Vision Research 51 (2011) 521–528

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Vision Research
journal homepage: www.elsevier.com/locate/visres

Dynamical systems modeling of Continuous Flash Suppression
Daisuke Shimaoka ⇑, Kunihiko Kaneko
Department of Basic Science, The University of Tokyo, Japana r t i c l e

i n f oArticle history:
Received 31 March 2010
Received in revised form 19 January 2011
Available online 4 February 2011
Keywords:
Binocular rivalry
Adaptation
Neural modelinga b s t r a c t
Continuous Flash Suppression (CFS) is a technique in which a stationary image in one eye can be reliably
suppressed by rapid presentation of different ﬂashing images in the other. In this paper we address why
ﬂashing stimuli modulate the visibility of the stimuli. We determine, in particular, which type of neuralnetwork is sufﬁcient for the modulation of the dominance duration, assuming that elemental units areendowed with reciprocal inhibition and adaptation. We show that the model introduced by Wilson(2007) reproduces ﬂash suppression, which is considered to be involved in CFS, but does not reproduce
CFS. We then extend the model by including a stimulus feature dimension. With this extension, we foundthat the model accounts for the modulation of visibility observed in CFS. In addition, this model captured
some deﬁning characteristics of CFS such as dependence on ﬂash interval and the depth of suppression.
Our ﬁndings suggest that a network with inhibition and adaptation including feature dimension provides
a crucial mechanism for the modulation of the dominance duration in CFS.
Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction
Binocular rivalry has shed light on the dynamics of visualawareness and its underlying neural basis. In the study of this phenomenon, a method termed Continuous Flash Suppression (CFS)
has been recently devised (Tsuchiya & Koch, 2005). This techniquedelivers continuously ﬂashing images to one eye while a stationary
image is presented to the other eye. A remarkable phenomenon in
CFS is that it elongates the dominance duration (DD) of the perceptof the ﬂashing stimulus in CFS regime compared to the DD in binocular rivalry (BR). At the same time, the ﬂashing stimulus shortens the DD of the percept of the stationary stimulus compared to
the DD in BR. As a result, the ﬂashing stimulus is perceived for alonger period than the stationary stimulus. The ﬂashing images
thereby modify the observer’s visual perception. While Tsuchiya
and Koch used fragmented and contour-rich visual stimuli in their
CFS experiment, it is known that features of stimulus patterns suchas color and high-dimensional texture are not critical for the suppression of a stationary image. Indeed, the suppression of a stationary image is induced even with the simple monochrome stimulus
set as adopted by Gilroy and Blake (2005). Rather, the ﬂash interval

Abbreviations: BR, binocular rivalry; FS, ﬂash suppression; CFS, Continuous
Flash Suppression; DD, dominance duration; sDD, dominance duration of thepercept of the stationary stimulus; fDD, dominance duration of the percept of theﬂashing stimulus.
⇑ Corresponding author. Address: Department of Basic Science, The University ofTokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan. Fax: +81 3 5454 6732.
E-mail address: shimaoka@complex.c.u-tokyo.ac.jp (D. Shimaoka).
0042-6989/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.visres.2011.01.009

profoundly modulates the perceptual dominance, as reported in
Tsuchiya and Koch (2005, Supplementary info).
CFS is now used as an established technique in psychophysicaland fMRI experiments for its strong effect on perceptual dominance(Fang & He, 2005; Gilroy & Blake, 2005; Jiang, Costello, Fang, Huang,& He, 2006; Jiang, Costello, & He, 2007; Pasley, Mayes, & Schultz,
2004), especially to investigate the relationship between subjective
perception and retinal aftereffect (Tsuchiya & Koch, 2005). However,few studies (Tsuchiya & Koch, 2005; Tsuchiya, Koch, Gilroy, & Blake,2006) have addressed why stimulus ﬂashing renders the stationary
image invisible. It is important to elucidate underlying neuralmechanisms of how the ﬂashing stimulus modulates perceptual
dominance, for determination of the relationships between visualawareness and neural dynamics. Since the retinal aftereffect isprimarily occurring in the retina, whereas the visual rivalry takes
place in the cortex, the neural mechanisms underlying these two
phenomena are distinct, and should be separately addressed.
In the study of multistable perception, several studies have proposed neural mechanisms which account for spontaneous rivalry
alternation. Most of models have employed both reciprocal inhibition between competing neural groups and adaptive suppressionof neural activity over time. By means of these mechanisms, agroup of neurons maintains transient dominance until the activityof the competing group can no longer be inhibited, leading to aswitch of perceptual dominance. These mechanisms are now supported in part by psychophysical and fMRI studies (Lankheet,
2006; Tong & Engel, 2001). Lankheet (2006) succeeded in extracting adaptation and mutual inhibition components in rivalry, while
Tong and Engel (2001) provided evidence for inhibition of activity
in V1 monocular neurons during rivalry.

