Modulation of spatiotemporal dynamics of binocular rivalry by collinear facilitation

Modulation of spatiotemporal dynamics of binocular rivalry by collinear facilitation
Journal of Vision (2010) 10(11):3, 1–15
http://www.journalofvision.org/content/10/11/3
1
Modulation of spatiotemporal dynamics
of binocular rivalry by collinear facilitation
and pattern-dependent adaptation
Min-Suk Kang
Department of Psychology and Vanderbilt Vision Research
Center, Vanderbilt University, Nashville, TN, USA
Sang-Hun Lee
Department of Brain and Cognitive Sciences, Seoul National
University, Seoul, South Korea
June Kim
Department of Brain and Cognitive Sciences, Seoul National
University, Seoul, South Korea
David Heeger
Randolph Blake
Department of Psychology and Center for Neural Science,
New York University, New York, NY, USA
Department of Psychology and Vanderbilt Vision Research
Center, Vanderbilt University, Nashville, TN, USA, &
Department of Brain and Cognitive Sciences, Seoul National
University, Seoul, South Korea
The role of collinear facilitation was investigated to test predictions of a model for traveling waves of dominance during
binocular rivalry (H. Wilson, R. Blake, & S. Lee, 2001). In Experiment 1, we characterized traveling wave dynamics using a
recently developed technique called periodic perturbation (M.-S. Kang, D. Heeger, & R. Blake, 2009). Results reveal that
the propagation speed of waves for a collinear stimulus increased regardless of whether that stimulus was suppressed
(replicating earlier work) or dominant; this latter finding is contrary to the model’s prediction. In Experiment 2, we measured
perceptual dominance durations within a localized region in the center of two rival stimuli that varied in degree of collinearity.
Results reveal that increased collinearity did not change average dominance durations regardless of the rivalry phase of the
stimulus, again contrary to the model’s prediction. Incorporating pattern-dependent modulation of adaptation rate into the
model accounted for results from both experiments. Using model simulations, we show how interactions between collinear
facilitation and pattern-dependent adaptation may influence the dynamics of binocular rivalry. We also discuss alternative
interpretations of our findings, including the possible role of surround suppression.
Keywords: binocular rivalry, computational modeling, collinear facilitation, perceptual organization, perceptual dynamics
Citation: Kang, M.<S., Lee, S.<H., Kim, J., Heeger, D., & Blake, R. (2010). Modulation of spatiotemporal dynamics of binocular
rivalry by collinear facilitation and pattern-dependent adaptation. Journal of Vision, 10(11):3, 1–15, http://www.journalofvision.
org/content/10/11/3, doi:10.1167/10.11.3.
Introduction
Neurons in primary visual cortex (V1) respond to
contours of specific orientations falling within their
receptive fields (De Valois, Albrecht, & Thorell, 1982;
Hubel & Wiesel, 1962, 1968), but the strength of those
responses can be substantially modulated by the concurrent presence of oriented contours falling outside the
receptive fields of those neurons (Allman, Meizin, &
McGuinness, 1985; Blakemore & Tobin, 1972; Maffei &
Fiorentini, 1976). This contextual property of V1 physiology implies that neural signals depend on stimulus
features beyond the boundaries of the conventionally
doi: 1 0. 11 67 / 1 0 . 11. 3
defined classical receptive field, and several mechanisms
mediating these contextual influences have been described.
One mechanism, termed surround suppression, entails a
reduction in neural response strength once an optimum
stimulus is enlarged beyond the boundaries of the classic
receptive field (e.g., see synopsis by Smith, 2006). Putative
perceptual consequences of surround suppression include
reductions in apparent contrast (e.g., Xing & Heeger, 2001)
and elevations in contrast detection and discrimination
thresholds (e.g., Yu, Klein, & Levi, 2002). Another
contextual mechanism, termed collinear facilitation, entails
enhancement in neuronal responses to a target contour
when that contour is accompanied by neighboring, collinear contours (Li, Piëch, & Gilbert, 2006), and here, too,
Received March 23, 2010; published September 2, 2010
ISSN 1534-7362 * ARVO
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
there are putative perceptual concomitants (e.g., Field,
Hayes, & Hess, 1993; Kovács & Julesz, 1993). In this
paper, we focus on contextual modulation within the
context of binocular rivalry, i.e., the unpredictable alternations in perceptual dominance that occur when dissimilar images are presented separately to the two eyes
(Levelt, 1965; Wheatstone, 1838).1
Numerous studies imply that rivalry dynamics are
influenced by the spatial context in which rival stimuli
appear. We know, for example, that multiple rival targets
located in different parts of the visual field can engage in
synchronized alternations when they share stimulus
features such as color or contour orientation (Alais &
Blake, 1999; Dörrenhaus, 1975; Kovács, Papathomas,
Yang, & Fehér, 1996; Whittle, Bloor, & Pocock, 1968).
Similarly, visual features located outside the boundaries of
a rival target can influence the predominance of that rival
target, as though the strength of the rival target were being
modulated by its neighbors (Alais & Blake, 1998; Fukuda
& Blake, 1992; Ichihara & Goryo, 1978; Mapperson &
Lovegrove, 1991; Paffen, Tadin, te Pas, Blake, &
Verstraten, 2006; Paffen et al., 2004). One particularly
salient example of spatial interactions is revealed by the
wave-like spread of dominance often seen during transition periods of binocular rivalry. To control where and
when these traveling waves of dominance arise, Wilson,
Blake, and Lee (2001) created spatially extended rival
targets that confined the spatial path of these wave-like
transitions and used brief contrast increments to control
the location and time at which the transitions arose. With
these stimuli and procedures, Wilson et al. were able to
measure the speed with which waves of dominance
emerged around a concentric rival target. They found that
traveling waves propagated more rapidly when the
contours forming the suppressed pattern were collinear
with respect to the path of the wave.
To account for collinearity’s effect on traveling waves,
Wilson et al. proposed a simple model (see Figure 1) in
which two layers of cortical units, each activated by one of
the two rival stimuli, engage in reciprocal inhibition. This
inhibition spreads through interneurons to adjacent units in
the other layer, an arrangement that promotes recurrent
disinhibition and, thereby, the spread of dominance over
time. The units within a layer are interconnected with one
another by reciprocal excitatory connections that mutually
reinforce activity within neighboring units of a given layer.
