Temporal and orientational effects on adaptation to the Necker cube

Outline

Abstract

Introduction

In a series of articles, Orbach and colleagues examined the influence of several factors on the reversibility of the Necker cube (Heath, Ehrlich, & Orbach, 1963; Orbach, Ehrlich, & Heath, 1963; Orbach, Ehrlich, & Vainstein, 1963). For the present purposes, we limit our focus to the study by Orbach, Ehrlich and Heath (1963) in which Necker cubes were presented repeatedly. The frequency of the presentation, together with the on/off duration of the stimuli were manipulated. Orbach et al. used the obtained reversal rates as support for their theory about the mechanism behind the adaptation phenomenon. They introduce the concept of 'satiation of orientation'. Prolonged viewing time gradually increases the 'satiation' of one interpretation of the Necker cube, up to a particular threshold where the perception of the cube reverses, after which the satiation for the original interpretation rapidly drops to zero. The theory of Orbach et al. is attractive in its simplicity, and, by considering residual satiation as a factor in the perception of the figure, the theory elegantly explains the data for the several conditions.

Most of the research started explicitly or implicitly from this same point of view, in which perceiving one interpretation of an ambiguous figure creates a tendency to see the subsequent stimulus the other way, due to an interpretation-dependent adaptation mechanism. Let us call this the selective adaptation hypothesis. This tacit assumption that only one interpretation at once influences subsequent perception seems to be supported by the data of Shulman (1993a, 1993b). The latter author presented Necker cubes and bistable apparent motion configurations as test stimuli for adaptation. Manipulating the allocation of attention by introducing a task in displays with opposing adaptive stimuli, he observed that participants only adapted to a stimulus they had attended.

Still one could ask the question how the visual system rules out the unperceived alternative. The research by Shulman does not necessarily address this problem. The mechanisms that bring only one interpretation of a bistable stimulus to awareness are probably different from those that are responsible for attentive selection. Maybe there are other possibilities than the selective satiation in the form Orbach et al. (1963) advocate. There is research suggesting that adaptation not only occurs for the 'chosen' alternative in the perception of the ambiguous figure, but also for the interpretation that is not actually perceived. This point of view will be referred to as the nonselective adaptation hypothesis.

In an experiment in which bistable biased motion quartets, obtained by varying the aspect ratio between horizontal and vertical elements, were followed by neutral bistable quartets, Hock, Schöner and Hochstein (1995) found that the organization of the neutral test quartets were influenced by the perception of the preceding quartets. The first presentation not only generated adaptation when the participant had organized the first stimulus in the biased motion direction, but also in the case in which the participant had seen the alternative, unbiased organization. In other words, the bias did not only affect the perception of the test stimulus when the adaptive perception was congruent with the bias, but also when the bias had not dominated the adaptive perception. This, however, only occurred when the adaptive stimulus was, though biased, still sufficiently ambiguous. Hock et al. came to the conclusion that the visual system builds the two competitive interpretations simultaneously, and that the adaptation applies to both possible organizations before determination of which of both will be consciously perceived.

The results of the experiment by Gepshtein and Kubovy (1997) with multistable dot lattices can also be interpreted as supporting the nonselective adaptation hypothesis. An adapting rectangular dot lattice ('A') was presented in a random orientation, in which the aspect ratio between rows and columns was manipulated so as to bias the perceptual organization by the participants. One of the directions, which will be called the 0-orientation, was aligned with one of the possible grouping directions in the subsequent hexagonal test lattice ('T'). Since A is a rectangular lattice, the direction perpendicular to the 0-orientation will be named the 90-orientation. The respective vector lengths indicating the distance between the dots in that direction, will be written as d₀ and d₉₀. The ratio between these distances is crucial for the probabilities of the organization in each of these directions. The results were as illustrated in Figure 1.

The remarkable parallelism between the curves indicates that the bias in A has similar effects regardless of whether A was seen in the 0-orientation or the 90-orientation. These results seem to confirm the nonselective adaptation hypothesis. Notice that logit[P(T->0|A->0)] is larger than 0 for the range of tested d₉₀/d₀-ratios. Superimposed on the adaptation effect, there is a hysteresis term; participants tend to organize their perception in the same direction in two subsequent presentations. One could crystallize the hysteresis- and adaptation paradigm in the following formula:

The matter of generalizability of the phenomenon of unselective adaptation was investigated in the following experiment.

