Neural field model of binocular rivalry waves
Paul C Bressloff1
1Department of Mathematics, University of Utah
July 10, 2014
Part I. What is binocular rivalry?
AMBIGUOUS FIGURES - BISTABLE PERCEPTION
STOCHASTIC PROCESS
A process of change governed byprobabilities at each step.
2 | JANUARY 2002 | VOLUME 3 www.nature.com/reviews/neuro
R E V I E W S
advantage in overall predominance, as indexed by thepercentage of total viewing time for which it is domi-nant. So, for example, a high-contrast rival figure willbe visible for a greater percentage of time than a low-contrast one19, a brighter stimulus patch will predominateover a dimmer one20, moving contours will enjoy anadvantage over stationary ones21, and a densely contouredfigure will dominate a sparsely contoured one17,22. Doesa ‘strong’ rival figure enjoy enhanced predominancebecause its periods of dominance last longer, on average,than those of a weaker figure, or because its periods ofsuppression are abbreviated, on average? The evidencefavours the latter explanation: variations in the stimulusstrength of a rival target primarily alter the durations of suppression of that target, with little effect on itsdurations of dominance17,23.
Can these unpredictable fluctuations in dominanceand suppression be arrested by mental will power?Hermann von Helmholtz, among others, believed thatthey could24. Observing rivalry between sets of orthogo-nally oriented contours presented separately to the twoeyes, Helmholtz claimed to be able to hold one set ofcontours dominant for an extended period of time byattending vigorously to some aspect of those contours,such as their spacing. Ewald Hering, Helmholtz’s long-standing scientific adversary, characteristically disagreedwith this claim, arguing that any ability to deliberatelymaintain dominance of one eye’s view could be chalkedup to eye movements and differential retinal adapta-tion25. Which view does the weight of evidence favour? Itdoes appear that, with prolonged practice, attention canbe used to alter the temporal dynamics of rivalry26 with-out resorting to oculomotor tricks. However, this evi-dence also indicates that observers cannot maintaindominance of one rival figure to the exclusion ofanother26, even when that temporarily dominant figurecomprises interesting, potentially personal visual mater-ial27 — an attended rival figure eventually succumbs tosuppression despite concentrated efforts to maintain itsdominance. In this respect, binocular rivalry differsfrom dichotic listening, in which a listener can maintainfocused attention indefinitely on one of two competingmessages broadcast to the two ears.
There is reason to believe that ‘top–down’ atten-tional modulation of rivalry operates by boosting theeffective strength of a stimulus during dominance. Ooiand He28 found that a dominant stimulus was less sus-ceptible to a perturbing event presented to the othereye when observers voluntarily focused attention onthat dominant stimulus. However, we know that volun-tary attention cannot be guided by visual cues pre-sented during suppression phases of rivalry29; evidently,then, voluntary attention does not have access to infor-mation portrayed in a suppressed figure. However,involuntary attention can be captured during suppres-sion: stimulus events known to capture involuntaryattention — such as the sudden onset of motion in apreviously stationary figure — are sufficient to rescuea stimulus from suppression, thrusting it into con-scious awareness at the expense of its competitor30–32.So, voluntary, ‘endogenous’ attention seems to operate
Definitive answers to these questions are not yetavailable, but this review summarizes what we know atpresent. We start with an overview of the hallmark per-ceptual properties of binocular rivalry, for these will illu-minate the search for its neural concomitants. From theoutset, it is important to keep in mind that rivalry prob-ably does not stem from a single, omnibus process; inour view, it is near-sighted to speak of ‘the’ neural mech-anism of binocular rivalry. Instead, multiple neuraloperations are implicated in rivalry, including: registra-tion of incompatible visual messages arising from thetwo eyes; promotion of dominance of one coherent per-cept; suppression of incoherent image elements; andalternations in dominance over time. These distinctoperations might be implemented by neural events dis-tributed throughout the visual pathways, an overarchingtheme that we shall develop in this review.
Perceptual characteristics of rivalryTemporal dynamics. Fluctuations in dominance andsuppression during rivalry are not regular, like the oscil-lations of a pendulum. Instead, successive periods ofdominance of the left-eye stimulus and the right-eyestimulus are unpredictable in duration, as if being gener-ated by a STOCHASTIC PROCESS driven by an unstable timeconstant9,17,18. It is possible, however, to bias this dynamicprocess by boosting the strength of one rival figure overanother. In this case, the ‘stronger’ competitor enjoys an
a
b d
c
Figure 1 | Examples of some well-known ambiguousfigures, the perceptual appearance of which fluctuatesover time despite unchanging physical stimulation.a | The Necker cube. b | Rubin’s vase/face figure. c | E. G.Boring’s old lady/young woman figure. d | Monocular rivalry, inwhich two physically superimposed patterns that are dissimilarin colour and orientation compete for perceptualdominance113. Readers are encouraged to view each figure fordurations sufficient to experience alternations in perception,which, for naive viewers, can take some time. Evidently, whenone views figures such as these, the brain vacillates betweenalternative neural states; for this reason, such multistablefigures offer a promising means to study the neural bases ofvisual perception.
© 2001 Macmillan Magazines Ltd
BINOCULAR RIVALRY
NA
TU
RE
RE
VIE
WS
|NEUROSCIE
NCE
VO
LUM
E 3
| J
AN
UA
RY
200
2 | 3
REVIE
WS
cont
ours
are
pitt
ed a
gain
st v
ertic
al,b
ut in
FIG
.2a,
the
ver-
tica
l com
petit
ors a
ppea
r in
a la
rger
,glo
bally
con
grue
ntco
ntex
t tha
t is n
ot p
rese
nt in
the
bott
om p
air o
ftar
gets
.W
hen
obse
rver
s ‘tr
ack’
peri
ods o
fdom
inan
ce a
nd su
p-pr
essi
on w
hile
vie
win
g di
spla
ys li
ke t
hese
,the
tar
get
embe
dded
in th
e m
eani
ngfu
l,co
ngru
ent c
onte
xt te
nds t
opr
edom
inat
e re
lativ
e to
the
sam
e ta
rget
in th
e in
cong
ru-
ent
cont
ext33
.Mor
eove
r,ob
serv
ers
need
not
be
con-
scio
usly
awar
e oft
he m
eani
ngfu
l str
uctu
re o
fa ri
val t
arge
tth
at p
rom
otes
its p
redo
min
ance
— th
e bo
ostin
g ef
fect
of
cont
ext o
pera
tes e
ven
whe
n ob
serv
ers d
o no
t rea
lize
that
a ta
rget
can
be
glob
ally
org
aniz
ed in
to a
mea
ning
ful p
at-
tern
(for
exa
mpl
e,a
Dal
mat
ian
dog)
34.W
ith d
ispla
ys su
chas
thos
e sh
own
in FI
G.2
,enh
ance
d pr
edom
inan
ce c
omes
abou
t thr
ough
a le
ngth
enin
g of
the
dura
tions
ofd
omi-
nanc
e of
that
targ
et,n
ot th
roug
h re
duct
ions
in th
e du
ra-
tion
s ofi
ts su
ppre
ssio
n ph
ases
.It i
s int
eres
ting
to n
ote
that
a si
mila
r pat
tern
ofr
esul
ts is
obs
erve
d fo
r var
iatio
nsin
pre
dom
inan
ce o
fthe
vas
e/fa
ce b
ista
ble
figur
e (F
IG.1
b)
— m
akin
g th
e fig
ure
mor
e fa
ce-l
ike
incr
ease
s the
dur
a-ti
ons o
f‘fa
ce’d
omin
ance
but
doe
s not
aff
ect t
heir
sup-
pres
sion
dur
atio
ns (t
he p
erio
ds fo
r whi
ch th
e fig
ure
isse
en a
s a v
ase;
D.A
.Leo
pold
,unp
ublis
hed
obse
rvat
ions
).T
he in
abili
ty o
fcon
text
to c
ount
erac
t sup
pres
sion
indi
cate
s tha
t neu
ral p
roce
sses
that
am
plify
the
salie
nce
ofa
dom
inan
t tar
get a
re n
ot e
ngag
ed d
urin
g su
ppre
s-si
on.T
he d
iffer
entia
l effe
ct o
fstim
ulus
stre
ngth
and
con-
text
on
the
perc
eptu
al p
redo
min
ance
of
a pa
tter
n is
stro
ng e
vide
nce
that
dom
inan
ce a
nd su
ppre
ssio
n re
ly o
ndi
stin
ct n
eura
l pro
cess
es,a
con
clus
ion
that
is su
ppor
ted
by e
lect
roph
ysio
logi
cal s
tudi
es in
mon
keys
repo
rtin
gbi
nocu
lar r
ival
ry (s
ee b
elow
).