To account for the dependence of dominance wave speed
on the spatial configuration of a rival target, these lateral,
excitatory connections are more spatially extensive among
units responsive to contours that are collinear, a property
that promotes faster waves. To incorporate transitions from
dominance to suppression states, the excitatory units in
both layers (but not the inhibitory units) undergo neural
adaptation in proportion to their activation levels. Simulations of that model were in good agreement with the
speeds of traveling waves measured by Wilson et al. With
the addition of an asymmetry in the interaction profiles
2
between the layers of neurons engaged in reciprocal
inhibition, Knapen, van Ee, and Blake (2007) were able
to generalize the Wilson et al. model to stimulus conditions
where traveling waves of dominance were emerging during
rivalry between stimuli rotating in opposite directions.
Although the model shown in Figure 1 was developed
to account for the relatively fast spread of dominance of a
collinear pattern as it emerged from suppression, this
model also makes predictions pertaining to the spread of
dominance within a suppressed pattern when the currently
dominant pattern is collinear (i.e., when the collinear
pattern is being engulfed by suppression over time). Based
on intuition, one would expect a wave emerging from a
suppressed stimulus to propagate more slowly when the
currently dominant stimulus has strong collinearity, because
neural responses within neurons representing that dominant
stimulus are stronger as a consequence of the relatively
strong excitatory connections among those neurons. Moreover, simulations of that stimulus condition using the
Wilson et al. model produce results consistent with that
intuition; those simulations are presented in the discussion
of results from Experiment 1.
We tested this model prediction by measuring wave
speed for different combinations of rival targets varying in
Figure 1. Schematic illustration of the model in which the two
layers of neurons represent the two rival stimuli, respectively. The
extent and the strength of recurrent excitation (shown in blue)
associated with different contour orientations (V = vertical; D =
diagonal; H = horizontal) are shown by the three Gaussian
curves. Inhibitory connections are shown in red.
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
relative collinearity, and the results were surprising: a
currently dominant rival pattern succumbs to suppression
more quickly when its contours are collinear. This
counterintuitive behavior can be accommodated within
the collinear facilitation model proposed by Wilson et al.
(2001) by the addition of a single parameter that governs
the rate of buildup of pattern-dependent adaptation. In the
General discussion section, we also explain why surround
suppression, the other form of contextual modulation
mentioned earlier, cannot explain the dependence of
traveling waves on collinearity.
Experiment 1
To measure pattern-dependent traveling wave dynamics
during binocular rivalry, we used the periodic perturbation
technique (Figure 2a) that was described and validated in
a previous paper (Kang, Heeger, & Blake, 2009). The
technique exploits the potency of a localized increment in
contrast of the suppressed stimulus to promote local
dominance of a small region of that suppressed pattern
and, moreover, for that local region of dominance to
spread over neighboring areas of the visual field (Wilson
et al., 2001). The traveling wave dynamics are inferred
based on the observer’s binary categorization of their
perceptual experiences within a restricted region of the
rival figure termed the monitoring region: perceptual
switches at the monitoring region are delayed but time-
3
locked to the triggers, implying the existence of wave-like
signals that propagate from the trigger site to the
monitoring region. Using this technique, Kang et al.
(2009) confirmed that the traveling waves of dominance
of a stimulus emerging from suppression were faster when
the contours comprising that stimulus were parallel to the
path of the wave, not orthogonal.
The aim of Experiment 1 was to examine the influence
of the collinear facilitation within the currently dominant
stimulus, and to achieve this aim, we prepared two
stimulus conditions that yielded one of four possible
traveling wave conditions (Figure 2b). A diagonal (D)
grating was always presented to one eye and either a
vertical (V) or a horizontal (H) grating was presented to
the other eye. Within these vertically elongated stimuli,
the V pattern comprised the high collinearity condition
whereas the H pattern comprised the low collinearity
condition. The lower region of Figure 2b illustrates two
different traveling wave conditions associated with each
pair of rival stimuli. Consider, for example, the situation
where the rival stimuli were the D and V patterns (shown
in the left column). In this situation, the emerging
traveling waves appeared in the V pattern when the D
pattern was perceptually dominant, and appeared in the D
stimulus when the V pattern was perceptually dominant.
Considering that the triggers were presented in anti-phase
at the upper and lower regions of the two rival stimuli
(Figure 2a), one of these waves propagated upward in
response to the trigger given within the lower region of
one eye and the other wave propagated downward in
Figure 2. Stimulus conditions and procedure for Experiment 1. (a) Stimulus sequence of periodic perturbation technique. Two vertically
elongated rival stimuli are presented to the two eyes. One trigger is presented at the upper region of the left eye and another trigger is
presented at the lower region of the right eye. These two triggers are presented in anti-phase and trigger period refers to the duration
between the two trigger presentations within the same eye. (b) Stimulus conditions. The D pattern is always presented to one eye and
either the H pattern or the V pattern is presented to the other eye (abbreviations designating these stimulus combinations are shown at the
top). These two stimulus conditions, together with their associated triggers, produced four traveling wave conditions shown designated as
follows: W(V/D), W(H/D), W(D/V), and W(D/H), where the bolded letter denotes the pattern receiving the trigger. Details are explained in
the text.
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
response to the trigger given within the upper region of
the other eye’s stimulus. Similarly, dominance waves
traveling in opposite directions could be triggered within
the D and H rival stimuli.
In referring to these four types of traveling wave
conditions, we will use the notation shown at the bottom
of Figure 2b, in which the first character indicates the
dominant stimulus and the second character indicates the
suppressed stimulus. To denote that the traveling waves
emerge from the suppressed stimulus, the character
denoting the suppressed stimulus is shown in bold, and
we refer to this suppressed stimulus as the carrier of the
wave. For example, W(D/V) refers to the condition where
the initially dominant stimulus is the D pattern and it is
the V pattern that emerges from suppression as a traveling
wave of spreading dominance.
4
Triggers were periodically presented at the upper and
lower regions of each stimulus, respectively. Each trigger
was presented for 200 ms with 100% contrast. Trigger
period was individually adjusted from 3 s to 6 s. Observers
were instructed to fixate the center of the stimuli (designated by the two white markers) and to monitor and track
fluctuations in rivalry dominance within this region by
pressing and holding either of two keys. Observers declared
dominance only when one or the other of the rival gratings
within the monitoring region was exclusively dominant,
with neither key being pressed when mixtures were
experienced. Each tracking session lasted 60 s and each
test condition was repeated eight times. For each pair of
rival stimuli (V and D; H and D), we randomly shuffled the
order of the two trigger positions (upper left/lower right
and lower left/upper right), the eye (left or right) receiving
the D pattern, and the tilt (clockwise or counterclockwise
from vertical) the D pattern.
