-
Vision Res. Vol. 34. No.8. pp. 1089-1104. 1994
CoPYlright ~ 1994 Elsevier Science LtdPrinted in Great Britain.
All rights reserved
0042-6989/94 $6.00 + 0.00
An analysis of the reset of visual cortical circuits responsible
for the binding or segmeilltation of visualfeatures into coherent
visual forms yields a model that explains properties of
visuall~rsistence. Thereset mechanisms prevent massive smearing of
visual percepts in response to rapidly moving images.The model
simulates relationships among psychophysical data showing inverse
relatiollS of persistenceto flash luminance and duration, greater
persistence of illusory contours than real cont()iurs, a
U-shapedtemporal function for persistence of illusory contours, a
reduction of persistence due to adaptation witha stimulus of like
orientation, an increase of persistence due to adaptation witlll a
stimulus ofperpendicular orientation, and an increase of
persistence with spatial separation of a m:asking stimulus.The
model suggests that a combination of habituative, opponent, and
endstopping meclflanisms preventsmearing and limit persistence.
Earlier work with the model has analyzed data about
boundaryformation, texture segregation, shape-from-shading, and
figure.-ground separation. ThIIJS, several typesof data support
each model mechanism and new predictions are made.
Vision Neural networks Visual cortex Visual persistence Feature
binding Illusory co~ Off-cells
INTRODUcrJON test stimulus decreases; but when subjects
adaptHumans and other animals form useful visual represen- to a
stim~lus of a pe~ndicular orienta~ion totations of rapidly changing
scenes, The visual system ~he test stimulus, persistence of the
test stimulusrapidly resets the segmentations of changing parts of
a Increases, , .scene to prevent image smearing, This article
explains .The subsequ.ent on~et of a maskIng s~lmulushow a neural
network theory of early visual processes greatly curtails
persistence of a target stimulus.
proposed by Grossberg and Mingolla (1985a, b, 1987) Before
presenting the details of model mechanisms, weaccounts for many of
the data on visual persistence, The briefly describe how the model
addresses each of thesetheory suggests that a key process governing
these data data sets,is the time taken to reset a segmentation, We
simulatereset dynamics that help to force a rapid return of the
Inverse relation of persistence to luminance and tonetwork to a
state unbiased by prior segmentation in stimulus durationorder to
better process incoming data, We explain how Figure I (a), taken
from Bo,wen, Pola and Matinhysteresis in the segmentation network
is a rate-limiting (1974), shows that, for each luminance curve,
persistencefactor in visual persistence, and show that properties
of is inversely related to stimulus dll1ration. Except for verythe
hysteresis match key psychophysical data. Psycho- short stimulus
durations, persi:>tence is also inverselyphysical studies of
visual persistence have revealed four related to stimulus
luminance, Similar results have beenkey sets of data, which are all
explainable by the model, found by many ~uthors (see Coltheart,
1980; Breitmeyer,
P ' '0 I I d . I 1984 for reviews)..erslstence IS Inverse y re
ate to stlmu us I h d f B ' (1974) b'd t' d t t . I I ' n t e stu y
0 owen et aj. , su ~ects were
ura Ion an 0 s Imu us ummance, "III t . t h I th I asked to
match the perceived offset of a target stimulus
.usory con ours persIs muc onger an rea, 0 0t d ' II t d t b th
with the perceived onset of ~i probe stimulus, The
con ours an I usory con ours 0 no 0 eye, , .,, I t ' h' b t ' t
d physical mterstlmulus Interval Ibetween the target andInverse re
a Ions IP e ween persIs ence an , ...t o I d t. h t ' t ' f I .mask
stimuli provided a measure of the target's persist-S Imu us ura Ion
c arac ens IC 0 ummance- L d G'
ld (1981) d h .
db d t ence, ong an I ea argue t at perceive
ase con ours. ffi ' d f .bWh b . t d t t t . I f th 0 set IS not
a goo measure 0 persistence ecause some
.en su ~ec s a ap 0 a s Imu us 0 e same ..," " parts of the
stimulus may continue to persist beyond the
onentatlon as the test stimulus, persistence of the 0 d t' I ffi
t S k' d L (1979) dperceive S Imu us 0 se, a III an ong an~-~ Long
and McCarthy (1982) sho,wed that when subjects
were told to attend to any resid"lal trace of the stimulus,and
not just perceived offset, the duration of totalpersistence was
directly related to stimulus luminance,
1089
.Center for Adaptive Systems and Department of Cognitive
andNeural Systems, Boston University, III Cummington Street.Boston,
MA 02115, U.S.A.
tTo whom all correspondence should be addressed.
GREGORY FRANCIS,* STEPHEN GROSSBERG,*t ENNIO MINGOLLA*
Received 4 August 1992; in revised form 8 February 1993; in
final form 15 June 1993
-
1090 GREGORY FRANCIS el 01.
(a)Stimuli(f'~f"ltC?'i) ..
~~...,.,~C)D....x D 0
400
uUII:
~UUCU
-.;:.~u~
Measures of total persistence have, in turn, beencriticized as
being the result of afterimages or iconicmemory (Coltheart, 1980;
Breitmeyer, 1984; DiLollo,1984). Perceived offset and total
persistence are thusdifferent features of the dynamic processing of
achanging stimulus. In this paper we model persistencedata based
upon perceived offset, and we do not considerthe properties of
total persistence until the conclusion.When we refer to persistence
we mean the time betweenphysical offset and perceived offset.
The inverse relationships between persistence andstimulus
duration and luminance imply that persistencecannot be modeled as a
simple decay of activity of someneural stimulus representation. The
initial strength ofsuch a representation at the moment of stimulus
offsetwould presumably increase with stimulus duration orluminance,
yielding a higher starting point from whichdecay would begin, and
thus longer persistence. FigureI(b) (solid lines) demonstrates that
persistence of signalsin the model is inversely related to stimulus
duration and
200
...."/~..- -. _ -::"::':':"::":::"'::'~- -
--
---c --
~" -.,-, -~ .~.'~~8:::,=;::,-=. ~ -:--=-. ~ ,~
I I I I I I I I I I I j'
100 200 300 400 500 600 700 800 9001000
50 250
Target duration (msec)
(b)350
Slimulir.. D..
.0
'U'u'"E
300
"
.~..~
£9_-
(a) Subject AC
:-A
~
~ 250-;=
.~ 200~.,...c 150..~
"Vi~ 100~
')--o.."""""oo ~:::::: ~ SO I 1~~~~~1~-t-"-i'0 100 200 300 400
500 600 700 800 900 1000
Target duration (simulated msec)
FIGURE 2. (a) Illusory contours persist longer than real
contours.Persistence of illusory contours is maximal at an
intermediate durationof the stimulus. [Reproduced with permission
from Meyer and Ming(1988).) (b) Computer simulation of real and
illusory boundarycontour persistence as a function of flash
duration. The boundariesproduced in response to the illusory
contours persist longer than theboundaries produced in response to
the real contours. Persistence ofillusory contours peaks at an
intermediate stimulus duration. as in thedata. The solid lines
connect the points sampled in the data of Meyer
and Ming (1988).
.c'"co;;:
,...u0
u 0
eQ.~ uOj
...'"~'-u'--0"'.c.~ '"u-~'-~
:?
.EwtI)~
TeSI flash luminancc
0 + 0.45 log fl -L
.-0.49 log fI .L
.-0.82 log fI -L
6-1.lllogfl-L.-1.42 log fI .L
I I I
100 200 300
Test flash duration (msec)
(b)350 r-
~ 300'"u
E
2: -----==~==~==~. +--If- + 0.45 log fl .LII Simulaled ICSI
.-0.49 log fl -L; flash luminance. -0.82 log fl -L
A-I.lllogfl-L-.1.42 log fl -L-8-
luminance at all but the shortest durations. The modelachieves
this close match with the psychophysical databy generating an
active reset signal at stimulus offsetwhich inhibits the persisting
signals~ of the stimulus(Grossberg, 1991). We later quantitatively
analyze howthe strength of the reset signal increases with
stimulusduration and luminance.