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D. Shimaoka, K. Kaneko / Vision Research 51 (2011) 521–528

Based on these observations, we hypothesize that adaptation
and reciprocal inhibition are involved in CFS, especially in the modulation of DD by the ﬂashing stimuli. By using reciprocal inhibition
and adaptation as composition elements, we investigate the requisite neural-network architecture of CFS. Here we adopt a simple
model, because our focus is on capturing the essential neuralmechanism underlying CFS. Speciﬁcally, we examine a minimalmodel introduced by Wilson (2007) that implements reciprocal
inhibition and self-adaptation.
First, we examine the response of Wilson model to a single ﬂashof stimulus to test whether the model reproduces ﬂash suppression (FS), which is thought to be required in CFS. FS is the perceptual suppression of a monocularly presented image, upon ﬂashingof a new stimulus to the contralateral eye (Wolfe, 1984). We subsequently examine the model’s response to a continuously ﬂashing
input as employed in CFS experiments and show that the responseis opposite to that observed in CFS experiments. To reproduce theexperimental results, we extend the original model so that it candeal with feature dimensions of visual stimuli. In the extended
model, the ﬂashing stimulus remarkably elongates its DD andshortens that of the stationary image. In addition, we show that
this extension also reproduces the dependence of DD on the ﬂash
interval, a unique characteristic of CFS. Finally, we investigate amodel with more units, conﬁrming that the suppression of CFS is
far stronger than that of FS and BR.
2. Model2.1. Wilson model of binocular rivalry
Wilson (2007) introduced one of the simplest models of BR that
implemented competitive inhibition and self-adaptation. Despite
its simplicity, the model can reproduce Levelt’s second and fourthlaw (Levelt, 1966). This model is composed of four valuables,
namely EL, ER, HL and HR. EL (ER) represents the activity level of neurons driven by the left (right) monocular inputs L(t) (R(t)), whereas
HL (HR) stands for a slow hyperpolarizing current injected into EL(ER). The temporal evolution of the four valuables are described assdEL¼ EL þ M  maxðLðtÞ  aER þ eEL  gHL ; 0Þ;
dt

sH
s

dHL
¼ HL þ EL ;dt

dER
¼ ER þ M  maxðRðtÞ  aEL þ eER  gHR ; 0Þ;
dt

sH

dHR
¼ HR þ ER :dt

The meanings of the parameters are fully explained in Wilson(2007), Table 1. Unless otherwise noted, the system parameters
are set as dt = 0.01 ms, s = 20 ms, sH = 900 ms, and R = 1. Most of
the other parameters are chosen as supplied in Table 1 of Wilson(2007).
We brieﬂy review basic behaviors of the model with constantstimuli (L(t) = L, R(t) = R). The system has two ﬁxed points andtheir stability depends on the parameter values (Wilson, 2007).
Fig. 1 illustrates how the model’s behavior depends on values ofthe parameters L/R and a. The boundaries of the regions in theparameter space depicted in the ﬁgure are represented by the following inequalities (The deviation of these inequalities and fulldescription of the behavioral patterns are found in Wilson (2007)),



1
R
;
a> gþ e
M
L

ð1Þ



1
L
a> gþ e;MR

ð1 equationÞ

0

1
 e;
M


1
L
e
;
a>
M
R
a<


a>


1
R
e
:
M
L

ð2Þ

ð3Þ

0

ð3 equationÞ

Since switching of perceptual dominance in binocular rivalry ishighly stochastic even under well-controlled experimental
conditions, effects of noise should be taken into account. Indeed,
under the presence of noise, Wilson model reproduces a gamma
distribution of DD, which is a typical characteristic of binocular riv-

