The effects of noise on binocular rivalry waves: a stochastic neural field model

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ournal of Statistical Mechanics:J Theory and Experiment

The effects of noise on binocular rivalrywaves: a stochastic neural field model

Matthew A Webber1 and Paul C Bressloff1,2

1 Mathematical Institute, University of Oxford, 24–29 St Giles’,Oxford OX1 3LB, UK2 Department of Mathematics, University of Utah, 155 South 1400 East,Salt Lake City, UT 84112, USAE-mail: matthew.webber@queens.ox.ac.uk and bressloff@math.utah.edu

Received 18 May 2012Accepted 25 June 2012Published 12 March 2013

Online at stacks.iop.org/JSTAT/2013/P03001doi:10.1088/1742-5468/2013/03/P03001

Abstract. We analyze the effects of extrinsic noise on traveling waves of visualperception in a competitive neural field model of binocular rivalry. The modelconsists of two one-dimensional excitatory neural fields, whose activity variablesrepresent the responses to left-eye and right-eye stimuli, respectively. The twonetworks mutually inhibit each other, and slow adaptation is incorporated intothe model by taking the network connections to exhibit synaptic depression. Wefirst show how, in the absence of any noise, the system supports a propagatingcomposite wave consisting of an invading activity front in one network co-movingwith a retreating front in the other network. Using a separation of time scalesand perturbation methods previously developed for stochastic reaction–diffusionequations, we then show how extrinsic noise in the activity variables leads to adiffusive-like displacement (wandering) of the composite wave from its uniformlytranslating position at long time scales, and fluctuations in the wave profilearound its instantaneous position at short time scales. We use our analysis tocalculate the first-passage-time distribution for a stochastic rivalry wave to travela fixed distance, which we find to be given by an inverse Gaussian. Finally,we investigate the effects of noise in the depression variables, which under anadiabatic approximation lead to quenched disorder in the neural fields duringpropagation of a wave.

Keywords: neuronal networks (theory)

c© 2013 IOP Publishing Ltd and SISSA Medialab srl 1742-5468/13/P03001+30$33.00

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Contents

1. Introduction 2

2. Binocular rivalry waves and the structure of V1 4

2.1. Functional architecture of V1 . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2. Binocular rivalry waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1. The network model of rivalry oscillations. . . . . . . . . . . . . . . . 7

2.2.2. The network model of rivalry waves. . . . . . . . . . . . . . . . . . . 11

3. The stochastic neural field model of binocular rivalry waves 12

3.1. Rivalry waves in a deterministic neural field model . . . . . . . . . . . . . . 13

3.2. Effects of extrinsic noise in the fast activity variables . . . . . . . . . . . . . 15

3.3. Explicit results for a Heaviside rate function . . . . . . . . . . . . . . . . . . 20

4. Numerical results 22

5. Discussion 27

Acknowledgments 28

References 28

1. Introduction

One of the major challenges in neurobiology is understanding the relationship betweenspatially structured activity states observed in both normal and pathological brain regions,and the underlying neural circuitry that supports them. This has led to considerablerecent interest in studying reduced continuum neural field models in which the large-scaledynamics of spatially structured networks of neurons is described in terms of nonlinearintegro-differential equations, whose associated integral kernels represent the spatialdistribution of neuronal synaptic connections [2, 83, 84]. Such models provide an importantexample of spatially extended excitable systems with nonlocal interactions. As in the caseof nonlinear partial differential equation (PDE) models of diffusively coupled excitablesystems [46, 49], neural fields can exhibit a diverse range of spatiotemporal dynamics,including solitary traveling fronts and pulses, stationary pulses and spatially localizedoscillations (breathers), spiral waves, and Turing-like patterns; see the reviews [16, 26,29]. In recent years, neural fields have been used to model a wide range of neurobiologicalphenomena, including wave propagation in cortical slices [65, 66] and in vivo [40],geometric visual hallucinations [20, 30], EEG rhythms [57, 63, 67, 76], orientation tuningin primary visual cortex (V1) [6, 75], short term working memory [24, 51], control of headdirection [86], and motion perception [38]. One particularly useful feature of neural fieldsis that analytical methods for solving these integro-differential equations can be adaptedfrom previous studies of nonlinear PDEs. These include regular and singular perturbationmethods, weakly nonlinear analysis and pattern formation, symmetric bifurcation theory,

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Evans functions and wave stability, homogenization theory and averaging, and stochasticprocesses [16].

Recently, we have developed a continuum neural field model for binocular rivalrywaves [21]. Binocular rivalry is the phenomenon whereby perception switches back andforth between different images presented to the two eyes. The resulting fluctuations inperceptual dominance and suppression provide a basis for non-invasive studies of thehuman visual system and the identification of possible neural mechanisms underlyingconscious visual awareness [7, 8]. Various psychophysical experiments have demonstratedthat the switch between a dominant and suppressed visual percept propagates as atraveling front for each eye [44, 53, 85]. In our previous paper, we showed how such atraveling front could arise in a neural field model consisting of two mutually inhibitory, one-dimensional (1D) excitatory networks with slow adaptation [21]. The combination of cross-inhibition paired with a slow adaptive process forms the basis of most competitive networkmodels of binocular rivalry [45, 48, 52, 71, 72, 85]. However, these studies have neglectedspatial effects or have treated them computationally3. The advantage of a continuumneural field model is that it provides an analytical framework for studying perceptualwave propagation. In particular, we were able to derive an analytical expression for thespeed of a binocular rivalry wave as a function of various neurophysiological parameters,and to show how properties of the wave were consistent with the wave-like propagation ofperceptual dominance observed in experiments [21]. In addition to providing an analyticalframework for studying binocular rivalry waves, we also showed how neural field methodsprovide insights into the mechanisms underlying the generation of the waves. In particular,we highlighted the important role of slow adaptation in providing a ‘symmetry breakingmechanism’ that allows waves to propagate.

