A model of transparent motion and 
non-transparent motion aftereffects 
Alexander Grunewald* 
Max-Planck Institut fiir biologische Kybernetik 
Spemannstral]e 38 
D-72076 Tiibingen, Germany 
Abstract 
A model of human motion perception is presented. The model 
contains two stages of direction selective units. The first stage con- 
tains broadly tuned units, while the second stage contains units 
that are narrowly tuned. The model accounts for the motion af- 
tereffect through adapting units at the first stage and inhibitory 
interactions at the second stage. The model explains how two pop- 
ulations of dots moving in slightly different directions are perceived 
as a single population moving in the direction of the vector sum, 
and how two populations moving in strongly different directions are 
perceived as transparent motion. The model also explains why the 
motion aftereffect in both cases appears as non-transparent motion. 
I INTRODUCTION 
Transparent motion can be studied using displays which contain two populations of 
moving dots. The dots within each population have the same direction of motion, 
but directions can differ between the two populations. When the two directions are 
very similar, subjects report seeing dots moving in the average direction (Williams 2 
Sekuler, 1984). However, when the difference between the two directions gets large, 
subjects perceive two overlapping sheets of moving dots. This percept is called 
transparent motion. The occurrence of transparent motion cannot be explained by 
direction averaging, since that would result in a single direction of perceived motion. 
Rather than just being a quirk of the human visual system, transparent motion is 
an important issue in motion processing. For example, when a robot is moving its 
* Present address: Caltech, Mail Code 216-76, Pasadena, CA 91125. 
838 A. GRUNEWALD 
motion leads to a velocity field. The ability to detect transparent motion within 
that velocity field enables the robot to detect other moving objects at the same time 
that the velocity field can be used to estimate the heading direction of the robot. 
Without the ability to code multiple directions of motion at the same location, 
i.e. without the provision for transparent motion, this capacity is not available. 
Traditional algorithms have failed to properly process transparent motion, mainly 
because they assigned a unique velocity signal to each location, instead of allowing 
the possibility for multiple motion signals at a single location. Consequently, the 
study of transparent motion has recently enjoyed widespread interest. 
STIMULUS 
PERCEPT 
Test 
Figure 1: Two populations of dots moving in different directions during an adap- 
tation phase are perceived as transparent motion. Subsequent viewing of randomly 
moving dots during a test phase leads to an illusory percept of unidirectional motion, 
the motion aftereffect (MAE). Stimulus and percept in both phases are shown. 
After prolonged exposure to an adaptation display containing dots moving in one di- 
rection, randomly moving dots in a test display appear to be moving in the opposite 
direction (Hiris & Blake, 1992; Wohlgemuth, 1911). This illusory percept of motion 
is called the motion aftereffect (MAE). Traditionally this is explained by assuming 
that pairs of oppositely tuned direction selective units together code the presence 
of motion. When both are equally active, no motion is seen. Visual motion leads 
to stronger activation of one unit, and thus an imbalance in the activity of the two 
units. Consequently, motion is perceived. Activation of that unit causes it to fa- 
tigue, which means its response weakens. After motion offset, the previously active 
unit sends out a reduced signal compared to its partner due to adaptation. Thus 
adaptation generates an imbalance between the two units, and therefore illusory 
motion, the MAE, is perceived. This is the ratio model (Sutherland, 1961). 
Recent psychophysical results show that after prolonged exposure to transparent 
motion, observers perceive a MAE of a single direction of motion, pointing in the 
vector average of the adaptation directions (Mather, 1980; Verstraten, Frederick- 
sen, & van de Grind, 1994). Thus adaptation to transparent motion leads to a 
non-transparent MAE. This is illustrated in Figure 1. This result cannot be ac- 
counted for by the ratio model, since the non-transparent MAE does not point in 
the direction opposite to either of the adaptation directions. Instead, this result 
suggests that direction selective units of all directions interact and thus contribute 
to the MAE. This explanation is called the distribution-shift model (Mather, 1980). 
However, thus far it has only been vaguely defined, and no demonstration has been 
given that shows how this mechanism might work. 
A Model of Transparent Motion and Non-transparent Motion Aftereffects 839 
This study develops a model of human motion perception based on elements from 
both the ratio and the distribution-shift models for the MAE. The model is also 
applicable to the situation where two directions of motion are present. When the 
directions differ slightly, only a single direction is perceived. When the directions 
differ a lot, transparent motion is perceived. Both cases lead to a unitary MAE. 
2 OUTLINE OF THE MODEL 
The model consists of two stages. Both stages contain units that are direction 
selective. The architecture of the model is shown in Figure 2. 