Spat
ial a
ttri
bute
s ofr
ival
ry.P
erce
ptua
l dom
inan
cedu
ring
riv
alry
can
take
on
a ‘p
atch
y’ap
pear
ance
whe
nth
e in
duci
ng
figu
res
are
rela
tive
ly la
rge,
as if
riva
lry
wer
e oc
curr
ing
sim
ulta
neou
sly
wit
hin
zone
s dis
trib
-ut
ed o
ver t
he v
isua
l fie
ld35
;thi
s ten
denc
y is
par
ticul
arly
stro
ng fo
r fov
eally
vie
wed
riv
al ta
rget
s36.H
owev
er,t
hedo
min
ance
pha
ses o
floc
ally
dis
trib
uted
riv
al ta
rget
sca
n no
neth
eles
s bec
ome
entr
aine
d,th
ereb
y cr
eatin
g an
over
all p
atte
rn o
fcoh
eren
t per
cept
ual d
omin
ance
11,1
2,37
.R
emar
kabl
y,th
e co
nso
lidat
ion
of
loca
l riv
alry
into
glob
al d
omin
ance
occ
urs r
eadi
ly e
ven
whe
n th
e co
m-
pone
nt fe
atur
es a
re d
istr
ibut
ed b
etw
een
the
two
eyes
,as
can
be e
xper
ienc
ed u
sing
the
pair
ofr
ival
figu
res r
epro
-du
ced
in F
IG.2
c.It
is te
mpt
ing
to c
oncl
ude
that
per
cep-
tual
gro
upin
g du
rin
g ri
valr
y re
sult
s fr
om t
he s
ame
coop
erat
ive/
com
petit
ive
inte
ract
ions
that
pro
mot
e fig
-ur
al g
roup
ing
duri
ng n
orm
al v
isio
n38,3
9 .A
seco
nd st
riki
ng sp
atia
l fea
ture
ofr
ival
ry c
once
rns
the
tran
siti
on p
erio
ds w
hen
one
figu
re o
vert
hrow
san
othe
r to
achi
eve
perc
eptu
al d
omin
ance
.Typ
ical
ly,t
hese
tran
siti
ons
are
not
inst
anta
neou
s,lik
e su
cces
sive
lyex
pose
d sn
apsh
ots
ofon
e im
age
and
then
the
oth
er.
Inst
ead,
dom
inan
ce e
mer
ges i
n a
wav
e-lik
e fa
shio
n,or
igi-
natin
g at
one
regi
on o
fa fi
gure
and
spre
adin
g fr
om th
ere
thro
ugho
ut th
e re
st o
fthe
figu
re.W
ilson
et a
l.10w
ere
able
to e
stim
ate
the
spee
d w
ith
whi
ch d
omin
ance
spre
ads
by u
sing
riv
al ta
rget
s in
whi
ch d
omin
ance
was
forc
edto
spre
ad a
long
a g
iven
pat
h;an
exa
mpl
e of
thei
r ri
val
effe
ctiv
ely
only
dur
ing
dom
inan
ce,w
here
as in
volu
n-ta
ry,‘
exog
enou
s’at
tent
ion
cont
inue
s to
wor
k du
ring
supp
ress
ion.
Besid
es st
imul
us st
reng
th a
nd a
ttent
ion,
visu
al co
ntex
tca
n al
so in
fluen
ce th
e pr
edom
inan
ce o
fa fi
gure
dur
ing
riva
lry.
Look
at t
he tw
o pa
irs o
friv
al ta
rget
s in
FIG
.2a,
b.W
ithi
n th
e ci
rcul
ar re
gion
s ofb
oth
pair
s,ho
rizo
ntal
a b c d
Figu
re 2
|B
inoc
ular
riva
lry.T
hese
riva
l tar
gets
— d
esig
ned
to b
e vie
wed
by
cros
sing
the
eyes
tosu
perim
pose
the
two
half-
imag
es b
inoc
ular
ly (in
set)
— ill
ustra
te s
ever
al h
allm
ark
char
acte
ristic
s of
bino
cula
r riva
lry. a
,b| U
sing
the
two
pairs
of r
ival t
arge
ts in
aan
d b,
com
pare
the
inci
denc
e of
dom
inan
ce o
f the
cen
tral r
ed/g
reen
gra
ting
whe
n it
appe
ars
in a
con
siste
nt g
loba
l con
text
(a) a
ndw
hen
it do
es n
ot (b
). c
| Thi
s riv
al ta
rget
clo
sely
rese
mbl
es a
figur
e de
vised
by
Dia
z-C
anej
a11(fo
r atra
nsla
tion
of h
is pa
per,
whi
ch w
as w
ritte
n in
Fre
nch,
see
REF
.118
). N
otic
e ho
w fr
eque
ntly
you
expe
rienc
e a
com
plet
e ‘b
ullse
ye’ o
r a c
ompl
ete
set o
f hor
izont
al c
onto
urs,
per
cept
ual o
utco
mes
that
indi
cate
inte
rocu
lar g
roup
ing.
d| A
riva
lry ta
rget
illus
tratin
g th
e te
nden
cy o
f dom
inan
ce to
emer
ge lo
cally
and
then
spr
ead
glob
ally.
Onc
e th
ese
two
half-
imag
es h
ave
been
fuse
d, fix
ate
the
cent
ral ‘
bulls
eye’
, but
obs
erve
the
alte
rnat
ions
in d
omin
ance
bet
wee
n th
is p
air o
f riva
l gra
tings
(one
a s
pira
l gra
ting
and
the
othe
r a ra
dial
gra
ting)
. In
parti
cula
r, no
te h
ow th
e ra
dial
gra
ting
emer
ges
from
sup
pres
sion
at a
sin
gle
poin
t, w
ith d
omin
ance
radi
atin
g in
bot
h di
rect
ions
from
this
loca
tion.
Wils
on e
t al.10
used
riva
l figu
res
such
as
this
to e
stim
ate
the
rate
at w
hich
dom
inan
cesp
read
s. R
athe
r tha
n w
ait f
or d
omin
ance
spo
ntan
eous
ly to
em
erge
at u
npre
dict
able
loca
tions
, the
yin
trodu
ced
abru
pt c
ontra
st in
crem
ents
to d
isrup
t sup
pres
sion
loca
lly, a
nd m
easu
red
the
spee
d at
whi
ch th
e re
sultin
g do
min
ance
wav
e tra
velle
d ar
ound
this
esse
ntia
lly o
ne-d
imen
siona
l figu
re.
Rea
ders
can
vie
w fu
rther
dem
onst
ratio
ns o
f riva
lry b
y na
vigat
ing
to R
.B.’s
Bin
ocul
ar R
ivalry
web
page
. Par
t dm
odifie
d w
ith p
erm
issio
n fro
m R
EF.1
0©
200
1 M
acm
illan
Mag
azin
es L
td.
© 20
01 M
acm
illan
Mag
azin
es L
td
Binocular rivalry: perception switches back and forth betweendifferent images presented to the two eyesTwo images compete for perceptual dominance; one dominatesfor a few seconds before switching to the other
CHARACTERISTICS OF RIVALRY IIncreasing the strength of one rival figure (brighter, movingrather than stationary, densely contoured) increases thepercentage of time that it is dominant.