Another important feature of binocular rivalry is that it has a significant stochasticcomponent. For example, recordings from the brain and reports by subjects duringbinocular rivalry tasks show dominance time statistics that may be fit to a gammadistribution [58]. In addition, statistical analysis of such data shows little correlationbetween one dominance time and the next [54, 55, 58]. This suggests that the switchingbetween one eye’s dominance and the next may be largely driven by a stochastic process.Some previous models have accounted for this by presuming that the input arriving at thenetwork encoding rivalry is stochastic, so the noise is extrinsic [36, 54]. Recent modelingefforts have examined the effects on dominance switching of additive noise terms in eitherthe activity or adaptation variables [13, 48, 61, 71]. In this paper, we investigate the effectsof noise on the propagation of binocular rivalry waves by considering a stochastic versionof our neural field model. We analyze the model by extending our recent work on thetheory of fluctuating fronts in neural fields with extrinsic noise [17]. The latter, in turn,adapts methods developed previously for PDEs [1, 4, 32, 68]. Such methods exploit aseparation of time scales in which there is a diffusive-like displacement (wandering) of the

3 Several previous studies have modeled the spontaneous switching between rivalrous oriented stimuli in terms of apair of ring networks (neural fields on a periodic domain) with slow adaptation and cross-inhibition, representing apair of hypercolumns for the left and right eyes, respectively [48, 52]. In these models, the rivalrous states consistof stationary activity bumps coding for either the left-eye or right-eye stimuli. (Rivalry effects in a spatiallyextended model have also been examined in a prior study by Loxley and Robinson [59], in which rivalrous stimuliare presented to a single one-dimensional network.) However, these models were not used to study binocularrivalry waves and, as formulated, were on the wrong spatial scale since they only considered short-range spatialscales comparable to a single hypercolumn (see footnote 4).

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retinaLGN

cortex(a) /2

0

x- /2π

π

πA

BR B

AR

(b)

RETINA VISUAL CORTEX

yleftright

yx

Figure 1. (a) Visual pathways from the retina through the lateral geniculatenucleus (LGN) of the thalamus to the primary visual cortex (V1). (b) Schematicillustration of the complex logarithmic mapping from the retina to V1. The fovealregion in the retina is indicated by a gray disk. Regions AR and BR in the visualfield are mapped to regions A and B in the cortex.

front from its uniformly translating position at long time scales, and fluctuations in thefront profile around its instantaneous position at short time scales. The precise form ofnoise in large-scale neural populations is still poorly understood. Therefore, for the sakeof generality, we take the noise to be multiplicative rather than additive, and use theStratonovich formulation since we consider extrinsic rather than intrinsic noise. The factthat we can carry over stochastic PDE methods to neural field equations is consistent witha number of studies that have shown how neural fields can be reduced to an equivalentPDE for particular choices of the weight kernel [28, 50, 67].

2. Binocular rivalry waves and the structure of V1

Before introducing the basic theory of binocular rivalry waves, we first review severalimportant properties of neurons and their functional organization within primary visualcortex (V1).

2.1. Functional architecture of V1

V1 is the first cortical area to receive visual information from the retina (see figure 1).The output from the retina is conveyed by ganglion cells whose axons form the opticnerve. The optic nerve conducts the output spike trains of the retinal ganglion cells to thelateral geniculate nucleus (LGN) of the thalamus, which acts as a relay station betweenretina and primary visual cortex (V1). Prior to arriving at the LGN, some ganglion cellaxons cross the midline at the optic chiasm. This allows the left and right sides of thevisual fields from both eyes to be represented on the right and left sides of the brain,respectively. Note that signals from the left and right eyes are segregated in the LGN andin input layers of V1. This means that the corresponding LGN and cortical neurons aremonocular, in the sense that they only respond to stimuli presented to one of the eyes butnot the other (ocular dominance).

One of the striking features of the early visual system is that the visual world ismapped onto the cortical surface in a topographic manner. This means that neighboringpoints in a visual image evoke activity in neighboring regions of visual cortex. Moreover,

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Figure 2. (a) Schematic illustration of an orientation tuning curve of a V1 neuron.Average firing rate is plotted as a function of the orientation of a bar stimulusthat is moved back and forth within the receptive field (RF) of the neuron. Thepeak of the orientation tuning curve corresponds to the orientation preference ofthe cell. (b) Schematic illustration of iso-orientation (light) and ocular dominance(dark) contours in a region of primate V1. A cortical hypercolumn consists of twoorientation singularities or pinwheels per ocular dominance column.

one finds that the central region of the visual field has a larger representation in V1than the periphery, partly due to a non-uniform distribution of retinal ganglion cells. Theretinotopic map is defined as the coordinate transformation from points in the visual worldto locations on the cortical surface, and can be approximated by a complex logarithm [69].Superimposed upon the retinotopic map are additional maps reflecting the fact thatneurons respond preferentially to stimuli with particular features [78]. Neurons in theretina, LGN and primary visual cortex respond to light stimuli in restricted regions of thevisual field called their classical receptive fields (RFs). Patterns of illumination outside theRF of a given neuron cannot generate a response directly, although they can significantlymodulate responses to stimuli within the RF via long-range cortical interactions (seebelow). The RF is divided into distinct ON and OFF regions. In an ON (OFF) regionillumination that is higher (lower) than the background light intensity enhances firing.The spatial arrangement of these regions determines the selectivity of the neuron todifferent stimuli. For example, one finds that the RFs of most V1 cells are elongated, sothe cells respond preferentially to stimuli with certain preferred orientations (see figure 2).Similarly, the width of the ON and OFF regions within the RF determines the optimalspacing of alternating light and dark bars for eliciting a response, that is, the cell’s spatialfrequency preference. In recent years much information has accumulated about the spatialdistribution of orientation selective cells in V1 [39]. One finds that orientation preferencesrotate smoothly over the surface of V1, so approximately every 300 µm the same preferencereappears, i.e. the distribution is π-periodic in the orientation preference angle. One alsofinds that cells with similar feature preferences tend to arrange themselves in verticalcolumns, so to a first approximation the layered structure of the cortex can be ignored. Amore complete picture of the two-dimensional distribution of both orientation preferenceand ocular dominance in layers 2/3 has been obtained using optical imaging techniques[9]–[11]. The basic experimental procedure involves shining light directly onto the surfaceof the cortex. The degree of light absorption within each patch of cortex depends on thelocal level of activity. Thus, when an oriented image is presented across a large part of the

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visual field, the regions of cortex that are particularly sensitive to that stimulus will bedifferentiated. The topography revealed by these methods has a number of characteristicfeatures [64]—see figure 2(b): (i) orientation preference changes continuously as a functionof cortical location, except at singularities or pinwheels, (ii) there exist linear zones,approximately 750× 750 µm2 in area (in primates), bounded by pinwheels, within whichiso-orientation regions form parallel slabs; (iii) linear zones tend to cross the borders ofocular dominance stripes at right angles, while pinwheels tend to align with the centers ofocular dominance stripes. These experimental findings suggest that there is an underlyingperiodicity in the microstructure of V1 with a period of approximately 1 mm (in catsand primates). The fundamental domain of this approximate periodic (or quasiperiodic)tiling of the cortical plane is the hypercolumn [41, 42, 56], which contains two sets oforientation preferences θ ∈ [0, π) per eye, organized around a pair of singularities—seefigure 2(b). Within each hypercolumn, neurons with sufficiently similar orientations tendto excite each other whereas those with sufficiently different orientations inhibit eachother, and this serves to sharpen a particular neuron’s orientation preference [6, 34].Moreover, anatomical evidence suggests that inter-hypercolumn connections link neuronswith similar orientation preferences [3, 73].