Stage 2 
Stage 
Figure 2: The model contains two stages of direction selective units. Units at stage 
I excite units of like direction selectivity at stage 2, and inhibit units of opposite 
directions. At stage 2 recurrent inhibition sharpens directional motion responses. 
The grey level indicates the strength of interaction between units. Strong influence 
is indicated by black arrows, weak influence is indicated by light grey arrows. 
Units in stage 1 are broadly tuned motion detectors. In the present study the precise 
mechanism of motion detection is not central, and hence it has not been modeled. It 
is assumed that the bandwidth of motion detectors at this stage is about 30 degrees 
(Raymond, 1993; Williams, Tweten,  Sekuler, 1991). In the absence of any visual 
motion, all units are active at a baseline level; this is equivalent to neuronal noise. 
Whenever motion of a particular direction is present in the input, the activity of 
the corresponding unit (vi) is activated maximally (vi - 9), and units of similar 
direction selectivity are weakly activated (vi = 3). The activities of all other units 
decrease to zero. Associated with each unit i at stage 1 is a weight wi that denotes 
the adaprational state of unit i to fire a unit at stage 2. During prolonged exposure 
to motion these weights adapt, and their strength decreases. The equation governing 
the strength of the weights is given below: 
dwi 
dt -- R(1 - wi) - viwi, 
where R = 0.5 denotes the rate of recovery to the baseline weight. When wi - 1 
the corresponding unit is not adapted. The further wi is reduced from 1, the more 
840 A. GRUNEWALD 
the corresponding unit is adapted. The products viwi are transmitted to stage 2. 
Each unit of stage I excites units coding similar directions at stage 2, and inhibits 
units coding opposite directions of motion. The excitatory and inhibitory effects 
between units at stages I and 2 are caused by kernels, shown in Figure 3. 
1 
0.8 
0.6 
0.4 
0.2 
0 
Feedforward kernels 
I I I- 
excitatory 
inhibitory 
1 
0.8 
0.6 
0.4 
0.2 
0 
Feedback kernels 
excitatory 
inhibitory 
-180 0 180 -180 0 180 
Figure 3: Kernels used in the model. Left: excitatory and inhibitory kernels between 
stages I and 2; right: excitatory and inhibitory feedback kernels within stage 2. 
Activities at stage 2 are highly tuned for the direction of motion. The broad ac- 
tivation of motion signals at stage I is directionally sharpened at stage 2 through 
the interactions between recurrent excitation and inhibition. Each unit in stage 2 
excites itself, and interacts with other units at stage 2 through recurrent inhibition. 
This inhibition is maximal for close directions, and falls off as the directions be- 
come more dissimilar. The kernels mediating excitatory and inhibitory interactions 
within stage 2 are shown in Figure 3. Through these inhibitory interactions the 
directional tuning of units at stage 2 is sharpened; through the excitatory feedback 
it is ensured that one unit will be maximally active. Activities of units at stage 2 
are given by Mi = max4(mi, 0), where the behavior of rni is governed by: 
= -rni + (1 - rni)(F/+ + B/+) - (1 + rni)(F/-- + B-). 
F/+ and F? denote the result of convolving the products of the activities at stage 
I and the corresponding adaptation level, vjwj, with excitatory and inhibitory 
feedforward kernels respectively. Similarly, B/+ and B- denote the convolution of 
the activities Mj at stage 2 with the feedback kernels. 
3 SIMULATIONS OF PSYCHOPHYSICAL RESULTS 
In the simulations there were 24 units at each stage. The model was simulated 
dynamically by integrating the differential equations using a fourth order Runge- 
Kutta method with stepsize H = 0.01 time units. The spacing of units in direction 
space was 15 degrees at both stages. Spatial interactions were not modeled. In 
the simulations shown, a motion stimulus is present until t - 3. Then the motion 
stimulus ceases. Activity at stage 2 after t = 3 corresponds to a MAE. 
A Model of Transparent Motion and Non-transparent Motion Aftereffects 841 
3.1 UNIDIRECTIONAL MOTION 
When adapting to a single direction of motion, the model correctly generates a 
motion signal for that particular direction of motion. After offset of the motion 
input, the unit coding the opposite direction of motion is activated, as in the MAE. 
A simulation of this is shown in Figure 4. 
Stage 1 Stage 2 
act  act 8 
4 4 4 
60 '`-120` 't2 time 60 -l11l/l2 time 
_. .8o __ _8o 
di er ction 240 ""' direction 240' 
300 300 
360 360 
Figure 4: Simulation of single motion input and resulting MAE. Motion input is 
presented until t = 3. 
