We estimated the effective number of trajectories that amblyopic observers could track with their amblyopic eyes and their non-amblyopic eyes using stimuli and methods described in S. P. Tripathy, S. Narasimhan, and B. T. Barrett (2007). The stimuli consisted of dots moving along straight-line trajectories. In Experiment 1, one of the *T* trajectories (the target) deviated clockwise or counterclockwise by ±19°, ±38°, or ±76°, halfway through the trajectory. In Experiment 2, *D* of the *T* trajectories deviated, all in the same direction and with the same magnitude of direction change. In both experiments, we varied *T* and the angle of deviation. In Experiment 2, we also varied *D*. Amblyopic observers reported the direction of deviation of the target trajectories and, for each eye, the effective number of tracked trajectories was estimated. This number increased systematically with increasing magnitude of deviation of the targets. On average, the effective numbers of tracked trajectories were approximately 15% smaller for the amblyopic eyes for each of the three magnitudes of deviation. A comparison with data previously published for normal eyes failed to reveal any deficit in the effective number of trajectories tracked by the non-amblyopic eyes of amblyopic observers for the current task.

^{2}, speed of 4°/s, and were presented on a background of luminance 5.0 cd/m

^{2}. The trajectories on a trial were presented for 51 frames (850 ms). Any trajectory deviations could only occur between the vertical lines demarking the screen midline (see stimuli in Figures 1 and 2), i.e., on frame 27. In the first experiment, on each trial, only one deviating trajectory (the target) was presented, along with zero or more non-deviating trajectories (the distractors); the magnitude of deviation and the total number of trajectories remained fixed within an experimental block and were varied across experimental blocks. In the second experiment, on each trial, one or more target trajectories were presented, with the total number of trajectories held constant across blocks; the magnitude of deviation and the number of deviating trajectories remained fixed within an experimental block and varied across experimental blocks. In trials with more than one target trajectory, all target trajectories deviated by the same angle and in the same direction (i.e., clockwise (CW) or counterclockwise (CCW)). On half the trials, the deviation of the targets was CW, and on the remaining trials the deviation was CCW. Following each stimulus presentation the observer, using appropriate keys on the keyboard, reported the perceived direction (CW or CCW) of deviation of the target. Performance was measured as the proportion of correct responses in each experimental condition, i.e., for each tested combination of total number of trajectories, number of deviating trajectories, and magnitude of deviation. Random jitter of the orientations of the trajectories over the range ±80° ensured that parallelism cues could not be used for identifying the direction of deviation of the target.

*T*) presented on each trial. For each combination of deviation angle and

*T,*each amblyopic observer completed two blocks with the amblyopic eye (AE) and two blocks with the non-amblyopic eye (NAE); for each condition, the order in which the two eyes of an observer were tested was randomized. Percentages of correct responses were determined separately for the AE and the NAE for each observer for each condition tested. For each of the three magnitudes of deviations employed (±19°, ±38°, and ±76°), data were collected for the AE and the NAE, for values of

*T*= 1, 2, 3, 4, 6, 8, and 10 trajectories.

*A*of the

*T*trajectories present in the stimulus and completely ignored the remaining (

*T*−

*A*) trajectories. The percentage of correct responses

*P*for this LCHO is given by the equations:

*T*), for different values of

*A*. The experimentally measured percentage of correct responses for any particular value of

*T*can be converted into an

*effective number of tracked trajectories*(

*E*) by interpolating between these curves for the corresponding value of

*T*.

*T*) was held constant, and the number of deviating trajectories in each stimulus (

*D*) varied between experimental blocks. Figure 2 schematically shows two instances of 6-trajectory stimuli (

*T*= 6) with two trajectories deviating (top panel;

*D*= 2) and four trajectories deviating (middle panel;

*D*= 4); with all the target trajectories deviating by +38° in each panel. Within each block of 100 trials, the magnitude of deviation of each target trajectory was fixed, as were the total number of trajectories (

*T*) and the number of deviating trajectories (

*D*) presented on each trial. Observers knew in advance how many trajectories would be present and how many of these would deviate in each trial within a block. For each combination of deviation angle,

*T*and

*D*that was tested, each amblyopic observer completed two blocks with the AE and two blocks with the NAE; for each condition, the order in which the two eyes of an observer were tested was randomized. Percentages of correct responses were determined separately for the AE and the NAE of each observer for each condition tested. For each of the three magnitudes of deviations employed (±19°, ±38°, and ±76°), data were collected for each eye, for values of

*T*= 6 trajectories, and

*D*= 1, 2, 3, 4, and 6. The experiment was repeated with

*T*= 8;

*D*= 1, 2, 3, 5, 6, and 8 and with

*T*= 10;

*D*= 1, 2, 3, 5, 7, and 10.