Experiment 1

Method

Participants

Stimuli

The adapting cube was biased by thickening the lines of one of the possible front planes. This manipulation biases towards seeing this plane closer to the perceiver. The value this bias variable will take for the analyses is equal to the number of pixels the thickened lines are broader than the other lines in the cube, ranging from 0 (no bias in the adapting stimulus) to 4. By convention, the perceptual organization that corresponds to the bias is assigned the number 1. The alternative is called organization 2.

The orientation of the cubes was random and identical over the two presentations per trial.

Immediately after the two cubes were presented, the two respective response screens were offered. Each response screen consisted of two disks in which both competing organizations were indicated by the corresponding occluded cubes. The participants could respond using a mouse to click on the occluded cubes referring to the organizations they had perceived, first for the adapting (biased) Necker cube and then for the second (unbiased) one.

Apparatus

Design

Procedure

Results

In Figure 3 this dependent variable is plotted against bias. The linear regression line is shown. The intercept is fixed at 0 because an unbiased stimulus is expected to keep the odds-ratio of the alternatives at 1. The statistical aspects of the regression analysis are described in Table 1 and Table 2.

Table 1. Variance-table of the linear regression of logit[P(A1)] against bias. Data are aggregated over participants, sessions and orientations.

Source         df  SS      MS      F(1)          Rē
Model          1   0.3928  0.3928  261.30******  0.9849
Error          4   0.0060  0.0015
Uncorr. total  5   0.3988

Table 2. Parameter-estimates for the linear regression of logit[P(A1)] against bias. Data are aggregated over participants, sessions and orientations.

Predictor(1)  Regression weight  t(1)
bias          0.1144             16.17 ****** 

The simple linear regression on the data summed over participants, sessions and orientations shows a strongly significant effect of the bias on ln[P(A1)/P(A2)], with F_(1,4)=261.30, p<0.0001, which means that we have been successful in influencing the participants' perception of the Necker cubes by manipulating line width. Now that this has been established, we can move on to comparing the data obtained with Necker cubes with the dot lattice data of Gepshtein and Kubovy (1997).

To that purpose the data are conditionalized on the response for the adapting stimulus. In Figure 4 one can see how this intervention reorganizes the data. This figure is to be compared with Figure 1. The parallelism between both curves is broken. The adaptation effect does not seem to appear after the alternative (i.e. not congruent with bias) interpretation of the adapting stimulus has been perceived. The statistical analyses support this observation (see Table 3 to 6).

The simple linear regression of logit[P(T1|A1)] against bias for the data summed over participants, sessions and orientations yields statistically significant results with F_(1,3)=12.37, p<0.05. Both the slope and the intercept are significant, with t-values of respectively 21.02 and -3.52, p<0.0005 and p<0.05. The situation looks different for the data conditionalized on A2. With alpha=0.05, the linear regression of logit[P(T1|A2)] against bias is not significant. When calculated, the slope is even positive, which is against the logic of adaptation. Only the intercept term is significant, with a t=-41.23, p<0.0001.

Table 3. Variance-table of the linear regression of logit[P(T1|A1)] against bias. Data are aggregated over participants, sessions and orientations.

Source       df  SS      MS      F(1)     Rē
Model        1   0.3258  0.3258  12.37 *  0.8048
Error        3   0.0790  0.0263
Corr. total  4   0.4048

Table 4. Parameter-estimates for the linear regression of logit[P(T1|A1)] against bias. Data are aggregated over participants, sessions and orientations.

Predictor(1)  Regression weight  t(1)
Intercept     2.6430             21.02 *****
bias         -0.1805             -3.52     *

Predictor(1)  Regression weight  t(1)
Intercept     -2.5313            -41.23******  
bias           0.0526              0.13  n.s.

