A Continuous Attractor Neural Network Model of
Divided Visual Attention
Dominic I. Standage and Thomas P. Trappenberg

Raymond M. Klein

Faculty of Computer Science
Dalhousie University
Halifax, NS, Canada
E-mail: {standage, tt}@cs.dal.ca

Department of Psychology
Dalhousie University
Halifax, NS, Canada
E-mail: klein@or.psychology.dal.ca

Abstract— The biologically realistic model of selective visual
attention by Deco et al uses a continuous attractor neural network
to simulate a saliency map in posterior parietal cortex. We test
the ability of the model to explain experimental evidence on
the distribution of spatial attention. The majority of evidence
supports the view that attention is a unitary construct, but recent
experiments provide evidence for split attentional foci. We simulate two such experiments. Our results suggest that the ability to
divide attention depends on sustained endogenous signals fromshort term memory to the saliency map, stressing the interplay
between working memory mechanisms and attention. Our results
also point to a mechanism whereby inhibitory endogenous signals
may play a role in dividing attention, suggesting a possible
mechanism for inhibition of return.

I. I NTRODUCTION
Attention is an old concept in psychology correlated withenhanced processing of objects or regions in space [1]. Whileattention is a multi-modal phenomenon [2], [3], the majority of
research has focused on selective visual attention (SVA). The
limited capacity of the visual system necessitates a mechanism
to select stimuli from the visual field, and Tsotsos pointed
out that attention solves the complexity problem of sensory
processing [4].
A distinction can be drawn between pre-attentive and attentive visual processing [5]. Pre-attentive processing refers
to bottom-up (BU) feature saliency of visual stimuli whereby
items that differ from their surroundings ‘pop out’ to theviewer. Attentive processing refers to top-down (TD) influences on perception of stimuli determined by object andlocational bias such as task instructions or foreknowledge ofstimulus characteristics. Determining saliency, then, is both a
BU and TD requirement, and computational models of SVA
include maps that integrate BU salience across object features[6], TD bias [7], and the interplay of both [8], [9].
Koch and Ullman [6] provide a neural network modelof SVA in which topographical feature maps are integrated
by a winner-take-all (WTA) saliency map of BU stimuli. In
their model, inhibiting the selected location causes a shift tothe next most salient location. Wolfe [8] builds on Neisser’s
pre-attentive/attentive distinction [5], integrating BU and TD
saliency criteria in his Guided Search model. Treisman [7]provides a model of spatial attention to solve the Binding
Problem, in which a TD saliency map determines object

features selected for further processing, and suggests parietal
cortex as the biological correlate of her ‘master’ map. Deco etal [9] use inhibition to mediate BU and TD influences in an
instantiation of Duncan and Humphreys’ biased competition
model [10], simulating saliency in posterior parietal cortex(PP) with a Continuous Attractor Neural Network (CANN).
Spatial saliency in PP interacts with BU feature maps toconverge on a winning location. See [11] and [12] for a reviewof these and other models.
There is long-standing debate about the distribution of SVA.
Many cognitive models propose a unitary focus of attention,
likened to a roving spotlight over the visual field [1]. Variantsof the spotlight metaphor include gradient [13], [14] and zoom
lens [15] models, suggesting that attention may be a gradedphenomenon, attenuated around a central focus. A large bodyof evidence supports such unitary models [1], [16], but several
more recent experiments have provided evidence for noncontiguous allocation of SVA [17], [18], [19].
Here we study how split attention can be achieved by adynamic implementation of a WTA map. Despite their WTA
nature, CANNs are able to account for split attention when
network dynamics facilitate long transition states between
regimes [20] and when dominated by sustained inputs. We
simulate the experiments of Müller et al [19] and Awh andPashler [18] with a CANN model, demonstrating mechanisms
by which the model can (and cannot) account for theirfindings, and make predictions for further study.
II. M ETHODSIn simulations, we used a fully connected recurrent rate
model with N nodes. The average state ui of a node withindex i is given byτX
dui (t)= −ui (t) +wij rj (t)∆x + Iiext (t),
dt
j

(1)

where τ is a time constant, Iiext is external input to the
network, ∆x = 2π/N is a scale factor, and ri is a normalized
square of ui given byg(ui ) =ui 2.
1P
1 + 2 j uj 2 ∆x

(2)

We use this normalization through divisive normalization(shunting inhibition) to force more biologically realisticsmooth (Gaussian) bubbles [21].
Weight matrix w is determined by Gaussian function22

wij = Aw e−d /2σw − C

for all connections to node i from node j (for all i, j
where d is given by
d = min(|i − j|∆x, 2π − |i − j|∆x).