*A*of the

*T*trajectories presented, when

*D*of these

*T*trajectories deviated. The percentage of correct responses

*P*for this LCHO is given by the equations:

*y*items from a pool of

*x*items.

*T*= 6, the predicted performance of the LCHO as a function of the number of deviating trajectories in the stimulus (

*D*) for different values of

*A*. The experimentally measured percentage of correct responses for any particular value of

*D*can be converted into an

*effective number of tracked trajectories*(

*E*) by interpolating between these curves for the corresponding value of

*D*. The interpolation procedure has been elaborately discussed in Tripathy, Narasimhan, et al. (2007). Similar predictions can be plotted for

*T*= 8, and for

*T*= 10 (see Figure 4 in Tripathy, Narasimhan, et al., 2007), and the effective number of tracked trajectories (

*E*) can be determined for any combination of

*T*and

*D,*for any magnitude of angle of deviation.

Observer | Age (years) | Gender | Type | Strabismus (at 6 months) | Eye | Refractive error | Line letter acuity (single-letter acuity)* |
---|---|---|---|---|---|---|---|

JS | 22 | F | Strabismic | L EsoT 6–8 ^{Δ} &
L Hyper 4–6 ^{Δ} | R | +1.25 | 20/16 |

L | +1.00 | 20/40 (20/32 ^{+1}) | |||||

AP | 19 | F | Strabismic | L EsoT 4 ^{Δ} and
L Hyper 2 ^{Δ} | R | −1.50/−0.50 × 180 | 20/12.5 ^{−2} |

L | −0.75/−0.25 × 5 | 20/50 (20/32 ^{+1}) | |||||

JT | 52 | F | Strabismic | L EsoT 5 ^{Δ} | R | −1.00/−0.50 × 10 | 20/16 ^{+2} |

L | −0.75/−0.50 × 80 | 20/63 ^{−1} (20/25 ^{−2}) | |||||

SC | 27 | M | Anisometropic | None | R | +0.50 | 20/16 ^{+2} |

L | +3.25/−0.75 × 60 | 20/63 ^{−2} (20/63 ^{+2}) | |||||

JD | 19 | M | Strabismic and anisometropic | L EsoT 3 ^{Δ} | R | +2.50 | 20/16 |

L | +5.00 | 20/125 (20/125 ^{+2}) | |||||

SP | 36 | F | Strabismic and anisometropic | R EsoT 14 ^{Δ} | R | +1.25/−0.50 × 147 | 20/52 ^{−2} |

L | −0.50/−0.25 × 180 | 20/20 |

The acuities listed in Table 1 were determined using a Bailey–Lovie chart. We specify both the full line letter acuity and, when available, the single letter acuity is shown in parenthesis.

*T*trajectories deviating

*T*), when there was only one trajectory deviating. The figure shows the AE and the NAE data separately, for deviations of ±19°, ±38°, and ±76° (triangles, squares, and circles, respectively), with each observer plotted in a separate panel. Also superimposed on each panel is the predicted performance of the LCHO (dotted lines) as illustrated in the bottom panel of Figure 1. Figure 4 shows the same data averaged across the six observers, with each panel showing the data for one of the three magnitudes of deviation tested. Again the predicted performance of the LCHO has been presented in the background. Also shown in Figure 4 is the average performance of the three normal observers (green lines and symbols) in Experiment 1 of Tripathy, Narasimhan, et al. (2007).

*T*= 6), and the largest relative difference resulted when the magnitude of deviation was ±38° and there were four trajectories in the stimulus (

*T*= 4).

