Ординатура / Офтальмология / Английские материалы / Progress in Brain Research Visual Perception, Part I Fundamentals of Vision Low and Mid-Level Processes in Perception_2006

Time to Filling-in (Secs)

At first sight, the finding that time to filling-in is predicted by cortical projection size might suggest that perceptual filling-in reflects a slow neural fillingin process that takes many seconds to interpolate the cortical projection area. Such a slow neural interpolation process would be unlikely to contribute to normal surface perception, and an experiment was designed to reject this hypothesis. Different stimuli of which the projection areas in retinotopical cortex were rectangular with different widths and lengths

were devised (De Weerd et al., 1998). If filling-in proceeded slowly across the rectangular projection area, one would predict that the time from stimulus onset to filling-in would correlate with the shortest distance across. However, the data showed that the shortest distance across the rectangular projection area was not a good predictor of the time required for filling-in. Among the factors that were correlated with the time required for filling-in was the total length of the cortical projection of the figure’s

boundary. Computational work suggests that the representation of a longer boundary is more robust than that of a shorter boundary, because of the larger recurrent network that would be involved in encoding a longer boundary (Francis et al., 1994; Francis and Grossberg, 1996). The stabilization of a boundary inside the RF of boundary detectors might lead, therefore, to slower adaptation the longer the boundary is. Assuming that perceptual filling-in occurs because of a weakening of boundary representations owing to a stabilization of boundaries on the retina and adaptation of boundary detectors, the robustness of the boundaries of figures with longer cortical boundary projections would explain why such figures take longer to fill in. This proposal goes back to the original ideas by Gerrits and Vendrik (1970), who suggested that boundary detectors inhibit the spread of activity related to the surfaces, and models from Grossberg (1987a, b; 2003a, b) in which interactions between a boundary system and a surface feature system form the basis of perception. The idea that the adaptation of an inhibitory signal related to boundary representations is required for filling-in was supported by a psychophysical experiment in which the gray figure was moved by a few tenths of a degree at a rate of 1 Hz, keeping the average position constant (De Weerd et al., 1998). This subtle manipulation was sufficient to significantly increase the time required to observe perceptual filling-in, suggesting that neurons with small RFs may contribute to the boundary representation, and that the jittering of the boundaries delayed the adaptation of neurons encoding the boundaries. Thus, any condition in which the inhibitory signal from the boundary would remain intact longer would delay perceptual filling-in, while any condition that would promote adaptation and weakening of the inhibitory signal (e.g., very accurate fixation) would make filling-in happen sooner. This implies that the time that elapses before filling-in is related to the adaptation of boundary detectors, while the time required for the filling-in process itself might be much briefer. This hypothesis is compatible with the idea that the neural filling-in processes producing the filling-in illusion might also be relevant for normal surface perception.

Evidence for the existence of fast perceptual filling-in processes came from several elegant

233

experiments by Paradiso and colleagues. In one such experiment (Paradiso and Nakayama, 1991), the presentation of a homogeneous bright disk on a dark background was followed by a dark ‘masking’ display with a bright circle of a diameter smaller than that of the bright disk. When the masking display with the circle was presented at an appropriate, very brief time interval after presentation of the bright disk, observers temporarily perceived the inner part of the circle as dark, as if the circle had blocked the spread of brightness from the inner side of the disk’s contour toward the middle. By manipulating the size of the circle and the time interval between the disk and masking stimulus, Paradiso and colleagues arrived at a speed of brightness spread on the order of 1001/s. A number of other experiments confirmed that brightness spread, though fast, is in fact a time-consuming event that takes place on a time scale of milliseconds for objects that span few visual degrees on the retina (Todorovic, 1987; Paradiso and Hahn, 1996; Rossi and Paradiso, 1996). Biologically plausible, computational models have provided theoretical support for fast brightness interpolation (Grossberg and Todorovic, 1988; Arrington, 1994; Neumann et al., 2001).