150 :==--6--~---6 ~'-'100
Parado.\'ical increase of persistence of illusor)' contours
Figure 2(a) (taken from Meyer & Ming, 1988) showsthat
illusory contours have different persistence proper-ties than
contours defined by luminance edges. Not onlydo illusory contours
persist substantially longer than realcontours, but persistence of
an illusory contour peaks atan intermediate stimulus duration. In
contrast, thepersistence of a stimulus defined by luminance
edgescontinually decreases as stimulus duration increases.These
data place strong constraints on the source of
so
! I I I I I I I I0 ' , .., ., ,
0 SO 100 ISO 200 2S0 300 3S() 400
Test flash duration (simulated msec)
FIGURE I. (a) Persistence is inversely related to flash
luminance andflash duration. [Reproduced with pennission from Bowen
1'101. (1974).)(b) Computer simulation of boundary signal
persistence as a functionof flash duration and flash iuminance.
Dashed lines simulate modelperfonnance ,,'ithout habituative
transmitter gates that fonn the basis
of the reset mechanism required to explain data on
persistence.
300
3UU I
200
100
-
CORTICAL DYNAMICS OF BINDING AND RESET 1091
signals used to reset a changing visual segmentation. Inour
model, only changes in luminance-derived edgesgenerate reset
signals. Thus, figural boundaries that in-clude illusory contours
persist longer than contours ofcorresponding length that are
defined entirely by lumi-nance edges, because the former contain
luminance edgesas a smaller proportion of the total contour. Figure
2(b)shows that the model's responses to illusory contourspersist
longer than real contours. Persistence of illusorycontours in the
model is not inversely related to stimulusduration at short
durations because illusory contourstake some time to fully develop
(Reynolds, 1981) and, ifnot fully developed, they can more quickly
disappear.
(a)ISO
dp
";;..'"~..
uc.,;;.~.,=-
--
Effects of orientation-specific adaptationFigure 3 (hatched
bars) shows that adaptation to
stimuli can also influence persistence duration (Meyer,Lawson
& Cohen, 1975). This figure demonstrates thatwhen subjects
adapted to a stimulus of the sameorientation as the test stimulus,
persistence of the teststimulus decreased; but when subjects
adapted to astimulus of a perpendicular orientation to the
teststimulus, persistence of the test stimulus increased. Ineach
case, persistence could be changed by as much as:i: 20 msec. These
data provide two clues about thehypothesized reset signal. First,
it suggests thatadaptation or habituation drives the reset signal.
Second,it indicates that opponent interactions between
pathwayssensitive to opposite orientations regulate the
inhibitionthat forms the reset signal. Below we explain how aneural
circuit consistent with these observations cangenerate a transient
response at stimulus offset that actsas a reset signal. Figure 3
(shaded bars) shows thatadaptation influences persistence of
signals in the modelin the same way revealed by psychophysical
studies.
I I I I I
0.12 0.24 0.36 0.48 0.72
Spatial separation (6x) in deg
(b)
200
U'OJ'"e~OJ
";-;e~OJUCOJ~
";n0;eo
40
'-'uCD
~'0..'"'c.."Vi-OJ
:;C-
-:..CDCIV
.:U
so
20
0
-20
-40I I I I I
0.12 0.24 0.36 0.48 0.72
Spatial separation (Ax) in simulated deg
FIGURE 4. (a) The persistence of thin lines moving in
stroboscopicmotion depends on the spatial separation between
successive images.[Reproduced with permission from Farrell el 01.
(1990).) (b) Computersimulation of boundary signal persist'~nce as
a function of the spatialseparation between contours of a target
and a mask. The dashed linesimulates model performance without the
spatial competition. Notethat the size of our simulation plan'~ did
not permit testing spatial
separations larger than shown.
Same Orthogonal
Orientation of adaptation stimulus
FIGURE 3. Hatched bars: change in persistence depending
onwhether the adaptation stimulus had the same or orthogonal
orien-tation as the test grating. [plotted from data in Meyer el
al. (1975).]Shaded bars: computer simulation of boundary signal
persistencedepending on whether the adaptation stimulus had the
same or
orthogonal orientation as the test grating.
100
50
~- D -Ct -0- -0--0,IIII 8
8
: I'I 8150
100
-
1092 GREGORY FRANCIS et at.
Shortening of persistence by a spatially proximal mask
Figure 4(a) from Farrell, Pavel and Sperling (1990)shows that
the influence of a mask on the persistence ofa target depends on
the spatial distance between thestimuli, with closer masks
decreasing the persistence ofthe target. Other studies (Farrell,
1984; DiLollo &Hogben, 1987) have found similar results.
Mostresearchers interpret this result as being due to
spatialinhibition, which prevents smearing of moving stimuli.The
model contains this type of inhibition and Fig. 4(b)(solid line)
demonstrates that the persistence of signalsin the model correlates
well with the psychophysicaldata.
The model that we use to simulate persistence datawas originally
developed to explain many other types ofpsychophysical and neural
data, such as data aboutboundary completion, illusory contour
fonnation, tex-ture segregation, shape-from-shading,
three-dimensionalfigure-ground pop-out, brightness perception,
andfilling-in of three-dimensional surface percepts(Grossberg,
1987a,b, 1994; Grossberg & Mingolla,1985a, b, 1987; Grossberg
& Todorovic, 1988). Modelmechanisms have also been derived from
several basicprinciples about visual infonnation processing
+-
-
-~.:!: '/ '-/
/ """"""" ~.~~ +. -
-
CORTICAL DYNAMICS OF BINDING AND RESET 1093
not implement the inhibition in a specific architecturecapable
of explaining the properties of visual persistence.
Simulations of the model generated Figs l(b), 2(b), 3,and 4(b)
with a single set of parameters (except where themodel is
intentionally "dissected" for analysis). All theequations and
parameters governing the model behaviorare described in the
Appendix. These simulationsdemonstrate the model's competence to
explain qualitat-ive relationships between persistence and
variousstimulus qualities. The simulations do not in every
caseprovide a quantitative match with the psychophysicaldata. Why
this is so is discussed in the concludingremarks.
(a)
(b)
FEATURE BINDING AS A SOURCE OF VISUALPERSISTENCE
In this section we describe how the boundary contoursystem (BCS)
(Grossberg & Mingolla, 1985a, b, 1987)addresses the
psychophysical properties of visual persist-ence. Figure 5
schematizes the model, with each cell'sicon drawn to indicate its
receptive field structure.
We base our first observation on the adaptationstudies of Meyer
et al. (1975), which show that persist-ence can be specifically
modified according to stimulusorientation. These data are explained
using model cellsthat locate and represent oriented boundaries.
Thus,although cells in the first level contain
unorientedcenter-surround receptive fields, the remaining cells
inthe network respond best to a boundary of a specificorientation.
In agreement with neurophysiological dataon receptive field
properties in visual cortex (Hubel &Wiesel, 1965), the first
level of oriented cells in the BCScorrespond to cortical simple
cells. These cells respondto oriented luminance edges of a specific
polaritycentered at a particular point on the retina. Cells in
thenext level, corresponding to complex cells in visualcortex,
receive rectified inputs from like-oriented simple
with distance. Grossberg and Mingolla (1985a, b) callthis
process theftrst competitive stage. This competitionoccurs across
positions in the direction of preferredorientational tuning
(endstopping) as well as acrosspositions lateral to the preferred
direction. Activitiesfrom the first competitive stage feed into a
process ofcross-orientation inhibition at each position called
thesecond competitive stage, which partially accounts for
theopponent processing implied by the results of Meyer etal. (1975)
in Fig. 3.
