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

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logi

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keys

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11,1

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this

to e

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the

rate

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can

vie

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onst

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vigat

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to R

.B.’s

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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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T T T T

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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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