Periods of suppression are decreased rather than period ofdominance increased - Levelts propositions
Fluctuations in dominance and suppression are irregular:switching times given by a Gamma distribution (Logethetis et al1996)
CHARACTERISTICS OF RIVALRY II
Perceptual dominance can take on a ’patchy’ appearance whenthe inducing figures are relatively large (Kovacs et al 1996)
CHARACTERISTICS OF RIVALRY IIIAttending to one of the rivalry figures can increase itsdominance - not possible to suppress other image completely
Perceptual dominace transitions are not instantaneous.
Instead, dominance emerges in a wave-like fashion, originatingat one region of a figure and spreading from there throughoutthe rest of the figure (Wilson et al 2001)
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I# ! ,.S 95:?J B=5:9@ /3XW<< 5=5863 #D<9 :;4B6546B 2989 :?;B94B; 6?56 6?9 <;@9K 2;7K@ =8;@7:9 =566984 B7=98=;BD6D;4 F;8 6?9 Z8B6,X/<B ;F B6D<7K56D;4G 6?7B 89=KD:56D4> 89B7K6B ;4 ?7<54 8DC5K8L-.3)4:9 6?9 1 4978;4B 2989 @;<D4546G 5 K;:5K =7KB9 6; 6?9 B7==89BB9@$4978;4B 68D>>989@ 5 25C9 6?56 B=895@ 56 :;4B6546 B=99@ 58;74@ 6?98D4> IED>3 .AJ3 1D<7K56D;4B ;F 6?9 85@D5K =566984 2D6?;76 :;KKD4958F5:DKD656D;4 =8;@7:9@ 5 @;<D454:9 25C9 B=99@ ;F -3-+ :< B!,3d;KKD4958 F5:DKD656D;4 25B D468;@7:9@ F;8 :;4:9468D: =566984BG 54@6?DB D4:895B9@ 25C9 B=99@ 6; +3+/ :< B!,3 1=D85K 658>96B 2D6? BD>W4DZ:546KL 89@7:9@ :;KKD4958 F5:DKD656D;4 =8;@7:9@ 5 B=99@ ;F-3SN :< B!,3 E786?98<;89G BD<7K56D;4 ;F 85@D5K 658>96 =566984B 2D6?>5=B B?;29@ 6?56 <;@9K @;<D454:9 25C9B 2989 AK;:]9@ AL 5 6?899W:L:K9W2D@9 >5= A76 :;7K@ _7<= 5 ;49W:L:K9 >5=3 #?7BG 5KK BD<7K56D;489B7K6B 5>899 2D6? ;78 @5653#?DB <;@9K >;9B A9L;4@ =89CD;7B 8DC5K8L <;@9KB-+G-X AL =8;CD@D4>
5 49785K <9:?54DB< F;8 @;<D454:9 25C9 =8;=5>56D;4 54@ D4D4:;8=;856D4> :;KKD4958 F5:DKD656D;4G 2?D:? ?5B A994 B?;24 6;94?54:9 @;<D454:9,S3 E;8 85@D5K >856D4>BG @;<D454:9 25C9 =8;=5W>56D;4 DB >9498569@ AL 89:788946 @DBD4?DAD6D;4 2D6? 5 B=56D5K B=895@5==8;TD<56D4> 6?9 @DB654:9 A962994 ?7<54 ;:7K58W@;<D454:9:;K7<4B,N3 ^6 DB ]4;24 6?56 BD4>K9WK5L98 89:788946 D4?DAD6;8L 496W2;8]B :54 B7==;86 25C9 =8;=5>56D;4XG B; ;78 4962;8] =8;CD@9B 5F786?98 <9:?54DB< F;8 D4?DAD6;8L 25C9 =8;=5>56D;43 #; ;A65D4 6?962;F;K@ B=99@ D4:895B9 F;8 :;KKD4958 =566984B D4 6?9 <;@9KG D6 25B49:9BB58L 6; 9T694@ 6?9 B=56D5K 854>9 ;F 89:788946 9T:D656D;4 6; 62D:96?56 ;F 6?9 89:D=8;:5K D4?DAD6D;4R <;89 K;:5KD`9@ 89:788946 9T:D656D;4=8;@7:9@ <7:? B<5KK98 B=99@ D4:895B9B 5B 9T9<=KDZ9@ AL 6?9 B=D85KBD<7K56D;4B3 d;469T6W@9=94@946 9FF9:6B D4 (, ?5C9 A994 9T=K5D49@AL BD<DK58 K;4>W854>9 :;KKD4958 9T:D656D;4--3)78 @565 :K958KL D<=KD:569 5 896D4;6;=D:5KKL ;8>54D`9@ CDB75K 5895
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T T T T
S S S S
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S
a
b
T
IT IS
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© 2001 Macmillan Magazines Ltd
NEURAL CORRELATES OF BINOCUAR RIVALRY
Visual evoked potentials recorded from scalp electrodes onoccipital cortex
NATURE REVIEWS | NEUROSCIENCE VOLUME 3 | JANUARY 2002 | 5
R E V I E W S
To achieve this kind of tight linkage, Brown andNorcia73 repeatedly modulated the contrast of twodichoptically viewed, orthogonally oriented gratings atslightly different rates, thereby ‘tagging’ the VER wave-forms associated with the two rival gratings.VERs wererecorded while observers pressed buttons to track fluc-tuations in dominance and suppression between thegratings. The resulting tagged waveforms associatedwith the two gratings showed conspicuous, inverselyrelated modulations in amplitude: when the amplitudeof one grating was large, that of the other grating wasinvariably small. Moreover, these modulations weretightly phase-locked to the observers’ perceptual reportsof dominance and suppression (FIG. 3).
These VER measurements, although establishing afirm coupling between brain signals and perceptionduring rivalry, do not tell us where within the visualpathways these signals are arising — electrodes placedover the occipital pole could be registering neural sig-nals arising from any of the multiple visual areas con-tained within the folds of the occipital cortex. To get atthe question of neural locus requires the deployment ofbrain-activity measurements with considerably greaterspatial resolution. With that end in mind, we turn nextto studies using functional brain-imaging techniques.
Functional magnetic resonance imaging. In the past 4 years, several groups have used functional magneticresonance imaging (fMRI) to identify brain regions inwhich blood oxygen level dependent (BOLD) signalsfluctuate in synchrony with binocular rivalry alterna-tions. One study74 documented the existence of multiplecortical areas in which levels of brain activity (inferredfrom modulations in the BOLD signal) were reliablyassociated with spontaneous changes in rivalry statewhile viewing dichoptically presented face and gratingstimuli. Bilateral transient activation was observed in aregion of the fusiform gyrus that is implicated in the pro-cessing of facial information, and in the frontoparietalareas of the right hemisphere, which are implicated inspatial attention. This study focused on transitions in
reduced in magnitude by rivalry suppression — theseare aftereffects attributable to global, rather than local,motion adaptation63,64. All of these results fully supportthe idea that the mechanisms responsible for suppres-sion are cortical. The extent to which these adapta-tion results indicate the involvement of differentvisual areas in suppression clearly requires furtherpsychophysical investigation, in concert with imagingexperiments in humans and electrophysiologicalexperiments in animals.
Visual priming during suppression. Exposure to a visualstimulus can make other, related stimuli easier to iden-tify, as indexed by faster performance and improvedaccuracy — the initial stimulus, in other words,‘primes’visual processing of the subsequent stimulus. Doespriming occur if the priming stimulus is rendered invis-ible by binocular suppression? For visual tasks involvinghigher-level cognitive processes, including picture prim-ing65 and semantic priming66, the answer clearly is ‘no’— suppression renders normally effective priming stim-uli impotent. These results are not too surprising, forboth of these priming paradigms call for relativelyrefined analyses of visual information, of the sort con-ventionally attributed to high-level visual processingoutside the domain of early visual areas. Evidently, dur-ing suppression phases of rivalry, input to those process-ing stages is effectively blocked.
Direct evidenceVisual evoked responses. A handful of studies has usedscalp electrodes placed over the occipital lobe to recordvisually evoked responses (VERs) while observers expe-rience binocular rivalry; with one exception67, thesestudies have found reductions in the amplitude of theVER signal associated with the suppressed target68–72.These findings, however, were based on time-averagedrecordings pooled over the left and right eyes, makingit impossible to link fluctuations in VER amplitudewith shifts in dominance and suppression measured in real time.
