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Visual backward masking: Modeling spatial
and temporal aspects
Frouke Hermens
1
and Udo Ernst
2
1
Laboratory of Psychophysics, Brain Mind Institutem, École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
2
Group for Neural Theory, Département d’´Etudes Cognitives (DEC), École Normale, Supérieure (ENS), Paris, France
Keywords
visual backward masking
2007 • volume 3 • no 1-2 • 93-105
Received 31.08.2006
Accepted 27.12.2006
Correspondence concerning this article should be addressed
to Dr. Frouke Hermens, Laboratory of Psychophysics, Brain
Mind Institute, École Polytechnique Fédérale de Lausanne
(EPFL), Station 15, CH-1015 Lausanne, Switzerland. E-mail:
frouke.hermens@ep.ch
VISUAL BACKWARD MASKING:
MODELING SPATIAL AND
TEMPORAL ASPECTS
In visual backward masking, a target stimulus is fol-
lowed by a mask, which impairs performance on the
target. Although visual masking is often used as a tool
in cognitive and behavioral sciences, its underlying
mechanisms are still not well understood. The focus
of masking research has been on understanding how
it is possible that for some combinations of target and
mask, a delay of the mask yields stronger masking
than having the mask immediately follow the target.
This phenomenon is known as ‘B-type masking’ or ‘U-
shape masking,’ of which the latter refers to the shape
of the curve linking stimulus onset asynchrony (SOA)
between the target and the mask to performance.
Explanations of B-type masking are either based on a
single process (e.g. Anbar & Anbar, 1982; Bridgeman,
1978; Francis, 1997) or on a combination of two proc-
esses (e.g. Neumann & Scharlau, in press; Reeves,
1986). Most models which use a single process apply
a mechanism which was termed ‘mask blocking’ by
Francis (2000). The basic idea of this mechanism is
that a relatively strong target can block the mask’s
signal at short SOAs, but fails to do so at intermedi-
ate SOAs due to the decaying trace of the target. The
two process theories assume that the U-shape curve
in B-type masking actually consists of two parts, both
of which are monotonic. The two underlying processes
might relate to the accounts of ‘integration’ and ‘inter-
ruption’ masking (Scheerer, 1973), or to ‘peripheral’
and ‘central’ processes (Turvey, 1973).
While the focus of visual backward masking has been
on temporal aspects, the effects of the spatial layout of
the target and the mask have received much less inter-
est (but, see Cho & Francis, 2005; Francis & Cho, 2005;
Hellige, Walsh, Lawrence, & Prasse, 1979; Kolers, 1962).
If spatial aspects were investigated, they mainly involved
low-level aspects, such as the spatial distance between
the target and the mask, and the spatial frequencies of
ABSTRACT
In modeling visual backward masking, the focus
has been on temporal effects. More specically,
an explanation has been sought as to why stron-
gest masking can occur when the mask is delayed
with respect to the target. Although interesting
effects of the spatial layout of the mask have
been found, only a few attempts have been made
to model these phenomena. Here, we elaborate a
structurally simple model which employs lateral
excitation and inhibition together with different
neural time scales to explain many spatial and
temporal aspects of backward masking. We argue
that for better understanding of visual masking,
it is vitally important to consider the interplay of
spatial and temporal factors together in one sin-
gle model.
Advances in Cognitive Psychology
DOI: 10.2478/v10053-008-0017-0
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Frouke Hermens and Udo Ernst
the stimuli. Recently, Herzog and colleagues (Herzog,
Schmonsees, & Fahle, 2003a, b; Herzog & Fahle, 2002;
Herzog & Koch, 2001; Herzog, Harms, Ernst, Eurich,
Mahmud, & Fahle, 2003c; Herzog, Koch, & Fahle, 2001;
Herzog, Fahle, & Koch, 2001) started to investigate the
effects of the spatial layout of the mask systematically,
while keeping the target (a vertical Vernier) constant.
Even though the mask consisted of simple bar elements
only, slight changes in the layout of these elements
resulted in large differences in masking strengths. For
example, adding two collinear lines to a grating mask
strongly impaired performance on the Vernier target
(Herzog, Schmonsees, & Fahle, 2003a).