*T*= 2, 3, or 4. When the target trajectory deviated by ±76°, better performance was evident in the normal observers only when there were few trajectories in the stimulus, i.e., for

*T*= 2, 3, or 4. Interestingly, one might have anticipated that for deviations of ±76° differences in performance would have been more evident for larger values of

*T,*on account of saturation effects (i.e., percentages of correct responses being close to 100%) for small values of

*T*; however, the data obtained did not follow this anticipated pattern. The normal observers consistently performed better than the amblyopic observers did with either eye when there were few trajectories in the stimulus. The largest absolute difference between the performance of the normal observers and that of the amblyopic eyes was 13.7% for a target deviation of ±19° with

*T*= 2, and the largest relative difference was 15.8% which occurred for a deviation of ±19° with

*T*= 4. The largest absolute difference between the performance of the normal observers and that of the amblyopic eyes was 13.9% for a target deviation of ±19° with

*T*= 2, and the largest relative difference was 15.7%, which also occurred for the same values of deviation and

*T*. In summary, only small differences were found between AEs and NAEs, but the differences between the performance of the normal observers and that of either eye of amblyopic observers were substantially larger.

*T*trajectories deviating

*D*), when there were 6 trajectories in the stimulus. The figure shows the AE and NAE data separately for deviations of ±19°, ±38°, and ±76° (triangles, squares, and circles, respectively), with each observer plotted in a separate panel. Also superimposed on each panel is the predicted performance of the LCHO (dotted lines) as illustrated in the bottom panel of Figure 2. Figures 6 and 7 show, using an identical format, the data for the same three observers for 8- and 10-trajectory stimuli, respectively (i.e.,

*T*= 8, and

*T*= 10); the performance of the LCHO has also been recalculated using the appropriate values of

*T*and plotted for

*A*= 1, 2, …, 5.

*T*= 6, 8, and 10) yields a different pattern of results for the three observers. Observer JD's performance was better in the NAE compared to the AE in almost all of the conditions tested. Observer JS showed little or no difference between the two eyes on almost all of the conditions tested, while observer SC showed a consistent difference for many of the ±38° deviation conditions and to a lesser extent for some of the ±19° deviation conditions. The largest absolute difference in performance between the two eyes for JD, JS and SC was 17.50, 11.50, and 9.50%, respectively; the largest relative difference in performance between the two eyes was 24.65%, 16.65%, and 13.38%, respectively, with reference to the NAE. However, these large differences were the outliers—in most conditions we tested, the inter-eye differences in performance were just a few percent, with the AE occasionally performing better than the NAE.

*D,*particularly when the deviation was ±76°. Likewise, data were not evaluated in some conditions with

*D*of 1 because the performance of one or more observers was close to chance.

*D*) and the total number of trajectories (

*T*); this is also true for the AEs and the NAEs. Since

*D*only had a small influence on the effective number of tracked trajectories, for each of the three magnitudes of deviation tested, for the AEs and NAEs, we averaged the effective number of tracked trajectories over the different values of

*D*. Table 2 shows the mean number of trajectories tracked in the different experimental conditions. The effective number of tracked trajectories changed only slightly for the different values of

*T*tested, and hence Table 2 also shows the effective number of trajectories tracked, averaged over the three values of

*T*. Our observers showed idiosyncratic differences in the different conditions in which an amblyopic deficit was observed. However, when performance is averaged across observers for different values of

*T,*the amblyopic deficit, estimated from the last two columns of Table 2, corresponds to a reduction in the effective number of tracked trajectories by 14.7%, 14.3%, and 15.1% for deviations of ±19°, ±38°, and ±76°, respectively. On average, for a given angle of deviation, the effective numbers of trajectories tracked by the amblyopic eyes were approximately 15% lower than those tracked by the fellow eyes. Paired 2-tailed

*t*-tests comparing the mean of the effective numbers of trajectories tracked by the NAEs to that for the AEs found the amblyopic deficit to be significant for ±19° deviations (

*t*= −2.675,

*df*= 35,

*p*= 0.011) and for ±38° deviations (

*t*= −2.680,

*df*= 26,

*p*= 0.013) but not for ±76° deviations (

*t*= −1.694,

*df*= 8,

*p*= 0.123). The data for (

*T*= 6,

*D*= 3, deviation of ±38°, AE) and (

*T*= 10,

*D*= 1, deviation of ±76°, NAE) conditions were dropped when performing the paired

*t*-test because the effective number of tracked trajectories was not calculated for the other eye (the percentage of correct responses being close to 100% and 50%, respectively). The non-significant difference between the two sets of eyes for the ±76° deviation is probably because the number of data points for which the effective number of trajectories could be estimated was small on account of saturation effects.

