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. Author manuscript; available in PMC: 2007 Aug 13.
Published in final edited form as: Vision Res. 2006 Nov 1;46(28):4726–4734. doi: 10.1016/j.visres.2006.09.006

Target Recovery in Metacontrast: The Effect of Contrast

Haluk Öğmen 1,4, Bruno G Breitmeyer 2,4, Steven Todd 3,4, Lynn Mardon 3,4
PMCID: PMC1945166  NIHMSID: NIHMS14579  PMID: 17081585

Abstract

The visibility of a target stimulus (T) can be reduced by an aftercoming and spatially non-overlapping mask stimulus (M1), a phenomenon known as metacontrast masking. Interestingly, the visibility of the masked target can be recovered when a secondary mask (M2) is added to the T-M1 sequence. We analyzed a computational model of retino-cortical dynamics (RECOD) and derived the prediction that contrast dependence of metacontrast and target recovery should parallel the contrast dependence of afferent magnocellular and parvocellular pathways, respectively. In a psychophysical experiment, we tested this prediction by systematically varying a) M2′s contrast and b) the M1-M2 onset asynchrony (SOA). At the optimal M1-M2 SOA, target recovery effect increased with M2′s contrast without saturating, but at the optimal M1-M2 metacontrast SOA, reduction of M1′s visibility saturated very rapidly as M2′s contrast increased. Quantitative comparisons of psychophysical results with model simulations provide support for our prediction. We conclude that metacontrast masking is driven by signals originating from the magnocellular pathway and target recovery in metacontrast is driven by signals originating from the parvocellular pathway.

Keywords: Metacontrast, visual masking, dissociation, parvocellular, magnocellular

1. INTRODUCTION

The visibility of a target stimulus can be strongly reduced when it is followed in time by a mask stimulus, a phenomenon known as backward masking (Stigler, 1910, 1926; Fry, 1934; reviews: Bachmann, 1994; Breitmeyer & Öğmen, 2000, 2006). The plot of target visibility as a function of stimulus onset asynchrony (SOA) between the target and the mask is known as the masking function. Three generic types of masking functions have been reported in the literature: monotonic (or Type A), where peak masking occurs at SOA=0; non-monotonic Type B, where peak masking occurs at a single positive value of SOA; and finally non-monotonic multi-modal (oscillatory) masking function where peak masking occurs at two or more values of SOA (Breitmeyer & Öğmen, 2000, 2006). These different types of masking functions reflect the fact that backward masking is not a unitary phenomenon but can result from a variety of mechanisms. The choice of stimulus parameters determines critically which mechanisms will prevail in a given stimulus configuration. Operationally, four generic types of backward masking can be distinguished: 1) Backward masking by light, produced by a mask consisting of a uniform light field, 2) Backward masking by noise, produced by a mask consisting of a noise field, 3) Backward masking by structure, produced by a mask that shares structural similarities with the target, and finally, 4) a specific type of backward masking by structure is metacontrast, where the mask is adjacent to the target but does not spatially overlap with it. In general, backward masking by light and noise generate monotonic masking functions, whereas backward masking by structure and metacontrast can generate non-monotonic masking functions, especially when the mask- to-target (M/T) energy ratio is less than or equal to unity (Weisstein, 1972; rev. Breitmeyer & Öğmen, 2006).

