurtz 2002). These thalamic areas receive projections from the prepositus and vestibular nuclei (Asanuma et al. 1983; Lang et al. 1979; Warren et al. 2003) and have widespread projections to both the frontal and parietal cortexes (Huerta and Kaas 1990; Huerta et al. 1986; Kaufman and Rosenquist 1985; Shook et al. 1990, 1991).
We propose that a similar viewing distance–dependent vestibular signal to that used in the TVOR, perhaps in the form of an efference copy of the suppressed oculomotor drive, is used as an extraretinal compensation for the geometrical consequences of motion parallax by scaling both memory saccades and reflexive eye movements during subject motion through space. Such signals might then be used to completely update the cortical representation of space during either passive or active motions. It is important that future studies address the origins and pathways of extraretinal signals to the sensorimotor centers in the cortex and their involvement in reconstructing a neural map of objects in the environment during movement.
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FOOTNOTES |
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Address for reprint requests and other correspondence: D. Angelaki, Dept. of Anatomy and Neurobiology, Box 8108, Washington University School of Medicine, 660 South Euclid Avenue, St. Louis MO 63110 (E-mail: angelaki{at}pcg.wustl.edu)
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near) and divergence (near 