Fig. 1. Basic behavioral patterns of Wilson model in response to stationary inputs in L/R-a parameter space. The time course of each behavioral pattern is demonstrated in theleft four ﬁgures. (i) ER- or EL-ﬁxed point (winner take all) at which ER or EL continues ﬁring. (ii) Both-ﬁring ﬁxed point (fusion) at which both ER and EL continue ﬁring. (iii) Limit
cycle (binocular rivalry) in which dominance switches when HR or HL reaches close to its asymptote. (iv) The other limit cycle in which dominance switches long before HR or
HL reaches its asymptote.

D. Shimaoka, K. Kaneko / Vision Research 51 (2011) 521–528

alry. The importance of noise in binocular rivalry is further discussed in Moreno-Bote, Rinzel, and Rubin (2007). For these reasons, we examine the model with weak Gaussian white noise in
the following simulations.
3. Results
3.1. Time-lagged input to Wilson model switches dominance over wideparameter region
Before investigating the model response to repetitive ﬂash, we
inquire whether a single ﬂash of input switches the dominanceof neural activity in Wilson model. For this purpose, we ﬁrst present a monocular (stationary) stimulus to one eye of the model andthen give another monocular stimulus to the other eye. The interval between the two stimulus presentations is 2000 ms, which is
sufﬁcient for the slow variable, i.e. HR or HL, to reach its asymptote.
We examine whether the response of neural activity to the second
stimulus is dominant compared to that to the ﬁrst stimulus. In particular, the parameter region is examined in which the dominanceswitches immediately after the second stimulus.
Fig. 2 illustrates the parameter region in which the model behaves like FS (without the loss of generality, we presented the ﬁrst
stimulus to the right eye and the second stimuli to the left). The FSlike behavior is observed over a wide area including where binocular rivalry takes place in response to a normal stationary stimulus.
The parameter region of FS-like behavior can be captured by thefollowing arguments obtained with the use of the adiabatic approximation before and after the time-lagged input. If a > ðg þ M1 Þ RL is satisﬁed, the ER-ﬁxed point vanishes upon the delivery of the second
input to EL. Once the system leaves the vanished ﬁxed point, it moves
toward either of the newly appearing ﬁxed points (EL, ER) = (ML, 0)2

2

Lð1þMgÞ MLð1þMgÞaM R
and ðMRaM;
Þ, depending on the parameter values.
ð1a2 M2 Þð1þMgÞ ð1a2 M 2 Þð1þMgÞ
1R
is satisﬁed, the system is attracted to the EL-ﬁxed
If a > Mð1þMgÞ
L
1
R
1
and 1þMg
< RL are satisﬁed, the system
point. However, if a < Mð1þMgÞ
L

is attracted to the bi-ﬁring ﬁxed point where EL is larger than ER.
3.2. Repetitive ﬂashing input shortens fDD of Wilson model
We next investigate whether repetitive ﬂashing of input modiﬁes DD as observed in experiments of CFS. In CFS experiments of