Although the previous experiments suggest that brightness perception is related to an interpolation process that proceeds from the inner side of a surface’s edges toward its middle, the perceived brightness is related to luminance ratios between neighboring surfaces (Rossi and Paradiso, 1996). Thus, in a display consisting of a gray patch flanked by two dark patches, the gray patch will be perceived as brighter compared to the same gray patch flanked by two bright patches. Accordingly, ramping up the luminance values of the flanking patches from dark to bright will decrease the perceived brightness of the gray patch, and the opposite will occur when the luminance values of the flanking patches are ramped down, in the absence of any physical changes inside the gray patch. This shows that perceptual filling-in represents more than just an interpolation of physical features in the image, and that what is interpolated depends on global computations of brightness ratios (contrasts) in the image. These computations come down to a discounting of the illuminant, and have

234

been mathematically described by Grossberg (1997, 2003a, b). Grossberg’s models include an on-center off-surround input network, which normalizes incoming luminance signals and discounts the illuminant. Boundary signals are constructed by a boundary-processing system based on the discounted signals, which are also used as a seed for a diffusion-like interpolation process in a surface-processing system.

Neurophysiological evidence

If one could find neurons with RFs inside the gray patch whose firing rate would be correlated with the perceived brightness of the patch while the luminance values of the flanking patches in the RF surround would be ramped up and down, then one would have evidence for neural brightness interpolation being directly related to brightness perception. Exactly that type of responses was found in a small proportion of V1 neurons recorded in cat visual cortex (Rossi and Paradiso, 1999). A strict correspondence between variations in brightness perception and variations in neuronal responses across a large set of experimental conditions was found in 10% of the recorded neurons. These data indicate that V1 may be involved in the interpolation of brightness, but do not exclude that other visual areas getting input from V1 may participate as well. In addition, the brightness-related responses observed in V1 could at least in part reflect feedback from extrastriate cortex.

The experiments from Paradiso and colleagues show that global computations during which the illuminant is discounted determine the brightness values that are interpolated between surface contours. Likewise, the adaptation process preceding filling-in of a figure by its background during maintained fixation depends in part on global computations determining what region in the image is a figure, and what region is the background. In an experiment in which the amount of texture surrounding a gray region was manipulated (De Weerd et al., 1998), two observations were made. First, to produce effective filling-in of the gray region, a minimal amount of surrounding texture

was required, and up to a certain limit, adding texture increased the percentage of trials with fill- ing-in and reduced the time before filling-in took place. Second, when the amount of texture was reduced, such that the stimulus started to look like a narrow frame made of texture on a homogeneously gray background (a narrow, textured frame with the gray background showing on the inside and on the outside), it was the frame that became filled-in by the gray background, rather than the opposite. For square sizes up to 41, the ratio of the width of the surrounding frame to the size of the square had to exceed about 0.25 before the gray square would be filled in by the texture on a majority of trials. This suggests that what becomes filled-in is always what is perceived as the figure. These observations show that the direction of fill- ing-in is fundamentally influenced by global computations in the image that determine what is perceived as a foreground and what as a background.

The above studies suggest that perceptual fillingin may involve a slow adaptation of figure-ground segregation mechanisms (in which both the boundaries and the size of the figure play a role) followed by a fast neural interpolation of the figure by its background. To directly test the presence of active, neural interpolation in retinotopic visual cortical areas, neurophysiological recordings were carried out in areas V1, V2, and V3 of the Rhesus monkey. Two monkeys were trained to fixate a point away from a gray square surrounded by dynamic texture, which in humans would produce perceptual filling-in, while recording from neurons with RFs that overlapped with the gray square. The monkeys were trained to maintain fixation for up to 14 s. Average population histograms of neurons in V2 and V3 (showing activity averaged over trials and neurons from the two monkeys) showed an initial onset transient followed by reduced activity. After a number of seconds, however, neuronal activity increased and reached a level that matched activity measured when the RF was physically filled in with texture (Fig. 3a). The activity increases occurred in the absence of any physical changes to the stimuli. Increasing square size delayed the time point at which the increasing activity reached a level that was indistinguishable from

the activity produced by physical filling-in of the RF. From that time point on, one might expect that the neurons would not be able to distinguish the stimulus with the square from the stimulus with the square physically filled in with texture. Accordingly, across a range of square sizes, that time point accurately predicted when human subjects exposed to the same stimuli viewed by the monkey, on average, perceived perceptual filling-in (Fig. 3b). In addition, stimulus conditions that prevented perceptual filling-in in humans did not produce activity increases in V2 and V3 neurons, or produced activity increases that did not reach the activity level obtained when the square was physically filled in (Fig. 3a, 12.81 panel). These observations strongly suggested that V2 and V3 contribute to neural interpolation processes associated with perceptual filling-in.