Signals surviving the competitive stages input tocooperative
bipole cells that are sensitive to more globalproperties of the
configuration of image contrasts.
-
1094 GREGORY FRANCIS el al.
Figure 5, for example, shows a horizontally tuned bipolecell,
which received excitatory inputs from a horizontalrow of
horizontally tuned cells and inhibitory inputsfrom vertically tuned
cells at the same locations. Thisorientation-specific inhibition
helps to explain the orien-tation-specific data of Meyer et al.
(1975) in Fig. 3.Grossberg and Mingolla (1985b) showed that this
inhi-bition provides the network with a property of
spatialimpenetrability, which prevents boundary linkings
fromforming across intervening boundaries of roughlyperpendicular
orientation. Every bipole cell has twoindependent lobes to its
receptive field, and each lobemust receive a sufficient amount of
excitatory input fromthe second competitive stage for the bipole
cell togenerate a response. A bipole cell triggers a responseonly
if its receptive fields are each stimulated by one ormore boundary
components. For example, a bipole cellwhose receptive field center
is at a corner of a boundarycannot generate a response because the
contourstimulates only one side of its receptive field. On the
other hand, a bipole cell centered between two illusorycontour
inducers as in, say, a Kanizsa square, willgenerate a bipole
response if the inputs are sufficientlystrong. In this fashion,
parallel arrays of bipole cells cangenerate an illusory contour.
von der Heydt, Peterhansand Baumgartner (1984) have found evidence
support-ing the existence of bipole cells in area V2 of
monkeycortex.
Bipole-to-hypercomplex feedback carries out a spatialsharpening
process similar to the first competitive stage.Here, each bipole
cell feeds back on-center, off-surroundsignals to hypercomplex
cells of the same orientationsensitivity. Spatial sharpening is
another part of themodel that correlates with t.he spatial
inhibitionindicated in Fig. 4(a).
The positive feedback from the bipole cells to lowerlevel
hypercomplex cells provides the rate-limitingsource of persistence
in the modc~l because the activitiesgenerated by the feedback from
one bipole cell can exciteparallel arrays of other bipole cells,
which in turn feed
(b)(a)
(c) (d)
FIGURE 7. (a) Stimulus input to the system, a bright square on a
dark background. (b) Boundary response to the squareshortly after
the input returns to the background level. (c) Boundary signals
start to erode from the comers of the square toward
the middle or the contours. (d) Boundary erosion is almost
complete.
-
CORTICAL DYNAMICS OF BINDING AND RESET 1095
back signals that can excite the original bipole cell,
asindicated in Fig. 6(a). A self-sustaining feedback loop
isgenerated by the cooperative interactions of thepathways marked
by heavy lines. Thus, within themodel, persistence is due the
positive feedbackinteractions that choose a coherent boundary
segmenta-tion from among many possible groupings, and
inhibitpotential groupings that are weaker. This positive feed-back
loop causes hysteresis that is controlled by othermodel
mechanisms.
For example, as indicated in Fig. 6(b), the end of asurface
contour does not support these feedback inter-actions. In this
figure the leftmost boundary signal ofFig. 6(a) has been deleted.
Without this boundary signal,the left bipole cell does not fire
because only one side ofits receptive field receives stimulation.
Upon stimulusoffset, the effect of this organization is that, even
withoutan active reset signal, the boundary signal at the end ofa
contour passively decays away and exposes a newcontour end [as in
Fig. 6(a,b)]. At this new contour end,the process repeats itself
for another bipole cell and soon from the ends of the surface
contour inward towardthe middle of the contour [as in Fig. 6(b,c)].
It is thisinward erosion of boundary signals that we correlatewith
the persisting visual percept beyond stimulus offset.In the
simulations described below, we show that, with-out an active reset
signal, this passive erosion is insuffi-cient to model the
psychophysical data on persistence.
Figure 7 summarizes a simulation of boundary signalerosion.
Figure 7(a) shows the stimulus presented to
2.5
2
1.5
~~~~9(I)
~~~0~
05
0
FIGURE 8. Time plot of boundary signal activities at a
cross-section of Fig. 7 (lower horizontal contour). The signals
dropmarkedly upon stimulus offset as the bottom-up input is
removed. The top-down signals maintain the boundaries for a
substantia! length of time. but the boundaries erode away from
the ends toward the middle of the contour.
the system, a bright square on a dark background.Figure 7(b-
-
GREGORY FRANCIS el al.1096
FIGURE 9. At stimulus offset, a gated dipole circuit produces a
transient rebound of activity in the non. stimulated opponent
pathway.
a transient rebound of activity in the previously non-stimulated
pathway.
The time plot next to each cell or gate describes thedynamics of
this circuit. In the case shown, the sharpincrease and then
decrease of the time plot at the lowerright of Fig. 9 indicates
that an external input stimulatesthe horizontal pathway. In
response to the strongersignal being transmitted to the next level,
the amount oftransmitter in the gate inactivates during stimulation
andthen rises back toward the baseline level upon stimulusoffset.
Notice that the inactivation and reactivation oftransmitter occur
more slowly than changes in the activi-ties of the neural cells.
Each slowly habituating transmit-ter multiplies, or gates, the more
rapidly varying signal inits pathway, thereby yielding net
overshoots and under-shoots at input onset and offset,
respectively. Duringstimulation, the horizontal channel wins the
opponentcompetition against the vertical channel as indicated inthe
top right time plot. However, upon offset of thestimulation to the
horizontal channel, the input signalreturns to the baseline level
but the horizontal transmit-ter gate remains habituated below its
baseline value. Asa result, shortly after stimulus offset, the
gated tonicinput in the horizontal channel has a net signal below
thebaseline level. Meanwhile, the vertical pathway main~
Tonicinput
PhasicOn-input
tains the baseline response at all cells and gates beforethe
opponent competition. Thus, when the horizontalchannel is below the
baseline activity, after stimulusoffset, the vertical channel win:s
the opponent compe-tition and produces a rebound 01" activity as
shown in thetop left time plot. As the horizontal transmitter
gaterecovers from its habituated sta1:e, the rebound signal inthe
vertical channel weakens and finally disappears.
Figure 10 shows how this rebound of activity acts as areset
signal in the full BCS arlchitecture. Figure 10(a)schematizes how
inputs in a "horizontal pathway excite ahorizontal bipole cell. As
described above, thesehorizontal bipole cells can generate a
hysteresis thatcorresponds to persistence. Due to the interactions
of thegated dipole circuit, offset of the horizontal inputgenerates
a rebound of activity in the vertical pathway,which, as Fig. 10(b)
demonstral:es, inhibits the horizon-tal bipole cell. This
inhibition greatly speeds up theerosion of boundary signals al1ld
decreases persistence.Note that, apart from its crucial role in
explainingpersistence data, the inhibition of bipole cells by
offsetsignals from perpendicularly oriented pathways of thegated
dipole circuit has been shown to play an equallycrucial role in
preventing unwanted boundary groupingsacross intervening surfaces
(spatial impenetrability).
-
CORTICAL DYNAMICS OF BINDING AND RESET 1097
(a)
(b)
FIGURE 10. (a) A horizontal input excites a horizontal bipole
cell,which supports persistence. (b) Upon offset of the horizontal
input, arebound of activity in the vertical pathway inhibits the
horizontalbipole cell. This inhibition resets the hysteresis of the
feedback loop
and reduces persistence.