523

Tsuchiya and Koch (2005), fragmented, contour-rich patterns(known as Mondrian pattern) were exploited as ﬂashing stimuli.
However, rather simple stimuli of a monochrome pinwheel andits contrast reversed version induce basically the same featuresof CFS, including suppression of visibility of the stationary stimulus
and reduction of after-image intensity for the stationary stimulus(Gilroy & Blake, 2005). Since such ﬂashing inputs have distinct
patterns at every ﬂash, optimality of pattern for eliciting ﬁring bya group of neurons changes from pattern to pattern. Thus, after aﬂash at a given time elicits the maximum response of a group of
neurons, the subsequent ﬂash often yields little or no response.
Therefore, focusing on the temporal structure of the stimulus used,
we set the external stimuli (L(t) and R(t)) so that the stimuli to one
eye remain unchanged while those to the other eye continuechanging between on and off.
To examine the response of the model to the ﬂashing input, theinterval of the ﬂash should be taken into account, since the previous experiment (Tsuchiya & Koch, 2005) reported that the ﬂash
interval substantially affected DD. Fig. 3a is a graph adopted from
Supplementary info of Tsuchiya and Koch (2005) showing that the
DD of the ﬂashing input is longest when the stimulus ﬂash interval
is 80–320 ms, and that the ﬂashing input does not otherwise increase the DD. To replicate this experimental condition, Wilson
model was presented with ﬂashing input for 60 s in four different
trials. We set stimulus strength for the ﬂashing input weaker thanthat for the stationary input, and measured the mean and total DDof ﬂashing and stationary stimuli.
Fig. 3b shows how ﬂash interval modulates DD of ﬂashing andstationary stimuli, which are hereafter referred to as fDD andsDD, respectively. In contrast to Fig. 3a, fDD (blue lines) is always
shorter than sDD (red lines). Although we ran the simulations overa wide range of the L/R-a parameter values, we did not observe themodulation of DD observed in CFS experiments. The same tendency is observed when we employ other input patterns such as
sinusoidal wave instead of the rectangular input. These results
indicate that, in Wilson model, continuous ﬂash does not lead tothe suppression of visibility of a stationary stimulus. Note that thisfailure is not surprising given the fact the model deals with only
the subset of neurons that responds to a part of the ﬂashing input.
In the simulations we frequently observed that dominanceswitches from the ﬂashing stimuli to the stationary stimulus during a stimulus-off period of the ﬂashing input sequence. Thisobservation suggests that fDD is not so large because the unit thatreceives the ﬂashing input cannot maintain dominance during thestimuli-off period.
3.3. Extension of Wilson model accounts for visibility modulation by
the ﬂashing stimulus

Fig. 2. The responses of Wilson model to time-lagged input in the L/R-a parameter
space. The region where Wilson model changes dominance from right (unit withthe ﬁrst stimulus) to left (unit with the second stimulus) within 100 ms is marked
with gray. Transient switches in the dominance are included to be FS without
reference to the subsequent behavior. Noise = 1.0  106. With a deterministic
simulation, nearly the same diagram is obtained.

In this section we add a slight extension of Wilson model. To
check whether the extension of the model explains the corresponding experiment, we particularly investigate dependence ofDD on the ﬂash interval. To verify that the ﬂashing input modulates
DD over a wide parameter region, the extended model is examined
under the L/R-a parameter domain.
We have thus far focused on the temporal structure of the stimuli in CFS experiments, ignoring the effect of stimulus feature. This
stimulus feature, however, may play a crucial role. For example,
Mondrian ﬂashing patterns, which are often employed to elicit
CFS, have a fragmented, non-uniform spatial structure. One way
to introduce stimulus feature in Wilson’s framework is to add
other units that respond to different stimuli than that preferredby existing units. As a simple extension of the model, the case of
two units in each eye is depicted in Fig. 4a.
This extension leads to the following differential equations

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D. Shimaoka, K. Kaneko / Vision Research 51 (2011) 521–528

Fig. 3. Comparison of DDs between (a) data from a psychophysical experiment (Tsuchiya and Koch (2005), Supplementary note) and (b) the numerical results for Wilson
model. In each ﬁgure, the red lines represent sDD while the blue lines represent fDD. The rightmost two points are for the case of binocular rivalry. In (a), four subjects tracked
the visibility of a stationary stimulus during 1 min in four different trials viewing when any part of the stationary stimulus was visible (red) or invisible at all (blue). Error bars
represent standard error across the trials. In (b), stimulus strength for the left unit is 0.9, while that for the right unit is 1.0. Error bars represent standard deviation. The
system parameters are: a = 4.0, g = 3.5, e = 0.0, M = 1.0 and Noise = 5.0  107. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the
web version of this article.)dEL1¼ EL1 þ M  maxðL1 ðtÞ  aER1  a2 ER2  gHL1 ; 0Þ;
dt
dHL1
sH
¼ HL1 þ EL1 :
dt