Only a few modeling attempts have been made to
explain spatial aspects of visual masking. The aspects
that were modeled include the effect of the distance of
the mask to the target (modeled by Breitmeyer & Ganz,
1976; Bridgeman, 1971; Francis, 1997), and the distri-
bution of the mask’s contour (modeled by Francis, 1997).
Several of the existing masking models (Anbar & Anbar,
1982; Di Lollo, Enns, & Rensink, 2000; Weisstein, 1968)
are constructed in such a way that they cannot account
for spatial aspects of the target and the mask.
Here, we describe a structurally simple model that
can explain several spatial aspects of visual backward
masking as well as temporal aspects. The model we use
is inspired by the basic structures found in the visual
cortex, with excitatory and inhibitory neurons driven by
feed-forward input, and exchanging action potentials via
recurrent horizontal interactions. We describe neural ac-
tivity in terms of population ring rates, whose dynam-
ics are similar to the classical Wilson-Cowan differential
equations (Wilson & Cowan, 1973) for spatially extended
populations. Here, we will present new simulations of the
effects of a shift of the mask either in space or time,
embedded in an overview of results earlier presented
by Herzog et al. (Herzog, Ernst, Etzold, & Eurich, 2003;
Herzog, Harms et al. 2003c).
SETUP OF THE MODEL
The general structure of our model is illustrated in
Figure 1. The input I(x,t) is ltered by a Mexican hat
kernel and fed into an excitatory and an inhibitory
layer. The activation of both layers is updated over
time, where activation from both layers is mutually
exchanged via the coupling kernels W
e
and W
i
. The
activation dynamics of the model are determined by
two coupled partial differential equations for the ring
rates of neuronal populations, originally introduced by
Wilson and Cowan (1973). We modied the original
equations in order to match more recent work (Ben-
Yishai, Bar-Or, & Sompolinsky, 1995; Ernst, Pawelzik,
Sahar-Pikielny, & Tsodyks, 2001) on the simulation of
neural populations in the visual cortex, by dropping
the shunting factors and using piecewise linear activa-
tion functions h
e
and h
i
, which do not saturate for high
inputs,
hI
sI I
ei
ei
,
,
()=
>
ì
í
î
for
otherwise
0
0
(1)
with neuronal gain constants s
e
and s
i
. The activation
in the excitatory (A
e
) and in the inhibitory layer (A
i
) is
updated according to
t
e
e
e
eeee eiei i
Axt
t
Axt
hw AW xt wAWxtI
¶
¶
=- +
+*-* +
(,)
(,)
()(,)( )( ,) ((,)xt
[]
(2)
t
i
i
i
ieie eiii i
Axt
t
Axt
hw AW xt wAWxtI
¶
¶
=- +
+*-* +
(,)
(,)
()(,)( )( ,) ((,)xt
[]
(3)
In these equations, τ
e
and τ
i
denote time constants,
and w
ee
, w
ie
, w
ei
, w
ii
are weighting coefcients for the
interactions. x denotes the position of the neuronal
population in the corresponding layer, and t denotes
time. We assume an approximate retinotopical map-
ping of the visual input onto the cortical layer, such
that x also describes position in the visual eld.
Recurrent interaction between the layers is mod-
eled by
Wxx
xx
ei
ei ei
,
,,
() exp
()
-
¢
=-
-
¢
æ
è
ç
ç
ö
ø
÷
÷
1
22
2
2
2
ps s
, (4)
Excitatory layer
Inhibitory layer
W
i
W
e
W
e
W
i
V
V
Stimulus
Figure 1.
The general setup of the model. The input, which is coded
as an array of ones and zeros is fed into an inhibitory and
an excitatory layer via a Mexican-hat lter. The activation
of these layers is updated over time.
Model visual backward masking
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for excitatory and inhibitory interactions, respectively.