The conclusion seems to be straightforward: adaptation only applies to the consciously perceived organization. These data are not without problems, however. First of all, there is a large variability among subjects, which is not visible in these analyses. Another factor of variability is the effect of orientation. The influence of orientation is a well-known factor in the perception of Necker cubes, sometimes used to illustrate likelihood-against-simplicity. Of course, this was why we randomized this factor in the first place, but, reanalyzing the results, we came to the conclusion that orientation was too large a source of systematic variance to ignore and leave uninvestigated.

In order to have a sufficient number of observations per datapoint to calculate the proportions, we group the frequencies per 10 degrees. Given the fact that cubes without bias are invariant over rotations of 180°, we add the frequencies for identical cubes. The plotted values for the 0-bias condition, summed over participants and sessions, are presented in Figure 5. The data clearly show a periodic pattern. An analysis confirms this.

Periodicity is described by a sinusoidal function. Here we actually use a cosine, for the sake of simplicity, since the graph reaches a maximum at 0. Both approaches are equivalent, however. The curve follows the course of 1.5 wave in 180°; this implies a peridocity ù of 3. The equation

Table 7. Gauss-Newton parameter-estimates for the nonlinear regression of logitP(A1)] with amplitude and phase, according to the model described earlier. Data are aggregated over participants and sessions for the condition with unbiased adapting stimuli.

Parameter  Value
Amplitude  1.6602    
Phase      0.6566

Table 8. Variance-table of the linear regression of logit[P(A1)] against the model's prediction for bias=0. Data are aggregated over participants and sessions.

Source         df  SS       MS       F(1)           Rē
Model          1   24.8057  24.8057  183.02 ******  0.9150
Error	       3    2.3041   0.1355
Uncorr. total  4   27.1097

With an R²-value of 91.50%, we can conclude that the previously described model fitted the data well for the bias=0 condition, with data summed over participants and sessions. Similar analyses of this model with other bias levels, that we will not discuss here, prove that it is also appropriate for data from other bias conitions than 0. The model was adjusted by adding an intercept term to include the 'upwards' shift towards choosing the A1-alternative. Earlier in this paper we have established the linear relationship between bias and the logistic tendency to respond with A1. One final model, including both bias and orientation at the same time, can be formulated as

Table 9. Gauss-Newton parameter-estimates for the nonlinear regression of logit[P(A1)] with amplitude, phase and bias slope, according to the model described earlier. Data are aggregated over participants and sessions.

Parameter   Value
Amplitude   1.6399
Phase       0.7191
Bias slope  0.1566

Table 10. Variance-table of the linear regression of logit[P(A1)] against the model's prediction for bias=0. Data are aggregated over participants and sessions.

Source          df   SS        MS        F(1)            Rē
Model            1  244.3094  244.3094  1195.07 ******  0.8813
Error          161   32.9135    0.2044
Uncorr. total  162  277.2228

The R² value of 0.8813 confirms that we have developed a rather solid descriptive model for predicting the odds of the possible organizations of a Necker cube given its orientation and bias. In the light of the research on adaptation, the strong determination of the disambiguation by the orientation invokes one hypothesis concerning the absence of an adaptation effect after perception of the unbiased configuration, A2. Most of the cubes presented were not maximally ambiguous because their orientation forces the visual system into one particular interpretation. Possibly the pattern of data is analogous to the findings of Hock, Schöner and Hochstein (1995), in that nonselective adaptation only occurs if the adapting stimuli are ambiguous enough. The evident solution is to conduct an experiment with Necker cubes in conditions that elicit maximal entropy. However, there seems to be very solid experimental evidence about the influence of twodimensional properties of the Necker cube on its perceptual organization. Most experiments using Necker cubes involve stimuli in orientations parallel to the main viewing axes (in terms of the previous analyses: 0°, 90°, 180°, and 270°). To resolve the questions of the effects of twodimensional configuration and orientation on the disambiguation of the Necker cube, we have executed an additional experiment.

Experiment 2

Method

Participants

Stimuli

The second manipulation involved the points at which the square parts of the cubes intersected (Figure 6, Table 11).