The inverse relationships between persistence andstimulus
duration and luminance, as shown in Fig. l(a),follow from the
properties of the gated dipole. Thelonger the stimulus duration, or
the stronger (moreluminous) the input to a gated dipole, the more
habitu-ated becomes the transmitter gates and, thus, thestronger
becomes the reset signals. The significance ofthe strength of the
reset signals is evident in Fig. l(b),which shows that persistence
of signals in the model isinversely related to stimulus duration
and luminance,except at very short stimulus durations. At short
stimu-lus durations, there is only a very weak reset
signalgenerated by the gated dipoles. However, because thestimulus
presentation is so brief, the BCS does notestablish a strong
hysteresis in the feedback loop.Indeed, because stimuli of a
greater duration or a higher
luminance create stronger boundary signals and canmore quickly
establish strong activities in the feedbackloop, at the shortest
stimulus durations persistence isdirectly related to stimulus
duration and luminance.Haber and Standing (1970) reported that
persistenceincreases with stimulus duration v{hen the stimuli
arebriefer than 20 msec. At longer stimulus durations, thegated
dipoles begin to generate stronger reset signals,which more quickly
remove the persisting boundarysignals in the feedback loop. Thus,
the inverse relation-ships between persistence and lumimance and
durationoccur when a strong hysteresis has been established inthe
BCS and when the gated dip-ole circuits producestrong reset
signals.
To emphasize the role of the gated dipole circuit andits reset
signal, we re-ran simulatiotilS for two luminancevalues and several
different stimulus durations butmodified one parameter so that the
transmitter gates didnot habituate. The dashed lines in Fig. l(b)
showthat without the transmitter habituation, persistenceincreases,
or does not change, with stimulus duration.Similarly, persistence
increases with stimulus luminancewithout the gated dipole circuit;
thus, it is the behaviorof habituative transmitters in the gated
dipole circuitthat explains the data of Bowen et al. (1974) within
theBCS model.
Because the habituative transmitters of the gateddipole circuit
are located outside the feedback loop ofthe BCS, only the offsets
of lumi:rlance-derived edgesgenerate reset signals. Therefore, only
the reset signalsgenerated by illusory contour inducers inhibit
thepersisting illusory contours. Inducers of illusorycontours have
few luminance edges and so, at stimulusoffset, generate fewer reset
signals than boundariesdefined entirely by real contours. With
fewer reset signalsavailable to break the hysteresis of the
feedback loop,illusory boundaries persist longer than real
boundaries.We show the results of simulations of these properties
inFig. 2(b). The luminous-based stimulus for these simu-lations was
an outline square, while the illusory contourinducers were L-shaped
stimuli at the corners of asquare. Our choice of inducer folmlS was
limited bycomputational resources as explained in detail in
theAppendix. This choice suffices to illulstrate the
dynamicalproperties of contours that are formed at positionswithout
luminance contrast. Other research concerningthe BCS has more fully
analyzed the relationshipsbetween boundary signals and perceived
illusorycontours through computer simulation (Gove,Grossberg &
Mingolla, 1993) and psychophysicalexperimentation (Lesher &
Mingollla, 1993). Grossbergand Mingolla (1985a) also analyzed! why
some inducersproduced stronger illusory contours than others.
Because luminance edges define ,only a small part ofan illusory
contour, the boundary representation takeslonger to become strongly
established than the boundaryrepresentation of the outline square~.
Therefore. illusorystimuli of short duration do not generate strong
hys-teresis and can more quickly erode at stimulus offset.Stimuli
of an intermediate duration generate strong
-
1098 GREGORY FRANCIS el at.
hysteresis, but do not produce strong reset signals atoffset.
Thus, these stimuli persist longest. Stimuli of longduration
generate a strong hysteresis, but they alsogenerate stronger reset
signals, which shorten persist-ence. Thus, over the same range of
stimulus durationsthat show an inverse relationship with
persistence for theoutline square, persistence of boundary signals
for anillusory contour is not inversely related to stimulusduration
at short durations, but peaks at someintermediate value, as shown
in Fig. 2(b). The solid linesin Fig. 2(b) connect simulation data
points of thestimulus durations sampled in the psychophysical
studiesof Meyer and Ming (1988). A comparison of these curveswith
the data in Fig. 2(a) shows that they are very similarin shape,
although the absolute persistence values aredifferent.
Finally, because the opponent processing andhabituative pathways
are orientation-specific, the modelexplains the adaptation results
of Meyer et al. (1975)shown in Fig. 3. Within the model, adaptation
to, say,a horizontal stimulus habituates the transmitter
inhorizontal pathways but leaves the vertical pathwaysunadapted. If
one then tests persistence of a horizontalstimulus, the horizontal
pathways further habituate thetransmitter during the test
presentation. The strongerthan usual habituation of the horizonal
pathways meansthat, at stimulus offset, the reset signals will be
strongerthan usual and persistence will decrease. On the otherhand,
if one tests persistence of a vertical stimulus, thenboth oriented
pathways become habituated: the horizon-tal pathways from the prior
adaptation and the verticalpathway from the habituation due to the
target presen-tation. As a result of the opponent competition
betweenthese habituated signals (Fig. 9), the reset
signalsgenerated at offset of the vertical test stimulus will
beweaker than usual and persistence will increase.
So far we have accomplished two major goals. First,we explained
how interactions of a tonic input, habitua-tive transmitter gates,
opponent processing, and rectifiedoutput signals in a gated dipole
circuit generate a resetsignal upon offset of an oriented luminance
edge.Second, we showed that the properties of this reset signalat
the second competitive and bipole stages of the modelaccount for
the inverse relationships between persistenceand stimulus luminance
and duration (Bowen et al.,1974), the prolonged non-monotonic
persistence proper-ties of illusory contours (Meyer & Ming,
1988), and theopposite influences on persistence of
orientation-specificadaptation (Meyer et al., 1975). .
It remains to show that the spatial inhibition, orendstopping
process, within the first competitive stage ofthe model can account
for the decrease in persistenceshown in Fig. 4(a). The solid line
in Fig. 4(b) demon-strates that the persistence of boundary signals
doesdepend on the separation between target and maskstimuli. For
comparison, the dashed line shows thepersistence of boundary
signals when we removed theoriented spatial competition of the
first competitive stagefrom the model (setting one parameter equal
to zero).The second source of spatial inhibition, in the
feedback
pathway of the bipole cells, remains intact. In the
currentsimulations it is of a smaller range than the interactionsof
the rrst competitive stage. The dasl1led line in Fig. 4(b)shows
that without the spatial competition there are nochanges in
persistence of the target stimulus until thetarget and mask
boundaries are within the range of theinhibition in the feedback
pathway. Thus, the spatialcompetition accounts for the abilit:f of
the maskingstimulus to reduce persistence of the talrget (Farrell
et al.,1990).
NEUROPHYSIOLOGICAL CORRELA TIS ANDPREDICTIONS
Grossberg (I 987a) reviews neural analogs of all stagesof the
present model in visual cortex. In particular, vonder Heydt et al.
(1984) found analogs of bipole cells inarea V2 of monkey visual
cortex, 1Nhich Cohen andGrossberg (1984), Grossberg (1984), and
Grossberg andMingolla (1985a) had modeled before these data
werepublished. It may be possible to use additional
neuro-physiological studies to verify the dynamic properties
ofthese cells. For example, it should be possible to observethe
inward erosion of boundary sign:als by observing asingle bipole
cell in visual cortex. Aftelr finding the centerof the bipole
cell's receptive field, one could run a seriesof experiments
varying the position clf luminance edgesrelative to the center of
the cell's receptive field. Becausethe model predicts that boundary
sigI1Lals erode from thecontour ends, the cell should show its
greatest persistingresponse when its receptive field is centered on
thecontour and should show less persistence as the exper-imenter
shifts the contour center to one side or the otherof the receptive
field center. Note that this predictionfollows despite another
prediction, derived from ananalysis of the BCS in response to
:;tatic stimuli, thatbefore stimulus offset the amplitude of
activity should benearly identical for bipole cells all allong the
contour,regardless of the cell's distance from inducers beforereset
occurs. Properties such as the inverse relationshipsbetween
persistence and stimulus duration andluminance, greater persistence
for illusory than realcontours, the effects of adaptation, alild
the influence ofa masking stimulus should also be observable in
aninvestigation of these cells.
