A Model of Spatial Representations in 
Parietal Cortex Explains Hemineglect 
Alexandre Pouget 
Dept of Neurobiology 
UCLA 
Los Angeles, CA 90095-1763 
alex@salk.edu 
Terrence J. Sejnowski 
Howard Hughes Medical Institute 
The Salk Institute 
La Jolla, CA 92037 
terry@salk.edu 
Abstract 
We have recently developed a theory of spatial representations in 
which the position of an object is not encoded in a particular frame 
of reference but, instead, involves neurons computing basis func- 
tions of their sensory inputs. This type of representation is able 
to perform nonlinear sensorimotor transformations and is consis- 
tent with the response properties of parietal neurons. We now ask 
whether the same theory could account for the behavior of human 
patients with parietal lesions. These lesions induce a deficit known 
as hemineglect that is characterized by a lack of reaction to stimuli 
located in the hemispace contralateral to the lesion. A simulated 
lesion in a basis function representation was found to replicate three 
of the most important aspects of hemineglect: i) The models failed 
to cross the leftmost lines in line cancellation experiments, ii) the 
deficit affected multiple frames of reference and, iii) it could be 
object centered. These results strongly support the basis function 
hypothesis for spatial representations and provide a computational 
theory of hemineglect at the single cell level. 
I Introduction 
According to current theories of spatial representations, the positions of objects 
are represented in multiple modules throughout the brain, each module being spe- 
cialized for a particular sensorimotor transformation and using its own frame of 
reference. For instance, the lateral intraparietal area (LIP) appears to encode the 
location of objects in oculocentric coordinates, presumably for the control of sac- 
cadic eye movements. The ventral intraparietal cortex (VIP) and the premotor 
cortex, on the other hand, seem to use head-centered coordinates and might be 
A Model of Spatial Representations in Parietal Cortex Explains Hemineglect 11 
A B 
50 
Stimulus 
 Left 
I I  [ I 
C1 C2 C1 C2 C3 
FP 
TarNt Distractors 
x'e  e  C1 
 x C2 
   C3 
Figure 1' A. Retinotopic neglect modulated by egocentric position. B. Stimulus- 
centered neglect 
involved in the control of hand movements toward the face. 
This modular theory of spatial representations is not fully consistent with the be- 
havior of patients with parietal or frontal lesions. Such lesions causes a syndrome 
known as hemineglect which is characterized by a lack of response to sensory stim- 
uli appearing in the hemispace contralateral to the lesion [3]. According to the 
modular view, the deficit should be behavior dependent, e.g., oculocentric for eye 
movements, head-centered for reaching. However, experimental and clinical studies 
show that this is not the case. Instead, neglect affects multiple frames of reference 
simultaneously, and to a first approximation, independently of the task. 
This point is particularly clear in an experiment by Karnath et al (1993) (Fig- 
ure 1A). Subjects were asked to identify a stimulus that can appear on either side 
of the fixation point. In order to test whether the position of the stimuli with 
respect to the body affects performance, two conditions were tested: a control con- 
dition with head straight ahead (C1), and a second condition with head rotated 
20 degrees on the right (or equivalently, with the trunk rotated 20 degrees on the 
left, see figure) (C2). In C2, both stimuli appeared further to the right of the trunk 
while being at the same location with respect to the head and retina than in C1. 
Moreover, the trunk-centered position of the left stimulus in C2 was the same than 
the trunk-centered position of the right stimulus in C1. 
As expected, subjects with right parietal lesions performed better on the right 
stimulus in the control condition, a result consistent with both, retinotopic and 
trunk-centered neglect. To distinguish between the two frames of reference, one 
needs to compare performance across conditions. 