s

ð4Þ

(EL2, HL2, ER1, HR1, ER2 and HR2 are deﬁned in the same fashion.).
The point of the extension is that each unit on one side receives
different input (such as L1 and L2). Thereby the extended model canrepresent each unit as tuned to different stimulus features such as
spatial phase, orientation, and color. For example, when external
input (Li, Ri) stands for sinusoidal images of different spatial phase,the corresponding unit can be interpreted as a group of V1 simple
cells in which the neurons respond to particular spatial phase(Fig. 4a).
In this model, each unit receives inhibitory synaptic input withstrength a2, in addition to the input with strength a in the originalmodel. In this study, we adopt a2 = a for simplicity. The effect of a2
on DD will be discussed in the subsequent section.
To examine the effect of stimulus ﬂashing in our extended model, we set the input to one eye as stationary and that to the othereye as anti-phase ﬂashing inputs as illustrated in Fig. 4c. Each of
the ﬂashing inputs has the form of an on–off rectangular wave,
with constant ﬂash intervals and strength. We test whether fDD
is longer than the corresponding DD in BR, based on the deﬁnition
that the sum of the activities of the two units for one eye stands forthe activity that determines the dominance of the eye.
Results of simulation for a typical parameter set are demonstrated in Fig. 4b and c. The ﬁgure shows that the activities withﬂashing inputs oscillate at the frequency of the external inputs(10 Hz). The sum of these activities maintains their dominance(fDD) for twice as long as DD in BR (blue, right most point). At
the same time, sDD decreases to less than half of DD in BR (red,right most point). Consequently, fDD is much larger than sDD, eventhough the stimulus strength of a ﬂashing stimulus (0.9) is set
weaker than that of the stationary stimulus (1.0).
We subsequently examine the dependence of DD on ﬂash
interval in the extended model. The model was presented withﬂashing input for 60 s in four different trials to replicate theexperimental conditions, while the mean and total DDs were
measured. Fig. 5 shows results of simulation for a typical parameter set. Dependence of DD on the ﬂash interval is characterizedby two features. First, when the ﬂash interval is less than 50 ms,
the total and mean fDD (blue lines) are nearly four times larger
than DD in BR. At the same time, the mean sDD (red line) is held
constant below DD in BR. Consequently, the fDD is longer thanthat of sDD, even though the stimulus strength of the ﬂashing input is weaker than that of the stationary input. Second, when
ﬂash interval increases more than 50 ms, fDD decreases to beclose to DD in BR designated at the right edge. Similarly, sDD increases to approach DD in BR.
These two effects of ﬂashing input on DD are more pronounced,
when the strength of inhibition between the diagonal units a2 is
set larger. If a2 is small enough, the effects are diminished since
the system is reduced to the original Wilson model. To conclude,
the modulation of DD as a function of ﬂash interval is captured
by a simple and minimal extension to Wilson model. The only

D. Shimaoka, K. Kaneko / Vision Research 51 (2011) 521–528

525

Fig. 6. (a) A schematic diagram of the ﬂashing and the stationary equivalent input.
(b) Mean DDs of the extended model plotted as a function of ﬂash interval. Greenline represents the approximate analytic solution (Eq. (5)), while blue circles
represent mean DDs during 1 min in 1 trial of simulation (ﬂash interval = 100 ms,
no noise condition). The stimulus strengths of ﬂashing stimuli are set to: L1,
L2 = 0.8 or 1.0, while those of the stationary equivalent are L1 = L2 = 0.7256
according to Eq. (6) provided in Supplementary data. In both cases, the systemparameters are: a = a2 = 2.4, g = 3.0, e = e2 = 0.0, M = 1.0. (For interpretation of thereferences to color in this ﬁgure legend, the reader is referred to the web version ofthis article.)

Fig. 4. (a) Architecture of the extended model. Two units on one side represent
position in feature space. (b and c) Time courses of response in the extended modelto (a) stationary inputs to both sides and (b) stationary input to one side andﬂashing stimuli to the other side. Note that ER2 and HR2 are almost always zerothroughout. The system parameters are: a = a2 = 4.0, g = 3.5, e = 0.0, M = 1.0, e2 = 0.0,
noise = 5.0  107 and ﬂash interval = 100 ms.

discrepancy from the experiment lies in the condition of verysmall ﬂash intervals (10 ms), which will be considered in
Discussion.
We additionally test whether DD is modulated by the ﬂashing
input over a wide range of parameter values. The results indicatethat the stimulus ﬂashing effect is valid all over the BR phase.
The stimulus ﬂashing effect for the ﬂashing input is particularly
strong around the upper right or the upper left boundary for fDD
or sDD, respectively.