2.2. Binocular rivalry waves

One way to observe and measure the speed of perceptual waves in psychophysicalexperiments of binocular rivalry [53, 85] is to take the rival images to be a low-contrastradial grating presented to one eye and a high-contrast spiral grating presented to the othereye. Each image is restricted to an annular region of the visual field centered on the fixationpoint of the observer, thus effectively restricting wave propagation to the one dimensionaround the annulus. Switches in perceptual dominance can then be triggered using a briefrapid increase in stimulus contrast within a small region of the suppressed low-contrastgrating. This induces a perceptual traveling wave in which the observer perceives thelocal dominance of the low-contrast image spreading around the annulus. The observerpresses a key when the perceptual wave reaches a target area at a fixed distance from thetrigger zone, and this determines the wave speed [53, 85]. Since the rival images consist oforiented gratings, one might expect that primary visual cortex (V1) plays some role in thegeneration of binocular rivalry waves. Indeed, it has been shown using functional magneticresonance imaging that there is a systematic correspondence between the spatiotemporaldynamics of activity in V1 and the time course of perceptual waves [53]. However, it hasnot been established whether the waves originate in V1 or are evoked by feedback fromextrastriate cortex. Recently Kang et al [44, 45] have developed a new psychophysicalmethod for studying binocular rivalry waves that involves periodic perturbations of therival images and linear gratings. An observer tracks rivalry within a small, central regionof spatially extended rectangular grating patterns, while alternating contrast triggers arepresented repetitively in the periphery of the rival patterns. The basic experimental setupis illustrated in figure 3. A number of interesting results have been obtained from thesestudies. First, over a range of trigger frequencies, the switching between rival perceptswithin the central regions is entrained to the triggering events. Moreover, the optimaltriggering frequency depends on the natural frequency of spontaneous switching (in theabsence of triggers). Second, the latency between triggering event and perceptual switchingincreases approximately linearly with the distance between the triggering site and the

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Figure 3. Schematic diagram illustrating the experimental protocol used to studybinocular rivalry waves [44]. High-contrast triggers are presented periodically inanti-phase within the upper extended region of one grating pattern and withinthe lower region of the rival pattern. The subject simply reports perceptualalternations in rival dominance within the central monitoring region indicatedby the horizontal black lines on each pattern. The monitoring region is a distance∆d from the trigger region, which can be adjusted. If ∆t is the latency betweenthe triggering event and the subsequent observation of a perceptual switch, thenthe speed c of the wave is given by the slope of the plot ∆d = c∆t.

central region being tracked by the observer, consistently with the propagation of atraveling front. Third, the speed of the traveling wave across observers covaries with thespontaneous switching rate.

2.2.1. The network model of rivalry oscillations. The above psychophysical experimentssuggest that binocular rivalry waves consist of two basic components: the switchingbetween rivalrous left/right-eye states and the propagation of the switched state across aregion of cortex. Let us first focus on the switching mechanism by neglecting spatial effects.Suppose, for the sake of illustration, that a horizontally oriented grating is presented to theleft eye and a vertically oriented grating is presented to the right eye. This triggers rivalrydue to the combination of orientation and eye specific cross-inhibition in V1. During left-

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Figure 4. Schematic diagram of a competitive network architecture for rivalryoscillations [52, 71, 85], consisting of two homogeneous populations of cells,one driven by left-eye stimuli and the other by right-eye stimuli. Recurrentconnections within each population are assumed to be excitatory, whereasconnections between the two populations are inhibitory (cross-inhibition). Eachnetwork is described by two variables: a fast activity variable and a slowadaptation variable.

eye stimulus dominance, it is assumed that a group of the left-eye neurons that respondto horizontal orientations are firing persistently, while right-eye neurons are suppressedby cross-inhibitory connections. Of course, there may still be some low rate firing of theright-eye neurons, but it will be less than the firing rate of the left eye, horizontally tunedneurons [8]. Following this, some slow adaptive process causes a switch and so right-eye,vertical orientation neurons fire persistently, suppressing the left-eye neurons, resultingin a repetitive cycle of perceptual dominance between the left-eye and right-eye stimuli.The competitive network architecture of reciprocal inhibition paired with slow adaptation(figure 4) has been used extensively to model oscillations in binocular rivalry [35, 48, 52,71, 72, 74, 80, 85]. (In some versions of the model, recurrent excitation is omitted.) In mostcases a firing rate model appears sufficient to capture the elevation in neuronal spikingassociated with the dominant stimulus.

It remains an open question as to which slow adaptive process is most responsible forthe eventual switching of one stimulus dominance to the other [71]. The mechanism ofspike frequency adaptation has been suggested, since it can curtail excitatory activity in asingle neuron [52, 85]. Spike frequency adaptation is the process by which a hyperpolarizingcurrent is switched on due to a build-up of a certain ion, like calcium, within thecell due to repetitive firing [77]. The maximal firing rate of a neuron is lowered asa result. In the case of binocular rivalry, this may cause the dominant population toeventually drop its firing rate with the result that cross-inhibition suppressing the otherpopulation is then low enough for the suppressed populations to rekindle its firing rateinto dominance. Since the recently released population is not adapted, it can then remainin dominance and suppress the other population for a period of time roughly equal tothe time constant of spike frequency adaptation [52, 59, 85]. It is also possible for thesuppressed population to ‘escape’ from inhibition rather than be ‘released’ [60]. Another

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proposed switching mechanism is that the inhibitory synapses from one eye’s neurons tothe other’s undergo synaptic depression4. This is the process by which synaptic resourcessuch as neurotransmitters, vesicles, and scaffolding proteins are exhausted due to theircontinuous use [25, 82]. If inhibitory synapses remain repeatedly active, due to one eye’sneurons suppressing the others, eventually most of those synapses’ resources will be usedup, the effect of inhibition will be weakened and the suppressed population will escape [48,52, 71]. Synaptic depression could also occur in the excitatory recurrent connections. Forconcreteness, we will take slow adaptation to arise from synaptic depression in all synapticweights; however, the specific choice of adaptation does not affect the main results of ourpaper.