0 20 40 60 80 100 120
4
2
Time (s)
0
VE
P
Figure 3 | Visually evoked potentials recorded during rivalry. Observers view a pair of orthogonally oriented gratings, one imaged to each eye. The bars of thegrating flicker repeatedly in counterphase, producing reliable, time-locked modulations in the amplitude of the visually evoked potential (VEP) recorded from scalpelectrodes placed over the occipital pole. The flicker rate of the left eye’s grating differs from that of the right eye, such that each grating produces its own distinctwaveform that can be teased apart from the other, followed over time, and correlated with the observer’s record of rivalry alternations. This reveals robust, reliablemodulations in the VEP amplitudes of the two waveforms, both highly correlated with the perceptual state of the evoking grating73. This pattern of results clearlyreveals a neural signature of binocular rivalry arising within the occipital cortex, but it is not possible to pinpoint definitively from which visual area(s) these signalsarise. Modified with permission from REF. 73 © 1997 Elsevier Science.
© 2001 Macmillan Magazines Ltd
Functional magnetic resonace imaging (fMRI) - used to identifybrain regions in which blood oxygen level dependent (BOLD)signals fluctuate in synchrony with binocular rivalry
Flutuating BOLD signals that are highly correlated withobservers’ perceptual reports have been found in primary visualcortex (grating stimuli) and higher-order visual areas (faces,buildings)
WAVES MEASURED BY FMRI (LEE ET AL 2005)
Red (blue) circles indicates subregion of V1 corresponding to upper(lower)-right quadrant of low-contrast stimulus
SINGLE UNIT RECORDINGS IN AWAKE PRIMATES
8 | JANUARY 2002 | VOLUME 3 www.nature.com/reviews/neuro
R E V I E W S
suppression described earlier. Overall, both striate andearly extrastriate areas (such as areas V4 and MT) showedactivity changes during rivalry, but for most cells theactivity modulations, although highly significant in astatistical sense, were modest compared with the percep-tual changes experienced during rivalry. Moreover,almost none of the neurons ceased to fire completelyduring suppression.
Responses were markedly different in the temporallobe. The inferotemporal cortex, a region starting just infront of area V4 and continuing almost up to the tempo-ral pole, has an essential role in higher visual functions,including pattern perception and object recognition89.Inferotemporal neurons respond with high selectivity to complex, two-dimensional visual patterns, or even toentire views of natural and artificial objects. Damage tothe inferotemporal cortex typically produces severedeficits in perceptual learning and object recognition,even in the absence of significant changes in basic visualcapacities. It was natural to query the types of responsechange observed during rivalry in this high-level area.The monkeys participating in these experiments reportedwhether they perceived a sunburst-like pattern, or imagesof animate or man-made objects89. Recordings showedthat most of the inferotemporal neurons were active onlywhen their preferred stimulus was perceived. In otherwords, in contrast to neurons in areas V4 and MT, infero-temporal neurons showed essentially no activity duringthe perceptual suppression of the stimulus, indicatingthat the studied areas represent a stage of processingbeyond the resolution of perceptual conflict.
The intriguing complexity and diversity of responsesin early extrastriate cortex during rivalry hints at its role inperceptual organization. The areas surrounding the pri-mary visual cortex, including V4 and MT, are in ananatomical position to integrate information fromascending and descending visual streams, and to interactwith structures that are crucial for object vision.Responses in these areas can be considerably enhancedor inhibited when the monkey attends to the cell’s pre-ferred or non-preferred stimulus, respectively90,91, evenwhen there is no concomitant change in the stimulusitself, and the mechanisms underlying such changes arealso competitive in nature92. Damage to area V4 andposterior inferotemporal cortex disrupts the top–downinput to early areas, strongly interfering with the ani-mal’s ability to ignore distracters in the lesioned areas93
and to detect less salient stimuli94,95. In short, the diverseactivity observed in early extrastriate cortex mightreflect the competitive interactions that characterize allthose selection processes involved in image segmentationand grouping, interactions that are greatly accentuatedduring binocular rivalry.
Finally, it should be noted that, in any area of the brain,the absence of changes in firing rate should not be inter-preted as an absence of perceptual state changes, as popu-lations of neurons can increase and decrease the coher-ence of their firing as a function of time. Such increasesin coherence have significant effects on the next stage ofprocessing, as synchronized inputs produce higher andmore steeply depolarized membrane excursions for
the LGN of the alert, fixating monkey provided no evi-dence for rivalry inhibition at the subcortical level in thegeniculostriate system85.
Neurons in the cortex behave differently. Experimentswith monkeys reporting rivalry showed that inhibitionof responses during binocular suppression is evident asearly as the primary visual cortex86. In these experiments,the animals reported the perceived orientation ofrivalling gratings by pulling levers, while maintaining fix-ation on a central light spot for several seconds. Notably,the psychophysical performance of these trained mon-keys was similar to that obtained from human observers,indicating that similar neural mechanisms mightunderly rivalry in the two species.
The extent to which neural activity was modulated in phase with the animal’s perceptual report increased insuccessive stages of early visual cortical areas. Curiously,however, some extrastriate neurons were excited onlywhen their preferred stimulus was visible, whereas otherswere excited when it was suppressed87,88. The latter neu-rons, the activity of which is in reverse correlation withthe animals’ perception of their preferred stimulus,might be part of an inhibitory mechanism that is sepa-rate from and, to some extent, independent of the mech-anisms of perception. Such an independent mechanismwas predicted by psychophysical measurements of theeffects of the strength of a stimulus on its predomi-nance17,22,23. It also offers a possible explanation for thedifferential effects of stimulus strength and context on
40
20
0
Spi
kes
s–1
Figure 5 | Single-cell recordings during rivalry. Usingoperant conditioning techniques, monkeys are trained tooperate a lever to indicate which one of two competingmonocular images is dominant over time (see REF. 13 for detailsof training). Activity recorded from single cells in the awake,behaving monkey can be correlated with the animal’sperceptual reports, thereby identifying brain regions in whichcellular activity mirrors perceptual experience. The bar alongthe x axis indicates alternating awareness of the two images.
© 2001 Macmillan Magazines Ltd8 | JANUARY 2002 | VOLUME 3 www.nature.com/reviews/neuro
R E V I E W S
suppression described earlier. Overall, both striate andearly extrastriate areas (such as areas V4 and MT) showedactivity changes during rivalry, but for most cells theactivity modulations, although highly significant in astatistical sense, were modest compared with the percep-tual changes experienced during rivalry. Moreover,almost none of the neurons ceased to fire completelyduring suppression.
Responses were markedly different in the temporallobe. The inferotemporal cortex, a region starting just infront of area V4 and continuing almost up to the tempo-ral pole, has an essential role in higher visual functions,including pattern perception and object recognition89.Inferotemporal neurons respond with high selectivity to complex, two-dimensional visual patterns, or even toentire views of natural and artificial objects. Damage tothe inferotemporal cortex typically produces severedeficits in perceptual learning and object recognition,even in the absence of significant changes in basic visualcapacities. It was natural to query the types of responsechange observed during rivalry in this high-level area.The monkeys participating in these experiments reportedwhether they perceived a sunburst-like pattern, or imagesof animate or man-made objects89. Recordings showedthat most of the inferotemporal neurons were active onlywhen their preferred stimulus was perceived. In otherwords, in contrast to neurons in areas V4 and MT, infero-temporal neurons showed essentially no activity duringthe perceptual suppression of the stimulus, indicatingthat the studied areas represent a stage of processingbeyond the resolution of perceptual conflict.