While about half of the recorded neurons in V2 and V3 showed activity that correlated with fillingin, no such neurons were found in V1. A conclusion that V1 neurons did not contribute to perceptual filling-in would be flawed if the V1 recordings had been done at the same eccentricities used for the recordings in V2 and V3. At matched eccentricities, RFs in V1 are much smaller than those in V2 or V3, such that small fixation errors would affect visual processing in V1 much more than in V2 or V3. However, the V1 recordings were carried out at an average eccentricity of 24.51, where human subjects experienced strong filling-in, and where the average RF size (2.51) was comparable to that of cells in V2 (21) and V3 (3.71), at the smaller eccentricities studied in these areas (9.11 and 7.61, respectively). This suggests that the absence of activity increases related to filling-in in V1 was not due to smaller RF size, which could have interfered with the adaptation of boundary detectors in V1. Thus, because the relationships between the size of fixation errors and RF size were similar in V1, V2, and V3, it is reasonable to conclude that V1 plays little role in the filling-in of dynamic texture (see De Weerd et al., 1995).

The human psychophysical data from De Weerd et al. (1998) suggest the importance of accurate fixation for filling-in to occur, and so do the data collected in the monkeys. The average standard deviation of the fixation errors was around 0.41 in

235

one monkey and 0.81 in the other (De Weerd et al., 1995). The activity increases related to filling-in were less prevalent in the second monkey. Further, an analysis of eye movements in a subset of trials indicated that small deviations away from fixation within the electronic fixation window abolished the activity increases associated with filling-in.

It would be interesting to repeat the above experiments after training monkeys to signal the moment of filling-in. The correlation between perceptual filling-in responses in humans and neuronal responses in monkeys measured under identical stimulus conditions, however appealing, remains an indirect way of linking a neural response to a percept. It has been shown that monkeys can be trained to report the filling-in of the blind spot (Komatsu and Murakami, 1994), so there is no reason in principle why monkeys could not also be trained to report filling-in produced by prolonged stabilization of the image (through fixation). Measuring the monkey’s ability to fill in would require that the monkey be trained first to respond to the physical filling-in of a gray square embedded in texture, choosing stimulus parameters that would make perceptual filling-in impossible (e.g., choosing a square that is too large). Interspersed among a majority of these trials, a minority of test trials would present stimuli that do produce filling-in, to which the monkey would then be expected to respond as if the perceived filling-in were physical. Unfortunately, a test trial approach is difficult to combine with recording experiments such as those performed by De Weerd et al. (1995), which yielded no more than a single successfully completed trial per minute (due to fixation errors that prematurely ended the trial).

The slowly increasing activity level visible in the average histograms in neurons with their RFs centered on the gray square suggests that the neural interpolation process is slow, in contradiction to what was argued earlier. This raises doubts again as to whether such a slow process could be relevant for normal surface perception. However, when considering neurons with strong activity increases (and strong responses to physical fillingin), individual trials suggest that the activity increases in fact occur at a discrete moment in time. In a typical trial in an example of such neurons

(Fig. 4), a number of seconds with limited activity was followed by one or two discrete periods of increased activity, very much similar to filling-in reports. Thus, the gradual activity increases in average histograms correspond to an increase in the probability of perceptual filling-in rather than to a slow neural filling-in process. The activity in individual trials is in agreement with the idea that a long process of adaptation (in which boundary representations and figure-ground segregation weaken) is followed by a sudden filling-in event. The sudden onset of the increased activity suggests the presence of a fast filling-in process, which might be similar to the processes that can be inferred from the psychophysical experiments on brightness filling-in. This confirms that the neural interpolation process associated with dynamic texture filling-in revealed in V2 and V3 might also play a role in normal surface perception.