Several studies showed that the visibility of a target (T) masked by a primary mask (M1) can be recovered if an appropriately timed secondary mask, M2, is added to the T-M1 sequence. This phenomenon is known as “target disinhibition” or “target recovery”. Since mechanisms involved in backward masking depend on stimulus parameters, different types of target recovery are obtained according to the types of masks used in the experiments. For example, Robinson (1966) used disks of diameters 0.23 deg, 0.46 deg, and 0.92 deg for T, M1, and M2, respectively, all of 20 ms duration. He fixed the SOA between M1 and M2 to 20 ms (M2 was presented 20 ms after M1), varied the SOA between T and M1, and compared the masking functions obtained with and without M2. Both cases produced Type-A functions, however, masking was weaker in the presence of M2. This stimulus configuration is expected to produce mainly masking by light and, due to the proximity of the target’s and masks’ contours, some additional masking by structure. The recovery of the target was accompanied by a masking of M1, suggesting that M2 acts by masking M1, which in turn eliminates the masking-by-light effect of M1 on T. In a follow up study, Robinson (1968) showed that target recovery in Type-A masking did not occur in dichoptic viewing conditions, i.e. when T and M1 were presented to one eye while M2 was presented to the other eye. This result would be expected since masking by light (here produced by M2) is greatly reduced or eliminated under dichoptic viewing (Battersby, Oesterreich & Sturr, 1964). The finding that a secondary mask that temporally follows the primary mask can lead to target recovery has been replicated and generalized to other stimulus configurations and tasks (e.g., Dember & Purcell, 1967; Schiller & Greenfield, 1969; Long & Gribben, 1971; Purcell & Stewart, 1975; Dember, Schwartz, & Kocak, 1978; Kristofferson, Galloway, & Hanson, 1979; Byron & Banks, 1980; Tenkink & Werner, 1981; Purcell, Stewart, & Hochberg, 1982; Briscoe, Dember, & Warm, 1983).

Breitmeyer, Rudd, and Dunn (1981) investigated the timing of target recovery in metacontrast using the stimulus configuration shown in Figure 1. In their study, both the primary mask M1 and the secondary mask M2 were nonoverlapping, surrounding masks. First, as a baseline condition, they measured the visibilities of T and M1 in the T-M1 sequence at the optimal metacontrast SOA (45 ms). They then measured the visibilities of T and M1 in the presence of the secondary mask M2. The change in visibility for T (M1) was computed by subtracting the perceived contrast of T (M1) in the baseline condition from the perceived contrast of T (M1) in the presence of M2. The changes in the visibility of T and M1 relative to their respective baseline values as a function T-M2 SOA are plotted in Figure 2.When M2 precedes the T-M1 sequence by 150ms or longer time intervals, it has little effect on the visibility of T or M1. When T-M2 SOA is within the -130ms to +30ms range, an increase in the visibility of the target is observed (target recovery). Another important point to notice is that while the visibility of the target is recovering, there is no concomitant change in the visibility of the primary mask M1. This indicates a dissociation between the visibility of M1 and its masking effectiveness. As M2 is delayed further (T-M2 SOAs larger than 30ms), the opposite dissociative effect is observed, viz., while the visibility of M1 decreases there is no concomitant change in the visibility of the target. Taken together, target recovery in metacontrast exhibits two properties not observed in previous studies using different combinations of mask types:

  1. Maximum target recovery occurs when M2 temporally precedes M1. For example, in the data shown in Fig. 2, maximum target recovery occurs when M2 precedes M1 by 75ms to 15ms (depending on observers), corresponding to T-M2 SOA values of -30ms to +30ms.

  2. There is a double dissociation between target recovery and the primary mask’s visibility.

Figure 1.

Figure 1

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Spatial configuration of the stimulus used in Breitmeyer et al. (1981).

Figure 2.

Figure 2

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Change in the visibility of the target (open symbols) and the primary mask M1 (filled symbols) as a function of T-M2 stimulus onset asynchrony (SOA). The change in visibility for T (M1) plotted in the ordinate was computed by subtracting the perceived contrast of T (M1) in the baseline condition, where M2 was not presented, from the perceived contrast of T (M1) in the presence of M2. Different symbols correspond to different observers. Replotted from Breitmeyer et al. (1981).