Fig. 5. Total and mean DDs for the extended model are plotted as a function of ﬂash interval. In each ﬁgure, the red lines represent sDD while the blue lines represent fDD. The
rightmost two points are for the case of binocular rivalry. Numerical results are obtained from simulation for 60 s  4 trials. Error bars represent standard deviation across thetrials. Stimulus strength for the left units is 0.9, while that for the right units is 1.0. The system parameters are: a = a2 = 4.0, g = 3.5, e = e2 = 0.0, M = 1.0 and noise = 5.0  107.
(For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)526

D. Shimaoka, K. Kaneko / Vision Research 51 (2011) 521–528

In summary, we introduce a minimal extension to Wilson
model that includes stimulus feature dimension. In this model,
the ﬂashing stimulus substantially elongates fDD and shortens
sDD, which are key characteristics of CFS. In addition, the stimulus
ﬂashing effect is enhanced as the ﬂash interval decreases. Thiseffect is observed all over the BR region, and is further enhancednear the borders of the region.
3.4. How stimulus ﬂashing affects dominance duration in the extendedmodel
We consider the mechanisms of CFS in the previous section. For
this purpose, an analytic solution for fDD is approximated as Eq.
(5). It is derived in the parameter region where normal binocular
rivalry takes place (marked light green in Fig. 1). The detailed derivation is found in Supplementary material.




1
2 þ Mg Lþ g  ða þ a2 ÞT L ¼ sH lnðgÞ  ln
M
1 þ Mg R

ð5Þ

The resulting DD derived from this formula is plotted as thegreen line in Fig. 6b, with the results of simulation (when ﬂash
interval is 100 ms) as blue circles. The approximated solution isin good agreement with the simulation result. This solution is comparable to DD in BR, which was obtained in Wilson (2007) as




1
L
þga
:
T L ¼ sH lnðgÞ  ln
M
R
In these equations, the last term in the latter logarithm stands forthe contribution of external inputs. One can interpret the reason
for the prolongation of DD by ﬂashing stimulus, as the effective inL
crease of input strength from a RL to ða þ a2 Þ 2þMg. Note that the1þMg Requation conﬁrms our observation from simulation that the larger
a2 is, the longer is the DD.
3.5. Dependence of depth of suppression of CFS on the number of units
The extended model examined above is of course too simpliﬁed
to represent a spatially extended system. To consider more realistic
cases, we investigate a model with more than two units. The modelis represented as:NX
dEs Li ¼ ELi þ M  max 0; Li ðtÞ  gHi  ai ERj
dtj!