Methods
All aspects of this study were approved by the
Vanderbilt University Institutional Review Board. Six
observers participated in this experiment (5 males, 1 female,
average age 34 years old). Except for the first author and
the last author of this paper, all other observers were naive
to the purpose of the study. All had normal or correctedto-normal vision, and all gave informed consent after
thorough explanation of the procedures.
All stimuli and trial-related events were controlled by a
Macintosh G4 computer (Apple, CA) running Matlab (Mathworks, MA) in conjunction with the Psychophysics Toolbox
(Brainard, 1997; Pelli, 1997). Stimuli were presented on the
screen of a Sony E540 21-inch monitor (1024 H ! 768 V
resolution; 120-Hz frame rate; 21.67 cd/m2 mean luminance) in a dimly illuminated room. Stimuli were viewed
against a gray background (21.67 cd/m2) through a mirror
stereoscope placed 90 cm from the monitor.
Vertically elongated rival gratings (0.8- ! 5- visual
angle, 4 c/deg) were presented to the left and right eyes
(Figure 2b). To promote stable binocular alignment, each
rival stimulus was bordered with a black rectangular
frame (3.6- ! 8-), the width of which was 0.25-. Five
pairs of horizontal line segments (0.5- in length) were
presented at both sides of rival stimuli. Two pairs of them
indicated the trigger locations (at T1.5- with respect to the
center of the stimuli) and the central pair (colored white)
designated the monitoring region (described below).
Before the experiment, the contrast values of the rival
stimuli were adjusted for each observer (15%–30%) to
produce reasonably slow perceptual alterations (targeting
a mean dominance duration of È4 s) and reliable traveling
waves (Kang et al., 2009). As an aside, we also collected
pilot data using horizontally elongated rival stimuli and
obtained the same pattern of results as those produced by
the vertically elongated stimuli; we chose to use vertical
stimuli to minimize horizontal eye movements and to
avoid hemifield differences.
Results
Collinear contours promote faster traveling waves.
Figure 3a shows the switch functions of the four traveling
wave conditions, averaged across six observers. The
switch function was obtained from the tracking record of
the periodic perturbation technique, which represents the
mean perceptual states as a function of trigger phase. The
detailed procedure for obtaining this switch function is
described elsewhere (Kang et al., 2009). In brief, the
switch function is derived by averaging the time series of
tracking records from the onset of one trigger to the onset
of the other one. As shown in Kang et al. (2009), the
switch function representing W(D/V) (blue solid line) is
shifted leftward compared to the switch function representing W(D/H) (red solid line)2. This means that a pattern
comprised of collinear contours exhibits faster traveling
waves as it emerges from suppression, compared to a pattern
composed of weakly collinear contours. This finding is
entirely consistent with previous studies (Knapen et al.,
2007; Wilson et al., 2001). However, the traveling waves
emerging within the initially suppressed D pattern were also
faster when that pattern was in rivalry with the collinear V
pattern (condition W(V/D), blue dotted line), compared to
when the D pattern was in rivalry with the H pattern
(condition W(H/D), red dotted line). A collinear pattern, in
other words, more quickly succumbs to suppression, contrary to expectation.
For purposes of statistical analysis, the latency of the
waves was obtained from the switch function by estimating
the threshold of a sigmoid fitted to the switch function (see
details in Appendix A). In this measure, the latency reflects
the time at which the perceptual states of both rival stimuli
are balanced. Figure 3b shows latency associated with the
four traveling wave conditions. Consistent with visual
inspection of the switch functions (Figure 3a), the latencies
obtained from those four traveling wave conditions show
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
5
Figure 3. Results from Experiment 1. (a) Averaged switch functions from six observers. Four traveling wave conditions are indicated as
follows. The blue lines represent the traveling waves arising from V and D rival patterns, and red lines represents the traveling waves
arising from H and D patterns. Solid lines indicate the traveling waves emerging from V/H carrier whereas the dotted lines indicate the
traveling waves emerging from D carriers (the term “carrier” refers to the pattern that is emerging from suppression into dominance
consequent to presentation of a trigger at one end of that pattern). (b) Latencies for each of the four traveling wave conditions. The
orientation of the fill pattern within each histogram signifies the stimulus pattern rivaling with D pattern. Colored circles indicate the
individual data points, with a given color referring to a given observer. Error bars equal T1 SE.
that traveling waves propagated faster when rivalry was
between the D and V patterns relative to rivalry between
the D and H patterns, irrespective of which stimulus was
the carrier of the traveling waves. A repeated measure of
two-way ANOVA with the factors of carrier pattern (V/H
and D) and collinearity (V and H) revealed that the effect of
collinearity was statistically significant [F(1, 5) = 23.90, p G
0.01], but the effect of the carrier pattern was not significant
[F(1, 5) È 0, p È 1]. This result was evident for all
observers (Figure 3b, each color represents a different
observer). Latencies obtained by estimating the time point
at which the switch function passes the mean perceptual state
(0.5) produced similar results, again contrary to expectation.
Experiment 1 confirms that the speed of the emerging
traveling waves depends on the degree of collinearity of
the rival targets, but this is true regardless of the
perceptual state of the collinear pattern. We find, in other
words, that a stimulus dominant in rivalry is more
susceptible to spreading suppression when that stimulus
consists of contours that are collinear. In the following
paragraphs, we will show that this outcome is inconsistent
with predictions from the collinear facilitation model
proposed by Wilson et al. (2001). Moreover, we will show
how a simple modification to that model brings its
predictions in line with these seemingly counterintuitive
results. That modification involves the addition of a
parameter that governs the rate of pattern-dependent
adaptation. To start, we need to explain the rationale for
pattern-dependent adaptation in the original model.
First, it is well-known that exposure, or adaptation, to a
pattern temporarily reduces the effective contrast of that
pattern, with the magnitude of this reduction in effective
contrast proportional to the contrast experienced during
adaptation (Blakemore, Muncey, & Ridley, 1973; Greenlee,
Georgeson, Magnussen, & Harris, 1991). Stronger contrast,
in other words, produces more robust adaptation, presumably attributable to the stronger neural responses associated
with higher contrast stimulation. In contemporary parlance,
contrast adaptation is a by-product of contrast gain control
(e.g., Heeger, 1992), a component of which seems to
operate prior to the level of binocular rivalry (Watanabe,
Paik, & Blake, 2004). It is reasonable to expect, therefore,
that collinear facilitation could boost the stimulus strength
of a pattern and, thereby, increase the magnitude of selfadaptation associated with viewing that pattern. Second, it
is commonly believed that contrast adaptation plays an
important role in triggering switches in dominance during
binocular rivalry (Shpiro, Moreno-Bote, Rubin, & Rinzel,
2009; Wilson, 2007), and we know that adaptation to a
pattern prior to the onset of rivalry (Blake & Overton,
1979; Wade & de Weert, 1986) or intermittently during
ongoing rivalry (Kang & Blake, 2010) reduces the
dominance of that pattern during rivalry.