Table 11. Parameter values for the manipulation of the twodimensional configuration of the Necker cube. The values represent the distance of the corners of both squares of the twodimensional cube relative to the length of the side, in horizontal and vertical directions.

Cube  Horizontal distance /side  Vertical distance /side
C1    1/2 (=3/6)                 1/2 (=3/6)
C2    1/3 (=2/6)                 1/3 (=2/6)
C3    1/6                        1/6
C4    1/2 (=3/6)                 1/3 (2/6)
C5    1/2 (=3/6)                 1/6

Since this was not an adaptation experiment, the trial consisted of only one cube presentation, with a duration of 300 ms, followed by the response screen after 100 ms. The latter, as in the first experiment, showed occluded equivalents of both possible organizations. The participants were asked to click the appropriate cube with the mouse.

Apparatus

Design

Procedure

Results

The periodic function previously introduced, logit[P(A1)]=A*cos(3*orientation + phi), was applied to the data generated by this experiment. The analysis was performed for each of the cube configurations separately. The parameter estimates per condition are summarized in Table 12. An evaluation of the fit of this model per cube configuration would lead us too far from the scope of this paper; the statistics for the model in Table 13 are based on the level of discrepancy between the predicted values and the observed values for all the cube configurations together (35 observations).

Table 12. Gauss-Newton parameter-estimates for the nonlinear regression of logit[P(A1)] with amplitude and phase, according to the model described earlier. Data are aggregated over participants and sessions.

Cube configuration  A (amplitude)  phi (phase)
C1                  1.7216         0.6128
C2                  1.6697         0.6141
C3                  1.2627         0.5659
C4                  1.5857         0.4380
C5                  1.5873         0.1638

Table 13. Variance-table of the linear regression of logit[P(A1)] against the model's prediction for all cube configurations taken together. Data are aggregated over participants and sessions.

Source         df  SS       MS       F(1)           Rē
Model           1  42.5749  42.5749  136.85 ******  0.8010
Error          35  10.5777   0.3111
Uncorr. total  36  53.1526

With a coefficient of determination of 80.10%, we can state that the model accounts for the observed findings to a sufficient degree. Looking at Table 12, we can conclude that, within the group of symmetric cubes, the amplitude decreases with decreasing distance between the corners of the two square parts of the twodimensional configuration (evolution C1->C2->C3). The asymmetry of the cube seems to decrease the phase factor (evolution C1->C4->C5). Asymmetry has no straightforward effect on amplitude. Neither is there a monotonic relation between the 'depth' of the cube and the phase.

The theoretical implications of these findings are not directly relevant for the adaptation study currently discussed. This experiment, however, has given valuable information for the selection of the stimuli for the next experiment.

Experiment 3

Now the nonselective-adaptation hypothesis can be tested for the Necker cube in conditions that can be assumed not to impose one organization to an extent that it would destroy the effect. In addition, if the intervention was successful in reducing the effect of orientation, the distance between the upper and lower curves, respectively referring to the strength of adaptation after biased interpretation and unbiased interpretation (cf. Figure 4), will be less. The adapting and test stimuli have the same rotation during the trial, so orientation is an important covariate in the organization of both figures, which enlarges the hysteresis. Neutralizing the effect of orientation can be expected to reduce the absolute value of the intercept of both lines.

Now suppose we replicate experiment 1 with the most ambiguous orientations, and we find the same pattern of data as in the experiment with dot lattices (Figure 1), then have we proven the existence of adaptation to the unperceived in Necker cubes? No, since a completely different account for the data is possible (Gepshtein and Kubovy, 1997). This theory, described in the next few paragraphs, is called the internal state hypothesis.

First of all, consider the possibility that the module processing information in order to select an appropriate percept is subject to random noise factors. The endogeneous random activity in this module will bias it to one or another interpretation in the case of multistable perception. Together with the noise, this 'preference' of the module will randomly fluctuate in time, with a certain inertia. In other words, there will be a positive correlation between the states at time a and time b (ta, tb). This correlation will decrease to zero over temporal distance between ta and tb. So if this module processes an unbiased adapting stimulus at ta, and produces a specific response, there will be a chance higher than 0.5 that the stimulus at tb will cause the same output. There will be hysteresis. Figure 7 illustrates the reasoning.