Likewise, the ability of masking stimuli to reducepersistence of
the target should be measurable at asuitable population of
hypercomplex cells. Other hyper-complex cells should exhibit
habitualtive gating of theirresponses, as well as opponent rebounds
to offset ofstimuli that are oriented perpendicular to their
receptivefields. The firing of these rebounding hypercomplex
cellsshould correlate with diminished pe:rsistence in targetbipole
cells that are tuned to a perpendicular orientation.
Lesher and Mingolla (1993) have tested the BCSmodel through
psychophysical ex]periments on theinduction of illusory contours in
a Kanizsa square usingvariable numbers of line ends that are
perpendicular tothe illusory contour. They found that an inverted
Ufunction relates illusory contour clarity to the number of
-
CORTICAL DYNAMICS OF BINDING AND RESET1099
line end inducers. The BCS explains the inverted U asfollows.
More line end inducers at first produce more cellactivations at
hypercomplex cells of the second competi-tive stage whose receptive
fields are centered at andperpendicular to the line ends. These
responses are calledend cuts (Grossberg & Mingolla, 1985b).
They are dueto an interaction between the first and second
competi-tive stages. The first competitive stage inhibits
thosehypercomplex cells at a line end whose orientationalpreference
is parallel to the line. These inhibited cellsdisinhibit (via the
second competitive stage) hypercom-plex cells at the line end whose
orientational preferenceis perpendicular to the line (see Fig. 5).
These greaternumbers of end cuts, in turn, collinearly cooperate
tomore strongly activate the bipole cells that generate theillusory
contour.
As more line inducers are used, however, they getcloser
together. They then inhibit each others' targethypercomplex cells
via lateral inhibitory signals from thefirst competitive stage.
Their net responses are herebyweakened by lateral inhibition, as
are their end cuts andtheir illusory contours. Lesher and Mingolla
(1993)noted that other models of illusory contour formationcannot
explain this inverted U effect.
The BCS model suggests that an inverted U functionmay also
relate illusory contour persistence to thenumber of line end
inducers under these experimentalconditions, since weaker bipole
cell activations shoulderode more quickly after stimulus offset and
since agreafer number of reset signals would be generated. Thatis,
as the spatial density of line inducers increases beyondthe point
where illusory contour clarity reverses, thenillusory contour
persistence should also reverse inresponse to line inducers that
are on for a fixed amountof time. Stimuli in the rising portion of
the Lesher andMingolla (1993) study provide both stronger bipole
cellactivity and stronger reset signals, and so present a
moreambiguous case. This type of experiment would probethe
interactions between first and second competitivestages, as well as
between the spatial and temporalproperties of emergent
segmentation.
Grossberg (1987a, Section 30) linked properties of theBCS
simple, complex, and hypercomplex cells to exper-imentally reported
properties of spatial localization andhyperacuity. Badcock and
Westheimer (1985a, b) usedflanking lines to influence the perceived
location of a testline. They varied the position of the flank with
respectto the test line as well as the direction-of-contrast
offlank and test lines with respect to the background. Theyfound
that two separate underlying mechanisms wereneeded to explain their
data: a mechanism concernedwith the luminance distribution within a
restrictedregion, and a mechanism reflecting interactions
betweenfeatures. Within the central zone defined by the
firstmechanism, sensitivity to direction-of-contrast wasfound, as
would be expected within an individual recep-tive field. On the
other hand, a flank within the surroundregion always caused a
repulsion which is independentof direction-of-contrast. Thus "when
flanks are close toa target line, it is pulled toward the flank for
a positive
flank contrast but they push eacl1l other apart if the flankhas
a negative contrast. A flank in the surround regionalways causes
repulsion under th,e conditions presented"(p. 1263). To further
test independence of direction-of-contrast due to the surround,
they also found that "theeffect of a bright flank on one side can
be cancelled bya dark flank on the other. Withiin the central zone
thisprocedure produces a substantia:! shift of the mean of
apositive contrast target line towards the positive contrastflank"
(p. 1266).
Badcock and Westheimer (1985a) noted that theaverage of
luminance within the 4~entral zone is sensitiveto
amount-of-contrast and dir(:ction-of-contrast in away that is
consistent with a difference-of-Gaussianmodel. Such a computation
also occurs at the elongatedreceptive fields, or simple cells, of
the BCS (Fig. 5). Pairsof simple cells with like positions and
orientations butopposite directions-of-contrast t:hen add their
rectifiedoutputs at complex cells which are, as a
consequence,insensitive to direction-of-contra.st (Fig. 5). Such
cellsprovide the inputs to the first (:ompetitive stage.
Theoriented short-range lateral il1lhibition at the
firstcompetitive stage is thus insensitive to
direction-of-contrast, has a broader spatial Jrange than the
centralzone, and, being inhibitory, would al\\rays
causerepulsion-all properties of the Badcock and West-heimer
(1985a) data. In summary, all the main effects inthese data mirror
properties of the circuit in Fig. S.
In further tests of the existence and properties of
thesedistinct mechanisms, Badcock and Westheimer (1985b,p. 3) noted
that "in the surround zone the amount ofrepulsion obtained was not
influenced by verticalseparation of the flank halves, even when
they wereseveral minutes higher (or lower) than the target line.
Inthe central zone attraction was only obtained when thevertical
separation was small enough to provide someoverlap of lines in the
horizontal direction". These datafurther support the idea that the
central zone consists ofindividual receptive fields, whereas the
surround zone isdue to interactions across receptive fields which
are firstprocessed to be independent of direction-of-contrast, asin
Fig. 5. In our computer simulations of boundarycompletion and
segmentation (Grossberg & Mingolla,1985a, b), it was assumed
that the lateral inhibitionwithin the first competitive stage is
not restricted to anypreferred orientation, as is also true of the
surroundrepulsion effect in the Badcock and Westheimer
(1985b)data.
This theoretically predicted correlation betweenproperties of
hyperacuity, persistence, and illusorycontour formation presents an
opportunity to designnew types of psychophysical experiments with
which tofurther test the model. Other new experimental
opportu-nities are summarized in Grossberg (1994). Some ofthese are
now being explored in our laboratory.
RELATED FINDINGS AND CONCLUDING REMARKS
Because the equations presented in the Appendix arefor a
"single-scale" BCS, they cannot explain findings by
-
1100 GREGORY FRANCIS et a/.
Meyer and Maguire (1981) that the persistence of agrating
percept increases with spatial frequency. However,the multiple
scale BCS interactions described in Gross-berg (1994) can account
for this finding without affectingthe explanations described in
this paper. With multiplescales of oriented filters, a grating of a
low spatialfrequency excites both large and small filters, whereas
agrating of a high spatial frequency excites only smallscaled
filters. The net result of this skewed excitationdistribution
across scales is that the low spatial frequencygrating creates more
reset signals than the high spatialfrequency grating. All activated
filters of similar orienta-tional and disparity sensitivity at each
position input tothe same set of bipole cells. As in the case of
real vs il-lusory contours, more reset signals imply less
persistence.