The equations for HLi and HRi are identical to those of Eq. (4). For
simplicity, we set ai = a for any i. As in the two-unit model, every
unit in this model has different response selectivity. We have considered that the local feature of each ﬂash used in CFS experimentis composed of only one feature (e.g., only 180° of spatial phase instead of the mixture of the 0° and 180°). In this situation the input
is applied to only one unit at one time. To model this situation, we
adopted the circular stimulation manner, i.e., the ﬂashing input
travels over the N units one by one. At a given time, the on-input
is thereby applied only to one of the units and the off-inputs areapplied to the other units. Note also that the simulation result fromthis manner gives a conservative estimate of the effect of stimulus
ﬂashing compared to other regimes such as on–off switching.
With this model, we investigate the ‘‘depth of suppression’’associated with BR, FS, and CFS. This topic was psychophysicallyinvestigated to elucidate the source of suppression in CFS (Tsuchiya et al., 2006). First, the effect of the ﬂashing input on DD ismeasured by the ratio of DD in CFS and DD in BR. We computethe dependence of the ratio on the number of units in the extended
model (Fig. 7a). In this ﬁgure, the black and gray lines indicate
these ratios against the number of units exposed to ﬂashing stimuli
and those exposed to stationary stimulus, respectively. As thenumber of units increases, the former increases while the latter decreases, suggesting that the ﬂashing effect on DD is enhanced.
Subsequently, the depth of suppression of BR, FS and CFS isinvestigated by simulating the probe detection task, which has
been intensively employed to estimate the depth of suppressionin BR (Blake & Camisa, 1979; Fox & Check, 1972; Nguyen, Freeman,& Alais, 2003; Nguyen, Freeman, & Wenderoth, 2001; Norman,
Norman, & Bilotta, 2000; Watanabe, Paik, & Blake, 2004). In thistask, transient monocular stimulus (‘‘probe’’) is brieﬂy superimposed on a rival stimulus. The depth of suppression is estimatedas the minimal probe strength that makes the observer visible ofit. In our model framework, being visible of stimuli is supposed
to correspond to getting dominance of neural activity over units
in the opposite eye side. The minimal probe strength that makesa group of neurons get dominance would thereby correspond to
the depth of suppression. Fig. 7b shows the dependence ofFig. 7. (a) Ratios of the mean DD in CFS compared to the mean DD in BR, as a function of number of units in the extended model. Flash interval = 100 ms. Black and gray lines
indicate the ratios of mean fDD and sDD, respectively. (b) The threshold probe strengths for BR (gray, square), FS (black dot, star) and CFS (black, circle), as a function of thenumber of units in the extended model. The threshold probe strength was obtained through numerical simulation of probe detection task. The strength of the test stimuli for
BR, FS and CFS was held constant to be 1.0 during a trial, while the strength of the probe varied as probe = b + hexp(t2/r2), where a baseline stimulus strength b = 0.1 andstandard deviation r = 100 ms. During the smooth change of the probe, occurrence of dominance switch is detected. For each probe amplitude, we conduct 100 trials andcount the number of the dominance switching from the test stimuli to the probe stimuli. The fraction of the dominance switches consequently produces a sigmoid curve
against the probe amplitude. From this curve, we estimate the threshold amplitude at which 50% of trials exhibit dominance switching by applying logistic regression. For thecase of FS, the interval between peak time of the probe and the onset of the lagged-stimuli was set to be 150 ms, which was most effective to suppress the probe stimuli. For
the case of CFS, the ﬂashing interval was set to be 100 ms in this ﬁgure. Simulation with the ﬂash interval of 1.0 ms yielded the basically the same result as 100 ms.
dt = 0.25 ms. In both cases, the system parameters are: ai = 2.4, g = 3.0, e = 0.0, M = 1.0 and noise = 5.0  107.

D. Shimaoka, K. Kaneko / Vision Research 51 (2011) 521–528

threshold probe strength on the number of units. When N is lessthan 10, the threshold for FS is higher than that for CFS, probably
due to differences in timing of the probe. In CFS, after the high,transient response of the ﬂashed units, the units in the opposite
eye are stimulated with the probe stimulus. The probe can therefore switch the dominance. On the other hand in FS, the probe
comes during the transient response of the ﬂashed unit.
When N is sufﬁciently large, however, the threshold is highest
for CFS, followed by FS and weakest for BR, consistent with theresults of the probe detection test by Tsuchiya et al. (2006). Herewe qualitatively explain the mechanisms of the phenomena. As
explained in the previous paragraph, whether the probe switches
the neural dominance depends on amplitude of the transient activity driven by the rival stimulus. The amplitude of the transient
response thus gives an approximation of the threshold probe
strength. With this approximation, one can get a picture of why
the depth of suppression increases in CFS as follows. The transient
response is strong when the units are not sufﬁciently adapted tothe external stimuli, i.e., Hs are low. When N is large, each unit receives external input for short period, thereby the unit is lessadapted to the external stimuli. This is why threshold probe
strength increases with N. When N is increased further, Hs cannot
be lower than 0 by deﬁnition. Probe strength in CFS therefore saturates at a large N. Next, let us consider the other extreme, when N
is less than 10. In CFS, the probe comes when the ﬂashed units arein steady-state after the transient peak response. The probe therefore easily switches the dominance. In FS, however, the probe stimulus comes during the transient response of the ﬂashed unit. This is
why probe strength against CFS is smaller than that against FS.