Then if stimulus bias towards interpretation 1 is involved, the noise will have favor interpretation 2 to a higher extent than some critical point to overcome the bias induced by the stimulus and cause configuration 2 to arise. If the noise state in the adapting phase is favoring interpretation 2 to a high degree, there is an increased chance that the output in the test phase will also be 2, since both moments are not far apart in time. If this reasoning is translated to a plot of logit[P(T1|A2)], we obtain a line under 0, decreasing in function of the amount of bias. Exactly the same pattern as in Figure 1.

A similar, but inverse reasoning is to be applied to logit[P(T1|A1)]. When stimulus bias is present, there will be a bigger chance to have 1 as output, because the critical value the noise needs to cause interpretation 2 is shifted. The critical value for the test stimulus remains the same however, since there is no stimulus bias. On a certain number of trials, the module will be in a state that would generate a 2 response if no stimulus bias would be present, but gives 1 as output only because of stimulus bias. Then, if the state is inert enough to remain at the same value, the module will respond 2 in the test trial. Conditionalizing on A1 will again produce a plot decreasing in function of bias, above 0 this time.

Adaptation is not an actor in this play. The predictions given by the adaptation hypothesis and the internal state hypothesis are identical in most situations. Differential predictions are possible however, by manipulating the time variables in the adaptation paradigm. A longer adaptation phase is assumed to invoke a larger adaptation effect. In the internal state paradigm, longer exposure to the first stimulus does not affect any variable, as long as the time interval between the onset of the first and the second stimulus remains the same. For the internal state hypothesis, a presentation time of 300 ms with a ISI of 300 ms would have the same effect as an 'adaptation' interval of 450 ms and an ISI of 150, that is, if the participants are explicitly asked to respond with their initial perception on both stimuli, even if their interpretation shifts towards the alternative while viewing the adapting stimulus.

Experiment 3 will test this prediction. The first hypothesis states that the intercepts of the conditional curves will be lower than in experiment 1 because only the most ambiguous orientations were selected. If only the high stability of the cubes in the first experiment was the cause of the flat lower curve, then the two curves might be parallel in the outcome of this experiment. The second test involves the differential prediction from the perspectives of the internal state hypothesis and the nonselective adaptation hypothesis. Note that, if the lower curve remains flat, neither the nonselective adaptation nor the internal state can explain this. If both curves are parallel, the internal state hypothesis is correct if the slopes are unaffected by changes in adaptation duration, as long as the delay between the adapting and the test stimulus remains constant.

Method

Participants

Stimuli

To test for the differential prediction, four different time schedules were used in presenting the adapting and test cubes (Table 14). The 300-300 and the 450-150 scenario give the same value for the slope in the internal state hypothesis. The 450-150 scenario will have a steeper slope according to the adaptation hypothesis. After the presentation of the test cube, which lasted 300 ms, there was a delay of 300 ms before the response screen came on, in order to give processes the opportunity to finish instead of being halted by the appearance of the response screen. This intervention was inspired by the fact that, in the case of the adapting stimulus, processing of the image is still possible during the ISI.

Table 14. Representation of the different time schedules. SI is the on-screen duration of the adapting stimulus. The ISI is the interval between adapting and test stimuli.

          |  ISI=150 ms | ISI=300 ms |
-------------------------------------|
SI=300 ms |             |            |
-------------------------------------|
SI=450 ms |             |            |
-------------------------------------

Apparatus

Design

Within a session, all trials were completely randomized with respect to the other variables (orientation and bias).

Procedure

There was at least 4 hours between every two sessions.

Results

Table 15. Parameter-estimates for the linear regression of logit[P(A1)] against bias. Data are aggregated over participants and orientations.

Predictor(1)  Regression weight  t(1)
bias          0.6201             15.84 ******

Source         df  SS       MS       F(1)           Rē
Model           1  21.5309  21.5309  251.15 ******  0.9436
Error          15   1.2859   0.0857
Uncorr. total  16  22.8169

Note: (1) ******: significant with p<0.0001.