The intriguing complexity and diversity of responsesin early extrastriate cortex during rivalry hints at its role inperceptual organization. The areas surrounding the pri-mary visual cortex, including V4 and MT, are in ananatomical position to integrate information fromascending and descending visual streams, and to interactwith structures that are crucial for object vision.Responses in these areas can be considerably enhancedor inhibited when the monkey attends to the cell’s pre-ferred or non-preferred stimulus, respectively90,91, evenwhen there is no concomitant change in the stimulusitself, and the mechanisms underlying such changes arealso competitive in nature92. Damage to area V4 andposterior inferotemporal cortex disrupts the top–downinput to early areas, strongly interfering with the ani-mal’s ability to ignore distracters in the lesioned areas93
and to detect less salient stimuli94,95. In short, the diverseactivity observed in early extrastriate cortex mightreflect the competitive interactions that characterize allthose selection processes involved in image segmentationand grouping, interactions that are greatly accentuatedduring binocular rivalry.
Finally, it should be noted that, in any area of the brain,the absence of changes in firing rate should not be inter-preted as an absence of perceptual state changes, as popu-lations of neurons can increase and decrease the coher-ence of their firing as a function of time. Such increasesin coherence have significant effects on the next stage ofprocessing, as synchronized inputs produce higher andmore steeply depolarized membrane excursions for
the LGN of the alert, fixating monkey provided no evi-dence for rivalry inhibition at the subcortical level in thegeniculostriate system85.
Neurons in the cortex behave differently. Experimentswith monkeys reporting rivalry showed that inhibitionof responses during binocular suppression is evident asearly as the primary visual cortex86. In these experiments,the animals reported the perceived orientation ofrivalling gratings by pulling levers, while maintaining fix-ation on a central light spot for several seconds. Notably,the psychophysical performance of these trained mon-keys was similar to that obtained from human observers,indicating that similar neural mechanisms mightunderly rivalry in the two species.
The extent to which neural activity was modulated in phase with the animal’s perceptual report increased insuccessive stages of early visual cortical areas. Curiously,however, some extrastriate neurons were excited onlywhen their preferred stimulus was visible, whereas otherswere excited when it was suppressed87,88. The latter neu-rons, the activity of which is in reverse correlation withthe animals’ perception of their preferred stimulus,might be part of an inhibitory mechanism that is sepa-rate from and, to some extent, independent of the mech-anisms of perception. Such an independent mechanismwas predicted by psychophysical measurements of theeffects of the strength of a stimulus on its predomi-nance17,22,23. It also offers a possible explanation for thedifferential effects of stimulus strength and context on
40
20
0
Spi
kes
s–1
Figure 5 | Single-cell recordings during rivalry. Usingoperant conditioning techniques, monkeys are trained tooperate a lever to indicate which one of two competingmonocular images is dominant over time (see REF. 13 for detailsof training). Activity recorded from single cells in the awake,behaving monkey can be correlated with the animal’sperceptual reports, thereby identifying brain regions in whichcellular activity mirrors perceptual experience. The bar alongthe x axis indicates alternating awareness of the two images.
© 2001 Macmillan Magazines Ltd
Primate trained to operate a lever to indicate which of twocompeting monocular stimuli is dominant over timeActivity recorded in single cells correlated with animal’sperceptual reports.Experiments showed that inhibition of reponses is evident asearly as primary visual cortex (Leopold and Logethetis 1996).
Part II. Models of binocular ri-valry
COMPETITIVE NETWORK MODEL (NO SPACE)
right eye
recurrent excitation
cross
inhibition
fast activity u(t)
slow adaptation q (t)u
fast activity v(t)
slow adaptation q (t)v
left eye
Two populations of cells driven by left and right eye stimuli,respectivelyRecurrent excitatory connections within a population, mutualinhibition between populationsEach population exhibits some form of slow adaptation - spikefrequency adaptation or synaptic depression
COMPETITIVE NETWORKS IIMany studies of rivalry in competitive networks
Laing and Chow (2002)Taylor, Cottrell and Kristan (2002)Wilson (2003)Shpiro et al (2007,2009)Moreno-Bote, Rubin and Rinzel (2007)Kilpatrick and Bressloff (2010)Seely and Chow (2011)Diekman et al (2012)
Issues include
noise vs. adaptationtype of adaptationLevelt’s propositionsescape vs. release
COMPETITIVE NETWORK WITH SYNAPTIC DEPRESSIONPopulation activity variables u, v (current,voltage)
u̇(t) = −u(t) + aequ(t)F(u(t))− aiqv(t)F(v(t)) + IL
v̇(t) = −v(t) + aeqv(t)F(v(t))− aiqu(t)F(u(t)) + IR
Depression variables qu, qv represent the short-term depletion ofpresynaptic resources (slow recovery τs � 1)
τsq̇j(t) = (1− qj(t))− βqj(t)F(uj(t)), j = u, v,
FIXED POINTS FOR HEAVISIDE RATE FUNCTIONOff state U∗ = V∗ = I and Q∗u = Q∗v = 1
On-state or fusion state
(U∗,V∗) =
(ae − ai
1 + β+ I,
ae − ai
1 + β+ I), (Q∗u,Q
∗v) =
(1
1 + β,
11 + β
),
Left eye dominant state:
(U∗,V∗) =
(ae
1 + β+ I, I − ai
1 + β
), (Q∗u,Q
∗v) =
(1
1 + β, 1)
Right eye dominant state
(U∗,V∗) =
(I − ai
1 + β,
ae
1 + β+ I), (Q∗u,Q
∗v) =
(1,
11 + β
)
BIFURCATION DIAGRAM (KILPATRICK/PCB 2010)
0.1 0.2 0.3-0.4
-0.2
0
0.2
0.4
input I
u
WTA
fusion
rivalry
Off
Coexistence of rivalry oscillations and the fusion state(τs = 500, β = 5, ae = 0.4, ai = 1.0, κ = 0.05)
DOMINANCE TIMES
1200 1400 1600 1800
0.2
0
0.2
0.4
t
qL qR
uRuL
0.2 0.25 0.3 0.350
200
400
600
I
T L,R
5 80
200
400
600
β
TL,R
6 7
Oscillations arise through an escape rather than a releasemechanism – suppressed population’s activity crosses thresholdbefore dominant population ceases firing
Can construct explicit solution and determine dominance timesTL and TR.
1D RIVALRY WAVES (KANG ET AL 2009)
SPATIALLY EXTENDED COMPETITIVE NETWORK (1D)recurrent excitation w
cross inhibition wi
e
u m u
vm
n
vn
τdum
dt= −um + Iu +
∑n
([we]mnqu,nF(un)− [wi]mnqv,nF(vn))
τdvm
dt= −vm + Iv +
∑n
([we]mnqv,nF(vn)− [wi]mnqu,nF(un))
τsdqj,m
dt= 1− qj,m − βqj,mF(vm), j = u, v
SPATIALLY EXTENDED COMPETITIVE NETWORK (1D)
[we]mn is the strength of excitation from the nth to the mthpopulation with the same eye preference
[wi]mn is the strength of cross-inhibition between populationswith opposite eye preferences.
The weights are assumed to decrease with distance of separation|m− n| according to an exponential or Gaussian distribution.
Sigmoidal firing rate function
F(u) =F0
1 + e−η(u−κ)
Discrete model useful for simulations. For analytical insightstake a continuum limit =⇒ neural field model
NEURAL FIELD MODEL (PCB AND WEBBER 2012)Left eye network:
τ∂u(x, t)∂t
= −u(x, t) +
∫ ∞−∞
we(x− x′)qu(x′, t)F(u(x′, t)))dx′
−∫ ∞−∞
wi(x− x′)qv(x′, t)F(v(x′, t)))dx′ + Iu(x, t)
τs∂qu(x, t)
∂t= 1− qu(x, t)− βqu(x, t)F(u(x, t))
Right eye network:
τ∂v(x, t)∂t
= −v(x, t) +
∫ ∞−∞
we(x− x′)qv(x′, t)F(v(x′, t)))dx′
−∫ ∞−∞
wi(x− x′)qu(x′, t)F(u(x′, t)))dx′ + Iv(x, t)
τs∂qv(x, t)∂t
= 1− qv(x, t)− βqv(x, t)F(v(x, t)).
Part III. Traveling fronts
TRAVELING FRONT WITH SLOW ADAPTATION I
Suppose system is initially in a right dominated WTA state.