4. Discussion
To understand CFS at a neural circuit level, we studied a modelwith adaptation and mutual inhibition, revealing that it accounts
for the modulation of DD observed in psychophysical experimentsof CFS. Other aspects of CFS, such as the effect on afterimage, needfurther investigation.
One model for CFS was previously proposed by Tsuchiya andKoch (2005), in which both binocular rivalry and ﬂash suppression
were considered. Compared to their model, our extended model
describes neural circuits underlying CFS explicitly. In addition,
our model is based solely on mechanisms of adaptation and inhibition without ad hoc assumptions regarding relationships between
stimulus strength and a ﬂash interval.
One may consider CFS results from bottom-up attention elicitedby high saliency of the ﬂashing stimuli. Attentional effects havebeen generally considered to arise from the network with a hierarchical structure. However, the modulation of DD was successfully
reproduced in our model without including any hierarchical network architecture. Our ﬁndings imply that hierarchical mechanisms are not necessarily required for the modulation ofvisibility in CFS.
Our results indicate that the effect of the stimulus ﬂashing on
DD depends on the strength of the ﬂash. If the ﬂashing stimulus
is weaker than the stationary stimulus to the other eye, sDD ismarkedly shortened by the ﬂashing stimulus. If the ﬂashing stimulus is stronger than the stationary stimulus, however, the ﬂashingstimulus signiﬁcantly prolongs fDD. Our study thereby suggeststhat the modulation of DD is easily detected when the stimulus
strengths are not balanced between the two eyes.
In exploring the model composed of multiple units, we foundthat the depth of suppression induced by repetitive ﬂashing is considerably stronger than those by a single ﬂash and stationary stimulus when the number of units is sufﬁciently large. This strong
suppression is probably mediated by maintenance of dominance527

without sufﬁcient adaptation of the units. The simulation thatinvestigates the depth of suppression (Fig. 7b) yields results consistent with the psychophysical experiment by Tsuchiya et al. (2006).
Using a probe detection task, they measured the suppressive effects of ﬂashing inputs on stimuli given to the other eye, in comparison with those associated with binocular rivalry and ﬂash
suppression, and found that the threshold contrast for CFS is muchhigher than those for FS and BR.
Although our model accounts for the principal features of CFS, a
discrepancy still remains between the model and psychophysicalexperiments. In the experiment by Tsuchiya and Koch (2005),
stimulus ﬂashing at extremely high frequencies no longer suppressed the stationary stimulus. Speciﬁcally, the DD for extremely
short ﬂash intervals (10 ms) were nearly equal to those associated with a stationary stimulus (see Fig. 3a, leftmost points). In
our model, however, the fDD is larger than the sDD for extremely
short ﬂash intervals. Our study modeled only groups of neurons
in cortex where neural rivalry takes place. However, when a pathway to the cortex is considered, it is probable that external stimuli
of high-frequency do not reach the cortex. Therefore, increasing
ﬂash frequency probably decreases neural excitability, resulting
in a decrease in DD. Some physiological studies of cat and monkey
LGN and early visual cortex (Hamilton, Albrecht, & Geisler, 1989;
Hawken, Shapley, & Grosof, 1996; Movshon, Thompson, & Tolhurst,
1978) support this picture. In these studies, majority of examinedneurons exhibited high-frequency cut-off. In addition, the cut-off
frequency decreased when the recording cortical area was far fromthe retina.
The possibility of extending models for BR to stimulus feature
space has been proposed elsewhere. For example, Freeman(2005) proposed a model with four parallel visual channels, two
driven by the left eye and the others by the right, to account for
the timing of dominance intervals in distributed neural processing
in rivalry. Unlike this study, our purpose here was to determine theessential requirements for accounting for CFS. Our study demonstrates the importance of including stimulus feature space inexamination of CFS for the ﬁrst time.
Acknowledgments
We thank S. Ishihara and A. Nakajima for helpful discussion and
N. Tsuchiya and J. Ito for useful comments on an earlier version of
the manuscript. This work was supported in part by Grants-in-Aid
for scientiﬁc research (No. 21120004) from MEXT Japan.
Appendix A. Supplementary material
Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.visres.2011.01.009.
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