Deviation | T = 6 | T = 8 | T = 10 | Average | ||||
---|---|---|---|---|---|---|---|---|

NAE | AE | NAE | AE | NAE | AE | NAE | AE | |

19 deg | 1.28 ± 0.47 | 1.10 ± 0.23 | 1.15 ± 0.22 | 0.94 ± 0.29 | 1.00 ± 0.25 | 0.89 ± 0.23 | 1.14 ± 0.36 | 0.98 ± 0.27 |

38 deg | 3.05 ± 0.55 | 2.53 ± 0.56 | 2.79 ± 0.34 | 2.51 ± 0.61 | 2.84 ± 0.58 | 2.37 ± 0.67 | 2.87 ± 0.52 | 2.46 ± 0.65 |

76 deg | 4.48 ± 0.67 | 4.26 ± 0.66 | 5.04 ± 0.65 | 4.24 ± 1.45 | 5.06 ± 0.93 | 4.00 ± 2.00 | 4.91 ± 0.99 | 4.17 ± 1.57 |

*T,*and for deviations of ±76° for the NAEs for

*T*= 10), straight lines were fit to the data for the different values of

*D*for each combination of angle of deviation and

*T*. Visual inspection of the figures indicates that the ranges of the effective numbers of trajectories tracked when

*T*= 6, 8, and 10 are largely overlapped, suggesting that the effective numbers of trajectories tracked are largely independent of

*T,*for either set of eyes, particularly in comparison to the influence that the angle of deviation has on the effective number of trajectories tracked. For this reason, the effective numbers of trajectories tracked for the different values of

*T*were combined when performing the

*t*-test described above and in the analysis that follows.

*D*= 1. So the change in the pattern of results, in particular the improvement in performance of the fellow eye, is unlikely to be a consequence of the stimulus used and is more likely to reflect practice effects. This might be a fundamental difference between our study and some of the earlier studies. For example, Ho et al. (2006) tested each of their observers with 16 practice trials followed by 64 test trials, whereas each observer in our study performed several thousands of trials. It is likely that the performance of the fellow eyes of the amblyopic observers improved substantially on account of prolonged testing in Experiment 1, so that by the time they were tested in Experiment 2 their ability to track with their non-amblyopic eye was comparable to that of normal observers. It is possible that amblyopic observers find monocular viewing with their non-amblyopic eye to be substantially different to how things appear when they view things with both eyes, and many thousands of trials are needed to shake off the effects of habituation. In short, some of the fellow eye deficits in amblyopic observers may result from the novelty of the monocular percept compared to the percept when viewing with both eyes.

*describes*) this lower level of performance by assigning to it a smaller magnitude for the effective number of trajectories tracked.

*T*(1 ≤

*T*≤ 10) trajectories in the stimulus and all of them deviated in the same direction and by the same amount. If pooling of information was efficient over

*A*trajectories, thresholds should decrease by a factor of the square-root of

*T,*until

*T*>

*A,*and then remain constant. But the data show no decrease in thresholds as

*T*is increased from 1 to 10. Thus, we find no evidence of efficient pooling of information across a subset of trajectories, even when there were as few at 2 trajectories (

*T*= 2) in the stimulus. The absence of any integration of information from the two trajectories when

*T*= 2, suggests that trajectories are monitored individually.

*T,*

*A*and the angle of deviation were fixed, and the angle of deviation was substantially suprathreshold. A few trials of our stimulus will convince observers that on some trials within a block they can confidently identify some of the deviating trajectories (presumably because these were individually monitored or tracked) and on other trials

*within the same block*they cannot identify with confidence any of the deviating trajectories (presumably because none of these were monitored or tracked). It is obvious that observers do indeed have the ability to identify a subset of the deviating trajectories on a trial-by-trial basis. In fact, a couple of studies have investigated the shapes of the deviating trajectories in stimuli with more than one trajectory (Tripathy & Barrett, 2003, 2006).