This double dissociation supports the hypothesis that the visibility of the metacontrast mask M1 and its masking effectiveness are carried out by different processes. Single channel/single process models of metacontrast attribute the visibility of a stimulus and its masking effectiveness to the same process and therefore cannot explain this double-dissociation. An explanation of target recovery in metacontrast can be based on the dual process RECOD model (Öğmen, 1993; Breitmeyer & Öğmen, 2000, 2006). For simplicity, we will only highlight here three aspects of the model that are critical in explaining this phenomenon. First, as shown in Figure 3, the input is first processed by two parallel afferent pathways, one with a short-latency transient response, associated with the magnocellular pathway, and the second with longer latency sustained response, associated with the parvocellular pathway. Second, the model has a lumped representation of post-retinal areas that receive their dominant inputs from parvocellular and magnocellular pathways. While there are both feed-forward and feedback connections in the model, the critical interaction for target recovery in metacontrast is the reciprocal inhibition between these two systems, viz., sustained-on-transient and transient-on-sustained inhibition. And finally, as a linking assumption, the model uses time-integrated activities in the sustained post-retinal areas as a correlate of perceived brightness, the dependent variable used in Breitmeyer et al.’s study. Figures 4 and 5 depict schematically how double dissociation occurs in this model.

Figure 3.

Figure 3

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A simplified schematic of the RECOD model. Filled and empty synaptic symbols depict inhibitory and excitatory connections, respectively. The input is conveyed to cortical areas through afferent sustained parvocellular (P) and transient magnocellular (M) pathways. The cortical areas receiving their dominant inputs from P and M pathways are represented as lumped networks. We also refer to these pathways and their cortical targets as sustained and transient channels. The model assumes reciprocal inhibition (sustained-on-transient and transient-on-sustained) between these channels. This reciprocal inhibition is called inter-channel inhibition to contrast it with inhibitory interactions within each channel, which in turn is called intra-channel inhibition (Breitmeyer, 1984). The reciprocal inter-channel inhibition is critical in explaining metacontrast and target recovery effects. Modified from Öğmen (1993).

Figure 4.

Figure 4

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Schematic depiction of how the secondary mask M2 suppresses the visibility of the primary mask M1. The transient activity generated by M2 inhibits the sustained activity generated by M1. This transient-on-sustained inhibition is most effective when M2 follows in time M1. Note that under this timing condition, the transient activity of M1 does not receive any inhibition from M2. As a result, the inhibition of T by M1 remains intact and there is no target recovery.

Figure 5.

Figure 5

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Schematic depiction of how the secondary mask M2 leads to the recovery of the target’s (T) visibility. The sustained activity generated by M2 inhibits the transient activity generated by M1. This sustained-on-transient inhibition is most effective when M2 precedes in time M1. Inhibition of M1′s transient activity reduces M1′s inhibition on the target, leading to target recovery. Note that under this timing condition, the sustained activity of M1 does not receive any inhibition from M2. As a result, M1′s visibility remains intact.

In the current experiment, the target and the primary mask are again presented at the optimal metacontrast SOA. In the model, the main factor for metacontrast masking is the inhibition the transient activity of M1 exerts on the sustained activity of the target. In Figures 4 and 5, this inhibitory process is shown by a thick vertical arrow from the transient activity generated by M1 pointing toward the sustained activity generated by T. First consider the primary mask M1′s visibility (Figure 4). According to the model, M1′s visibility is correlated with the strength of its sustained activity. The major reduction of this activity will be due to the inhibition from M2′s transient activity when it overlaps in time with the sustained activity of M1. Therefore, the reduction of M1′s visibility is predicted to follow a Type B function, with M2 temporally following M1. As shown in Figure 4, within this M1-M2 SOA range, the transient activity of M1 does not receive any inhibition. As a result, the inhibition that M1 exerts on T remains intact and the model predicts that while the visibility of M1 is reduced, there will be no concomitant recovery of T. Next, consider what happens to the visibility of T when M2 temporally precedes M1 (Figure 5). According to the model, a reduction in the transient activity of M1 will in turn reduce the inhibition it exerts on the target, leading to a recovery of the target’s visibility. Moreover, the reduction in M1′s transient activity occurs mainly due to the inhibition from the sustained activity of M2. According to the dynamics of the inter-neuron network that carries the inhibition of sustained activity on transient activity (see Appendix), M2 has to be presented before M1 to make the sustained-on-transient inhibition effective. Notice, however, that when M2 precedes M1, no metacontrast masking of M1 occurs. Therefore, M1′s visibility should not change. Note that in this analysis we did not take into account a direct influence from M2 on T. This is because masking effects in our model are distant-dependent (as supported by extant data (Alpern, 1953; Bridgeman & Leff, 1979; Breitmeyer & Horman, 1981; Weisstein & Growney, 1969)) and, due to the large spatial separation between M2 and T, no direct masking is predicted to occur.