(3)
N ),
(4)

C is an inhibition constant describing the activity dependent
inhibition of an inhibitory pool of neurons, and Aw is a scale
factor.
External input Iiext is Gaussian shaped around input locationj, determined by
2
2(5)
Iiext = e−d /2σext
where d is given by Equation 4.
In all simulations, N = 100, C {0.1, 0.3}, Aw = 10,t = 1, τ = 10, and σw , σext {0.2, 0.4, 0.8}.
We classify our inputs along exogenous (exo) and endogenous (endo) dimensions. Exo inputs refer to neural responses
to stimuli, here representing visual cues. Endo inputs refer tovoluntary control of attention, here representing task instructions in behavioural studies.
Simulations are run with transient and sustained inputs.
Under both paradigms, we equate network activity with SVA.
Because transient inputs elicit WTA behaviour in CANN
models, we start by demonstrating one-bubble attractor states
as models of a unitary attentional focus. Transient input stimuli
are the norm in biological networks, as evidenced by highfiring rates at stimulus onset followed by lower rates whenstimuli are sustained in experimental settings. In the exo case,this initial burst of activity serves as input to higher corticalareas such as PP. Sustained firing after transient stimulation isa property of highly specialized neural assemblies [22], and assuch is the exception among biological networks, not the norm.
We interpret sustained endo inputs as STM representations oftask instructions in PFC.
In simulations with sustained inputs, we model exo inputs astransient when stimuli in behavioural experiments are stable,
as described above, and model these stimuli as sustained
when the features of spatially static stimuli are changing. We
interpret these changes as providing continual ‘refreshment’ ofneural representations due to novelty effects [23]. Endo inputs
are always sustained in this set of simulations.
We use a Gaussian shaped input profile to approximate
typical tuning curves of neurons, so their firing profiles arewell approximated by smooth curves. In the case of transient
input, the specific shape of localized input is unimportantbecause the network dynamic dominates after cessation of
input. With sustained input, a Gaussian input profile leads toa good approximation of a Gaussian output profile, achieving
the biological realism of our input profile described above.
We compare our CANN model of SVA to one with no lateralinteraction, modelling the latter by simply adding together its
Gaussian inputs.

III. S IMULATIONS
Müller et al [19] provided evidence for sustained divisionof visual attention by recording steady state visual evoked
potentials (SSVEP) while subjects attended to two of fourcontiguous horizontal locations. The SSVEP is the electrophysiological response in visual cortex to a rapidly flickering
stimulus, and has been shown to increase in amplitude whenattention is paid to the location of the stimulus [19]. They
found that target detection rates (TDR) and SSVEPs werelower at the location between separated targets in a detection task. Additionally, they showed that split locations wereattended just as well as adjacent locations in their experiment.
We model this experiment in Simulations 1 and 2.
Awh and Pashler [18] used a partial report procedure to test
subjects’ ability to divide spatial attention. Subjects viewed a
5x5 array of alpha-numeric characters containing 23 lettersand 2 digits. They fixated at a central location before thepresentation of two cues indicating the probable location of thedigits. The character array was then provided and the subjects’task was to identify the digits. Digits appeared between thecued locations on twenty percent of trials and performance at
the cued and intervening locations was compared. If SVA canbe divided, subjects should perform better at the cued locations
than in between. Subjects’ ability to divide SVA was foundto depend on the presence of a subsequent noise mask, but
the removal of array noise alone was sufficient to significantly
reduce division of attention, regardless of subsequent masking.
We model this work in Simulation 3.
A. Simulation 1
Adopting Müller’s terminology, we refer to the locationsof stimuli as 1, 2, 3 and 4, where 1 is the left-most locationand 4 is the right-most location. A 1+2 trial refers to trials
in which subjects were instructed to direct their attention tolocations 1 and 2, a 2+4 trial refers to instructions to attend tolocations 2 and 4, and so forth for other combinations of thefour locations. We equate activity in the CANN with spatialattention.
The network was given exo and endo inputs for 300
iterations of dt/τ, followed by 300 iterations without either
source of input. These times were sufficient for the network tostabilize under both dynamic regimes (both during and after
input). The input profile, final network state, and the network
state over time are shown in Figure 1A, B, and C respectively.
In 1+2 and 3+4 (adjacent) trials, bubbles corresponding to
the attended locations merge into a single bubble, predicting aunitary focus of attention maximal between the two locations.
In 1+3 and 2+4 (split) trials, one bubble wins at the expenseof the other, predicting that only one of the locations should
be attended. These results conflict with Müller’s findings, andare shown in Figure 1.
B. Simulation 2
Network configuration and the shape and location of inputs
is identical to Simulation 1. Exo and endo inputs are sustained