The convolution, represented by *, describes the ac-
cumulation of synaptic inputs from other populations
in the same or in a different layer. In the limit of large
neuron numbers, it can be written as a spatial integral
wAWxtw AxtW xxdx
ee ee ee ee
()(,)(,) ()*=
¢
-
¢¢
-¥
+¥
ò
. (5)
The feed-forward ltered input into both layers is
computed by
IxtSVxtSxtVx xdx(,)( )( ,) (,)( )=* =
¢
-
¢¢
-¥
+¥
ò
. (6)
using an input kernel dened as a difference of
Gaussians (DOG)
Vx x
xx
xx
E
e
I
I
() exp
exp
-
¢
=-
-
¢
()
æ
è
ç
ç
ö
ø
÷
÷
-
--
-
¢
()
1
2
2
1
2
2
2
2
2
2
ps
s
ps
s
ææ
è
ç
ç
ö
ø
÷
÷
(7)
SPATIAL ASPECTS
Size of the grating
In their experiments, Herzog et al. (g 3. Herzog,
Fahle, & Koch, 2001) presented a Vernier target fol-
lowed by a grating mask of a variable number of ele-
ments. Participants were asked to determine the off-
set direction (left or right) of the vertical Vernier. The
mask consisted of an array of aligned vertical Verniers
(as illustrated in Figure 2A). Masking was strongest
when the grating consisted of 5 elements (about 58%
correct decisions with a 20 ms Vernier duration), and
weakest for gratings with more than 11 elements
(about 91% correct).
First we will focus on an explanation of why the
5 elements yield stronger masking, while a larger
mask (25 elements) yields weaker masking. Figure
2B shows the time evolution (vertical dimension)
of the spatial activation in the excitatory layer
(horizontal dimension). During the first 20 ms, the
Vernier is presented, which results in a central ac-
tivation of the layer. After these 20 ms, the Vernier
input is ended and immediately the mask enters
20 ms
time
A
20 ms
time
time
position
B
N=5 N=25
Figure 2.
Stimulus sequence (A) and simulation results (B) of data presented by Herzog et al. (2001). A Vernier target was masked by a
grating consisting of either ve (left) or 25 elements (right). The model correctly predicts that the ve-element grating masks
the Vernier much more strongly than the 25-element grating.
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Frouke Hermens and Udo Ernst
the system. For both gratings, this results in strong
activation at the edges of the grating, strong inhibi-
tion in the surround of these edges, and suppres-
sion of all other activations. Since the edges of the
five-element grating are much closer to the position
where the Vernier was displayed, due to strong in-
hibition the remaining activation from the Vernier
will decay faster than in the case of a 25-element
grating.
To understand the consequences of these dy-
namics for perception, let us consider how activity
in the model might be related to Vernier visibility. A
common hypothesis is that the stronger an activa-
tion caused by a particular feature of a stimulus is,
the better it can be detected by an observer of this
activity. Consequently, the stronger the activation of
the center column responding to visual input at the
target’s position is, the better we expect the target
to be visible, even it is blending over with the mask’s
appearance, as in the typical reported percept of an
observer in the 25-element condition. We therefore
assume that the duration of the trace of activity as-
sociated with the center column, being above some
threshold Θ, is monotonically related to visibility of
the target element (linking hypothesis). It is there-
fore not necessary to model explicitly Vernier off-
set, as this feature of the target in the experiment
is used only as a vehicle to quantify visibility. From
elementary considerations in signal detection theory,
it is obvious that the longer a noisy process is be-
ing observed, the better any estimation gets of some
of its underlying parameters. The threshold in our
case plays the role of an ad-hoc quantication of the
neuronal background noise: only when activation in-
creases beyond this noise level, may stimuli become
visible. In order to quantify the linking hypothesis,
one normally uses an experiment in which visibility
or detection performance changes with some control
parameter, and then ts a continuous function link-
ing performance to a model variable. Once xed, this
function then allows prediction from the model how
performance will be in other experimental conditions.
20 ms
time
A
20 ms
time
time
position
B
N=5 N=25
Figure 3.
Stimulus sequence (A) and simulation results (B) of data presented by Herzog et al. (2003). A Vernier target was masked by a
eld of light of the size either of either ve (left) or 25 elements (right). The model correctly predicts that the ve-element size
eld masks the Vernier much more strongly than that of the size of a 25-element grating, as indicated by the longer Vernier
trace for the 25-element grating in the center of the image of the network activation.
Model visual backward masking
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While in a previous publication we employed this
quantitative procedure, in this review article we only
use qualitative measures, as e.g., predicting the peak
performance in a specic condition, for evaluating the
model’s performance.