Combining these ideas about adaptation and applying
them to the conditions of rivalry used in the present paper,
we see that collinearity can produce two opposing influences
on the strength of a rival target. On the one hand, neural
activity associated with a collinear pattern (i.e., the V stimulus)
should be stronger owing to the recurrent excitatory
connections among neurons registering the presence of that
pattern. On the other hand, those stronger neural responses
should produce greater self-adaptation of that collinear
stimulus. To confirm how these two factors might operate
for perception of traveling waves during binocular rivalry,
we implemented the model proposed by Wilson et al. (2001)
to account for traveling waves. In the following section, we
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
present results from simulations of that model showing that
it fails to account for the results of Experiment 1. We then
show that those results can be accommodated by adding a
single parameter to the model, one that controls the time
course of adaptation in a pattern-dependent manner.
Simulation of traveling waves
We adapted the network model shown schematically in
Figure 1 to the conditions of rival stimulation used in our
studies. In our instantiation of the model, the dynamics of
traveling waves are governed by the following equations:
CE ¯t EAj ¼ jEAj þ
PAj ¼ SAj j )I
8Ajk
N
X
k¼1
½PAj %2
ð10 þ HAj Þ2 þ ½PAj %2
8Bjk IBk þ )E
N
X
;
(Ajk EAk ;
k¼1
!
jðk j jÞ2
;
¼ exp
4A28
ð1Þ
ð2Þ
ð3Þ
!
jðk j jÞ2
ðk m jÞ
(Ajk ð XÞ ¼ exp
4A( ðXÞ
in which X is the stimulus pattern;
ð4Þ
CI ¯t IAj ¼ jIAj þ EAj ;
ð5Þ
CH ¯t HAj ¼ jHAj þ )H + AH EAj ;
ð6Þ
+ AH ¼ 1 þ gðXÞ in which X is the stimulus pattern:
ð7Þ
Subscripts A and B represent the two eyes, respectively,
X is the stimulus pattern, and j indicates the jth neuron
among those N units (40 in our implementation of this
6
model) representing vertically elongated stimuli presented
to the left and right eyes. E represents the rate of neural
activity governed by Equation 1, in which the time
constant CE equals 20 tu (tu refers to time unit, which is
arbitrary); H is the term embodying spiking adaptation.
Each unit in the network receives an input P whose value
is determined by stimulus strength S (=30), by subtractive
inhibition from neighboring cells representing the stimulus presented to the contralateral eye and by additive
recurrent excitation from neighboring cells representing
the stimulus presented to the same eye. The notation [P] is
the maximum operator applied to the external input that
compares P and 0 and returns the larger value. In this
simulation, the extent of reciprocal inhibition is governed
by Equation 3, in which the constant )I is equal to 0.1 and
A 8 is equal to 1.5 irrespective of rival stimuli. In contrast,
the extent of recurrent excitation governed by Equation 4
changes depending on the pattern of the rival stimulus,
with A ( equaling 2 for H stimulus, 2.5 for D stimulus, and
3 for V stimulus. The constant )E was set to 0.04 for
this simulation. Equation 5 determines the inhibition for
which CI is set to 11 tu. Slow adaptation is governed by
Equation 6 in which CH was set to 900 tu and adaptation
rate )H equaled 0.3 (see other details in Appendix A).
In this model, as in the original version, the magnitude of
recurrent excitation decreases differentially depending on
the collinearity of rival stimulus, such that the magnitude of
collinear facilitation falls off more slowly with increasing
distance for the V pattern (collinear in our stimulus
configuration) compared to the H pattern (weaker collinearity). The extent of inhibition is symmetrical regardless
of the rival pattern. In the original Wilson et al. model,
adaptation magnitude, as embodied in the parameter H, is
dependent on activity level E (Equation 6), and E is
dependent on collinearity. Thus, increased activity level
accompanied by collinear facilitation increases the magnitude of adaptation. There was no variable, however,
governing rate of adaptation, meaning that rate was
independent of pattern collinearity. We have added a rate
parameter + (Equation 6) whose value can vary with
collinearity according to the variable g(X). Note that when
the parameter value for + is set to 1.0 (i.e., g(X) = 0), the
modified model is identical to the original Wilson et al.
model (i.e., rate is invariant with pattern). The time course
of adaptation is slower relative to the original Wilson
model when + is less than 1.0 (i.e., g(X) G 0) and is faster
when + is greater than 1.0 (i.e., g(X) 9 0).
We first verified that the model (when modified for our
stimulus conditions) produced traveling waves under
conditions embodied in the periodic perturbation technique. Figure 4a shows how simulated neuronal activity
fluctuates over time in response to repeated triggers
presented alternately to a pair of rival targets composed
of diagonally oriented contours oriented leftward for one
eye and rightward for the other eye. In this simulation, the
extent and magnitude of collinear facilitation for the two
rival stimuli are the same to simulate the traveling waves
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
7
Figure 4. Model simulations for Experiment 1. (a) Simulation result of periodic perturbation between two, orthogonally oriented D rival
patterns, with color representing the activity levels of the 40 neurons forming the rivalry network (recall Figure 1). Triggers are presented
every 2500 tu (tu means time unit, which is arbitrary) in anti-phase between the two eyes, resulting a 5000-tu trigger period. The white
dotted line illustrates the wave front that is obtained by connecting the same activity level of all neurons over time. This wave front
indicates that perceptual switching occurs sequentially across this array of 40 neurons, producing a traveling wave of changing
dominance state. (b–d) The wave fronts associated with four traveling wave conditions with varying degrees of adaptation rate. The
legend designating these four conditions is shown at the right side of (d).
arising from these two orthogonally oriented, diagonal
patterns. Triggers were given every 2500 tu, meaning that
the trigger period was set to be 5000 tu in this simulation.