Is there a correlate of the adaptation process in the data? An interesting hint is given by the neuronal recordings obtained with square sizes that were too large to produce perceptual filling-in (12.81 panel in Fig. 3a), but which still produced moderate activity increases toward the end of stimulus presentation. In that stimulus condition, stimulus onset was followed by several seconds of activity below the baseline, indicating that the low activity level after stimulus onset reflected the presence of an active inhibitory process. It is possible that the increase of activity observed toward the end of the trial with the 12.81 square was made possible only

237

because of a decline in inhibition. For smaller square sizes (e.g., 41), the decrease in activity immediately following the stimulus onset transient did not decrease below the baseline. Nevertheless, it is possible that this activity decrease reflected inhibitory inputs partially counteracting strong excitatory inputs from the surrounding texture, rather than just a passive decline in activity. Thus, activity increases after a number of seconds of fixation might be enabled by an adaptation of this inhibitory signal (related to mechanisms of figure-ground segregation). To test this hypothesis, we quantified the amount of inhibition revealed in the large-square condition (12.81; Fig. 3b), and correlated it with the magnitude of the activity increases during prolonged fixation for a smaller square size (41; Fig. 3b), both stimulus conditions having been recorded in each neuron. We found that the more inhibition there was in the beginning of the trial in the 12.81 square condition, the stronger the activity increases were later in the trial in the 41 condition. This indicates that a reduction of inhibition that permits existing excitatory inputs from the texture to become effective in driving neurons with RFs in the gray square is a plausible model for the adaptation process that precedes filling-in. Based on the psychophysical data outlined above, one could speculate that this inhibitory signal is related to several mechanisms that maintain figure-ground segregation, including signals related to boundary representations. One interesting question related to this interpretation is whether neurons were encountered that showed an

adaptation of their signal around the time observers perceive perceptual filling-in. We have encountered only a single neuron in which activity was high starting at stimulus onset, and then showed a strong decline at the time where other neurons would show activity increases. This neuron could have been an inhibitory neuron, whose signal could have been directly related to declines in the strength of boundary representations.

To interpret the activity increases of neurons with RFs over the gray square as a neural interpolation process associated with perceptual filling-in does

not answer the question as to what aspect of the dynamic texture surface is actually carried by that neural signal. Is it the brightness, the general statistics of the line texture, the motion signal, or all aspects of the stimulus together? This is a relevant question, because of the multiple features present in the texture surface used, and because behavioral data from Ramachandran and Gregory (1991) suggest that different features of a background might spread across a figure at slightly different times during peripheral fixation of the figure. It has been suggested (Gattass et al., 1998) that interactions

between boundaries and surface features might take place in different areas depending on the features that define the boundaries between surface areas (e.g., luminance, color, motion, or texture). However, although a differential contribution of different areas to different types of interpolation processes is likely, activity increases in single neurons correlated with perceptual filling-in are probably related to the interpolation of several features at once. Experiments in which neural responses are measured in different cortical areas during perceptual filling-in of figures defined in different feature domains will be required to address this issue.

During the adaptation process preceding perceptual filling-in and neural interpolation, a competition may occur between the texture background and the figure’s boundaries. Specifically, the dynamic texture could contribute to the weakening of boundary information, which is predominantly given by the static black/gray luminance boundary. In computational models from Grossberg and colleagues (e.g., see Grossberg, 2003a), cooperative groupings of iso-oriented and collinear line segments compete with orthogonally oriented surface contours through inhibitory mechanisms, which are compatible with anatomical and physiological data (McGuire et al., 1991; Kisvarday and Eysel, 1993). Thus, in the stimulus used to test perceptual fillingin, the grouping of line segments in the background may weaken the representation of square boundaries that are orthogonal to the background elements. Furthermore, the competition between groupings in the background and the square’s contours may be biased in favor of the elements in the background, because they are constantly refreshed and the square’s boundary is not.

In addition, surface features might compete or interact with each other during filling-in. In the domain of color filling-in, the perceptual result of filling-in across a figure has been reported to be determined by both the figure and the background color. Fujita (1993) has demonstrated that prolonged fixation of the middle of a colored disk on a background of another color does not lead to a replacement of the figure’s color with that of the background, but rather with the appearance of a color mix throughout the entire stimulus. This suggests that filling-in is a bi-directional process,

239

which might be revealed only when the two features that fill the figure and background region are about equally strong.