Uniform elds of light
In the previous paragraph, we saw that a grating
of ve elements masks a Vernier target much more
strongly than a grating consisting of 25 elements. This
nding was surprising, because the 25-element grat-
ing contains much more energy than the grating of 5
elements. The model suggested that the difference in
masking strength could be explained by the distance
of the nearest edge of the mask. If the distance to
the edge of the grating is indeed what determines the
masking strength, one would also expect a uniform
eld of light of the size of the ve element grating to
be a stronger mask than one of the size of the 25-ele-
ment grating. Figure 3 shows that the model indeed
predicts stronger masking for a small uniform eld of
light than for a large one. In the top part of this gure
the sequence of stimuli is shown. The energy of the
eld of light was set such that the overall energy of
the mask matched that of the corresponding grating
mask. Figure 3B shows the activation over time for
the two light masks. The pattern of results resembles
that obtained for grating masks (Figure 2B). The small
eld of light suppresses the Vernier activation more
strongly than the larger one.
Whether small elds of light mask more strongly
than larger ones was experimentally investigated by
Herzog, Harms et al. (2003c). Vernier offset discrimi-
nation thresholds indicated that the small light-eld
was indeed a stronger mask, although the difference
in thresholds between the two mask sizes was not as
large as for the grating masks. By using a function
that linked network activation to thresholds (the ‘link-
ing hypothesis’), Herzog, Ernst et al. (2003) showed
that the model could accurately predict the observed
thresholds.
Irregularities in the mask
Two ndings suggest that breaking up the regularity
of the mask increases its masking strength. Herzog et
al. (g. 4; 2001) introduced two gaps in the grating
20 ms
time
A
20 ms
time
time
position
B
Gaps Double intensity
Figure 4.
Stimulus sequences (A) and simulation results (B). A Vernier target was followed either by a grating with two gaps at offset
positions +/-2 from the Vernier, or two elements of double luminance at these positions. Experimental data showed that both
masks yield a strong increase in offset discrimination thresholds with respect to the standard grating. The simulations show
that the model can well detect the irregularities in the mask, and explain how these irregularities result in an increase in
masking strength. The irregularities are associated with strong network activation causing strong inhibition in their immediate
surroundings that suppresses activation of the target, because the irregularities were close to the target.
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Frouke Hermens and Udo Ernst
by removing two elements (illustrated in the left plot
of Figure 4A), which resulted in a grating consisting
of ve central elements and two more distant groups
of nine elements. The removal of the two grating ele-
ments strongly increased the strength of the mask.
Similarly, Herzog et al. (g. 7A; 2004) increased the
luminance of the two elements at position offsets +2
and −2 from the Vernier, as illustrated in the right part
of Figure 4A. Also, this slight change in mask layout
resulted in a strong increase in the masking strength.
The simulation plots of Figure 4B show how we can
understand the strong increase in masking strength
by the introduction of the gaps or the double lumi-
nance elements into our model. The model is sensi-
tive to irregularities in the grating, which yield high
activations in the neuronal layers. As the activation
induced by the gaps or by the elements with doubled
luminance is close to the preceding Vernier activity,
the decay of the Vernier activation will be faster, and
thus predicted performance will be low.
The simulations with the mask with the two gaps
show that not only the mask affects the target, but
also the target affects the mask. The inner edges at
the two gaps show weaker activation than the outer
edges, which can be understood as resulting from
stronger inhibition of the inner edges by the target
than the outer edges. Said differently, the target
forwardly masks the mask.
Masking is predicted to be slightly weaker for
the mask with the two gaps than for the mask with
double luminance lines. At this time, there is no
experimental data to determine whether this pre-
diction is correct. Thresholds were determined for
both masks, however, with different observers with
different amounts of training in the Vernier discrim-
ination task. It would be interesting, though, to test
this prediction in the future.
Center = 0" Center = 400" Center = 800"
Center = 1600" Center = 2000" Center = 2200"
Center = 2300"
Figure 5.
The activation in the excitatory population over time (vertical dimension) for different sizes of the shift of the center of the
grating to the right. The small red horizontal bar indicates where the activity at the center drops below a certain value. The
model predicts that when the grating’s edge approaches the Vernier, the Vernier’s trace is strongly reduced, implying much
worse performance on the Vernier.