Let u(t) and v(t) denote the activity variables of the left- and right-eye populationsat time t. (In this paper we use the version of neural field theory in which the activityvariables u, v represent a local population current or voltage. For a detailed discussion ofdifferent neural field representations see the reviews [16, 29]). The rate-based equationsfor a competitive network model with synaptic depression can be constructed as follows:

τdu(t)

dt= −u(t) + Iu(t) + wequ(t)f(u(t))− wiqv(t)f(v(t)) (2.1)

τsdqu(t)

dt= 1− qu(t)− βqu(t)f(u(t)) (2.2)

and

τdv(t)

dt= −v(t) + Iv(t) + weqv(t)f(v(t))− wiqu(t)f(u(t)) (2.3)

τsdqv(t)

dt= 1− qv(t)− βqv(t)f(v(t)), (2.4)

where the positive constants we and wi denote the strengths of recurrent excitatory andcross-inhibitory connections, and Iu, Iv denote the input stimuli from the left and righteyes. The nonlinear function f represents the mean firing rate of a local population andis usually taken to be a smooth, bounded monotonic function such as a sigmoid

f(u) =1

1 + e−η(u−κ), (2.5)

with gain η and threshold κ. The variables qu, qv are taken to be synaptic depressionvariables representing the fraction of baseline synaptic resources that are available tothe neuron for firing. These synaptic resources are depleted under continuous firing at arate proportional to βqf [5, 79, 81] and recover under no firing exponentially with timeconstant τs, τs τ . Previously [48] we have explicitly analyzed the existence and stabilityof fixed point solutions of equations (2.1) and (2.4) in the high gain limit η →∞ of (2.5)for which f becomes a Heaviside function

f(u) = H(u− κ) =

0 if u < κ

1 if u > κ.(2.6)

4 More precisely, synaptic depression tends to be associated only with excitatory synapses, so in our simplifiedmodel, depressing inhibitory connections would have to be mediated by excitatory connections innervating localinterneurons, for example.

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This choice of rate function proves particularly useful when analyzing traveling wavesolutions of continuum neural fields [2, 16, 27, 65]; see sections 3 and 4. Denoting ahomogeneous fixed point by (U∗, V ∗, Q∗u, Q

∗v), we have for Iu = Iv = I, with I constant,

U∗ = Q∗uweH(U∗ − κ)−Q∗vwiH(V ∗ − κ) + I

V ∗ = Q∗vweH(V ∗ − κ)−Q∗uwiH(U∗ − κ) + I

Q∗u =1

1 + βH(U∗ − κ), Q∗v =

1

1 + βH(V ∗ − κ).

There are four possible homogeneous fixed points and all are stable. First, there is theoff state U∗ = V ∗ = I, which occurs when I < κ, that is, the input is not strong enoughto activate either population. Second, there is the on state or fusion state, where the twopopulations are simultaneously active:

(U∗, V ∗) =

(we − wi

1 + β+ I,

we − wi

1 + β+ I

),

(Q∗u, Q∗v) =

(1

1 + β,

1

1 + β

),

which occurs when I > κ − (we − wi)/(1 + β). This case is more likely for verystrong depression (β large), since cross-inhibition will be weak, or when the localconnections are strong and excitation dominated. Finally there are two winner-takes-all (WTA) states in which one population dominates the other: the left-eye dominantstate

(U∗, V ∗) =

(we

1 + β+ I, I − wi

1 + β

),

(Q∗u, Q∗v) =

(1

1 + β, 1

)and the right-eye dominant state

(U∗, V ∗) =

(I − wi

1 + β,we

1 + β+ I

),

(Q∗u, Q∗v) =

(1,

1

1 + β

).

The WTA states exist provided that

I > κ− we

1 + β, I < κ+

wi

1 + β.

This will occur in the presence of weak depression (β small) and strong cross-inhibitionsuch that depression cannot exhaust the dominant hold that one population has on theother. It can also be shown that equations (2.1)–(2.4) also support homogeneous limit cycleoscillations in which there is periodic switching between left-eye and right-eye dominanceconsistent with binocular rivalry [48]. Since all the fixed points are stable, it follows thatsuch oscillations cannot arise via a standard Hopf bifurcation. Indeed, we find bistableregimes where a rivalry state coexists with a fusion state as illustrated in figure 5. (Suchbehavior persists in the case of smooth sigmoid firing rate functions, at least for sufficientlyhigh gain [48].) For a detailed discussion of the various bifurcation scenarios in competitiveneural networks see [60, 70, 71].

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Figure 5. (a) Bifurcation diagram showing homogeneous solutions for the leftpopulation activity u as a function of the input amplitude I. Solid lines representstable states, whereas circles represent the maximum and minimum of rivalryoscillations. It can be seen that there are regions of off/WTA bistability,WTA/fusion bistability, and fusion/rivalry bistability. The parameters are τs =500, κ = 0.05, β = 5, we = 0.4 and wi = 1. (b) Homogeneous oscillatory solutionin which there is spontaneous periodic switching between left-eye and right-eyedominance. A plot against time of the left-eye neural field u (solid red) and theright-eye neural activity v (solid blue) together with the corresponding depressionvariables qu (dashed red) and qv (dashed blue).

2.2.2. The network model of rivalry waves. In order to take into account the propagationof activity seen in binocular rivalry waves, it is necessary to introduce a spatially extendednetwork model. Therefore, suppose that the neuronal populations responding to stimulifrom the left eye, say, are distributed on a one-dimensional (1D) lattice and are labeledaccording to the integer n; a second 1D network responds to stimuli from the right eye—see figure 6. In terms of the functional architecture of V1, one can interpret the nthleft/right-eye populations as consisting of neurons in a given hypercolumn that respondmaximally to the distinct stimuli presented to the two eyes at a particular coarse-grainedlocation in space. In the case of the orientated grating stimuli used by Kang et al [44,45]—see figure 3—this would mean neurons whose orientation preference coincides withthe stimulus orientation presented to a given eye. Since the orientation does not changealong the length of the grating, all neurons receive the same external drive. Lettingun, vn, qu,n, qv,n denote the activity and depression variables of the nth left-eye and right-eye networks, we have

τdumdt

= −um + Iu +∑n

[we]mnqu,nf(un)− [wi]mnqv,nf(vn)] (2.7)

τsdqu,m

dt= 1− qu,m − βqu,mf(um) (2.8)

and

τdvmdt

= −vm + Iv +∑n

([we]mnqv,nf(vn)− [wi]mnqu,nf(un)) (2.9)

τsdqv,m

dt= 1− qv,m − βqv,mf(vm). (2.10)

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Figure 6. Schematic diagram of a competitive neural network consisting of twosets of populations distributed along a 1D lattice. Recurrent connections withineach 1D network are assumed to be excitatory, whereas connections between thetwo networks are inhibitory (cross-inhibition). Slow adaptation is incorporatedinto the model by taking the network connections to exhibit synaptic depression.