As mentioned above, the network comprised 40 neurons,
and for this simulation the triggers were introduced among
the first 3 and last 3 neurons by briefly (125 tu) increasing
the stimulus strength twofold within those units. In
response to each trigger given during the suppression phase
of a rival stimulus, a traveling wave was indeed induced as
evidenced by the successive, abrupt changes in activity
over serially connected neurons (note that the first trigger
was given at 2500 tu). We characterize the spatiotemporal
modulation in network activity as the wave front of the
traveling wave, and one example of such a wave front is
indicated by the white dotted line in Figure 4a. This wave
front was defined by the series of time points among the
array of neurons when their activity levels reach an
arbitrary, fixed value (E = 30). The regularity of this wave
front behavior seen in Figure 4a arises, of course, because
the perturbations producing the spreading state transitions
defining the wave fronts are themselves periodic. The time
elapsing between the presentation of a trigger within one
extreme of the network and the arrival of the resulting
wave front at the other extreme of the network defines the
speed of the traveling wave. The simulation shown in
Figure 4a confirms that this instantiation of the reciprocal
inhibition model proposed by Wilson et al. produces
reliable traveling waves under conditions of periodic
perturbation. We next will use this wave front concept to
test the effect of different stimulus configurations on the
speed of waves.
We start with simulations where the adaptation rate is
assumed to be constant regardless of pattern collinearity (i.e.,
where + = 1.0 because g = 0)Vthis assumption represents
the way adaptation is implemented in the original version
of the Wilson et al. model. The resulting wave fronts are
shown in Figure 4b. As produced by the original version of
that model (Wilson et al., 2001), traveling waves emerging
from the V carrier (solid blue line) are faster than those
emerging from the H carrier (solid red line); this outcome is
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Kang et al.
consistent with results from earlier traveling wave studies
(Kang et al., 2009; Wilson et al., 2001). Next, consider how
the model with invariant adaptation rate behaves when the
simulated traveling waves emerge from the D pattern
(dotted lines in Figure 4b). Here the model simulation
produces waves that emerge faster when the D pattern
rivals with the H pattern than when it rivals with the V
pattern. This outcome corresponds to the intuitive prediction described in the Introduction section, but it does not
correspond to the actual results: in fact, the D pattern
emerges more quickly when overcoming suppression
exerted by the V pattern (recall Figure 3b, right-hand pair
of histograms).
The model’s behavior can be brought into compliance
with the empirical results simply by varying adaptation rate
dependent on collinearity. To reiterate, adaptation rate is
defined as adaptation constant ()H) multiplied by +: + = 1 +
g(X), in which the term g(X) represents the modulation
factor dependent on the degree of collinearity of the pattern
X. The modulation factor reflects pattern-dependent adaptation such that g(V) always equals or is greater than 0, and
g(H) always equals or is less than 0, whereas g(D) equals 0
for all conditions. Implemented in this way, adaptation rate
is highest for V and lowest for H, with D being
intermediate. The consequence of this modification to the
model can be seen in Figures 4c and 4d, which correspond
to two different sets of values for the parameter g that
modulates the adaptation rate +. Notice that the model
continues to produce the well-established speed difference
between collinear and orthogonal contours (vertical and
horizontal in this case) emerging from suppression: the
wave front defined by the solid blue line always intersects
the x-axis (time) at a point well to the left of the
intersection time for the wave front defined by the solid
red line. Now, however, the traveling waves, represented as
the wave fronts, associated with emergence of the D pattern
into dominance changes when values of g(V) or g(H)
change across Figures 4b–4d: the D pattern waves are faster
(as indicated by the leftward shift in the blue dotted line)
when that D pattern is in rivalry with the V pattern but
slower (as indicated by the rightward shift in the red dotted
line) when the D pattern is in rivalry with the H pattern.
These simulation results thus show that incorporation of
pattern-dependent adaptation rate into the model shown
schematically in Figure 1 enables that model to produce
the seemingly counterintuitive influence of collinearity on
emerging waves of dominance. Now, from our earlier
work (Kang et al., 2009) we know that the spatiotemporal
dynamics of traveling waves are closely related to the
spontaneous perceptual alternations during binocular
rivalry. This led us to hypothesize that the interaction
between collinear facilitation and pattern-dependent adaptation should also be reflected in the durations of
dominance measured under conditions where perceptual
alternations occur spontaneously, with no external influences caused by perturbation triggers. Experiment 2 tests
this prediction.
8
Experiment 2
In binocular rivalry, a stronger (e.g., higher contrast)
stimulus typically enjoys enhanced perceptual dominance
(e.g., Levelt, 1965). Combining this property of binocular
rivalry with the putative strengthening influence of collinear facilitation, one might expect that the V pattern
competing with the D pattern would show stronger
perceptual dominance than the H pattern competing with
the D pattern. Yet there are published results contrary to
this expectation. Specifically, Alais, Lorenceau, Arrighi,
and Cass (2006) measured concurrent perceptual dominance of two pairs of rival stimuli, i.e., a pair of spatially
separated 1D Gabor patches presented to one eye in
competition with a pair of noise patterns presented to the
other eye. The concurrent perceptual dominance (i.e.,
synchronization of dominance states) of the two Gabors
increased with their degree of collinearity. However, Alais
et al. found that the perceptual dominance (indexed by
mean dominance duration) of the individual rival gratings
did not vary with collinearity, contrary to what would be
expected based on the putative enhanced strength of
collinear contours. Based on this odd result, Alais et al.
concluded that “the rivalry processes do not impinge upon
the contour association strengths” (p. 1485). In view of the
results and implications from Experiment 1, however, we
realized that this odd failure for collinear stimuli to
predominate in rivalry might have something to do with
interactions between collinear facilitation and stimulus
pattern-dependent adaptation, leading us to perform this
next experiment.
Methods
Elongated stimuli like those used in Experiment 1 were
used for these measurements. Five of six observers who
participated in Experiment 1 (including the first author
and the last author) also participated in Experiment 2. As
before, observers tracked periods of exclusive rivalry
dominance of a given rival target (H, D, or V) within the
monitoring region of the stimuli during 60-s tracking
periods. (These measurements were actually performed
in the context of a larger set of conditions in which
small gaps were placed in one of the two rival stimuli,
but those gap conditions are not relevant for the
predictions we are examining here.) Now, however,
there were no external perturbations to trigger traveling
waves; we measured successive durations of dominance
produced by spontaneous transitions in rivalry state.
Because the monitoring region was quite small, mixture
periods were brief and infrequent, and consequently, we
considered only the periods of exclusive dominance of
the gratings within the monitoring region for analysis. As
in Experiment 1, a D (diagonal) pattern was presented to
one eye and either a V pattern (vertical: high collinearity)
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
or an H pattern (horizontal: low collinearity) was
presented to the other eye. A given rival condition was
repeated four times, and the eye receiving the diagonal
stimulus was counterbalanced.