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Edge distance
Previous simulations suggest that it is mainly the dis-
tance of the closest edge to the Vernier rather than
the number of lines in the mask that determines the
strength of the mask. This leads to the prediction that
if the 25-element grating is shifted with respect to the
location of the Vernier (as illustrated in the left part of
Figure 6), masking strength will increase. This model
prediction is illustrated in Figure 5, where the different
subplots show the activation of the excitatory popula-
tion across time (vertical dimension) for different sizes
of the shift of the 25-element grating (the grating is
shifted to the right of the center). The small red hori-
zontal bar indicates where the activity at the center
drops below a certain value. In the plot, a 0” shift
indicates that both the Vernier and the grating were
centered around the middle of the screen. A 400” shift
indicates that the grating’s center was shifted 400”
to the right, which means that the left edge of the
grating is 400” closer to the Vernier compared to the
standard situation. The model predicts that shifts up to
800” have little effect, while shifts larger than 1600”
strongly affect the Vernier’s visibility. Note that merely
looking at the moment the central activation drops
below a certain value suggests a different pattern of
results. This is because at some point the activation
of the Vernier and that of the mask’s edge appear at
the same spatial location. To avoid this confusion of
activation, a different linking hypothesis might need to
be used, or some spatial representation of the offset of
the Vernier needs to be coded by the model.
Whether Vernier discrimination performance in-
deed decreases with an increasing shift of the mask
was determined with one observer (author FH).
This observer was presented with a sequence of a
Vernier presented for 12 ms (the optimal duration
for this observer), followed by a 25-element grating
for 300 ms. The center of the grating was shifted
from 0”, via 400”, 800”, 1600”, 2000”, 2200”’, to
2300” (edge close to the Vernier position), as is
illustrated in the left part of Figure 6. For the rest,
the experimental procedure was the same as in
earlier demonstrations of the shine-through ef-
fect (e.g., Herzog, Harms et al., 2003c). The right
part of Figure 6 shows the results. Thresholds start
to rise at a shift of about 1600” (≈ ±8 elements
offset), and reach a maximum for a shift of 2300”
(≈ ±11.5 elements offset), where no threshold
could be measured anymore.
The model was correct in predicting that thresh-
olds increase with an increase in the shift of the
mask. In addition, the model could well predict for
which shift thresholds would strongly rise, which
suggests that the model is correct in its assump-
tion that the distance to the mask’s nearest edge
determines the masking strength.
Alternative explanations
Of the existing models of masking, only few are imple-
mented in such a way that spatial information about
the target and the mask can be coded (Bridgeman,
1978; Francis, 1997; Öğmen, 1993). Other compu-
tational models represent target and mask in single
neurons (Anbar & Anbar, 1982; Di Lollo et al., 2000;
Weisstein, 1968), an approach which does not allow
spatial information to enter the model system.
Of the models that can code for spatial proper-
ties, only the model by Bridgeman can easily be
implemented. The remaining two models (Francis,
time
shift
1 Observer (FH)
0
100
200
300
0 500 1000 1500 2000 2500
Center shift (arc sec)
Threshold (arc sec)
Figure 6.
The sequence of Vernier and mask (left) and Vernier offset discrimination thresholds for observer FH (right) as a function of
the size of the shift of the center of the grating mask. The data conrm the model’s prediction that a close edge yields strong
inhibition of the Vernier’s signal, reected in higher offset discrimination thresholds.
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Frouke Hermens and Udo Ernst
1997; Öğmen, 1993) involve many stages and com-
plex processing. For example, the model by Francis
(1997), which is based on the boundary contour
system (Grossberg & Mingolla, 1985), consists of
six layers with many complex interactions. To simu-
late these models, one probably needs the help of
the authors to understand the full details in order
to correctly implement the model. Moreover, these
models often require simplifications of the model
to be able to perform the simulations. Due to these
restrictions, we will only present simulation results
of Bridgeman’s model here.
The model by Bridgeman makes use of the Hartline-
Ratliff equation that was originally developed to de-
scribe lateral inhibition in the Limulus eye. A network
of 30 neurons is used to describe the effects of a visual
mask. The ring rate of each neuron in the network
changes over time depending on the excitatory sensory
input and the inhibitory effect of neighboring neurons.
To compare the network activations with the visibility
of the target, the ring rates in the network are com-
pared for a run in which only the target is presented,
with one in which both the target and the mask are
presented.