The model also suggests why different experimentalmethods find
different properties of persistence (Sakitt &Long, 1979). The
results in this paper have correlated thedisappearance of boundary
signals with the perceivedoffset of the stimulus. However, due to
the slow timeconstants of the habituation in the gated dipole
circuit,reset signals generated at stimulus offset may
persistbeyond the offset of the boundary signals.
Perceptualawareness of these reset signals may be used by
subjectswhen the experimental instuctions tell them to observeany
residual trace of the stimulus, as in the studies ofSakitt and Long
(1979). In particular, stronger luminanceimplies greater
habituation, which implies a greaterlength of time for the
rebounding channels of the gateddipole to return to baseline. Thus,
the persistence of thereset signals may be directly related to
stimulusluminance, in agreement with the psychophysical studiesof
total persistence (Sakitt & Long, 1979; Nisly &Wasserman,
1989).
In reading the Appendix, note that a single set ofparameters was
used to simulate all the properties ofvisual persistence. While
modifying the parameters maychange the quantitative values given in
Figs 1, 2, 3, and 4,the relevant functional properties expressed by
the curvesremain the same. Specifically, regardless of the
specificchoice of parameters (excluding cases where boundarysignals
or reset signals are not created at all), persistenceof signals in
the model is inversely related to stimulusluminance and duration,
illusory contours persist longerthan real contours,
orientation-specific adaptation hasopposite influences on
persistence, and a spatial maskingstimulus inhibits target
persistence. The relationshipsbetween persistence and stimulus
properties are built intothe structure, or non-parametric design,
of the model. Inthe present simulations, our goal has thus been to
clarifyand illustrate the qualitative functional meaning
ofpersistence data. In much the same way, quantitativepersistence
values vary from subject to subject and withexperimental
conditions. This being said, it needs also tobe noted that no
alternative explanation of persistenceproperties has explained as
wide a range of data,provided the level of detail implemented
herein, orattributed these properties to fundamental
designconstraints on the dynamic balance that regulatesperceptual
resonance and reset.
Given that the entire structure and dynamics of themodel had
previously been derived and tested on otherdata than persistence
data, the model's ability to simulatethe important functional
properties of persistence datalends even greater support to the
neural reality of thesemodel mechanisms. The ability of this same
small set ofmodel mechanisms to explain data from several
percep-tual and neural paradigms also provides conceptuallinkages
across paradigms whereby new types of exper-iments can be designed
to further test these mechanisms,
In summary we have shown how an analysis of theBCS cortical
model can offer new mechanistic andfunctional explanations of data
on visual persistence,These explanations are consistent with the
theory'sprevious explanations of boundary completion(Grossberg
& Mingolla, 1985a), texture segregation(Grossberg &
Mingolla, 1985b), shape-from-shading(Grossberg & Mingolla,
1987), and three-dimensionalvision (Grossberg, 1987b, 1994), among
others, whileextending its explanatory range still further into
thedifficult temporal phenomena of visual persistence,
Thefunctional role of the feature binding and reset mechan-isms
that the model predicts to be responsible for persist-ence suggests
links between persistence and fundamentalissues in the formation
and breakup of perceptual group-ings in a dynamically fluctuating
environment. Thus, farfrom being esoteric laboratory phenomena,
data onpersistence afford important clues about some of themost
fundamental processes of preattentive vision,processes moreover
that are being increasingly wellcharacterized by neural network
architectures.
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was partially supported by the Air Force Office of
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APPENDIXNetwork Equations and Parameters
The simulations of the model are a simplification of the
system'sinteractions described elsewhere (Grossberg & Mingolla.
1985a, b,1987). These simplifications were necessitated by the
following fact.Simulations in the prior reports of the model
concerned the spatialinteractions of the segmentation system and so
had no need toimplement the temporal dynamics of resonance and
reset, whosejuxtaposition of fast and slow time scales greatly
increased thecomputational load in simulations. To manage this
load. theexperimental stimuli were also simplified. as indicated in
the text.without losing their essential characteristics. Thus. the
quantitativematches of simulations to data are less significant in
this paper thanthe analysis of how model mechanisms work together
to explain thecomplex qualitative pattern of persistence data.
These qualitativeproperties are, moreover, robust. In much the same
way. differentpsychophysical studies of persistence do not produce
identical quanti-tative values across observers, but do show
consistent relationshipsbetween persistence and stimulus
properties. The psychophysical stud-ies of persistence also do not
give absolute measures. but provide onlyrelative values that can be
ordered within an experimental paradigm.
The model contains a total of eight levels of model neurons
ortransmitter gates. Each level, except for the first. consists of
twoparallel pathways coding horizontal and vertical orientations at
eachpixel location. Within the second level. each
orientation-specificpathway contains two simple cells responsive to
opposite polarities ofluminance gradients. Thus. associated with
each pixel point of theimage are seventeen different cells and
transmitter gates. Since wecarried out all simulations on a 40 x 40
pixel array, the simulatednetwork contained 27,200 cells and
transmitter gates. A differentialequation describes the behavior of
each cell and gate.
We eased the computational requirements of integrating
27.200differential equations with a number of simplifications.
First, ratherthan integrate all the differential equations
explicitly, we algebraicallycomputed some cell activities at the
equilibrium value of the differentialequation. We made this
simplification for the unoriented on-center,off-surround cells in
Levell. the oriented polarity-specific simple cellsof Level 2, and
the hypercomplex cells undergoing lateral inhibition inLevel 4.
Computing the values of these cells at equilibrium makes
theassumption that they operate on a faster time scale than the
other cells.Since the rate-limiting time scales of the simulations
involve dynamicsof habituative transmitter gates and feedback
within the BCS, unfold-ing the dynamics of these cells would not
adversely affect our
-
1102 GREGORY FRANCIS el 01.
decreasing for inputs further away from the oriental center-line
of thein-field the parameter l' controls the rate of fall off. Then
a simple cellthat is selectively responsive to a bright-to-dark
luminance gradientobeys the differential equation
dx 2BD", 2BD +
d/= -X;" +[Fi,,-Gi,,) .(AS)
where [PJ+ = max(p. 0). A cell responsive to a dark-to-bright
lumi-nance gradient obeys the equation
dx 2DBi" WB +-= -X". +[G,.-F".) .(A9)
dl I,. f,. ,.
To save computation. the activities of these cells were computed
atequilibrium as:
X7JD = [Fijk -Gijk]+' (AIO)and
explanations of visual persistence but would greatly increase
thecomputation required to simulate the model.
A second simplification took advantage of symmetries in the
imageplane. If the images presented to the system on the left and
right sidesof the plane are mirror images, then activities produced
on the righthalf will have corresponding equal values on the left.
Thus, to savecomputation, we could compute only the values on the
right side ofthe image plane and extrapolate the results to account
for the fullimage plane. Similarly, when the top and bottom
quadrants of the rightside are mirror images, we only needed to
compute the cell responseson the bottom quadrant. Finally, if the
upper and lower diagonals inthe quadrant are mirror images, then
the values of the horizontal cellsin the lower (upper) diagonal
equal the values of the vertical cells inthe upper (lower)
diagonal. Thus, by restricting the input stimuli to
besquare-shaped, we only had to integrate the differential
equations forthe horizontal cells in the bottom right of the image
plane to accountfor the behavior of the entire system.
These simplifications reduced the system of 27,200
differentialequations to only 1600 differential equations. The
LSODA integratorroutine (Petzold & Hindmarsh, 1987) performed
the integration ofthese equations. We based all the network
equations upon thosedescribed in Grossberg and Mingolla
(1985b).
X7jB = (Gilt -F,It]+, (All)
Level 3: oriented complex cells
Each cell in Level 3 becomes insensitive to the polarity of
constrastby summing the rectified activities of the cells in Level
2 of the samelocation and orientation. Each Level 3 cell obeys the
differentialequation
Level 0: image plane
Each pixel has a value associated with retinal luminance.