Result
Figure 5 shows the mean dominance durations from
four stimulus patterns associated with two pairs of rival
stimuli (V and D; H and D). Pairwise comparisons reveal
no statistically significant differences in the dominance
durations associated with rivalry between the V and D
patterns [F(1, 4) = 0.05, p 9 0.5] nor for the H vs. D
patterns [F(1, 4) = 2.24, p È 0.2]. Of particular relevance
for our purposes, the average dominance durations of the
V pattern and of the H pattern are not significantly
different [F(1, 4) = 4.78, p È 0.09], even though both V
and H patterns were in rivalry with the same D pattern.
This pattern of results is reminiscent of the puzzling
finding reported by Alais et al. (2006).
It is important to note that our results do not necessarily
indicate that vertical, diagonal, and horizontal contours
are equally strong rival targets; indeed, there is mixed
evidence in the rivalry literature concerning the effect of
contour orientation on binocular rivalry. Wade (1974,
1975) found no significant differences between predominance of vertical over horizontal contours, while Fahle
(1982) found that horizontal rival gratings were signifi-
9
cantly weaker compared to vertical or, for that matter,
compared to diagonally oriented rival gratings. In those
earlier studies, however, vertical and horizontal rival
targets were equally collinear because the rival targets
were either single lines (Fahle, 1982; Wade, 1974) or were
circular gratings. Recall, too, that results from our pilot
experiment replicated the results shown in Figure 3 using
horizontally elongated rival target configurations in which
horizontally oriented contours are the most highly collinear. The effects described in this paper, in other words,
pertain to collinearity, not orientation.
Finally, our results do not mean that the predominance
of stronger rival target is no greater than the predominance of weaker rival target. It is a common assumption
in rivalry models, including variants of the one being
tested in our paper, that reciprocal inhibition between
neural representations of rival stimuli, in addition to
adaptation, play a role in governing predominance; the
strengths of those inhibitory interactions are modulated by
the adaptation states of neurons activated by the dominant
stimulus and by neurons activated by the suppressed
stimulus. Moreover, it is now generally thought that
“noise” also represents an important ingredient governing
rivalry alternations, a point we return to in the next section
of the paper. So it would be the interaction of collinearity
with those other factors that would govern dominance
durations of spatially extended rival patterns (like ours) or
spatially neighboring rival patterns (like those used by
Alais et al.). Are the interactions embodied in the
modified version of the Wilson et al. model able to
predict the dominance durations measured in our Experiment 2? The simulations described in the next section
provide an affirmative answer to that question.
Simulation of rivalry dynamics
Figure 5. Results from Experiment 2. Average dominance
durations for four stimulus conditions, corresponding to two pairs
of rival patterns (V and D; H and D). The orientation of the fill
pattern within each bar signifies the stimulus pattern. Errors bar
indicate T1 SE and colored circles indicate data for a given
observer.
Simulations were performed in which we varied the
extent of recurrent excitation and pattern-dependent
adaptation gain to learn how these two mechanisms
interact to influence predicted rivalry dominance values.
We used the same set of equations that embody
interactions among an array of 20 units representing the
vertically elongated region of the visual field within which
the two rival patterns were imaged. In this simulation,
unlike the previous one for Experiment 1, we did not
provide external triggers to promote periodic switches in
rivalry state because in Experiment 2 switching occurred
spontaneously, not because of periodic perturbation. To
produce realistic alternations in rivalry state in the
absence of externally triggered switches, we added a
Gaussian noise term in Equation 2 (see details in
Appendix A). The inclusion of noise to promote spontaneous alternations comports with the widely accepted
assumption that spontaneous alternations in dominance
involve both neural adaptation and noise (Kang & Blake,
2010; Lankheet, 2006; Lehky, 1988; Moreno-Bote, Rinzel,
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
& Rubin, 2007; Shpiro et al., 2009; but see Wilson, 2005,
for a discussion of models that, in principle, can produce
alternations without noise). Van Ee (2009) provides an
excellent description of the ways that noise might influence
rivalry alternations, including ideas about exactly where in
the process noise exerts its influence.
Figure 6 shows the results from our simulations.
Figure 6a shows how simulated average dominance
10
durations for the V rival pattern (solid lines) and the D
rival pattern (dotted lines) vary as a function of adaptation
rate of the V pattern, i.e., g(V); the adaptation rate for the
D pattern remains constant. The different colored lines
denote stimulus conditions in which the V pattern is
assigned different values of A(, i.e., different degrees of
recurrent excitatory spread of the V pattern (shown by the
colored gradients within the figures). When g(V) is 0,
Figure 6. Model simulations for Experiment 2. Simulated dominance durations for (a) a V pattern (circles connected by solid lines) and (b)
an H pattern (squares connected by solid lines) in rivalry with a D pattern, as a function of adaptation rate (expressed in terms of the value
of g plotted along the x-axis). Also plotted are simulated dominance durations for the D pattern (triangles connected by dotted lines) when
it is in rivalry (a) with the V pattern and (b) with the H pattern. The parameter is an extent of excitatory spread (A(), and the different
simulated values are plotted in different colors as signified by the figure inserts. Excitatory spread for the D pattern is invariant (A( = 2.5);
the different colored dotted lines denote simulated dominance values for the D pattern dependent on the spread parameter for the V and
for the H patterns in rivalry with the D pattern. The single, black diamond symbol in each panel shows the simulated dominance durations
when two orthogonally oriented D patterns are engaged in rivalry. (c, d). Simulated activity level and adaptation level for a model unit
located within the middle of the array representing a vertically elongated V rival pattern. (c) Activity of the model unit with a fixed collinear
facilitation parameter (A ( = 3.0), simulated using two different values for pattern-dependent adaptation rate (gray for g = 0.0 and black for
g = 0.2). (d) Adaptation level within this model neuron. The color (adaptation rate) and the time course are matched with (c).