Conditionalizing on the responses to the adapting stimulus produces a completely different pattern of results than obtained in the first experiment. Selecting the most ambiguous orientations for the stimuli seems give rise to a remarkable parallellism between the regression lines of the datagroups (Figure 9). The F_(1,14)-values for the linear regression analyses of logit[P(T1|A1)] and logit[P(T1|A2)] are 28.17 a        nd 13.18 respectively, with associated p-values <0.0001 and <0.005. The R²-values of 66.80 and 48.50 are not particularly bad, given that the time variables are not taken into account to explain the variance.

Table 17. Variance-table of the linear regression of logit[P(T1|A1)] against bias. Data are aggregated over participants and orientations.

Source       df  SS      MS      F(1)          Rē
Model         1  2.7347  2.7347  28.17 ******  0.6680
Error        14  1.3592  0.0971
Corr. total  15  4.0938

Table 18. Parameter-estimates for the linear regression of logit[P(T1|A1)] against bias. Data are aggregated over participants and orientations.

Predictor(1)   Regression weight  t(1)
Intercept      1.1052             8.48 ******
bias          -0.3698            -5.31 ******

Table 19. Variance-table of the linear regression of logit[P(T1|A2)] against bias. Data are aggregated over participants and orientations.

Source       df  SS      MS      F(1)       Rē
Model         1  2.3664  2.3664  13.18 ***  0.4850
Error        14  2.5125  0.1795
Corr. total  15  4.8788

Table 20. Parameter-estimates for the linear regression of logit[P(T1|A2)] against bias. Data are aggregated over participants and orientations.

Source       df  SS      MS      F(1)       Rē
Model         1  2.3664  2.3664  13.18 ***  0.4850
Error	     14  2.5125  0.1795
Corr. total  15  4.8788

The analysis of the effect of the manipulations of the time variable gives much less clear results, however.

Table 21. Parameter estimates for the linear regression of logit[P(T1|Ax)] on bias per timing condition. Data are aggregated over participants and orientations.

T1|A1  
       Intercept  ISI=150 ms    ISI=300 ms
       SI=300 ms  1.2895 **      1.3367 *
       SI=450 ms  0.7198  *      1.0749 *
       
       Slope      ISI=150 ms    ISI=300 ms
       SI=300 ms  -0.3000 *     -0.5706 n.s.
       SI=450 ms  -0.1519 n.s.  -0.4541 *

T1|A2  
       Intercept  ISI=150 ms  ISI=300 ms
       SI=300 ms  -1.1927 *   -1.7062 n.s.
       SI=450 ms  -0.7130 *   -1.1550 *

       Slope      ISI=150 ms  ISI=300 ms
       SI=300 ms  -0.3424 *   -0.1640 n.s.
       SI=450 ms  -0.6621 *   -0.2075 n.s.

In Table 21, the most important features of the data analysis are summarized. A complete description of each regression will not be given for apparent practical reasons. Notice that the cells with slopes for (SI=300;ISI=300) and (SI=450;ISI=150) are to be compared to distinguish between the predictions of the adaptation model and the internal state model. The data are very inconsistent, however. The linear regressions mostly generate low F-values, and are not or only marginally significant. Due to the excessive noise in the results per timing condition, the parameters of the regressions for this last analysis are not reliable enough to deduce theoretical implications.

Conclusion

The pattern of data is thus similar to what Hock et al. (1995) have found: adaptation is nonselective only if the bistable stimuli used are maximally unstable. The alternative hypothesis that the adaptation and hysteresis phenomena are actually the product of a system in which a slowly fluctuating internal state determines the response, could not be ruled out. The paradigm as used in experiment 3 is not sensitive enough to differentiate on the basis of time variables.

In hindsight, the fact that experiment 3 could not differentiate between the two hypotheses is not surprising. The possible adaptation durations and interstimulus intervals form a combinatorial space, and generate as such a twodimensional infinity from which the specific values for the experiment can be sampled. Little is known about the effects of manipulating these time variables yet. Moreover, one can expect that processes such as attention interfere with the perception of the stimuli. Future research in this field will have to aim at revealing the precise time course and codeterminants of the perception.

References

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