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Fig. 2. (A) Input profile as described in Figure 1A. (B) Network activity
conforms to the sustained input profile, where lateral inhibition sharpens
the areas between peaks. For clarity, input (dashed) and output (solid) arenormalized to 1 in the figure. (C) Network activity over time for sustained
1+2 trial. (D) 1+3 input profile as described in Figure 1D. (E) Activity profile
conforms to 1+3 sustained input. (F) Network states over time.
Time

Fig. 1. (A) Nodes 20, 37, 54 and 71 correspond to locations 1, 2, 3 and 4
respectively. Exo inputs are applied to all locations. Endo inputs are applied
to locations 1 and 2 only. Combined exo and endo input activity shown atbottom. Gaussian width factors σw = 0.4, σext = 0.2. (B) Stable bubble
following transient input. The bubble is centred on node 29, reflecting themerge between locations 1 and 2. (C) Network activity over time. Input is
stopped after 200 iterations, followed by transition to a one bubble (merged)
state by approximately 250 iterations. (D) All parameters are identical to
A except endo inputs are applied to locations 1 and 3. (E) Stable bubble
following transient input, centred on node 54 (location 3). (F) Network statesover time. Location 3 wins by approximately 250 ms.

simultaneously for 500 iterations, sufficient for the network tostabilize.
With sustained inputs, our model replicates Müller’s findings for both adjacent and split trials, as shown in Figure 2A.
Results show that network activity conforms to the shape of
the inputs, though areas receiving only exo inputs are partiallysuppressed by lateral inhibition from more excitatory areas
receiving both exo and endo inputs. Despite this sharpening of
the activity profile, the network activity under sustained inputs
resembles a simple addition of Gaussian curves (AOG).
To test if an AOG provides a model of CANN behaviour
under sustained inputs, we measured the reduction or ‘dip’ inactivity between bubbles at different distances between inputs,
comparing it to peak activity in the bubbles. Correspondingly,
we measured the height of the midpoint between two Gaussian
curves as a function of the distance between them, comparing
it to their maxima. The width of CANN bubbles is matchedto the width of Gaussians in AOG. The shape of these curves
is similar, but the AOG is shifted. Unfortunately, without ameans to relate distance to physical parameters, this shift
does not help to distinguish one curve from the other. Bothmodels predict that SVA should be easier to divide as the
distance between attended locations is increased, and it would
be difficult to choose one model over the other if behaviouraldata were to validate this prediction. Results are shown in
Figure 3A. As models of brain networks, the CANN suggests
a lateral interaction profile, while the addition of Gaussians
suggests a purely feed-forward topographic organization.
Because the width of Gaussian inputs effects the curves in
Figure 3A, we plot these curves for different values of σext
in Figure 3B. This figure shows curves for the AOG case.
The same effect was observed for CANN bubbles (results
not shown). We equate the overlap between Gaussians withthe overlap between neural representations driving the input
signals, and predict that the more stimuli overlap in space (orfeature space) the harder it is to divide SVA.
C. Simulation 3
Network configuration is identical to Simulations 1 and 2
except for σw = 0.8 and C = 0.3. The number and locationof inputs are adjusted to reflect experimental conditions (described in Figure 4). Inputs are centred on nodes 10, 20, 30,
40 and 50, hereafter referred to as locations 1, 2, 3, 4 and
5. Exo inputs at locations 1, 3 and 5 represent letters (arraynoise) in Awh and Pashlers’ study. Exo inputs at locations
2 and 4 represent digits. Endo inputs at locations 2 and 4
represent task instructions to attend to these locations. In thesesimulations, we model Awh and Pashlers’ experiments only
as far as the presentation of the character array. The effectof noise masks and subsequent identification of target digits
presumably involve STM and object recognition processing
not included in our model.
We provide an exo input to location 3 for 200 iterations,
simulating subjects’ fixation between the target locations. This