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
adaptation rate is 1 (i.e., the original version of the Wilson
et al. model); for these parameter values, the simulated
dominance durations associated with the V pattern
increase and dominance durations of the D pattern
decrease as the spatial extent of recurrent excitation
broadens. Of particular relevance, note that predicted
dominance durations for V are always longer than
predicted durations for D, an outcome that, in fact, is not
observed empirically (recall Figure 5 and the results from
Alais et al., 2006). For non-zero values of g(V), however,
simulated dominance durations of the V pattern systematically decrease whereas simulated dominance durations
of the D pattern increase. Note that there is a value of g(V)
where the dominance durations for the D and V patterns
are comparable, i.e., the pattern of results found by us and
by Alais et al. (2006). Figure 6b shows the same pattern of
results for simulations of dominance durations for rivalry
between the H pattern (solid lines) and the D pattern
(dotted lines). Now when g(H) is 0 (i.e., + = 1.0),
dominance durations for the H pattern are consistently
less than dominance durations for the D pattern, contrary
to our findings. For non-zero values of g(H), dominance
durations for D and for H patterns are inversely affected,
and once again there is an adaptation rate value at which
those two patterns produce comparable simulated dominance durations. The patterns of simulated results in
Figures 6a and 6b thus confirm that pattern-dependent
adaptation rate brings the behavior of this model in line
with empirical results. Moreover, it is worth noting that
the values of g(V) and g(H) that yield comparable
dominance durations for V and D patterns and for D and
H patterns, respectively, approximate the values of g(V)
and g(H) that reproduce the traveling wave speed results
from Experiment 1 (see Figure 4d).
To envision how collinear facilitation and patterndependent adaptation jointly influence the behavior of a
model neuron within the array representing a rival target,
we extracted the simulated activity of the 10th model
neuron (E10), representing the V pattern when paired
against the D pattern in rivalry. Figure 6c shows that
unit’s activity over time for two different values of g(V);
the value of the collinear facilitation parameter was held
constant at A ( = 3.0. The activity of this model neuron
with fast adaptation rate (black trace, for the condition g =
0.2) rises quickly upon a transition and then quickly
begins to descend to a level lower relative to the activity
of a model neuron with g = 0 (gray trace). This quick drop
in activity level in the model unit results from the time
course of adaptation within that unit, which is shown in
Figure 6d. Here it can be seen that the value of H (level of
spiking adaptation; see Equation 6) rises more quickly and
to a higher level for the non-zero value of g (black trace)
compared to adaptation when g is zero (gray trace). As a
consequence, the stronger adaptation and, hence, weaker
activity level of this unit renders it more susceptible to
abrupt state changes in the model because of its increased
vulnerability to intrinsic noise. This behavior is consistent
11
with the putative role of noise in the production of
dominance state changes during binocular rivalry (Kang
& Blake, 2010; Lankheet, 2006; Moreno-Bote et al., 2007;
Shpiro et al., 2009).
General discussion
To explain the counterintuitive results of Experiments 1
and 2, we have posited that pattern-dependent adaptation
operates together with collinear facilitation, and we
provided computational evidence showing that a model
incorporating those elements could produce rivalry
dynamics mirroring our counterintuitive results. The
incorporation of an additional model parameter governing
adaptation rate is an unusual proposal and, while not
without precedent (Rezec, Krekelberg, & Dobkins, 2004),
does place the onus on us to demonstrate that other,
potentially more parsimonious explanations can be ruled
out.
One referee of an earlier version of this paper pointed
out that the speed of apparent motion (AM) produced by
briefly presented, spatially separated visual elements
appears faster when those elements comprise collinear
stimuli compared to orthogonal ones and when those
implied speeds are very high (Georges, Seriès, Frégnac, &
Lorenceau, 2002). Our stimuli, of course, do not involve
AM, but we and others (Wilson et al., 2001) certainly do
find that traveling waves emerge faster from a suppressed
stimulus that comprises collinear contours. The speed of
those traveling waves, however, is almost an order of
magnitude slower than the optimal speed evoking the
illusory increase in speed of briefly flashed AM sequences
(Georges et al., 2002). Moreover, the present experiments
reveal that waves of dominance also emerge more rapidly
when the highly collinear rival target is the currently
dominant one that is succumbing to suppression. The
emerging wave, in other words, is being carried on a more
weakly collinear pattern, meaning that AM cannot explain
the faster traveling waves from V and D rival stimuli
regardless of carrier. In addition, to explain our result
requires assuming that the substrate determining wave
perception is independent of the eye receiving the pattern
in which the wave emerges. That seems highly unlikely
since the waves are being repetitively triggered in leftand right-eye stimuli by monocularly presented trigger
stimuli; if eye of origin did not matter, then the periodic
perturbation technique would not work in the first place.
Next we need to reconsider the other form of contextual
modulation mentioned in the Introduction section, surround suppression, which has been invoked in other
studies involving short duration binocular rivalry (Bonneh
& Sagi, 1999) and, possibly, implicated in studies of
extended duration binocular rivalry (e.g., Paffen et al.,
2006, 2004). The stimulus configuration and procedure
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Kang et al.
used in our experiments (Figure 2a) differ in important
ways from those previous studies. For one thing, our rival
stimuli were presented for many seconds, not a fraction of
a second as done by Bonneh and Sagi (1999). Moreover,
our stimuli allowed us explicitly to separate effects of
surround and collinearity; surround conditions in Paffen
et al.’s (2004) study always involved collinear contours.
Likewise, in condition b in Paffen et al. (2006) the
surround stimuli were identical in both eyes (only rival
targets differed in orientation), unlike in our present study.
Specifically, we required observers to monitor the rivalry
state of a central region flanked above and below by
additional rival stimulation identical to that within the
monitoring region. It is reasonable, therefore, to assume
that the effective contrast of the contours within this
central region could be affected by the surrounding
contours. However, as pointed out earlier, for all of these
rival configurationsVV, H, and DVthe contour orientation within the central region is identical to the contour
orientations within the flanking regions. So to the extent
that surround suppression is dependent on relative
orientations between center and surround, there should
be no differential influence on the effective contrast of the
central, monitoring region. In addition, to the extent that
surround suppression also includes a non-oriented component (e.g., Meese, Georgeson, & Baker, 2006), we would
expect the relative orientations of central and flanking
regions not to matter for the degree of suppression arising
from that component. In short, we see no way to predict
the pattern of results obtained in our two experiments
based on known properties of surround suppression. One
would have to posit the existence of orientation-selective
suppression that operates such that the surround portion
of one eye’s stimulus exerts its strongest effect on the
central region of the other eye’s stimulus. Single-unit
recordings from macaque V1 clearly contradict such a
suppression scheme (Webb, Dhruv, Solomon, Tailby, &
Lennie, 2005).
Moreover, it is striking to compare the results from
contrast matching experiments using stimulus configurations similar to those used in the present study, for those
studies indicate that surround suppression should be the
same for what we are calling “high collinearity” and “low
collinearity”. Specifically, Xing and Heeger (2001) show
evidence for surround suppression when test and surround
are equal in contrast (as they are in our study), and
furthermore, the magnitude of that suppression was the
same for a “high collinearity” configuration and a “low
collinearity” configuration. Similarly, Ejima and Takahashi
(1985) found no difference in suppression strength between
high and low collinearity conditions when the contrast of
the central portion of their test grating was equal to the
contrast within the flanking grating regions. Now it is
true that these suppressive effects can be replaced by
facilitation effects, but this typically occurs only at very
low flanker contrast levels, whereas our rival stimuli
12
always had the same contrasts throughout their entire
extent.