Perturbation of system initiates a propagating front thatgenerates a switch from right to left eye dominance
Adiabatic approximation (τs � 1): If L is size of domain and c iswavespeed then L/c� τs so that qj(x, t) ≈ Qj, j = u, v
Consider a traveling wave front solution of the form
u(x, t) = U(ξ), v(x, t) = V(ξ), ξ = x− ct
(U(ξ),V(ξ))→ XL = (Quae + I, I −Qvai) as ξ → −∞,
(U(ξ),V(ξ))→ XR = (I −Quai,Qvae + I) , as ξ →∞
TRAVELING FRONT WITH SLOW ADAPTATION IIThreshold conditions
U(0) = κ, V(X) = κ
If c > 0 then the front represents a solution in which activityinvades a suppressed left eye network and retreats from adominant right eye network.
ξ
U(ξ)V(ξ)
advancing suppressed
percept retreating dominant
percept
ANALYTICAL SOLUTION ISubstitute the traveling wave solution into NF equations forfixed Qu,Qv and F(u) = H(u− κ):
−cdUdξ
+ U = Qu∫ 0−∞we(ξ − ξ′)dξ′ −Qv
∫∞X wi(ξ − ξ′)dξ′ + I
−cdVdξ
+ V = Qv∫∞
X we(ξ − ξ′)dξ′ −Qu∫ 0−∞ wi(ξ − ξ′)dξ′ + I.
Rewrite equations in integral form
U(ξ) = eξ/c
[κ− 1
c
∫ ξ
0e−z/cΨX(z)dz− I(1− e−ξ/c)
], ξ > 0
V(ξ) = e(ξ−X)/c
[κ− 1
c
∫ ξ−X
0e−z/cΦX(−z)dz
],
−I(e(ξ−X)/c), ξ > X
ANALYTICAL SOLUTION II
ΨX and ΦX defined by
ΨX(z) = Qu
∫ ∞z
we(y)dy−Qv
∫ z−X
−∞wi(y)dy.
ΦX(z) = Qv
∫ ∞z
we(y)dy−Qu
∫ z−X
−∞wi(y)dy.
Boundedness of solution as ξ →∞ (assuming c > 0) implies thethreshold conditions
κ =
∫ ∞0
e−sΨX(cs)ds + I,
κ =
∫ ∞0
e−sΦX(−cs)ds + I.
SYMMETRY BREAKINGIf Qu = Qv = 1 (no synaptic depression) then
κ =
∫ ∞0
e−sΨX(cs)ds + I, κ =
∫ ∞0
e−sΨX(−cs)ds + I.
Subtracting equations shows that∫ ∞0
e−s [ΨX(cs)−ΨX(−cs)] ds = 0
No traveling wave solution, since if c 6= 0 then
ΨX(cs)−ΨX(−cs) = −∫ cs
−cswe(y)dy−
∫ cs−X
−cs−Xwi(y)dy < 0
for all s ∈ [0,∞).
Slow synaptic depression (Qu 6= Qv) breaks the symmetry of thethreshold crossing conditions, leading to a unique (stable)solution for c,X as a function of the network parameters.
WAVESPEED COVARIES WITH ALTERNATION RATE 1/T
0.24 0.25 0.26 0.27 0.28
input I
5
6
7
8
9
0.24 0.25 0.26 0.27
0.9
1.0
1.1
1.2
1.3
c T-1
input I
Default parameters: ai = 1, ae = 0.4, σi = 1, σe = 2, β = 5, κ =0.05, I = 0.24,Qu = 0.42,Qv = 0.25
NUMERICS
Fix units by setting σe = 2, τ = 1. Wavespeed c = 1 indimensionless units corresponds to c = σe/2τ in physical units.
Kang et. al. (2009) find speeds of order 10 mm/sec. This isconsistent with c = 1 if we take σe ∼ 200µm and τ ∼ 10msec.
Trigger stimulus switched on at time t0 and has duration∆t = 10, corresponding to 200ms as in Kang et. al. (2009)
Size of the excited region ∆x ∼ σe. This is consistent with the sizeof perturbation used in the experiments by Kang et. al., whichwas of size 0.2 degrees, corresponding to 0.8mm of cortical tissue.
SOLITARY BINOCULAR RIVALRY WAVE
100 200 300 400 500 600
-0.2
-0.1
0.1
0.2
0.3
0.4
time t
v
u
qvqu
10
5
-5
-10
0
10
5
-5
-10
0
300 302 304 306 308 310 312 314time t
sp
ace
x
300 302 304 306 308 310 312 314time t
sp
ace
x
303 304 305 306 307 308t
2
4
6
8
10
x
302
t
stimulus
u(x,t) = 0.3
u(x,t) v(x,t)
spontaneous oscillations
WAVESPEED DEPENDS ON PHASE OF STIMULUS ONSET
-5
5
10
15
c
θ0
qvqu
1 2 3 4 5 6
cL cR
Kang et al. (2009) used periodic trigger stimuli and averaged overmultiple cycles so lost phase information. Also need to take intoaccount the effects of noise.
PERIODIC TRIGGER STIMULI AND ADDITIVE NOISE
Introduce additive white noise to the depression dynamics:
τs∂qu(x, t)
∂t= 1− qu(x, t)− βqu(x, t)f (u(x, t)) + σξu(x, t)
τs∂qv(x, t)∂t
= 1− qv(x, t)− βqv(x, t)f (v(x, t)) + σξv(x, t)
with
〈ξu(x, t)〉 = 〈ξv(x, t)〉 = 0
and
〈ξi(x, t)ξj(x′, t′)〉 = δ(x− x′)δ(t− t′)δi,j, i, j = u, v
Consider alternating, periodic trigger stimuli
SPONTANEOUS VS. PERIODICALLY FORCED
SWITCHING
100 200 300 400 500 600
-0.2
-0.1
0.1
0.2
0.3
0.4
100 200 300 400 500 600
-0.2
-0.1
0.1
0.2
0.3
0.4
0.5
time t
time tvu
qv qu
Noise strength σ = 1
Spontaneous switching
Noise strength σ = 1
Periodically forced switching
T0 = 100
BREAKDOWN OF MODE–LOCKING AS NOISE
INCREASES
100 200 300 400 500 600
- 0.2
- 0.1
0.1
0.2
0.3
0.4
100 200 300 400 500 600
0.2
0.4
0.6
time t
time t
time t
100 200 300 400 500 600
- 0.2
0.2
0.4
0.6
0
0
0
σ = 1
σ = 2
σ = 3
∆ u = u(x0,t1) − u(x0,t1)
0.01 0.02 0.03
P(ω)
ω/2π
P(ω)
P(ω)
200
400
600
800
5
10
15
20
25
30
1
2
3
4
5
6
0.01 0.02 0.03
0.01 0.02 0.03
ω/2π
ω/2π
σ = 1
σ = 2
σ = 3
power spectrum of U
RIVALRY WAVES PERSIST FOR SIGMOIDAL FIRING RATE
FUNCTION
100 200 300 400 500 600
-0.1
0.1
0.2
0.3
0.4
time t
v u
qv qu
(a)
(b) (c)10
5
-5
-10
0
10
5
-5
-10
0
290 295 300 305 310time t
sp
ace
x
sp
ace
x
290 295 300 305 310time t
304 306 308
4
6
8
x
302t
(d)
298 300
0
Part IV. Moving stimuli
ROTATING STIMULIThe left (right) eye is presented with a low (high) contrast carrier(mask) stimulus rotating in an anti-clockwise (clockwise)direction.
A transient increase in the contrast of the carrier stimulusinduces a pair of counter propagating waves.
The wave traveling in the same direction as the stimulus reachesthe top first, indicating that it has a higher speed
A
Left Eye Right Eye
B
Left Eye Right Eye
stimulus motion wave propagation
NETWORK MODEL OF DIRECTION SELECTIVITYSimplified stimuli: The left (right) eye is shown an orientedgrating moving rightwards (leftwards).
Pair of 1D neural fields that represent the activity of neuronsresponding maximally to the orientation and motion of thecorresponding grating.