Wedescribe the pixel-luminance values of the different stimuli used
in thesimulations below.
Level I: center-surround cellsThe activity Xlj of a Level I cell
centered at position (i,j) obeys a
shunting on-center, off-surround equation
dX)j I I ~ I ~-d = -Xij+(A -XiJt;..Bi.b'f/,.-(X;j+
C)t;..Djj,.,I,., (AI)t pq pq
where Ipq is the retinal luminance at position (p, q), A is the
maximumactivity of the cell, -C is the minimum activity of the
cell, and
B;jpq = B exp( -a -2 log 2[(i -p)2 + (j -q)~ (A2)
D;jpq=D exp[-p-210g2[(i -pf+(j-qr']] (A3)
are excitatory and inhibitory Gaussian weighting functions,
respect-ively. The term log 2 means the parameters a and p set the
radius oftheir respective Gaussians at half strength. Parameters
Band D areconstant scaling terms.
To save computation, the equilibrium response of the
differentialequation is found by setting the left hand side of
equation (I) equalto zero. The resulting algebraic equation can be
solved to find
A L Bijpqlpq -C L DijpqlpqX)j= pq pq. (A4)
1.0 + L(B;jpq + Djjpq)/pqpq
The activities of cells at this level share some key propenies
with thosefound in ganglion cells or LGN (Grossberg, 1987a). No
off-centeron-surround cells were implemented in our
simulations.
Level 2: oriented simple cellsThe following equations define
oriented simple cells that are centered
at position (i,j) with preferred orientation k. To create a
verticallyoriented input field, or in-field, that is specific to
the polarity of theluminance gradient, divide an elongated region
into a left half L;j. anda right half Rjj.' Add up the weighted sum
of the Leve! I inputs withinthe range of the left side
Fi~ = L EiJpqX~.,eLi..
(AS)
and the right side
G/jk = L E;jpqX~...Hi..
(A6
of the region, with
orientation, the tenD NX'Jr is a feedback signal from the higher
levelcell of the same position and orientation, and the termX~jk
Lpq Pjjpq(X~ + J)X~ is the inhibitory input from the lower
levelcells of the same orientation and nearby spatial positions.
Theinhibitory weights fall off in strength as the spatial distance
betweencells increases, as in
Pjjpq = P exp[ -cS-2Iog 2[(; -p)2 + U -q)2D, (AI5)
where P scales the strength of the inhibition, and cS controls
the spread.E1jpq = exp( -y-21og 2(; -p)2j (A7)
dXl~ 3 2BD WBT=-X,~+H(Xi~ +Xi~ ). (AI2)
Parameter H scales the activities of the input signals to the
complexcell.
Let'el 4: habituative transmitter gates
The signal in each oriented pathway is gated, or multiplied, by
ahabituative transmitter which obeys the following
equation(Grossberg, 1972)
dX.-if' = K[L(M -X~) -(X:~ +J)X1p.). (AI3)
This equation says that the amount of available transmitter
X~accumulates to the level M, via term KL(M -X~), and is
inactivatedby mass action at rate K(X~" + J)X~, where J is the
tonic input ofa gated dipole and X:" is its phasic increment. We
always set the rateK much smaller than 1.0 so that these equations
operate on a slowertime scale than the equations describing cell
activities. At the beginningof each simulation, each transmitter
value is set to the non-stimulatedequilibrium value X:" = LM I(L +
J).
Level 5: first competitive stage of h}percomple.\" cells
The gated signals of a fixed orientation compete via
on-centeroff-surround spatial interactions. Along with the tonic
signal comingup through the habituative transmitters, each cell
also receives a tonicinput which supports disinhibitory activations
at the next competitivestage (see Grossberg & Mingolla, 1985a,
b). The activity of a Level 5cell obeys differential equation
dX~~ 5 , .IT = -X1~+J +(X;~ +J)Xi~ + NXi.A
-X~~LPijpq(X~k+J)X~, (AI4)H
where -X~~ models the passive decay, the parameter J establishes
anon-zero baseline of activity for the cell, the term (X:~ + J)X1jk
is thegated excitatory input from the lower level at the same
position and
-
.
CORTICAL DYNAMICS OF BINDING AND RESET 1103
.... For the simulations in this paper, the differential
equation was
solved at equilibrium as
X s J + (Xf" + J)X~ + NX~J;jk= .(AI6)/ 1.0 + L
P;jpq(X~+J)X~k
pq
Level 6: second competitive stage of hypercomple.1: cells
The output signals from the first competitive stage compete
acrossorientation at each position. The activity of a cell
receiving thiscompetition obeys the differential equation
dXfjk 6 5 5d/ = -Xijk + (X;;.- XijK) (A17)
where X~J; and X~K represent orthogonal orientations.
Level 7: cooperative bipole cells and spatial
impenetrability
The next level uses a simplified version of bipole cells. As in
LevelI, we divide the in-field of each horizontal bipole cell into
a left sideL;;. and a right side R;}: (top and bottom for
vertically oriented bipolecells). Each bipole cell then sums up
excitatory like-oriented signalsand inhibitory
orthogonally-oriented signals within each side. Aslower-than-linear
bounded function squashes the net signal of eachside. We then set
the output threshold of the bipole cells so thatboundaries must
stimulate both sides of the receptive field for the cellto generate
an output signal. The differential equation describing eachbipole
cell activity is
dxl;. 7 fl ~ I 6 )+ I 6 )+)-d = -Xi;, + ,:.. X,.,k -X,.,Kt
oR...
+flE. IX~)+ -IX':",,)+) (AI8)
where
(c)300
100
so
0
Level 8: spalial sharpening
Output signals from the bipole cells are thresholded to
preventfeedback unless inputs activate both sides. These output
signals thenundergo a spatial sharpening much as in the first
competitive stage ofLevel 5. The activities of cells in Level 8
obey the differential equation
~ = -x~~ + [Xl~ -R]+ -X~-" L T(X;'k -R]+ (A20)..oS"
where parameter R is the output threshold for bipole cells,
parameterTscales the strength of the spatial inhibition, and Sijis
the eight nearestneighbors to pixel (i,j). These signals are scaled
by parameter N beforefeeding back to the cells in Level 5 to close
the feedback loop.
FIGURE AI. Changes in persistence as parameters are modified.
Ineach case only one parameter is varied. The large solid diamonds
markpersistence with the default parameter value. (a) Parameter N
inequation (AI4) scales the strength of the bipole pathway
feedbacksignals. Increasing parameter N strengthens the feedback
signals togenerate a stronger hysteresis in the network without
affecting thestrength of the reset signals. Persistence increases
with N. (b) ParameterK in equation (AI3) controls the rate of
habituation of the transmittergates. Greater habituation makes
stronger reset signals, so persistencedecreases as K increases. (c)
Parameter H scales the inputs to thehabituative gates in equation
(AI2). Increasing H creates a strongerresonance and stronger reset
signals. Starting with small values,increasing H has a larger
effect on the hysteresis than on the strengthof the reset signals,
thus persistence increases. At larger values,increasing H has a
larger effect on the strength of the reset signals than
on the hysteresis, thus persistence decreases.
Computation of images and persistence
We operationally defined the boundaries of an image to be
persistingwhenever, after target offset, a cell in Level 6 at the
location andorientation of the target image edge (real or illusory)
had an activityvalue> 0.5. The computer checked the values every
0.5 time steps afterstimulus offset (I time unit in the simulation
is equivalent to 10 msec).For all simulated images a value at each
pixel in simulated ft-Lindicated luminance intensity. The
background luminance was always10-6 simulated ft-L.
The simulated luminances and durations of the target !lashes
forFig. l(b) are indicated in the figure and were all bright
squares (26 x 26
pixels) on a dark background.