In a previous study (Öğmen et al., 2006), we showed that simulations of the RECOD model agree with Breitmeyer et al.’s (1981) data. Inspection of Figures 4 and 5 shows that the reduction of M1′s visibility is due to the afferent transient magnocellular signal generated by M2, while the recovery of the target’s visibility originates from the sustained parvocellular signal generated by M2. It is known that magnocellular responses have high gain and saturate rapidly as contrast is increased, while parvocellular responses have lower gain and increase linearly as a function of contrast (Kaplan & Shapley, 1986; Purpura, Kaplan, & Shapley, 1988). Therefore, the model predicts that if we change the contrast of M2, M1′s visibility and target recovery should follow a rapidly saturating and a linear function, respectively. The goal of this study was to test this new prediction.

2. METHODS

2.1. Psychophysical methods

The stimuli were a 0.75 deg disk (T), surrounded by two rings (M1 and M2), of thickness 0.5 deg and separation of 0.05 deg centered 1.5 deg above and 1 deg to the right of fixation. A match stimulus of the same dimensions as T or M1 was positioned symmetrically to the left of fixation. The duration of each stimulus was 10 ms. The observers’ (N=3) task was to indicate which of the two, the match or the stimulus of interest (T or M1, in separate sessions), appeared brighter. The luminance of the match varied according to a staircase procedure and converged to the Point of Subjective Equality (PSE). The T-M1 SOA was set to 60 ms, a value optimal for producing suppression of T’s contrast visibility. T-M2 SOA varied from -180 ms to 210 ms, corresponding to M1-M2 SOAs ranging from -240 to 150 ms. The PSEs for T and M1 were also measured in the absence of M2. The contrast values of T and M1 were both fixed at 60%. M2′s contrast varied to produce M2/M1 contrast ratios from 0.125 to 1.5. All three observers (43-yr old male, 57-yr old male, 47-yr old female) had normal or corrected-to-normal vision and were highly practiced in making psychophysical brightness judgments.

2.2. Computational methods

The model consists of a system of ordinary nonlinear differential equations which were solved numerically with the CVODE package. This package uses variable-coefficient forms of the Adams and backward differentiation formula methods (Cohen & Hindmarsh, 1994). The programs were written in C and were run on SUN workstations. All equations and parameters of the model were identical to those used in our previous study (Öğmen, Breitmeyer, & Melvin, 2003) with the exception of a nonlinearity of the form f(x) = x/(0.05+x) which was introduced to the retinal M cells so as to match their contrast dependence to the contrast dependence observed in electrophysiological data (Kaplan & Shapley, 1986). Figure 6 shows the contrast responses for parvocellular and magnocellular afferents in the model along with experimental data from Kaplan & Shapley (1986).

Figure 6.

Figure 6

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Top: Contrast response of parvo (P) and magno (M) cells. Replotted from Kaplan & Shapley (1986). Bottom: Contrast response of corresponding cells in the model. For visual comparison purposes, the plotted model responses have been scaled to bring them to a range comparable to the firing rates in the data.

In order to obtain reasonable simulation times, the spatial aspect of the model was simplified to represent a one-dimensional field which was sampled at 500 discrete locations. At the foveal inter-receptor spacing of 23 s, this field represents a foveal region of 3.2 deg. Therefore, in our simulations we used “representative stimuli” consisting of one-dimensional versions of the actual stimuli designed to fit into the 3.2 deg field around the fovea. The target was placed at the center of the 500 positions. The target, M1, and M2 each occupied 19 positions. The duration of each stimulus was 1.25 simulation-time units which is equivalent to 10 ms real-time units. T-M1 SOA was 7.5 simulation-time units which is equivalent to 60 ms real-time units. The center-to-center target-M1 separation was 125 positions and the center-to-center M1-M2 separation was 60 positions. These approximations lead to some quantitative discrepancies between the model and data (see Section 3). The model predicted perceived brightness was computed as the time-integrated activity of post-retinal sustained cells responding to the target.