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Fig. 3. (A) Dip between CANN bubbles (max. - intervening min.) plottedagainst the distance between peaks (solid line). Dip between peaks of an AOG
plotted against distance between them (dashed line). (B) AOG distance vs.
dip (as in (A)). From left to right, σi = 5, 10, 15, 20 .

input is followed by 200 iterations of the same exo input, andendo inputs at locations 2 and 4. We then provide simultaneous
exo inputs to all locations (the full character array) and endo
inputs to locations 2 and 4 (the attended locations) for 100
iterations. These numbers of iterations are sufficient to modelthe duration of input screens in Awh and Pashlers’ study.
Our results replicate those of Awh and Pashlers’ Experiment
1. The sustained endo inputs dominate the network, facilitating
the two bubbles shown in Figure 4A.
In Simulation 3b, we model Awh and Pashlers’ Experiments
4 and 4a, simulating removal of non-target characters byremoving all exogenous input except at locations 2 and 4. Our
results do not coincide with those Awh and Pashler. Without
exogenous stimuli at locations 1, 3 and 5, bubbles are justas cleanly differentiated, predicting divided attention whereAwh and Pashler found greatly reduced division of attention
in Experiment 4, and unitary attention in Experiment 4a.
IV. D ISCUSSION
The parameter dependence of our model raises several
issues. We use a normalised square gain function to createsmooth output. In Simulation 1, bubbles at attended locations
are sustained until cessation of input, after which a single
winner dominates. In adjacent trials, the bubbles at input
locations merge, peaking between the two centres of input.
This activity profile suggests that attention should be maximal

0.8
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Fig. 4. (A) Nodes 10, 20, 30, 40 and 50 are referred to as locations 1, 2,
3, 4 and 5. Locations 1, 3 and 5 represent locations of letters. Locations 2
and 4 represent locations of digits. Exo input is applied to location 3 for 200
ms (fixation) followed by 200 ms of exo input (location 3) and endo inputs
at locations 2 and 4. Exo inputs are then applied to all locations, while endo
inputs are applied to locations 2 and 4 for 100 ms. Sustained endo inputs
dominate, predicting divided attention in Awh and Pashlers’ Experiment 1.
(B) Exo and endo inputs at locations 2 and 4 only (no array noise). The
CANN incorrectly predicts sharply divided attention in Awh and Pashlers’Experiments 4/4a.

between the two adjacent attended locations, tailing off oneither side per the gradient model. Our choice of parameters
leads to a steep curve, such that the instructed locations arepoorly attended. This effect can be manipulated by adjustments
to σw and σext , but is more readily overcome by the use of a
stiffer gain function (eg Sigmoid). In this case, output becomesflatter and sharper, predicting less of an attentive gradient.
Our choice of gain function also effects Simulation 2. Ourmodelling of adjacent trials reveals a dip in activity between
foci, suggesting that the subjects of Müller et al may havedivided their attention between adjacent locations. BecauseMüller et al didn’t test subjects’ attention between adjacentstimuli, this effect doesn’t conflict with their results, but it
further exposes the model’s parameter dependence. Flatterbubbles would cancel the effect. The model predicts that
attention should be easier to divide as foci become moredistant (within a reasonable visual area). With stiffer gain, themodel still predicts divided SVA under sustained input, but inall-or-none fashion. As such, the model no longer resembles
the AOG model shown in Figure 3.
A second parameter that effects our simulations is thestrength of connectivity in the network. By reducing theratio of input strength to connection strength from 1 to 0.1,
connectivity dominates sustained input and the WTA natureof the model re-emerges. In this case, model behaviour undersustained input resembles behaviour under transient input anda flatter bubble may again be desirable.
Despite the difficulties raised by these parameter dependencies, they suggest an encouraging symbiosis between themodel and behavioural and physiological experiments. Resultsof Simulation 1 predict that in adjacent trials, attention should
be maximal between instructed locations in Müller’s experiments, whereas Simulation 2 predicts divided attention in
adjacent trials. Results of Simulation 2 predict attention shouldbe easier to divide in a 1+4 trial than in 1+3 and 2+4 trials.
These predictions could readily be tested with a probe stimulus
in the former case, and by adding a 1+4 test in the latter. Theirresults may help tune the parameters.
Regardless of configuration issues, results of Simulations
2 and 3 show that the CANN model is able to account for
divided SVA under sustained inputs. As such, we believe that
divided SVA may be possible for as long as endo and/or exo
signals are provided to PP, and that differences in behavioural
findings may reflect differences in experimental conditions
rather than subjects’ ability to divide their attention. Thisconjecture echoes that of Schneider [24] that different experimental paradigms may facilitate measurements of different
attention-related phenomena.
Figure 3A shows that the more our inputs overlap, themore readily a single bubble dominates. By increasing the
spatial distance between locations, we reduce the overlap
and facilitate multiple bubbles. By extending this principle
to feature space, we conjecture that the overlap between
integrated object representations should increase as a functionof their number of common features. For example, neuralrepresentations of a red circle and a red square should overlap
more than a red circle and a blue square, and attention should
be more difficult to divide between them.
Our focus has been on stable attractor states in this paper.
Transitions between dynamic regimes tend to be rapid, andgiven the large number of parameters that effect the model,
stable states provide a better foundation for our simulations.
As we show in [20], parameter adjustments effect transitions
between regimes. A possible explanation of the findings ofMüller et al is that divided SVA on split trials corresponds tothe transition between two-bubble and one-bubble states in a
WTA model. If so, subjects should be able to divide attention
for the duration of these meta stable states. Given more time,
subjects may have been unable to divide their attention in
Müller’s experiments.
Awh and Pashlers’ experiments are difficult to explain with
the CANN model. Awh and Pashler found that division of SVA
was greatly reduced following removal of noise surrounding
target stimuli. We believe the model can account for this resultin two ways. Firstly, Awh and Pashlers’ methods of probing
subjects’ attention may have interfered with its spatial allocation. Secondly, if the overlap between inputs is sufficiently