If the deficit is purely retinocentric, the results should be identical in both condi- 
tions, since the retinotopic location of the stimuli does not vary. If, on the other 
hand, the deficit is purely trunk-centered, the performance on the left stimulus 
should improve when the head is turned right since the stimulus now appears fur- 
ther toward the right of the trunk-centered hemispace. Furthermore, performance 
on the right stimulus in the control condition should be the same as performance on 
the left stimulus in the rotated condition, since they share the same trunk-centered 
position in both cases. 
12 A. POUGET, T. J. SEJNOWSKI 
Neither of these hypotheses can fully account for the data. As expected from a 
retinotopic neglect, subjects always performed better on the right stimulus in both 
conditions. However, performance on the left stimulus improved when the head 
was turned right (C2), though not sufficiently to match the level of performance on 
the right stimulus in the control condition (C1). Therefore, these results suggest a 
retinotopic neglect modulated by trunk-centered factors. 
In addition, Karnath et al (1991) tested patients on a similar experiment in which 
subjects were asked to generate a saccade toward the target. The analysis of reaction 
time revealed the same type of results than the one found in the identification 
task, thereby demonstrating that the spatial deficit is, to a first approximation, 
independent of the task. 
An experiment by Arguin and Bub (1993) suggests that neglect can be object- 
centered as well. As shown in figure lB, they found that reaction times were faster 
when the target appeared on the right of a set of distractors (C2), as opposed 
to the left (C1), even though the target is at the same retinotopic location in 
both conditions. Interestingly, moving the target further to the right leads to even 
faster reaction times (C3), showing that hemineglect is not only object-centered but 
retinotopic as well in this task. 
These results strongly support the existence of spatial representations using multiple 
frames of reference simultaneously shared by several behaviors. We have recently 
developed a theory [6] which has precisely these properties and we ask here whether 
a simulated lesion would lead to a deficit similar to hemineglect. Our theory posits 
that parietal neurons computes basis function (BF) of sensory signals, such as vi- 
sual, or auditory inputs, and posture signals, such as eye or head position. The 
resulting representation, which we called a basis function map, can be used for per- 
forming nonlinear transformations of the sensory inputs, the type of transformations 
required for sensorimotor coordination. 
2 Model Organization 
The model contains two distinct parts: a network for performing sensorimotor trans- 
formations and a selection mechanism. 
2.1 Network Architecture 
We implemented a network using basis function units in the intermediate layer 
to perform a transformation from a visual retinotopic map to two motor maps 
in, respectively, head-centered and oculocentric coordinates (Figure 2). The input 
contains a retinotopic visual map analog to the one found in the early stages of 
visual processing, and a set of units encoding eye position, similar to the neurons 
found in the intralaminar nucleus of the thalamus. These input units project to a 
set of intermediate units shared by both transformations. Each intermediate unit 
computes a gaussian of the retinal location of object, rz, multiplied by a sigmoid of 
eye position, ez: 
_ (-r)  
e 2 ' 
oi = -i (1) 
l+e- 
These units are organized in a map covering all possible combinations of retinal 
and eye position selectivities. As we have shown elsewhere [6], this type of response 
function is consistent with the response of single parietal neurons found in area 7a. 
A Model of Spatial Representations in Parietal Cortex Explains Hemineglect 13 
A B 
$accadic Eye Movements 
oooooooooooooO 
Retinotopic map 
Reaching 
CoooooooooooooO 
Head-centered map 
(Superior Colliculus) (Premotor Cortex) 
, 00000000000000 
00000000000000 
 o, 00000000000000 
 00000000000000 
 s, 00000000000000 
- o, 00000000000000 
 -s, 00000000000000 
00000000000000 
-,s, 00000000000000 
0000000000000 
Retal position (o) 
Retinotoplc ma pOlO 
Cpoooooooddd '' cells 
(V1) halamus) 
BF map 
(7a) 
Retinal position () Head-centered position () 
: BF map 
 (7a) 
Retinal position (o) 
Retinal position (o) 
Figure 2: A. Network architecture B. Typical pattern of activity 
The resulting map forms a basis function map which encodes the location of objects 
in head-centered and retinotopic coordinates simultaneously. 