3. RESULTS

Figure 7 shows the experimental results along with model predictions. The x-axis is the SOA between the secondary mask M2 and the target T. The y-axis is the amount of change in visibility with respect to baseline T-M1 sequence (i.e. without M2) in logarithmic units. In agreement with Breitmeyer et al. (1981)’s results (see Figure 2 above), we observe that target visibility increases for primarily negative T-M2 SOAs (recovery) while the visibility of M1 decreases for primarily positive T-M2 SOAs. As predicted, the reduction in M1′s visibility follows a Type-B function. Target recovery is observed for a broad range of negative T-M2 SOAs extending up to -150 ms. The model is in good agreement with the data, with the exception of slightly lower target recovery and much stronger masking of M1. This quantitative difference can be rectified by changing the reciprocal inhibition weights between transient and sustained systems. However, we did not change the parameter values because this quantitative difference does not affect the relative changes of these effects as a function of contrast, which is the main focus of our study1. Inspection of the obtained data shows that, as a function of SOA, the rate of increase in target recovery for T-M2 SOAs less than -30 ms is similar to the rate of decrease for T-M2 SOAs greater than -30s, producing a target recovery function that is approximately symmetric around its peak value at T-M2 SOA of -30ms. However, target recovery in the model is asymmetric in that it rises gradually to reach its peak at T-M2 SOA of 0 ms followed by a very rapid drop. In this respect, our model predictions follow more closely the data of Breitmeyer et al. (1981) shown in Fig. 2. The differences in the morphology of target recovery curves are likely due to differences in stimulus parameters and a systematic investigation of the parameter space can clarify this issue.

Figure 7.

Figure 7

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Change in the visibility in log units of the target (T) and the primary mask (M1) relative to the baseline condition where the secondary mask M2 was not present. The abscissa is T-M2 SOA and the different curves plot the results for different values of M2/M1 contrast ratio. Top and bottom panels are psychophysical data and model simulations, respectively. The values that fall inside the rectangles are used to plot Figure 8. The data represent the average across the observers. To avoid clutter, only the average standard error of the mean for T and M1 are shown.

The data indicate that, as M2/M1 contrast ratio is increased, both target recovery and the suppression of M1′s visibility are enhanced. However, as predicted, the decrease in M1′s visibility saturates rapidly while the increase in target recovery is more gradual. To visualize the contrast dependence of these effects more directly, we plot in Figure 8 the data points at the optimal T-M2 SOA for target recovery (-30 ms, corresponding to a M1-M2 SOA of -90 ms in the data; and 0ms, corresponding to a M1-M2 SOA of -60 ms in the model, as shown by the rectangles in Fig. 7) and for metacontrast suppression of M1 (120 ms, corresponding to a M1-M2 SOA of 60 ms in both the data and the model, as shown by the rectangles in Fig. 7) as a function of M2/M1 contrast ratio. Open and filled symbols correspond to the data and the model, respectively. As predicted, the function relating target recovery to M2/M1 ratio is more or less linear while the function relating M1′s visibility to M2/M1 ratio saturates rapidly.

Figure 8.

Figure 8

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Data points from Fig. 7 at the optimal SOAs for target recovery and M1′s masking (shown by rectangles in Fig. 7) are plotted as a function of contrast to show contrast dependence of target recovery (triangular symbols) and M1s visibility (square symbols). Open and filled symbols correspond to data and model respectively.

4. DISCUSSION

In addition to replicating the double-dissociation in target recovery previously reported by Breitmeyer et al. (1981), our findings confirmed a specific prediction of the RECOD model, namely that target recovery and metacontrast parallel the contrast dependencies of parvocellular and magnocellular systems, respectively. Before discussing theoretical and empirical implications of our findings, we will first consider alternative explanations for our findings.