large to create a single winning bubble under sustained inputs,
inhibitory inputs may be able to divide the bubble, accounting
for results of all experiments in their study. One interpretationof such negative input is that subjects were ‘unattending’ to thenoise. We believe that purposefully ‘unattending’ intervening
locations is a possible strategy for dividing SVA, rather thanpurposefully attending to locations of interest. In Awh andPashlers’ experiments, it’s possible that subjects were able toignore invalid locations only when there was something tobe ignored. Removal of noise may have removed their meansof doing so. The use of negative inputs further suggests thepossibility that such activity could be involved in inhibitionof return (IOR), whereby the currently selected location in asaliency map receives or produces inhibitory activity, allowing
visual search to move to other salient locations.
Because we interpret endo inputs as subjects’ representations of task instructions in WM, our model predicts thatinterference with STM should abolish split attention in bothMüller’s and Awh and Pashlers’ experimental conditions. Thisprediction could be tested in a dual task paradigm. By equatingtask instructions in behavioural studies with STM representations in WM, and by modelling these representations assustained endo inputs to the CANN, we revisit the relationshipbetween WM and attention. Attention has often been cited as
the primary constraint on WM capacity [25], but here we view
WM representations as the driving force behind attention.

V. C ONCLUSIONS AND F UTURE W ORK
The model of SVA by Deco et al [9] implements a saliency
map in PP with a CANN network. This instantiation of biased
competition [10] integrates BU and TD influences in a biologically realistic computational architecture. Our simulations
test this promising model on its ability to explain behaviouraland physiological evidence on the spatial distribution of SVA.
Our results demonstrate that CANNs provide a model ofspatial attention in PP capable of explaining divergent experimental findings. With transient inputs, the model’s WTA
nature predicts a unitary attentional focus. With sustained
inputs, the model accounts for divided SVA. As such, ourpredictions depend on the nature of exo and endo signals
in attentive phenomena. Here, the use of sustained inputs
replicates the findings of Müller et al [19]. The findings of
Awh and Pashler [18] are difficult to reconcile with the CANN
model, but they suggest a role for inhibitory mechanismsthat may be modelled by negative inputs. The possibility thatnegative inputs provide a mechanism to simulate IOR in themodel warrants further research.
The interplay between WM and SVA is paramount to ourmodel. If divided attention is facilitated by STM representations providing endo inputs to PP, then disruption of STM
should abolish divided attention. We believe further research inthis area would improve our understanding of the relationshipbetween WM and attention.

ACKNOWLEDGMENT
This work was supported in part by the NSERC grant
RGPIN 249885-03.
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