The activity of the unit in the output maps is computed by a simple linear combi- 
nation of the BF unit activities. Appropriate values of the weights were found by 
using linear regression techniques 
This architecture mimics the pattern of projections of the parietal area 7a. 7a is 
known to project to, both, the superior colliculus and the premotor cortex (via the 
ventral parietal area, VIP), in which neurons have, respectively, retinotopic and 
head-centered visual receptive fields. Figure 2B shows a typical pattern of activity 
in the network when two stimuli are presented simultaneously while the eye fixated 
10 degrees toward the right. 
2.2 Hemispheric Biases and Lesion Model 
Neurophysiological data indicate that both hemispheres contain neurons with all 
possible combinations of retinal and eye position selectivities, but with a contralat- 
eral bias. Hence, most neurons in the right parietal cortex (resp. left) have their 
retinal receptive field on the left hemiretina (resp. right). The bias for eye position 
is much weaker but a trend has been reported in several studies [1]. 
Therefore, spatial representations in a patient with a right parietal lesions are biased 
toward the right side of space. We modeled such a lesion by using a similar bias in 
the intermediate layer of our network. The BF map simply has more neurons tuned 
to right retinal and eye positions. We found that the exact profile of the neuronal 
gradient across the basis function maps did not matter as long as it was monotonic 
and contralateral for both eye position and retinal location. 
2.3 Selection model 
We also developed a selection mechanism to model the behavior of patients when 
presented with several stimuli simultaneously. The simultaneous presentation of 
14 A. POUGET, T. J. SEJNOWSKI 
stimuli induces multiple hills of activity in the network (see for instance the pattern 
of activity shown in figure lB for two visual stimuli). Our selection mechanism 
operates on the peak values of these hills. 
At each time step, the most active stimulus is selected according to a winner-take- 
all and its corresponding activity is set to zero (inhibition of return). At the next 
time step, the second highest stimuli is selected while the previously selected item 
is allowed to recover slowly. This procedure ensures that the most active item is 
not selected twice in a row, but because of the recovery process, stimulus with high 
activity might be selected again if displayed long enough. 
This mechanism is such that the probability of selecting an item is proportional 
to two factors: the absolute amount of activity associated with the item, and the 
relative activity with respect to other competing items. 
2.4 Evaluating network performance 
We used this model to simulate several experiments in which patient performance 
was evaluated according to reaction time or percent of correct response. 
Reaction time in the model was taken to be proportional to the number of time 
steps required by our selection mechanism to select a particular target. Performance 
on identification task was assumed to be proportional to the strength of the activity 
generated by the stimuli in the BF map. 
3 Results 
3.1 Line cancellation 
We first tested the network on the line cancellation test, a test in which patients are 
asked to cross out short line segments uniformly spread over a page. To simulate 
this test, we presented the display shown in figure 3A and we ran the selection 
mechanism to determine which lines get selected by the network. As illustrated in 
figure 3A, the network crosses out only the lines located in the right half of the 
display, just as left neglect patients do in the same task. The rightward gradient 
introduced by the lesion biases the selection mechanism in favor of the most active 
lines, i.e., the ones on the right. As a result, the rightmost lines win the competition 
over and over, preventing the network from selecting the left lines. 
3.2 Mixture of frames of reference 
Next, we sought to determine the frame of reference of neglect in the model. Since 
Karnath et al (1993) manipulated head position, we simulated their experiment 
by using a BF map integrating visual inputs with head position, rather than eye 
position. We show in figure 3B the pattern of activity obtained in the retinotopic 
output layer of the network in the various experimental conditions (the other maps 
behaved in a similar way). In both conditions, head straight ahead (dotted lines) or 
turned on the side (solid lines), the right stimulus is associated with more activity 
than the left stimulus. This is the consequence of the larger number of cells in 
the basis function map for rightward position. In addition, the activity for the left 
stimulus increases when the head is turned to the right. This effect is related to the 
larger number of cells in the basis function maps tuned to right head positions. 