4.1. Alternative theories

In a previous simulation study, Francis (1997) compared the predictions of the Boundary Contour System (BCS) model (Grossberg & Mingolla, 1985) to Breitmeyer et al.’s (1981) disinhibition data. Qualitatively, the secondary mask produced target recovery, in agreement with the data. However, as noted by Francis (1997), in its current form, the model failed to explain the dissociation between the visibility and the masking effectiveness of the primary mask. Additional studies are needed to establish whether other parameter values in the model can produce a double dissociation and capture the specific contrast dependencies that we report.

According to another explanation of target recovery, M2 can act as a cue and/or activate a facilitatory nonspecific pathway (e.g., as in the perceptual retouch model [Bachmann, 1984, 1994]) and thereby enhance the visibility of T directly (rather than doing so indirectly by suppressing M1′s effectiveness as a mask). However, this explanation would also predict that the visibility of M1 should also increase for the range of T-M2 SOAs where target recovery is observed. This is because M1 is spatially located between M2 and T and therefore is closer to M2 than T is. Inspection of data in Figure 7 shows that this is not the case. Nevertheless, to test this possibility directly, we ran a control experiment in which only M2 and T were presented. The direct-facilitation hypothesis predicts that T’s visibility should increase even in the absence of M1. However, results in Figure 9 show that in the absence of M1, M2 fails to enhance the visibility of T, thus ruling out the facilitation hypothesis.

Figure 9.

Figure 9

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Change in the visibility of T in log units as a function of T-M2 SOA. The square symbols are the data taken from the previous experiment where the stimulus consisted of M2, T, and M1. The circles are from the control experiment where only M2 and T were presented. Data represent the average across the observers +/-1 SEM.

4.2. Broader implications: Experimental paradigms using rapid succession of brief stimuli

Several experimental paradigms use temporally successive presentations of brief stimuli to investigate various aspects of visual processing such as attention (e.g., Müller & Findlay, 1988; Nakayama & Mackeben, 1989; Potter, 2006), feature fusion (Herzog, Parish, Koch & Fahle, 2003; Herzog, Lesemann, and Eurich, 2006), response priming (Vorberg et al., 2003), perceptual microgenesis (e.g., Bachmann, 2000; 2006; Öğmen & Breitmeyer, 2006), and scene recognition (Bacon-Macé et al., 2005; VanRullen & Koch, 2003). There are efforts to link findings from these different paradigms, for example analyzing to which extent masking and attentional blink (Giesbrecht, Bischof & Kingstone, 2003; Grandison, Ghirardelli & Egeth, 1997; Seiffer & Di Lollo, 1997), masking and attentional cueing (Scharlau & Ansorge, 2003; Breitmeyer, Koç & Öğmen, 2006), masking and priming (Scharlau & Neumann, 2003; Schmidt, 2002) interact.

Taken together, our results and previous studies of target recovery/disinhibition show that even a simple sequence consisting of three stimuli can lead to complex interactions among the stimuli. These interactions depend on temporal, spatial, and figural characteristics of the stimuli making up the sequence. For example, when spatial and figural characteristics of the stimuli lead to masking by noise or light, a disinhibition can be obtained by an aftercoming stimulus. Moreover, the disinhibition obtained in this case is accompanied by the inhibition of the mask’s own visibility, thus leading to an association between a mask’s visibility and its effectiveness in suppressing the target. However in metacontrast, where two stimuli share contour similarities but do not spatially overlap, a double dissociation is obtained between a mask’s visibility and its effectiveness. A stimulus presented before the target-mask sequence leads to the recovery of the target stimulus without affecting the visibility of the mask stimulus, while a stimulus presented after the target mask sequence reduces the visibility of the mask stimulus without affecting the visibility of the target stimulus. In addition, metacontrast and target recovery differ in the way they depend on stimulus contrast. Thus, interpretation of findings obtained from experimental paradigms containing a temporal sequence of brief stimuli requires a careful analysis of how elements in the sequence interact based on their spatial, figural, and temporal characteristics.