Finding a neural interpolation response associated with perceptual filling-in for one feature does not imply the existence of a similar interpolation process for another feature. Von der Heydt et al. (2003) recorded in V1 and V2 of Rhesus monkeys that were trained to respond to color filling-in (see also Friedman et al., 2003). The monkeys were trained to respond to physical color filling-in, and appeared to respond in a similar fashion to conditions in which the stimulus remained physically identical, but which led to perceptual filling-in in human observers. This experiment was possible because stimulus conditions were chosen to lead to sufficient perceptual filling-in in trials lasting only 6 s. Recordings were performed from two types of neurons: recordings were made from neurons that were sensitive to large color stimuli filling the RF (surface-sensitive neurons), which existed in limited numbers in V1 and V2. For those cells, the color figure to be filled in was positioned inside the RF. Recordings were also made from neurons with optimal responses to oriented bars or edges (border-sensitive neurons), which are often sensitive to (color) contrast polarity, and which can also signal border ownership (Zhou et al., 2000). For those cells, the RF was placed along the border of the figure. In 92% of trials, the trials presented physical filling-in of the disk by the surrounding annulus. In 8% of the trials (test trials), monkeys responded, while there was no physical change suggesting that the responses were triggered by perceptual filling-in. It was found that during test trials, there were no changes in activity in the surface-response neurons, but there were declines in activity in the border-sensitive neurons which, with some assumptions, could be related to the filling-in responses of the monkeys. By assuming that border detectors sensitive to polarity and border ownership in fact have the information to trigger a symbolic ‘filling-in’ of spaces between borders (potentially with help from higher-order visual cortex), von der Heydt et al. (2003) interpreted the link between activity changes in border detectors and the filling-in reports as support for a symbolic filling-in theory. Thus, even if the

240

surface-sensitive cells contributed to normal perception of surface color, they would not contribute to perceptual filling-in of color induced by maintained fixation away from a figure. The experiments from von der Heydt et al. (2003), however, may not be entirely conclusive for several reasons. The correlation between filling-in responses and neural data was not tested under a variety of conditions that would lead to large differences in the timing of filling-in responses. In trials without physical filling-in, monkeys might have timed their responses to occur just before the end of the trial (after 6 s). It is, therefore, not entirely established to what extent the monkeys’ responses correlated with a percept of color filling-in. Furthermore, the activity of border-sensitive neurons was only recorded with their RF on the edge of the figure, and it is possible that activity increases during filling-in would have been present if the neurons’ RFs had been placed in the middle of the figure. In addition, the physical filling-in that stood as a model for the percept of perceptual filling-in was a very gradual transition from the figure color to the annulus color in the figure region. One could question whether this is an appropriate model for perceptual filling-in. If the perceptual change in the figure during maintained fixation was indeed so slow, one might wonder whether the conditions were optimal to induce a strong illusion. Despite these considerations, the data of von der Heydt et al. (2003) suggest the interesting possibility that color filling-in is a result of a high-level symbolic operation, rather than the result of an isomorphic interpolation process. Color perception (and not brightness perception) is strongly affected by lesions in infero-temporal cortex, which is also strongly involved in object recognition (Heywood et al., 1995; Huxlin et al., 2000; Cowey et al., 2001). Further, color (and not brightness) is an important surface feature that permits the evaluation of important qualities of an object (e.g., whether a fruit is ripe and eatable). Color may be an integral part of object meaning, more than brightness or other surface features, and may thus be coded at high levels in the visual system.

For surface features that do show a neural correlate of perceptual filling-in in retinotopic areas, such a finding does not exclude a contribution of

higher-level areas. When neurons in retinotopically organized visual areas signal that a region of visual space that was occupied by a given surface now belongs to another surface, this new ‘interpretation’ of the visual scene might be enhanced by feedback, similar to the way in which feedback mechanisms can enhance the interpolation of fragmentary contour information. Furthermore, even when there is evidence for a neural interpolation process in retinotopical areas associated with fill- ing-in, it is possible that the activity is not linked with perception per se, but rather, is used as a signal by higher-order areas to ‘re-interpret’ the arrangement of (perceived) surfaces. Thus, the experience of perceptual filling-in might rely on cooperative activity in a network of visual areas at several levels in the visual system. This point is reinforced by other findings reviewed above, showing that global computations across the image strongly contribute to the outcome of the filling-in process.