Here [we]mn is the strength of excitation from the nth to the mth population with thesame eye preference, and [wi]mn is the strength of cross-inhibition between populationswith opposite eye preferences. The weights are typically assumed to decrease withdistance of separation |m − n| according to an exponential or Gaussian distribution.Slow adaptation is incorporated into the model by taking the network connections toexhibit synaptic depression along the lines of Kilpatrick and Bressloff [48]. Note thata similar network architecture was previously considered in a computational model ofbinocular rivalry waves [45, 85], in which cross-inhibition was mediated explicitly byinterneurons and, rather than including depressing synapses, the excitatory neurons weretaken to exhibit spike frequency adaptation. Numerical simulations of the model showedthat the network supported traveling waves consistent with those observed numericallyunder physiologically reasonable parameter regimes.

3. The stochastic neural field model of binocular rivalry waves

Although the discrete lattice model presented in section 2.2.2 is directly amenable tonumerical simulations, it is difficult to derive any analytical results for the model thatgenerate, for example, explicit expressions for how the wave speed depends on networkparameters. Therefore, in this section we turn to a neural field model that can beconstructed by taking an appropriate continuum limit of equations (2.7) and (2.10).Previously, we analyzed the existence and stability of traveling front solutions (rivalrywaves) in such a model [21]. Here we show how a theory of fluctuating fronts in stochasticneural fields—see [17]—can be used to analyze the effects of noise on binocular rivalrywaves.

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3.1. Rivalry waves in a deterministic neural field model

We begin by considering the deterministic neural field model analyzed in [21]. This canbe interpreted as a continuum version of the discrete network model of figure 6. Introducea lattice spacing a, and write [we]nm = awe(na,ma), un(t) = u(na, t) etc. Substitutinginto equations (2.7)–(2.10) and taking the continuum limit a→ 0 such that na→ x andma→ y, the discrete sums can be replaced by integrals to give the neural field model

τdu(x, t)

dt= −u(x, t) + Iu +

∫ ∞−∞

we(x− x′)qu(x′, t)f(u(x′, t)) dx′

−∫ ∞−∞

wi(x− x′)qv(x′, t)f(v(x′, t)) dx′ (3.1)

τsdqu(x, t)

dt= 1− qu(x, t)− βqu(x, t)f(u(x, t)) (3.2)

and

τdv(x, t)

dt= −v(x, t) + Iv +

∫ ∞−∞

we(x− x′)qv(x′, t)f(v(x′, t)) dx′

−∫ ∞−∞

wi(x− x′)qu(x′, t)f(u(x′, t)) dx′ (3.3)

τsdqv(x, t)

dt= 1− qv(x, t)− βqv(x, t)f(v(x, t)), (3.4)

assuming that the weights only depend on the distance between interacting populations.For concreteness, the distributions we and wi are both taken to be Gaussians:

we(r) =we√2πσ2

e

e−r2/2σ2

e , wi(r) =wi√2πσ2

i

e−r2/2σ2

i . (3.5)

Following [21], we assume that σe > σi (longer-range excitation) and fix the length scaleby setting σe = 2, σi = 1. Assuming that excitation spans a single hypercolumn, σe shouldbe the same approximate size as a hypercolumn, that is, of the order of 200 µm. We alsotake τ = 1 in units of the membrane time constant, which is typically of the order of10 ms.

The next step is to interpret the binocular rivalry wave seen in the experiments of Kanget al [44, 45] as a traveling wave front solution of the neural field equations (3.1)–(3.4), inwhich a high activity state invades the suppressed left-eye network, say, whilst retreatingfrom the dominant right-eye network; see figure 7. Such a wave is defined as

u(x, t) = U(x− ct), v(x, t) = V (x− ct) (3.6)

where c is the wave speed and ξ = x− ct is a traveling wave coordinate, together with theasymptotic conditions

(U(ξ), V (ξ))→ XL as ξ → −∞,(U(ξ), V (ξ))→ XR as ξ →∞ (3.7)

with U(ξ) a monotonically decreasing function of ξ and V (ξ) a monotonically increasingfunction of ξ. Here XL (XR) represents a homogeneous left-eye (right-eye) dominant state.Given the asymptotic behavior of the solution and the requirements of monotonicity, we

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Figure 7. Sketch of a right-moving traveling wave solution in which a highactivity state invades the suppressed left-eye network whilst retreating from thedominant right-eye network. The wave is shown in a moving frame, ξ = x− ct.

see that U(ξ) and V (ξ) each cross the threshold at a single location, which may be differentfor the two eyes. Exploiting translation invariance we take U(0) = κ and V (X) = κ.

As we showed in our previous paper [21], it is possible to construct an exact travelingwave solution of equations (3.1)–(3.4) by taking f to be the Heaviside function (2.6)and making the adiabatic approximation that qu(x, t) = Qu, qv(x, t) = Qv with Qu, Qv

constants. The latter is based on the assumption that adaptation is sufficiently slow thatthe wave traverses the cortex in a time Tl τs. Substituting the traveling wave solution(3.6) into equations (3.1) and (3.3) with fixed Qu, Qv and f(u) = H(u − κ) leads to theequations

− cdU

dξ+ U = Qu

∫ 0

−∞we(ξ − ξ′) dξ′ −Qv

∫ ∞X

wi(ξ − ξ′) dξ′ + I (3.8)

− cdV

dξ+ V = Qv

∫ ∞X

we(ξ − ξ′) dξ′ −Qu

∫ 0

−∞wi(ξ − ξ′) dξ′ + I. (3.9)

where, for simplicity, we have set Iu = Iv = I. Multiplying both sides of equations (3.8)and (3.9) by e−ξ/c, integrating with respect to ξ and imposing the threshold conditionsgives

U(ξ) = eξ/c[κ− 1

c

∫ ξ

0

e−z/cΨX(z) dz − I(1− e−ξ/c)

], ξ > 0 (3.10)

and

V (ξ) = e(ξ−X)/c

[κ− 1

c

∫ ξ−X

0

e−z/cΦX(−z) dz − I(1− e−(ξ−X)/c)

], ξ > X (3.11)

with Ψ and Φ defined by

ΨX(z) = Qu

∫ ∞z

we(y) dy −Qv

∫ z−X

−∞wi(y) dy. (3.12)

ΦX(z) = Qv

∫ ∞z

we(y) dy −Qu

∫ z−X

−∞wi(y) dy. (3.13)

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Finally, requiring that the wave solution remain bounded as ξ → ∞ (assuming c > 0)yields the pair of threshold conditions

κ =

∫ ∞0

e−sΨX(cs) ds+ I, κ =

∫ ∞0

e−sΦX(−cs) ds+ I. (3.14)

In the particular case Qu = Qv = 1 (no synaptic depression), it can be shown that theabove equations have no solution for c 6= 0, that is, there does not exist a traveling wavesolution [21]. On the other hand, slow synaptic depression with Qu 6= Qv breaks thesymmetry of the threshold crossing conditions, leading to a unique solution for c,X as afunction of the network parameters. Note that one can also establish that the travelingfront is linearly stable [21]. The threshold conditions can also be used to simplify theexpressions for the wave profile, namely,