For the implementation of the Bridgeman model,
we assumed a network of 500 neurons centered
around the position where the Vernier was present-
ed. In the original version of the model, 30 neurons
were used, of which the first and the last neurons
were linked to avoid edge effects. We choose a dif-
ferent approach: Since computers have become
much faster, we could easily extend the number of
Cell activation
30
40
50
60
70
80
t = 1 t = 3 t = 5 t = 15
Vernier only
30
40
50
60
70
80
ρ = 0.97456 ρ = 1.9549 ρ = 2.047 ρ = 2.4488
Vernier +
5 grating
30
40
50
60
70
80
ρ = 0.97621 ρ = 1.9555 ρ = 1.9854 ρ = 2.0469
Vernier +
25 grating
30
40
50
60
70
80
ρ = 0.97087 ρ = 1.9516 ρ = 1.9746 ρ = 2.1862
Vernier +
25 grating
with gaps
Figure 7.
Cell activations in Bridgeman’s (1978) model for the conditions (1) Vernier only, (2) Vernier followed by a ve-element grat-
ing, (3) Vernier followed by a 25-element grating, (4) Vernier followed by a 25-element grating with gaps. The value p in the
subplot titles refers to the sum of the squared correlation over time between the activation for condition (1) and the respective
condition. The higher the value of, the higher the predicted per-formance. The values indicate that the model fails to explain
why a 5-element grating (2), and the 25-element grating with gaps (4) are much stronger masks than the 25-element grat-
ing (3).
Model visual backward masking
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neurons in the model, thereby avoiding edge ef-
fects (activation could not spread to the boundaries
within the simulation time), while also avoiding
neurons that were not close in retinotopic space af-
fecting each other.
The background activation of the model was set
to 50, additional activation of the target and mask
was 22.5. The standard error of the Gaussian noise
was assumed to be 0.1. The interaction param-
eters were the same as in earlier simulations by
Bridgeman (1971, 1978). To initialize the network,
500 iterations were run in which only background
activation was provided, before the stimuli were
presented to the network. The target was presented
for 2 time frames, the mask for the remaining 18
frames.
Figure 7 shows the activation of the neurons at
different points in time for different stimulus se-
quences. The top row shows the activation of the
neurons after presentation of the Vernier only, the
bottom three rows for a Vernier followed by one of
three gratings (5-element grating, 25-element grat-
ing, 25-element grating with gaps, respectively).
The value in the subplots’ title (ρ) shows the sum
over time of the squared correlation between the
neuronal activation with the mask and that of the
run without a mask. The sum is shown instead of
the commonly used average, to make the outcome
less dependent on the number of time steps in the
simulation. If the model’s predictions agree with
the data, we would expect to find a high value of ρ
for the 25-element grating, and low values for the
other two masks. This is not what is found: The
value of ρ for the 25-element grating is, in fact,
lower than that for the other two gratings, suggest-
ing that Bridgeman’s model cannot account for the
experimental findings.
TEMPORAL ASPECTS
Onset of context
As discussed before, a grating of ve elements is a
stronger mask than one consisting of 25 elements
time
position
positive
SOA duration
20 ms
time
negative
SOA duration
20 ms
time
A
B
SOA = −50 ms SOA = −30 ms SOA = −10 ms SOA = 0 ms
SOA = 10 ms SOA = 30 ms SOA = 50 ms SOA = −80 ms
Figure 8.
Stimulus sequence (A) and simulation results (B) of data presented by Herzog et al. (2001). The small red horizontal bars
indicate where the activity of the trace drops below a particular threshold. A Vernier target was masked by a grating consist-
ing of a ve-element center and a 20-element surround, which were presented at different onset times. Once presented, the
stimulus remained on the screen until 300 ms after target offset. The model correctly predicts that the target strength remains
strongest for simultaneous onset of the mask’s center and surround.
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Frouke Hermens and Udo Ernst
(Herzog, Fahle, & Koch, 2001). Here, we will show
simulation results in which the relative onset of the
ve central elements and the 20 surrounding elements
of a 25-element mask was varied. Figure 8A shows
the sequences used in the experiment by Herzog
et al. (2001). For negative SOAs, the 20 surround-
ing elements of the mask preceded the ve central
elements. For positive SOAs, the central ve elements
were presented before the surrounding 20 elements.