Gaussian excitatory recurrent connections, cross-inhibition andslow synaptic depression.
Introduce an asymmetric shift in excitation of size x0 in the leftnetwork and a shift of −x0 in the right network.
x0
Left Eye
Right Eye
x
x x
x
Excitation
Inhibition
TRIGGERING STIMULUS
Induction of a wave moving in (i) the opposite direction tostimulus motions and (ii) the same direction as stimulus motion.For annuli stimuli, a local increase in the contrast of the carriercan induce both waves
For linear gratings, a local increase in contrast has to be inducedseparately at either one end or the other of the stimulus
Left Eye Right Eye
stimulus motion wave propagation
(i) (ii)
(ii)(i)carrier mask
TRAVELING WAVEFRONTS (PCB AND SAM CARROLL)
Plot of profiles in the co-moving frame ξ = x− ct for wave frontstraveling in the positive (left) and negative (right) direction.
DIRECTIONAL SYMMETRY BREAKING IA Space-time plot of traveling fronts with positive wave speed
(left) and negative wave speed (right) with x0 = 3
Note that U−(x, t) has been reflected about the x = 0 axis forvisual comparison against U+(x, t).
B Plot of analytically computed positive and negative wave speedsagainst x0
DIRECTIONAL SYMMETRY BREAKING II
Introducing asymmetric shift in cross-inhibition rather thanexcitation does not generate observed directional symmetrybreaking
Same result holds if a different form of asymmetry is introducedeg. an asymmetric shift in the spatial rate of decay in theweights.
Sensitive to source of slow adaptation - asymmetriccross-inhibition works in the case of spike frequency adaptation
Part V. Stochastic rivalry waves
STOCHASTIC MODEL I (PCB AND WEBBER 2012)
Langevin equation (or stochastic PDE) for the stochastic activityvariables U(x, t) and V(x, t):
dU =
[−U + Qu
∫ ∞−∞
we(x− y)F(U(y, t))dy
−Qv
∫ ∞−∞
wi(x− y)F(V(y, t))dy + Iu
]dt + ε
12 dWu
dV =
[−V + Qv
∫ ∞−∞
we(x− y)F(V(y, t))dy
−Qu
∫ ∞−∞
wi(x− y)F(U(y, t))dy + Iv
]dt + ε
12 dWv
with Qu,Qv fixed.
STOCHASTIC MODEL II
Wu, Wv represent independent Wiener processes
〈dW{u,v}(x, t)〉 = 0,〈dWi(x, t)dWj(x′, t′)〉 = 2δi,jC([x− x′] /λ)δ(t− t′)dtdt′
where i, j = u, v and 〈·〉 denotes averaging with respect to theWiener processes.
λ is the spatial correlation length of the noise such thatC(x/λ)→ δ(x) in the limit λ→ 0, and ε determines the strengthof the noise, which is assumed to be weak
SEPARATION OF TIME-SCALES I
Fluctuations generate two distinct phenomena that occur ondifferent time–scales (Geier et al 1983,Sagues, Sancho andGarcia-Ojalvo 2007)
[A] Diffusive–like displacement of the wave from its uniformlytranslating position at long time scales
[B] Fast fluctuations in the wave profile around its instantaneousposition at short time scales
Decompose solution as (ξ = x− ct)
U(x, t) = U0(ξ −∆(t)) + ε1/2U1(ξ −∆(t), t),V(x, t) = V0(ξ −∆(t)) + ε1/2V1(ξ −∆(t), t).
where (U0,V0) is deterministic wave solution
SEPARATION OF TIME-SCALES IISubstitute into NF equations and expand to O(ε1/2)
dU1(ξ, t)− Lu(U1(ξ, t),V1(ξ, t)) = ε−12 U′0(ξ)d∆(t) + g(U0)dWu
dV1(ξ, t)− Lv(U1(ξ, t),V1(ξ, t)) = ε−12 V′0(ξ)d∆(t) + g(V0)dWv
with U1 = U1(ξ, t) and ∆(t) = O(ε1/2).
Lu,Lv are non-self-adjoint linear operators
Lu(A1,A2) = cdA1
dξ+ A1 + Qu
∫ ∞−∞
we(ξ − ξ′)F′(U0(ξ′))A1(ξ′)dξ′
−Qv
∫ ∞−∞
wi(ξ − ξ′)F′(V0(ξ′))A2(ξ′)dξ′
Lv(A1,A2) = cdA2
dξ+ A2 + Qv
∫ ∞−∞
we(ξ − ξ′)F′(V0(ξ′))A2(ξ′)dξ′
−Qu
∫ ∞−∞
wi(ξ − ξ′)F′(U0(ξ′))A1(ξ′)dξ′
SEPARATION OF TIME-SCALES III
Let L denote the vector-valued operator with components Lu,Lv.That, is
L(
A1A2
)=
(Lu(A1,A2)Lv(A1,A2)
)L has a 1D null space spanned by (U′0(ξ),V′0(ξ))T
Solvability condition for the existence of a nontrivial solution:the inhomogeneous part is orthogonal to all elements of the nullspace of the adjoint operator L∗.
The latter is defined with respect to the inner product∫ ∞−∞
B(ξ) · LA(ξ)dξ =
∫ ∞−∞
L∗B(ξ) ·A(ξ)dξ
SEPARATION OF TIME-SCALES IVFind that
L∗(
B1B2
)=
(L∗u(B1,B2)L∗v(B1,B2),
)where
L∗u(B1,B2) = −cdB1
dξ+ B1 + F′(U0)Qu
∫ ∞−∞
we(ξ − ξ′)B1(ξ′)dξ′
−F′(V0)Qv
∫ ∞−∞
wi(ξ − ξ′)B2(ξ′)dξ′
and
L∗v(B1,B2) = −cdB2
dξ+ B2 + F′(V0)Qv
∫ ∞−∞
we(ξ − ξ′)B2(ξ′)dξ′
−F′(U0)Qu
∫ ∞−∞
wi(ξ − ξ′)B1(ξ′)dξ′
SDE FOR ∆(t)The adjoint operator L∗ also has a one-dimensional null-spacespanned by V(ξ).Obtain solvability condition
0 =
∫ ∞−∞V1(ξ)
[U′0(ξ)d∆(t) + ε1/2dWu(ξ, t)
]dξ
+
∫ ∞−∞V2(ξ)
[V′0(ξ)d∆(t) + ε1/2dWv(ξ, t)
]dξ.
Thus ∆(t) is a Brownian process with
〈∆(t)〉 = 0, 〈∆(t)2〉 = 2D(ε)t
D(ε) = ε
∫ ∞−∞
(V1(ξ)2 + V2(ξ)2)U2
0(ξ)dξ[∫ ∞−∞
(V1(ξ)U′0(ξ) + V2(ξ)V′0(ξ)) dξ]2 .
RESULTS I: HEAVISIDE F(u) = H(u− κ)
Snapshots of a stochastic composite wave
5 10 15 20x
- 0.1
0.1
0.2
0.3
0.4
5 10 15 20x
- 0.1
0.1
0.2
0.3
0.4
(a) (b)
5 10 15 20x
- 0.1
0.1
0.2
0.3
0.4
(c)
5 10 15 20x
- 0.1
0.1
0.2
0.3
0.4
(d)
RESULTS IIDetermine the stochastic positions Xa(t) such that U(Xa(t), t) = a,for various level set values a ∈ (0.5κ, 1.3κ)
Mean and variance
X(t) = E[Xa(t)], σ2X(t) = E[(Xa(t)− X̄(t))2]
averaged with respect to a and over N trials.
X(t) ∼ cεt and σ2X(t) ∼ 2D(ε)t (after initial transients)
1 2 3 4
9
10
11
12
13
14
1 2 3 4
2
4
6
8
time t
X(t)_
σX(t)2
(a) (b)
time t
x 10-4
RESULTS IIILet TL denote the first passage time (FPT) for the wave to travel adistance L: cTL + ∆(TL) = L given ∆(0) = 0.