U(ξ) =1

c

∫ ∞0

e−z/cΨX(z + ξ) dz + I (3.15)

and

V (ξ) =1

c

∫ ∞0

e−z/cΦX(−z − ξ +X) dz + I. (3.16)

Example plots of the wave speed are shown in figure 8. Baseline parameter valuesare chosen such that spontaneous oscillations and traveling fronts coexist, as foundexperimentally [44, 45]. The model wave speed is of the order of c = 1 in dimensionlessunits, that is, c = σe/2τ where σe is the range of excitation and τ is the membrane timeconstant. Taking σe to be of the order of 200 µm and τ to be of the order of 10 msyields a wave speed of around 10 mm s−1, which is consistent with the speeds observedexperimentally. (In the psychophysical experiments of Kang et al, binocular rivalry wavestook approximately 0.8 s to traverse 2 of the visual field. The magnification factor inhumans throughout the foveal region is approximately 0.4 cm/deg, which corresponds to0.8 cm of cortex.) Figure 8 shows that the speed of the wave is a decreasing function of thethreshold κ and an increasing function of the input amplitude I; the latter is consistentwith what is found experimentally when the stimulus contrast is increased [44, 45]. Yetanother experimental result that emerges from the model is that the wave speed covarieswith the frequency of spontaneous rivalry oscillations [21].

3.2. Effects of extrinsic noise in the fast activity variables

Several recent studies have considered stochastic versions of neural field equations thatare based on a corresponding Langevin equation formulation [12, 17, 33, 43]. Motivatedby these examples and the adiabatic approximation used in the analysis of deterministicwaves, we consider the following Langevin equation (or stochastic PDE) for the stochasticactivity variables 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

+ ε1/2g(U) dWu (3.17)

dV =

[−V +Qv

∫ ∞−∞

we(x− y)f V (y, t) dy −Qu

∫ ∞−∞

wi(x− y)f U(y, t) dy + Iv

]dt

+ ε1/2g(V ) dWv. (3.18)

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Figure 8. Plot of wave speed c in units of σ2/(2τ) as a function of (a) thethreshold κ and (b) the external input strength I. The default parametersare taken to be wi = 1, we = 0.4, σi = 1, σe = 2, β = 5, κ = 0.05, I = 0.24, Qu =0.42, Qv = 0.25 and the corresponding wave speed is c = 1.2. For this set ofparameters, the network operates in a regime that supports both traveling wavesand homogeneous oscillatory solutions, that is, spontaneous switching occurs inthe absence of traveling waves.

with Qu, Qv fixed. We assume that Wu, Wv represent independent Wiener processes suchthat

〈dWu,v(x, t)〉 = 0, 〈dWi(x, t) dWj(x′, t′)〉 = 2δi,jC([x− x′] /λ)δ(t− t′) dt dt′, (3.19)

where i, j = u, v and 〈·〉 denotes averaging with respect to the Wiener processes. Hereλ is the spatial correlation length of the noise such that C(x/λ) → δ(x) in the limitλ→ 0, and ε determines the strength of the noise, which is assumed to be weak. For thesake of generality, we take the noise to be multiplicative rather than additive. Followingstandard formulations of Langevin equations [37], the multiplicative noise term is taken tobe of Stratonovich form in the case of extrinsic noise. Note, however, that an alternativeformulation of stochastic neural field theory has been developed in terms of a neural masterequation [14, 15, 22, 23], in which the underlying deterministic equations are recoveredin the thermodynamic limit N → ∞, where N is a measure of the system size of eachlocal population. In the case of large but finite N , a Kramers–Moyal expansion of themaster equation yields a Langevin neural field equation with multiplicative noise of theIto form [14, 15].

The effects of extrinsic noise on binocular rivalry waves can be analyzed usingmultiple-time-scale methods previously developed for reaction–diffusion equations [1, 4,32, 68], which we recently extended to scalar neural field equations [17]. In the case ofmultiplicative noise in the Stratonovich sense, it is first necessary to take into account asystematic shift in the speed of the front (assuming that a front of speed c exists whenε = 0). This is a consequence of the fact that 〈g(U) dWu〉 6= 0 and 〈g(V ) dWv〉 6= 0 eventhough 〈dWu,v〉 = 0. These averages can be calculated using Novikov’s theorem [62]:

〈g(U) dWu〉 = ε1/2C(0)〈g′(U)g(U)〉dt, 〈g(V ) dWv〉 = ε1/2C(0)〈g′(V )g(V )〉 dt (3.20)

An alternative way to derive the above result is to Fourier transform equation (3.17)and evaluate averages using the corresponding Fokker–Planck equation in Fourier space(see [68] and the appendix of [17]). Note that in the limit λ→ 0, C(0)→ 1/∆x where ∆x

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is a lattice cutoff, which can be identified with the step size of the spatial discretizationscheme used in numerical simulations. Following [4, 17], it is convenient to rewriteequations (3.17) and (3.18) in such a way that the fluctuating term has zero mean:

dU =

[h(U) +Qu

∫ ∞−∞

we(x− y)f U(y, t) dy −Qv

∫ ∞−∞

wi(x− y)f V (y, t) dy + Iu

]dt

+ ε1/2 dRu(U, x, t) (3.21)

dV =

[h(V ) +Qv

∫ ∞−∞

we(x− y)f V (y, t) dy −Qu

∫ ∞−∞

wi(x− y)f U(y, t) dy + Iv

]dt

+ ε1/2 dRv(V, x, t), (3.22)

where

h(U) = −U + εC(0)g′(U)g(U), (3.23)

and

dRu(U, x, t) = g(U) dWu − ε1/2C(0)g(U)g′(U) dt (3.24)

dRv(V, x, t) = g(V ) dWv − ε1/2C(0)g(V )g′(V ) dt. (3.25)

The stochastic processes Ru, Rv have zero mean (and so do not contribute to the effectivedrift, that is, the average wave speed) and covariance

〈dRu(U, x, t) dRu(U, x′, t)〉 = 〈g(U(x, t)) dWu(x, t)g(U(x′, t))dWu(x

′, t)〉+O(ε1/2),

〈dRv(V, x, t) dRv(V, x′, t)〉 = 〈g(V (x, t)) dWv(x, t)g(V (x′, t))dWv(x

′, t)〉+O(ε1/2)

〈dRu(U, x, t) dRv(V, x′, t)〉 = 〈g(U(x, t)) dWu(x, t)g(V (x′, t))dWv(x

′, t)〉+O(ε1/2).