5. CONCLUSIONS

In conclusion, the contrast dependencies of target recovery and metacontrast parallel the contrast dependencies of parvocellular and magnocellular systems, respectively. The RECOD model quantitatively captures these relative changes as a function of contrast. Our findings cannot be explained by single channel/process models nor can they be accounted by direct facilitatory processes. Taken together, our results show that metacontrast masking is driven by signals originating from the magnocellular pathway and target recovery in metacontrast is driven by signals originating from the parvocellular pathway. Furthermore, our model can provide a theoretical framework to analyze interactions in experimental paradigms where multiple stimuli are presented in rapid succession.

Acknowledgments

This work was supported by NSF grant BCS-0114533 and NIH grant R01-MH49892.

APPENDIX

The equations and parameters given in Öğmen et al. (2003) were used in our simulations, with the exception of the nonlinearity at the output of M cells, as described in Section 2.2. Because the interactions between cortical sustained and transient systems are critical in the results presented in this paper, here we reproduce for definiteness the equations where these interactions take place. For details, the reader is referred to Öğmen et al. (2003).

The post-retinal cells, mainly driven by the parvo-cellular pathway (“post-retinal sustained cells”), form a network wherein the activity of the ith cell, pi, is governed by the shunting equation

1τdpidt=−Appi+(Bp−pi){Φ(pi)+2νi(t−η)}−pi{∑j=i−npf;j≠ii+npfΦ(pj)+∑j=i−npi+npHj−ipiνj(t−η−κp)+∑j=i−npi+npQj−impmj} (A1)

where the excitation consists of the afferent parvocellular signal 2νi(t−η)and a feedback signal Φ(pi). The first two components of the inhibitory signal consist of feedback∑j=i−npf;j≠ii+npfΦ(pj) and intra-channel feed-forward ∑j=i−npi+npHj−ipiνj(t−η−κp) inhibition terms. The third component ∑j=i−npi+npQj−impmj represents the inter-channel transient-on-sustained inhibition. The inhibitory kernels Hkpi and Qkmp determine the spatial spread of intra- and inter-channel inhibition, respectively. Parameter η represents the relative delay between the parvocellular and magnocellular signals. Parameter κp reflects the relative delay of the inter-channel inhibitory signal with respect to the excitatory signal. The inter-channel transient-on-sustained inhibition is based on the activity mj, which represents the activity of jth cell in the network of post-retinal cells mainly driven by the magno-cellular pathway (“post-retinal transient cells”). These cells receive excitatory and inhibitory inputs from the magnocellular pathway (2[yj(t)]++ and ∑k=j−npj+npHk−jmi[yk(t−κm)]++ respectively) and a post-retinal sustained-on-transient inhibition via the kernel Q pmk yielding the shunting equation:

dmjdt=−Ammj+(Bm−mj)2[yj(t)]++−mj{∑k=j−npj+npHk−jmi[yk(t−κm)]+++∑k=j−npj+npQk−jpmqk} (A2)

The function [·]++ denotes full-wave rectification that generates the “on-off” response characteristics of transient cells. Parameter κm reflects the relative delay of the intra-channel inhibitory signal with respect to the excitatory signal. The post-retinal sustained-on-transient inhibition is delivered by a network of inter-neurons and qk represents the activity of the kth inter-neuron in this network. These inter-neurons obey the additive equation:

dqkdt=−Aqqk+Bqpk, (A3)

which states that the net inhibitory signal qk to the transient cells is a temporally low-pass filtered2 version of the post-retinal sustained activity pk.

Footnotes

1

Note also that for M1′s visibility the model exhibits paracontrast effects (i.e. reduction in visibility for negative SOAs). This is because we used the same parameters as in Öğmen et al. (2003) where the stimulus configuration produced such paracontrast effects.

2

Equation (A3) describes a leaky-integrator which produces low-pass filtering of its input.

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