FTP density is given by an inverse Gaussian or Walddistribution:
f (TL) = F(TL;Lc,
L2
D),
where
F(T;µ, λ) =
[λ
2πT3
]1/2
exp(−λ(T − µ)2
2µ2T
).
10 20 30 40x
0.1
0.2
0.3
0.4
u(x,0)
2.33 2.34 2.35 2.36 2.37 2.38 2.39Τ
10
20
30
40
f(T)
(a) (b)
Part VI. Future directions
OVERSIMPLIFIED MODEL OF V1
The model only represents neurons that fire maximally withrespect to the given monocular image
For example, in the case of a vertical grating, we only considerneurons that fire maximally to vertical orientations
Neglects orientation tuning due to recurrent connectionsbetween neurons with different orientation preferences.
1D model cannot handle annulus experiment of Wilson et al(2001) with varying orientations
Speed of wave depends on colinearity of image -orientation-dependence of long-range horizontal connections
ORIENTATION-DEPENDENCE OF WAVESPEED (WILSON
ET AL 2001)
letters to nature
908 NATURE | VOL 412 | 30 AUGUST 2001 | www.nature.com
increase in propagation time with distance around the annulus (Fig.1b). At the greatest distance around the annulus, two observersshowed a ¯attening of their radial data, which is attributable tospontaneous reappearance of the suppressed pattern before arrivalof the triggered dominance wave. For each observer propagationtimes, Tp(x), were therefore ®t with an equation incorporatingconstant-speed wave propagation (v) along with the gamma prob-ability, P(t), of spontaneous release from suppression at the targetsite before wave arrival:
Tp�x� � T0 �x
v1 2 #
x=v
0P�t� dt
� �� #
x=v
0tP�t� dt #
x=v
0P�t� dt
� ��1�
where x is the travel distance and T0 is a constant response latencyevident in the zero-distance data. The second and third termsin the equation are: (wave arrival time, x/v) ´ (probability of noprior spontaneous reappearance) + (expected time of spontaneousreappearance) ´ (probability of prior spontaneous reappearance).Least-mean-squares ®tting of v and P(t) parameters to the radialdata revealed an average propagation speed across observers of3.65 6 0.54 degrees s-1.
Recurrent excitatory connections in primate visual cortex pref-erentially interconnect cells with similar preferred orientations and
receptive ®elds that are roughly collinear12,13. Both psychophysical14
and transcranial magnetic stimulation15 studies provide supportingevidence for collinear facilitation in humans. Therefore, we repeatedour measurements using a low-contrast concentric target in place ofthe radial target. The same spiral pattern was again used, as it hadthe same local orientation difference (458) from both radial andconcentric contours. All observers again showed a linear increase inpropagation time with distance, but the slopes were much shal-lower, signifying a greater propagation speed (Fig. 1b). Collinearityof the suppressed target contours increased speed approximatelytwofold in three observers and even more in the fourth, averaging9.60 6 4.76 degrees s-1. This increase in speed is consistent withprevious evidence for facilitation between collinear gratings duringthe dominance phase of rivalry16.
We investigated three additional aspects of dominance wavepropagation. First, a low-contrast, spiral target pattern was usedwith a pitch angle orthogonal to the high-contrast spiral mask. Thismanipulation produced a speed of 5.8 degrees s-1, intermediatebetween radial and concentric patterns (Fig. 1b, bottom left).Second, we tested whether dominance waves could propagateacross a gap in the suppressed stimulus. Accordingly, a permanentgap (0.928 wide) in visual angle (three grating cycles) was intro-duced into the radial annulus at a point that was 67.58 distant fromthe marked arrival point. Dominance waves that were triggered67.58 beyond the gap (that is, 1358 from the arrival point) wereblocked by the gap for both of the observers tested, and propagationtimes rose from 1.60 6 0.055 s (S.L.) or 1.55 6 0.056 s (R.B.) with-out the gap, to 2.71 6 0.17 s (S.L.) or 2.28 6 0.10 s (R.B.) in thepresence of the gap. Both of these differences were highly sig-ni®cant (t126 . 45.0; P , 10-6 for each subject), and the increasedtimes correlate with the longer pathway (by 67%) in the oppositedirection around the annulus when the shorter pathway is blockedby the gap. Very small gaps, however, can be traversed bydominance waves: a gap width of only 0.318 (one radial gratingcycle) yielded equivalent propagation times for gap and no-gapconditions. Third, we determined whether eye movements woulddisrupt wave propagation. At the moment of wave initiation,observers shifted ®xation from the central region (bull's-eye) ofthe target to the marked arrival point itself. Arrival times werenow independent of distance around the annulus, implying thateye movements effectively abolished retinotopically-based wavepropagation.
To learn how wave speed varies with eccentricity, we scaled ourentire stimulus so that the mean annular radius doubled from 1.8 to3.68, spatial frequency being halved to compensate for reducedresolution at the greater eccentricity. Complete data using the radial
a
1
2
3
b
2.0
1.5
1.0
0.5
2.0
1.5
1.0
0.5
0° 1° 2° 3° 4° 5° 6° 0° 1° 2° 3° 4° 5° 6°
Pro
pag
atio
n tim
e (s
)
2.0
1.5
1.0
0.5
2.0
1.5
1.0
0.5
0° 1° 2° 3° 4° 5° 6°Distance
(visual angle around annulus)
0° 1° 2° 3° 4° 5° 6°Distance
(visual angle around annulus)
Pro
pag
atio
n tim
e (s
)
3.2 degrees s–1
5.8 degrees s–1
3.7 degrees s–1
6.6 degrees s–1
3.3 degrees s–14.4 degrees s–1
16.2 degrees s–19.9 degrees s–1
5.7 degrees s–1
Figure 1 Rivalry stimuli and data of propagation times for dominance waves. a, In all
experiments one eye viewed the high-contrast spiral grating (middle), while the other eye
viewed either the lower-contrast radial (left) or concentric grating (right). Viewers can
experience dominance wave propagation by free-fusing the bull's-eyes. (Anaglypic
versions of these stimuli and demonstrations of triggering are available at http://
www.psy.vanderbilt.edu/faculty/blake/rivalry/waves.html.) Typically, when the radial
grating is pitted in rivalry against the spiral, one small portion of radial grating achieves
local dominance, and this propagates around the annulus. b, Propagation times for four
observers (H.R.W., top left; R.B., top right; S.L., bottom left; K.S., bottom right) as a
function of distance in degrees of visual angle around the annulus. Propagation times
were signi®cantly longer for the radial grating (®lled circles) than for the concentric grating
(open circles), and times for the spiral grating (open triangles) were intermediate. Lines
are the best ®ts of equation (1), and standard errors are indicated.
1.0 cm
R = 1.8° R = 3.6°
Horizontal
Vertical
Vertical
Fovea
1.25°5.0°
a b
2.0
1.5
1.0
0.5
0 1 2 3 4Cortical distance (cm)
Pro
pag
atio
n tim
e (s
) H.R.W. 3.6°H.R.W. 1.8°S.L. 3.6°S.L. 1.8°
v = 2.24cm s–1
Figure 2 Dependence of propagation times on cortical distance. a, Best-®tting complex,
logarithmic approximation (dashed lines) to a ¯attened retinotopic map of human V1
reported previously17. Thick lines plot the mapping of half annuli with radii of 1.8 and 3.68.Distance around the annulus was converted into centimetres across cortex using the
formulae: 1.08 = 0.6 cm (1.88 radius); 1.08 = 0.3 cm (3.68 radius). b, Radial pattern data
for two subjects and two eccentricities indicate that propagation times are roughly
constant in cortical coordinates. The best ®t of equation (1) (thick line) produced an
estimate of cortical speed of 2.24 cm s-1.
© 2001 Macmillan Magazines Ltd
REFERENCES AND ACKNOWLEDGEMENTS1 P. C. Bressloff and S. Carroll. Binocular Rivalry Waves in a Directionally Selective
Neural Field Model. Physica D . In press (2014).
2 P. C. Bressloff and M. Webber. The effects of noise on binocular rivalry waves: astochastic neural field model. J. Stat. Mech: special issue (2012).
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