The main idea of the subsequent analysis is to assume that the fluctuating termsin equations (3.21) and (3.22) generate two distinct phenomena that occur on differenttime scales: a diffusive-like displacement of the binocular rivalry wave from its uniformlytranslating position at long time scales, and fluctuations in the wave profile around itsinstantaneous position at short time scales [1, 4, 17, 32, 68]. It is important to point outthat we are now considering, in contrast to traveling front solutions of scalar neural fieldequations [17], composite wave solutions consisting of an invading front in the left-eyenetwork, say, co-moving with a retreating front in the right-eye network. Thus in additionto the center of mass of the composite wave, which moves with speed c in the absenceof noise, there is an additional degree of freedom corresponding to the ‘width’ of thecomposite wave. (In the case of a Heaviside rate function, the width is determined by thethreshold crossing point X; see equation (3.14).) For simplicity, we assume that the widthof the composite wave is only weakly affected by the noise; this is consistent with what isfound numerically. We now express the solution (U, V ) of equations (3.21) and (3.22) asa combination of a fixed wave profile (U0, V0) that is displaced by an amount ∆(t) fromits uniformly translating position ξ = x− cεt and a time-dependent fluctuation (U1, V1) inthe wave shape about its instantaneous position:

U(x, t) = U0(ξ −∆(t)) + ε1/2U1(ξ −∆(t), t), (3.26)

V (x, t) = V0(ξ −∆(t)) + ε1/2V1(ξ −∆(t), t). (3.27)

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The wave profile (U0, V0) and associated wave speed cε are obtained by solving the modifieddeterministic equation

− cdU0

dξ− h(U0) = Qu

∫ ∞−∞

we(ξ − ξ′)f U0(ξ′) dξ′ −Qv

∫ ∞−∞

wi(ξ − ξ′)f V0(ξ′) dξ′+Iu

(3.28)

− cdV0

dξ− h(V0) = Qv

∫ ∞−∞

we(ξ − ξ′)f V0(ξ′) dξ′ −Qu

∫ ∞−∞

wi(ξ − ξ′)f U0(ξ′) dξ′+Iv.

(3.29)

Both cε and U0 depend non-trivially on the noise strength ε due to the dependence on ε ofthe function h; see equation (3.23). Thus, cε 6= c for ε > 0 and c0 = c, where c is the speedof the front in the absence of multiplicative noise. Equations (3.28) and (3.29) are chosensuch that to leading order, the stochastic variable ∆(t) undergoes unbiased Brownianmotion with a diffusion coefficient D(ε) = O(ε) (see below). Thus ∆(t) represents theeffects of slow fluctuations, whereas (U1, V1) represents the effects of fast fluctuations.

The next step is to substitute the decompositions (3.26) and (3.27) intoequations (3.21) and (3.22) and expand to first order in O(ε1/2):

−[cε + ∆]U ′0(ξ∆) dt+ ε1/2[dU1(ξ∆, t)− [cε + ∆]U ′1(ξ∆, t) dt

]= h(U0(ξ∆)) dt+ ε1/2h′(U0(ξ∆))U1(ξ∆, t) dt

+ Qu

∫ ∞−∞

we(ξ − ξ′)(f(U0(ξ′∆)) + ε1/2f ′(U0(ξ′∆))U1(ξ′∆, t)) dξ′ dt

− Qv

∫ ∞−∞

wi(ξ − ξ′)(f(V0(ξ′∆)) + ε1/2f ′(V0(ξ′∆))V1(ξ′∆, t)) dξ′ dt

+ ε1/2 dR(U0(ξ∆), ξ, t) +O(ε)

where we have set ξ∆ = ξ − ∆(t) and ξ′∆ = ξ′ − ∆(t). A similar equation holds for dV .We now use the steady state equations (3.28) and (3.29) for U0, V0, after making theshift ξ → ξ − ∆(t), to eliminate terms and then divide through by

√ε. This gives the

inhomogeneous equations to O(ε1/2):

dU1(ξ∆, t)− Lu(U1(ξ∆, t), V1(ξ∆, t)) dt = ε−1/2U ′0(ξ∆) d∆(t) + dRu(U0(ξ∆), ξ, t) (3.30)

dV1(ξ∆, t)− Lv(U1(ξ∆, t), V1(ξ∆, t)) dt = ε−1/2V ′0(ξ∆) d∆(t) + dRv(V0(ξ∆), ξ, t) (3.31)

with Lu, Lv non-self-adjoint linear operators:

Lu(A1, A2) = cdA1

dξ+ h′(U0)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+ h′(V0)A2 +Qv

∫ ∞−∞

we(ξ − ξ′)f ′(V0(ξ′))A2(ξ′) dξ′

− Qu

∫ ∞−∞

wi(ξ − ξ′)f ′(U0(ξ′))A1(ξ′) dξ′

for any functions A1(ξ), A2(ξ) ∈ L2(R). Finally, for all terms in equations (3.30) and(3.31) to be of the same order we require that ∆(t) = O(ε1/2). It then follows that

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U0(ξ −∆(t)) = U0(ξ) +O(ε1/2) etc, and equations (3.30) and (3.31) reduce to

dU1(ξ, t)− Lu(U1(ξ, t), V1(ξ, t)) = ε−1/2U ′0(ξ) d∆(t) + dRu(U0(ξ), ξ, t) (3.32)

dV1(ξ, t)− Lv(U1(ξ, t), V1(ξ∆, t)) = ε−1/2V ′0(ξ) d∆(t) + dRv(V0(ξ), ξ, t). (3.33)

Let L denote the vector-valued operator with components Lu, Lv. That is,

L

(A1

A2

)=

(Lu(A1, A2)

Lv(A1, A2)

). (3.34)

It can be shown that for a sigmoid firing rate function and Gaussian weight distributions,the operator L has a 1D null space spanned by (U ′0(ξ), V ′0(ξ))T [31]. (The factthat (U ′0(ξ), V ′0(ξ))T belongs to the null space follows immediately from differentiatingequation (3.29) with respect to ξ.) We then have the solvability condition for the existenceof a non-trivial solution of equations (3.32) and (3.33), namely, that the inhomogeneouspart is orthogonal to all elements of the null space of the adjoint operator L∗. The latteris defined with respect to the inner product:∫ ∞

−∞B(ξ) · LA(ξ) dξ =

∫ ∞−∞

L∗B(ξ) · A(ξ) dξ (3.35)

where A(ξ) and B(ξ) are two vectors with integrable components. We find that

L∗(B1

B2

)=

(L∗u(B1, B2)