That said, we realize that surround suppression can play
a role in modulating the effective contrast in regions
within spatially extensive rival configurations, and we
acknowledge that future modeling work should take into
account the effects that those modulations may have in
governing contrast gain control during binocular rivalry.
To end on a speculative note, it is natural to wonder
about the possible functional significance of patterndependent adaptation, i.e., the mechanism necessary to
bring the Wilson et al. model in line with our results.
Barlow and Földiák (1989) have proposed that adaptation
serves to decorrelate neural responses, thereby promoting
more efficient neural coding. We typically think of
adaptation in terms of statistical dependency over time
(e.g., the continued presence of a given pattern), but
collinearity also represents an example of statistical
dependence, in this case over space. Indeed, statistical
analyses of the co-occurrence of edge elements within
natural visual scenes reveal that neighboring collinear
contours abound (Geisler, Perry, Super, & Gallogly,
2001). To the extent that adaptation operates to promote
greater statistical independence within neural elements
responsive to collinear contours, it stands to reason that
strong spatial dependency among image contours would
affect adaptation. In addition, a rate-dependent parameter
on that adaptation process would insure that optimization
happens fastest for those highly likely configurations.
Conclusion
A number of stimulus factors influence perceptual
alternations during binocular rivalry, including contrast,
spatial frequency, orientation, motion, stimulus complexity,
and top-down influences including attention (see Blake &
O’Shea, 2009, for an overview of those factors, and Kang,
2009, for a detailed explication of the spatiotemporal properties of rival stimulation). The present results, including
model simulations, advance our understanding of how the
pattern features of a rival stimulus influence spatial interactions engaged by those features, thereby affecting qualitatively distinct aspects of binocular rivalry dynamics.
Specifically, we have shown how contour collinearity in
binocular rivalry influences two aspects of rivalry dynamics.
(1) Traveling waves of dominance during state transitions
occur on a relatively short time scale (i.e., a fraction of a
second). (2) Perceptual dominance associated with spontaneous perceptual alternations occurs on a time scale up to
an order of magnitude longer than that associated with
traveling waves. Both of these aspects of rivalry are
modulated by collinearity but in non-obvious ways as we
have learned. To account for these counterintuitive results,
we propose that pattern-dependent adaptation interacts with
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
collinear facilitation, and we have shown that this interaction can be accommodated within an extant model based
on reciprocal inhibition among a network of units interconnected so as to promote varying degrees of collinear
facilitation.
13
dominance state. To simulate spontaneous changes in
dominance, we added a noise term J to Equation 2, such
that the value P is given by
PAj ¼ SAj j )I
Appendix A
Latency of traveling waves
For statistical analyses, we estimated the latency of the
waves from the switch function. The latency reflects the
time at which perceptual states of both rival stimuli are
equally likely. If traveling waves occur in response to all
triggers, the latency would correspond to the time at
which the mean perceptual state M(t) equals 0.5. However, if the periods of perceptual dominance differ for the
two rival stimuli, and thus, one rival stimulus remains
dominant for longer periods of time, the mean perceptual
state will change accordingly over the entire trigger phase,
implying that the latency identified based on the mean
perceptual state M(t) = 0.5 is not adequate. For this
reason, a general procedure was devised in which the
switch function was modeled by a sigmoid function, ! +
"/(1 + exp(j(E j ET)/A) in which ! equals M(0), the
mean perceptual state at trigger phase equals 0; " equals
M(:) j M(0), the difference between the mean perceptual
states between the two trigger onsets; ET is the threshold
level of the trigger phase in the sigmoid function; and A is
the growth rate. Latencies of the traveling waves were
identified individually by obtaining values of the threshold
trigger phase ET and transforming those values to the
latency values in milliseconds by using the trigger period.
Simulation details: Traveling waves
We conducted simulations using Matlab (Mathworks,
MA) running on an Intel-based Macintosh computer with
OS 10.5. The Euler method, a first-order numerical
procedure for solving ordinary differential equations, was
used for numerical integration with step size of 0.25 tu for
20,000 tu. Trigger period was set to 5000 tu starting from
2500 tu producing three waves associated with the carrier
presented to the left eye and another three waves
associated with the carrier presented to the right eye.
Wave fronts shown in Figures 4b–4d were obtained by
averaging these three wave fronts.
Simulation details: Spontaneous perceptual
alternations
The procedure used in Experiment 2 did not involve
external, periodic perturbations to produce changes in
N
X
k¼1
8Bjk IBk þ )E
N
X
k¼1
(Ajk EAk þ SAj J:
ðA1Þ
Noise values among the array of neurons are uncorrelated and are defined by a Gaussian distribution whose
mean equals 0 and standard deviation equals 0.2. The
amplitude of noise is scaled with the strength of the
stimulus and noise values are updated every 25 tu. A
single simulation is conducted for 240,000 tu and repeated
four times for each condition (where a condition refers to
a given combination of rival stimuli, extent of recurrent
excitation, and pattern-dependent adaptation level). Each
simulation required considerable time to complete, and to
complete the simulations we reduced the size of the
neuronal array from 40 to 20; this reduction in array size
does not affect the results. The dominance duration for a
given stimulus was defined as the duration during which
all of the middle five model neurons (i.e., the units
representing the monitoring region) were all specifying
dominance of the same stimulus.
Acknowledgments
This work was supported by grants from the U.S.
National Institutes of Health (R01-EY013358, R01EY016752, and P30-EY008126), and RB and SL were
also supported by the WCU program administered by the
National Research Foundation of Korea and funded by the
Korean Ministry of Education, Science, and Technology
(R32-10142).
Commercial relationships: none.
Corresponding authors: Min-Suk Kang and Randolph Blake.
Emails: [email protected]; [email protected]
Address: Department of Psychology, Vanderbilt University, 111 21st Ave. South, 301 Wilson Hall, Nashville,
TN 37240, USA.
Footnotes
1
To avoid confusion associated with the dual use of the
term “suppression,” we will use the term “surround
suppression” when referring to that form of contextual
modulation and the term “suppression” on its own to refer
to perceptual suppression during rivalry.
2
Portions of data shown in Figure 3a (solid lines) are
reproduced from a previous paper (Kang et al., 2009).
Journal of Vision (2010) 10(11):3, 1–15
Kang et al.
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