During adaptation the motion stimulus excites the corresponding units at stage 1, 
which in turn activate units at stage 2. Due to recurrent inhibition only one unit 
at stage 2 remains active (Grossberg, 1973), and thus a very sharp motion signal 
is registered at stage 2. During adaptation the weights associated with the units 
that receive a motion input decrease. After motion offset, all units receive the same 
baseline input. Since the weights of the previously active units are decreased, the 
corresponding cells at stage 2 receive less feedforward excitation. At the same time, 
the previously active units receive strong feedforward inhibition, since they receive 
inhibition from units tuned to very different directions of motion and whose weights 
did not decay during adaptation. Similarly, the units coding the opposite direction 
of motion as those previously active receive more excitation and less inhibition. 
Through recurrent inhibition the unit at stage 2 coding the opposite direction to that 
which was active during adaptation is activated after motion offset: this activity 
corresponds to the MAE. Thus the MAE is primarily an effect of disinhibition. 
3.2 TRANSPARENT MOTION: SIMILAR DIRECTIONS 
Two populations of dots moving in different, but very similar, directions lead to 
bimodal activation at stage 1. Since the feedforward excitatory kernel is broadly 
tuned, and since the directions of motion are similar, the ensuing distribution of 
activities at stage 2 is unimodal, peaking halfway between the two directions of 
motion. This corresponds to the vector average of the directions of motion of the 
two populations of dots. A simulation of this is shown in Figure 5. 
During adaptation the units at stage 1 corresponding to the input adapt. As before 
this means that after motion offset the previously active units receive less excitatory 
input and more inhibitory input. As during adaptation this signal is unimodal. Also, 
the unit at stage 2 coding the opposite direction to that of the stimulus receives 
842 A. GRUNEWALD 
Stage 1 Stage 2 
act act  
60 120ll/g2 time 60 1202 time 
direction 240 30' direction 240' 
300 
360 360 
Figure 5: Simulation of two close directions of motion. Stage 2 of the network model 
registers unitary motion and a unitary MAE. 
less inhibition and more excitation. Through the recurrent activities within stage 
2, that unit gets maximally activated. A unimodal MAE results. 
3.3 TRANSPARENT MOTION: DIFFERENT DIRECTIONS 
When the directions of the two populations of dots in a transparent motion display 
are sufficiently distinct, the distribution of activities at stage 2 is no longer unimodal, 
but bimodal. Thus, recurrent inhibition leads to activation of two units at stage 2. 
They correspond to the two stimulus directions. A simulation is shown in Figure 6. 
Stagel Stage2 
act  act 6 
60 ;20 g 2 time time 
60 12011!'2 
.. .80 fi .. .80 
direction 240 30" direction 240'' 
300 
360 360 
Figure 6: Simulation of two distinct directions of motion. Stage 2 of the model 
registers transparent motion during adaptation, but the MAE is unidirectional. 
Feedforward inhibition is tuned much broader than feedforward excitation, and as a 
consequence the inhibitory signal during adaptation is unimodal, peaking at the unit 
of stage 2 coding the opposite direction of the average of the two previously active 
directions. Therefore that unit receives the least amount of inhibition after motion 
offset. It receives the same activity from stage I as units coding nearby directions, 
since the corresponding weights at stage I did not adapt. Due to recurrent activities 
at stage 2 that unit becomes active: non-transparent motion is registered. 
A Model of Transparent Motion and Non-transparent Motion Aftereffects 
4 DISCUSSION 
843 
Recently Snowden, Treue, Erickson, and Andersen (1991) have studied the effect 
of transparent motion stimuli on neurons in areas V1 and MT of macaque monkey. 
They simultaneously presented two populations of dots, one of which was moving 
in the preferred direction of the neuron under study, and the other population was 
moving in a different direction. They found that neurons in V1 were barely affected 
by the second population of dots. Neurons in MT, on the other hand, were inhibited 
when the direction of the second population differed from the preferred direction, 
and inhibition was maximal when the second population was moving opposite to the 
preferred direction. These results support key mechanisms of the model. At stage 
1 there is no interaction between opposing directions of motion. The feedforward 
inhibition between stages 1 and 2 is maximal between opposite directions. Thus 
activities of units at stage 1 parallel neural activities recorded at V1, and activities 
of units at stage 2 parallels those neural activities recorded in area MT. 
Acknowledgments 
This research was carried out under HFSP grant SF-354/94. 
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