Since network performance is proportional to activity strength, the overall pattern 
of performance was found to be similar to what has been reported in human patients 
A Model of Spatial Representations in Parietal Cortex Explains Hemineglect 15 
A 
B 
C 
#%  
I . $     
/ jx., ' 
+     Cl 
FP Target Distractors 
Stimulus Stimulus -[' O O  )e, C3 
Figure 3: Network behavior in line cancellation task (A). Activity patterns in the 
retinotopic output layer when simulating the experiments by Karnath et al (1993) 
(B) and Arguin et al (1993) (C) 
(figure 1A), namely: the right stimulus was better processed than the left stimulus 
and performance on the left stimulus increases when the head is rotated toward the 
right. Therefore, just like in human, neglect in the model is neither retinocentric 
nor trunk-centered alone, but both at the same time. 
3.3 Object-centered effect 
When simulating Arguin et al (1993) experiments, the network reaction times were 
found to follow the same trends than for human patients. Figure 3C illustrates the 
patterns of activity in the retinotopic output layer of the network when simulating 
the three conditions of Arguin experiments. Notice that the absolute activity asso- 
ciated with the target (solid lines) in conditions I and 2 is the same, but the activity 
of the distractors (dotted lines) differs in the two conditions. In condition 1, they 
have higher relative activity and thereby strongly delay the detection of the target 
by the selection mechanism. In condition 2, the distractors are now less active than 
the target and do not delay target processing as much as they do in condition 1. 
The reaction time decreases even more in condition 3, due to a higher absolute 
activity associated with the target. Therefore, the network exhibits retinocentric 
and object-centered neglect, just like parietal patients [2]. 
4 Discussion 
The model of parietal cortex presented here was originally developed by consider- 
ing the response properties of parietal neurons and the computational constraints 
inherent in sensorimotor transformations. It was not designed to model neglect, so 
its ability to account for a wide range of deficits is additional evidence in favor of 
the basis function hypothesis. 
As we have shown, our model captures three essential aspects of the neglect syn- 
drome: 1) It reproduces the pattern of line crossing reported in patients in line- 
cancellation experiments, 2) the deficit coexists in multiple frames of reference si- 
multaneously, and 3) the model accounts for some of the object-based effects. 
16 A. POUGET, T. J. SEJNOWSKI 
We can account for a very large number of studies beyond the ones we have con- 
sidered here, using very similar computational principles. We can reproduce, in 
particular, the behavior of patients in line-bisection experiments and we can ex- 
plain why neglect affects multiple cartesian frames of reference such as retinotopic, 
head-centered, trunk-centered, environment-centered (i.e. with respect to gravity), 
and object-centered. 
It must be emphasized that these results have been obtained without using ex- 
plicit representations of these various cartesian frames of reference (except for the 
retinotopy of the BF map). In fact, this is precisely because the lesion affected 
noncartesian representations that we have been able to reproduce these results. We 
have assumed that the lesion affects the functional space in which the basis functions 
are defined. This functional space shares common dimensions with cartesian spaces, 
but cannot be reduced to the latter. Hence, a basis function map integrating retinal 
location and head position is retinotopic, but not solely retinotopic. Consequently, 
any attempts to determine the cartesian space in which hemineglect operates is 
bound to lead to inconclusive results in which cartesian frames of reference appear 
to be mixed. 
This study and previous research [6] suggests that the parietal cortex represents 
the position of objects by computing basis functions of the sensory and posture 
inputs. It would now be interesting to see if this hypothesis could also account for 
sensorimotor adaptation, such as learning to reach properly when wearing visual 
prisms. We predict that adaptation takes place in several frames of reference simul- 
taneously, a prediction that is testable and would provide further support for the 
basis function framework. 
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