At zero SOA, all 25 elements were presented simulta-
neously. In the experiment, Vernier offset discrimina-
tion thresholds were found to be minimal for an SOA
equal to zero, and increased with SOA (either positive
or negative).
The Wilson-Cowan type model can explain why
masking is weakest at zero SOA and increases with
SOA. The activation plots that illustrate this are
shown in Figure 8B. Each subplot shows the activa-
tion in the excitatory layer over time (vertical axis).
The small red horizontal bars in the plots indicate
where the activity of the trace drops below a cer-
tain threshold. During the first 20ms, the Vernier is
presented to the network, followed by the sequence
of mask parts. The duration over which the acti-
vation at the center of the population (where the
Vernier was presented) survives is an indication of
how well the Vernier will be perceived. The figure
shows that the Vernier’s signal best survives for an
SOA of zero, while the length of the Vernier’s trace
decreases with increasing absolute SOA (either
negative or positive).
Verbally, the explanation of the results can be
phrased as follows. When the center and the sur-
round are presented simultaneously, the network
will consider the two parts as one object. The edges
of this object are determined, and since they are far
away from the Vernier target, they will hardly affect
the signal of the Vernier. If the surround is present-
ed earlier, the network will respond by detecting the
edges of the two parts of the surround. Since the
edges of these parts are much closer to the Vernier
location, they will inhibit the Vernier more strongly.
Similarly, if the center is presented before, its edg-
es will be detected, and since also these edges are
close to the Vernier, they will inhibit the Vernier’s
signal. The trace of the mask in the population can
change over time, as soon as other elements of the
grating enter the network. This explains why early
onset of the context elements results in a longer
trace of the Vernier than late onset.
The model predictions were compared quanti-
tatively with the experimental findings by Herzog
et al. by applying a linking function converting the
length of the suprathreshold trace of the Vernier
into predicted thresholds (see model section). The
model predictions closely matched the experimen-
tal results (Figure 6; Herzog, Ernst et al., 2003).
Optimal masking at a non-zero
SOA
In the introduction, we mentioned the relatively strong
focus of the masking research community on explain-
ing that masking can be strongest at a non-zero SOA
(B-type masking). The work by Francis (2000) sug-
gests that many models that apply a non-linearity
(rectication) and decay can explain B-type masking.
As our version of the Wilson-Cowan model contains
both properties, we would expect that a combination
of target and mask can be found for which the model
shows strongest masking at a non-zero SOA. Figure
9 shows such a combination (left), together with the
corresponding network responses (right). The small
red horizontal bars indicate where the activity of the
time
increasing SOA
Figure 9.
Stimulus sequence (left) and responses of the excitatory population (right) for which optimal masking at a non-zero SOA oc-
curs. The small red horizontal bars indicate where the activity of the trace drops below a particular threshold. The Vernier’s
trace is long for a zero SOA, then decreases in length for intermediate SOAs, and returns to full length again at long SOAs,
indicating that masking is strongest at intermediate SOAs.
Model visual backward masking
103
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trace drops below a particular value. For short SOAs,
the target’s trace is long. For intermediate SOAs, the
length of the trace decreases, to increase again with
longer SOAs. This pattern of trace lengths as a function
of SOA suggests a U-shaped dependence of predicted
performance on SOA.
GENERAL DISCUSSION
In this paper, we have argued that it is important to
study both spatial and temporal aspects of visual back-
ward masking. Temporal aspects have been studied
for a long time. Although some basic spatial aspects,
such as the distance between target and mask, and
their spatial frequencies have been studied in the past,
it is only recently that spatial aspects have started to
be investigated systematically. A similar trend can be
seen for models of visual masking. Most earlier mod-
els (Anbar & Anbar, 1982; Weisstein, 1968) could only
model temporal aspects of masking, simply because
spatial aspects could not be coded by the models. An
exception is the model by Bridgeman (1978), which al-
lows for a representation of stimuli in a spatial array.
However, we showed that this model can not account
for the difference in masking strength of the 25-ele-
ment grating (weak masking), the ve-element grating
and the grating with two gaps (strong masking). Later
models can represent the spatial layout of the stimuli,
even in two dimensions (Francis, 1997; Öğmen, 1993).