-
..
1104 GREGORY FRANCIS tl of.
....
The inducers for the illusory stimuli in Fig. 2(b) were
luminanceincrements (pixel values of 0.15 simulated ft-L) in the
shape of Lsoriented appropriately in each quadrant to line up the
inducer edges.The real stimulus was a bright outline (3 pixels
wide) square of thesame luminance and size (32 x 32 pixels) as the
illusory square.
We did not simulate the remaining stimuli shown in Fig. 2(a)
becausethe simplified BCS used in our simulations creates boundary
signalsbetween the inducing stimuli for all of the stimulus sets.
It is not thefocus of this paper to show that our simulations
accurately createillusory contours. Rather, we investigated the
persistence of boundariesgenerated without a luminance edge.
Illusory contours are one exampleof these types of boundary
signals. A full simulation of the BCS withmore orientations, and
whose bipole cell weights are modulated in twodimensions of spatial
position and one of orientation, can accuratelypredict the
generation of illusory contours (Gove el al., 1993). Such
asimulation is beyond the scope of this paper.
The test stimulus for Fig. 3(b) was a pair of horizontally
orientedluminance bars (10 x I pixels, pixel values of 0.15
simulated ft-L)separated by 3 pixel spaces. The adaptation stimulus
was eitheridentical to the test stimulus or six small vertical
lines (4 x I pixelseach) evenly spaced and placed to intersect with
the horizontal bars ofthe test stimulus. The adaptation and test
stimuli were both presentedfor 100.0 simulated msec.
The target for Fig. 4(b) was a bright square of 20 x 20 pixels
witha pixel luminance value of 0.323 simulated ft-L. The mask for
Fig. 4(b)consisted of a bar (16 x 2 pixels) along each edge of the
target withequal pixel luminance and an edge-to-edge separation
from the targetof 3-9 pixel spaces, each pixel space corresponding
to 0.05 deg of visualangle. Larger spatial separations could not be
used due to the limitedsize of the simulation plane. We always
presented the target flash fora duration of 50.0 simulated msec and
matched its offset with the onsetof the mask. We kept the masking
flash on until the boundaries of thetarget flash fell below
threshold.
Parameter' selection
Because integration of nonlinear differential equations is
computa-tionally expensive, we simplified the BCS equations as much
aspossible. As a result, we could not use the same parameters as
othersimulations of the BCS, which calculated the equilibrium
response ofthe system (Cruthirds, Gove, Grossberg, Mingolla, Nowak
&Williamson, 1992; Gove et al., 1993; Grossberg & Mingolla,
I 985a, b,1987; Grossberg, Mingolla, & Williamson, 1993). In
particular,whereas the present simulations used only vertically and
horizontallyoriented cells, other BCS simulations have used oblique
orientations aswell.
To remain consistent with earlier simulations and to explain
theproperties of persistence, the parameters used in our
simulations wererequired to meet several properties. First, the
parameter set had toallow the BCS to locate oriented boundaries.
For example, if P is settoo large, then spatial inhibition between
cells of like orientation andnearby positions can mutually inhibit
activities at the next layer somuch that no signal survives the
competition. Similarly, the thresholdfor the bipole cell
activities, R, cannot be set too large (relative to theparameter Q
and the strength of the inputs to the bipole cells) or thebipole
cells will never fire.
A second requirement of the parameter set was that the
activities inthe feedback loop, once activated, needed to be strong
enough topersist once the external inputs were turned off. The
parameters N, Q,R, and T in the bipole feedback pathway control
this property of thenetwork. These parameters were set to insure a
persisting activity inthe network and proper creation of boundary
signals.
Figure Al(a) shows how the strength of activities in the
feedbackloop influences persistence. Parameter N scales the
strength of thebipole feedback pathway to the lower stages.
Increasing N strengthens
the boundaries in the BCS without changing the strength of the
resetsignals. Figure AI shows the persistence of a 100 simulated
msec,0.323 ft.L stimulus. Figure AI(a) shows that as N increases,
persistencealso increases because the hysteresis in the feedback
loop is stronger.
A third requirement of the parameter set was that it had to
allowgeneration of reset signals upon stimulus offset. This was
realized byproperly choosing parameters, J, K, L, and M, of the
habituativetransmitter gates. Grossberg (1980, Appendix E) showed
that thestrength of the reset signal increases with parameter M and
decreasesas J or L are increased. These parameters also establish
the lower limit(equilibrium) of the gate strength. If the gate was
allowed to habituatetoo much, it would no longer pass on sufficient
input to the higherlevels. In such a case, boundaries could
disappear before the offset ofthe stimulus. It is easy to find
parameters which avoid this problem.Parameter K controls the
relative rate of habituation, and increasingK allows for faster
habituation and stronger reset signals. Figure AI(b)shows that
increasing K decreases persistence.
The final task of parameter setting was to control the value of
theinputs to the habituative gates so that they created a strong
feedbackloop and generated strong reset signals. The six parameters
of uvelI act to compress the cell response to luminous inputs.
Equation (I)could have been replaced with a function like 10g(/,J
to get similarresults, but we choose equation (I) to remain
consistent with othersimulations of the BCS (Gove et al., 1993;
Cruthirds et al., 1992). Thisstage of compression explains why the
two lower curves of Fig. I havesimilar persistence despite the fact
that the stimulus of the lower curveis significantly more luminous.
The parameters of uvels 2 and 3 simplyscale the activities of the
oriented filters. Increasing the activities ofthese cells has two
effects. First, stronger signals create strongeractivities in the
feedback loop, which act to increase persistence. At thesame time,
these stronger signals increase habituation to generatestronger
reset signals upon stimulus offset. The balance of these
factorsdetermines persistence. Figure AI(c) shows that as parameter
Hincreases from 0.005 to 0.025 persistence increases. This increase
inpersistence indicates that the influence of the additional
strength givento the activities in the feedback loop is greater
than the additionalhabituation caused by the stronger inputs. As H
increases still further,the additional habituation, and stronger
reset signals, tend to dominatethe increases in boundary signal
strength. This same analysis was usedto explain the inverted-U
shapes of the curves in Fig. I(b) andFig. 2(b), as a function of
stimulus duration.
For our explanations of persistence properties, the
only"disallowed" values of parameters are ones that would generate
absurdconsequences even outside the domain of visual persistence.
Forexample, parameters could be set to prevent bipole cells from
perform-ing boundary completion. However, once they are set so as
to permitcompletion, illusory contours persist longer than real
contours. Thatthe data curves can be explained through an analysis
of the modelnetwork architecture shows that the persistence
properties of the modelare robust.
The network parameters remained unchanged across all
simulations,only the image luminances, durations, or
spatiotemporally adjacentstimuli were varied. The following
parameters were used: A = 67.5,B =2.5, C=60.0, D =0.05, H=O.I, J
=20.0, K=0.0003, L = 3.0,M = 5.0, N = 13.0, P = 0.0005, Q = 0.5, R
= 0.61, T = 0.3, « = 0.5,P = 3.0, y = 1.5, IS = 3.0. Each side of
the oriented masks in uvel 2,Lip., Rip., were rectangles of 4 x I
pixels in size. Each side of a bipolecell was restricted to a
single column (vertical) or row (horizontal)extending 18 pixels
from the position of the bipolecell. For comparisonpurposes, the
dashed lines in Fig. l(b) we computed with K = 0.0; andthe dashed
line in Fig. 4(b) was computed with P = 0.0.
All simulations were carried out on an Iris 4/280 or an Iris
8/280Silicon Graphics Superminicomputer. The computation of
eachsimulated data point in Figs l(b), 2(b), 3, and 4(b), required
approxi-mately half-an-hour on a multi-user machine.