It is an interesting finding that, for the stimulus used in our experiments, a correlate of perceptual filling-in was present in extrastriate cortex, but not in V1. Although V1 lesions at best leave the possibility for unconscious processing of visual information (‘blind sight’; Weiskrantz, 2004), and although V1 therefore could be considered necessary for conscious perception, the presently discussed data suggest that V1 activity is closer to ‘physical’ reality than to perception. Furthermore, while V1 may be a required component to produce conscious perception, we may not be ‘aware’ of the presence or absence of V1 activity (Crick and Koch, 1995). This illustrates further that perception is a reconstructive process that hinges on interactions between multiple areas and processing streams. It is not surprising then (in retrospect) that an attempt to replicate the activity increases associated with perceptual filling-in in anesthetized monkeys failed. Recordings in a number of neurons in V2 and V3 of the anesthetized Cebus monkey, using stimulus conditions comparable to those used in Fig. 3, showed initial onset transients associated with stimulus presentation, followed by a lower, constant level of activity (De Weerd, Gattass, Desimone and Ungerleider, unpublished data). These data suggest that lateral

connectivity within areas V2 and V3 is insufficient to generate the neuronal processes associated with perceptual filling-in, and that interaction with higher levels of the system is required. In other words, the neural interpolation of surfaces might be tightly linked with conscious perception. Other studies in V1 have shown that anesthesia abolishes neural responses related to signaling figure ground segregation (Lamme et al., 1998), and to signaling the direction of plaid motion (Guo et al., 2004); responses that likely involve feedback from the extrastriate cortex.

Relationship of perceptual filling-in to attention, learning, and cortical plasticity

Given the role of feedback speculated about in the previous section, it is a natural question to ask what the role of attention might be during perceptual filling-in. In a previous study of fillingin during maintained fixation, Lou (1999) had reported that instructing subjects to pay attention to a set of peripherally presented figures made it more likely to perceive filling-in of the attended figures, compared to other unattended figures. This was interpreted as an inhibitory effect of attention on the perception of peripheral figures during maintained fixation. In one of our experiments (De Weerd et al., 2006), subjects fixated a red dot on a large dynamic texture background, while to the left and to the right of fixation (71 eccentricity), a gray square was presented. Fixation was measured, and trials aborted when fixation errors occurred. The luminance of the two squares as a whole or of a small cue in the middle of the squares was modulated slightly in an independent fashion on the left and on the right side, at a rate of 1 Hz on average. Subjects were instructed to report all of these small, regularly occurring luminance changes with button presses, and they were cued to do so on the left or on the right in blocks of about 20 trials. Furthermore, subjects were told that the disappearance of one (or sometimes both) of the squares (1.51 1.51) would signal the end of the trial. Subjects were instructed to press the space bar when this occurred, and then to indicate with the left and right arrow keys of the keyboard

241

which square(s) had disappeared. Subjects (N ¼ 5) were not aware that the filling-in of the squares was in fact the measurement of interest in this experiment. We found that filling-in was reported more frequently at the attended side (71%) than at the unattended side (23%). However, when fillingin happened to be perceived at the unattended side, it occurred after a time period that was indistinguishable from the time period required to perceive filling-in at the attended side (Fig. 5). This indicates that attention changed the probability of detecting filling-in, without modifying the disinhibitory processes that lead to filling-in. Hence, although conscious perception seems to be required to trigger the disinhibitory processes preceding filling-in, selective attention does not modulate the time course of disinhibition. Instead, attention might simply boost the response to inputs provided by the background texture to neurons with RFs over the gray square, similar to the boosting of responses to real stimuli placed inside the RF. Once neural interpolation has started, attention might have the effect of increasing the magnitude of the neural responses related to interpolation, in turn increasing the probability of detecting perceptual filling-in of the attended square.

There is ample evidence for the effects of selective attention on stimuli presented within the classical RF (e.g., Moran and Desimone, 1985; McAdams and Maunsell, 1999; Reynolds et al., 1999), and there is also evidence for attentional modulation of the effects of surround stimuli on stimuli in the RF (e.g., Ito et al., 1998; Ito and Gilbert, 1999). The former effects are related to mechanisms of selective attention, while the latter effects to attentional modulation of mechanisms of boundary perception and grouping. The findings reported in Fig. 5 are related to yet another effect of attention: the modulation of the strength of interpolation mechanisms contributing to surface perception. Computational models from Grossberg have proposed how the same cortical circuitry might explain all these attentional phenomena in an integrative manner. In an original study by Grossberg (1973), shunting inhibition was used to implement multiplicative dynamics in neuronal connectivity. This idea was the basis for adaptive resonance models in which