However, these models are so complex that a single
simulation can take a standard computer days to per-
form (see the appendix of Francis, 1997), while at the
same time preventing any analytical investigation of
the relevant mechanisms.
Here, we showed that a structurally simple corti-
cal model with excitatory and inhibitory interactions
can uncover putative mechanisms of several spatial
and temporal aspects of masking. The model can
explain why a grating of 5 aligned Vernier elements
masks a Vernier target more strongly than one con-
sisting of 25 elements. Similarly, it explains why
a smaller uniform light-field masks more strongly
than a large one. The model also correctly predicted
that shifting the 25-element grating with respect
to the Vernier target results in stronger masking.
In addition to these spatial aspects of masking, the
model could explain why a delayed onset of mask
elements results in stronger masking, and how a
non-monotonic relation between SOA and masking
strength can be obtained.
The mechanisms which enable the model to work
in the described way are easy to understand: The first
stage of processing is a pure feed-forward filtering of
the stimulus, realizing an edge enhancement (or de-
tection of inhomogeneities) on the length scale of a
typical double bar distance. The features of a stimu-
lus pronounced by this procedure are then enhanced
through a localized excitatory interaction, while two
features within the distance of the length scale of
the inhibitory interactions will compete for activa-
tion. A necessary condition hereby is that enhance-
ment and competition are governed by two different
time scales, a fast one for enhancement, and a slow
one for competition. Through these time scales, fea-
tures of mask and target are either superimposing
or canceling each other. The most important aspect
leading sometimes to counterintuitive effects is the
strong recurrency in the interactions: even when a
feature in the target, which leads to a pronounced
activation in the network, has just been switched
off, the excitatory interactions can sustain this acti-
vation for a prolonged period. During this period a
competing, nearby feed-forward input of a mask has
no chance to produce sufficient activation which in
turn could suppress the target’s sustained activity.
Only when this activity has decayed sufficiently, is
the mask rendered effective. This mechanism in our
model provides a putative neural basis for U-shaped
masking curves.
By systematically comparing model output and
experimental results, we can determine which as-
pects of masking can be explained with a simple
mechanism, and which aspects need a more elabo-
rate model. For example, the U-shaped dependence
of performance on SOA for certain targets and masks
can be explained with a single mechanism, and does
not necessarily require two processes. However,
Francis and Herzog (2004) showed that masking
curves can intersect, even if the target and the task
are kept constant, and just the mask is varied. This
result poses strong restrictions on plausible mod-
els, suggesting that two or more neural processes
underlie masking curves [as suggested by Reeves
(1986) and Neumann and Scharlau (in press)].
Computational models are also necessary to de-
termine which conclusions can be drawn from data,
as is illustrated by a recent contribution by Di Lollo
and colleagues (2000) that received several com-
ments. In their article, Di Lollo et al. suggested
that no existing model could account for their data,
and in particular for common onset masking, where
the mask is onset at the same time as the target,
but remains on the screen after target offset. They
furthermore suggested that recurrent connections
were needed to explain the results, instead of the
104
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Frouke Hermens and Udo Ernst
feed-forward structure applied by existing models.
The problem with their statements was that they did
not check with simulations whether existing theo-
ries could already explain their data. Not much later,
Francis & Hermens (2002) performed the necessary
simulations and found out that common onset mask-
ing could easily be accounted for by existing mod-
els. Additional simulations then suggested which
experiment would distinguish between the existing
models and the newly proposed model by Di Lollo
et al. (2000). This experiment confirmed that the
recurrent model by Di Lollo et al., in fact, outper-
formed all existing models (Francis & Cho, 2007).
The ultimate goal of modeling visual processing
will be to construct a predictive model of the visual
cortex. However, current computer capacities and
also our current knowledge of the visual system do
not allow this yet. Until the ultimate model of the
brain can be constructed, we will have to work with
much simpler models. The best strategy hereby is
to tightly combine experimental and modeling stud-
ies to test upcoming theories of visual information
processing, and to break down visual processing
as far as possible into distinct modules which can
under certain conditions be studied separately from
each other. In such an integrative approach, we
have demonstrated that a structurally simple cor-
tical network can explain a quite extensive set of
data in visual masking, which suggests that masking
phenomena can be easily understood through the
dynamics of network structures that are common to
many areas found in the visual cortex.
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