L∗v(B1, B2),

)(3.36)

where

L∗u(B1, B2) = −cεdB1

dξ+ h′(U0)B1 + f ′(U0)Qu

∫ ∞−∞

we(ξ − ξ′)B1(ξ′) dξ′

− f ′(V0)qv

∫ ∞−∞

wi(ξ − ξ′)B2(ξ′) dξ′ (3.37)

and

L∗v(B1, B2) = −cεdB2

dξ+ h′(V0)B2 + f ′(V0)Qv

∫ ∞−∞

we(ξ − ξ′)B2(ξ′) dξ′

− f ′(U0)Qu

∫ ∞−∞

wi(ξ − ξ′)B1(ξ′) dξ′. (3.38)

The adjoint operator L∗ also has a one-dimensional null space, that is, it is spanned bysome vector-valued function V(ξ). Thus taking the inner product of both sides ofequations (3.32) and (3.33) with respect to V(ξ) leads to the solvability condition

0 =

∫ ∞−∞V1(ξ)

[U ′0(ξ) d∆(t) + ε1/2 dRu(U0, ξ, t)

]dξ

+

∫ ∞−∞V2(ξ)[V ′0(ξ) d∆(t) + ε1/2 dRv(V0, ξ, t)] dξ. (3.39)

Thus ∆(t) satisfies the stochastic differential equation (SDE)

d∆(t) = −ε1/2∫∞−∞ (V1(ξ) dRu(U0, ξ, t) + V2(ξ) dRv(V0, ξ, t)) dξ∫∞

−∞ (V1(ξ)U ′0(ξ) + V2(ξ)V ′0(ξ)) dξ. (3.40)

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Using the lowest order approximations dRu(U0, ξ, t) = g(U0(ξ)) dWu(ξ, t) anddRv(V0, ξ, t) = g(V0(ξ)) dWv(ξ, t), we deduce that (for ∆(0) = 0)

〈∆(t)〉 = 0, 〈∆(t)2〉 = 2D(ε)t (3.41)

where D(ε) is the effective diffusivity

D(ε) = ε

∫∞−∞ (V1(ξ)2g(U0(ξ))2 + V2(ξ)2g(V0(ξ))2) dξ

[∫∞−∞ (V1(ξ)U ′0(ξ) + V2(ξ)V ′0(ξ)) dξ]2

. (3.42)

Note that since ∆(t) =O(ε1/2), equation (3.26) implies that U(x, t) = U0(x−cεt)+O(ε1/2).Hence, averaging with respect to the noise shows that 〈U(x, t)〉 = U0(x− cεt) +O(ε1/2). Asimilar result holds for V . Thus, in the case of weak noise, averaging over many realizationsof the stochastic wave generates a mean wave whose speed is approximately equal to cε.This is indeed found to be the case numerically; see below.

3.3. Explicit results for a Heaviside rate function

In order to illustrate the above analysis, we consider a particular example where the meanspeed cε and diffusion coefficient D(ε) can be calculated explicitly. That is, set g(U) = g0Ufor the multiplicative noise term and take f(U) = H(u − κ). (The constant g0 has unitsof√

length/time.) The deterministic equations (3.28) and (3.29) for (U0, V0) reduce to

− cdU0

dξ+ γ(ε)U0 = Qu

∫ 0

−∞we(ξ − ξ′) dξ′ −Qv

∫ ∞X

wi(ξ − ξ′) dξ′ + I (3.43)

− cdV0

dξ+ γ(ε)V0 = Qv

∫ ∞X

we(ξ − ξ′) dξ′ −Qu

∫ 0

−∞wi(ξ − ξ′) dξ′ + I, (3.44)

where

γ(ε) = 1− εC(0)g20. (3.45)

These equations can be solved along identical lines to equations (3.8) and (3.9), and leadto the modified threshold conditions

κ =1

γ(ε)

∫ ∞0

e−sΨX(cs/γ(ε)) ds+ I, κ =1

γ(ε)

∫ ∞0

e−sΦX(−cs/γ(ε)) ds+ I. (3.46)

It immediately follows that both the speed c and displacement X depend on the noisestrength ε. The corresponding wave profiles are (see equations (3.15) and (3.16))

U(ξ) =1

c

∫ ∞0

e−zγ/cΨX(z + ξ) dz + I (3.47)

and

V (ξ) =1

c

∫ ∞0

e−zγ/cΦX(−z − ξ +X) dz + I. (3.48)

In order to calculate the diffusion coefficient, it is first necessary to determine thenull vector V(ε) of the adjoint linear operator L∗. Substituting f(U) = H(U − κ) andg(U) = g0U in equations (3.37) and (3.38) reveals that the components of V satisfy the

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simultaneous equations

cdV1

dξ+ γ(ε)V1 =

δ(ξ)

|U ′0(0)|Qu

∫ ∞−∞

we(z)V1(z) dz

− δ(ξ −X)

|V ′0(X)|Qv

∫ ∞−∞

wi(z −X)V2(z) dz (3.49)

cdV2

dξ+ γ(ε)V2 = − δ(ξ)

|U ′0(0)|Qu

∫ ∞−∞

wi(z)V1(z) dz

+δ(ξ −X)

|V ′0(X)|Qv

∫ ∞−∞

we(z −X)V2(z) dz. (3.50)

Proceeding along similar lines to [18, 47], we make the ansatz that

V1(ξ) = A1e−γξ/cH(ξ)− B1e−γ[ξ−X]/cH(ξ −X),

V2(ξ) = −A2e−γξ/cH(ξ) + B2e−γ[ξ−X]/cH(ξ −X). (3.51)

Substituting back into the adjoint equations yields algebraic conditions for the constantcoefficients Aj,Bj:

A1 =Qu

|U ′0(0)|(A1Ωe[0]− B1Ωe[X]) (3.52)

B1 =Qv

|V ′0(X)|(B2Ωi[0]−A2Ωi[−X]) (3.53)

A2 =Qu

|U ′0(0)|(A1Ωi[0]− B1Ωi[X]) (3.54)

B2 =Qv

|V ′0(X)|(B2Ωe[0]−A2Ωe[−X]) , (3.55)

where

Ωj[x] =

∫ ∞0

e−zγ/cwj(x+ z) dz, j = e, i. (3.56)

Differentiating equations (3.47) and (3.48) with respect to ξ and using equations (3.12)and (3.13) reveals that

U ′0(0) = −QuΩe[0]−QvΩi[−X] < 0, (3.57)

and

V ′0(X) = QvΩe[0] +QuΩi[X] > 0. (3.58)

We have also used the fact that the weight distributions we(x), wi(x) are even functionsof x. Substituting these derivatives into equations (3.52) and (3.55) gives

B2 = −QvΩe[−X]

QuΩi[X]A2, B1 = −QvΩi[−X]

QuΩe[X]A1. (3.59)

It follows from equations (3.54) and (3.59) that A2,B1,B2 can all be expressed as constantmultiples of A1, with the latter determined by normalizing the null vector.

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Figure 9. Effects of multiplicative noise in the activity variables. Simulationof the activity variables u(x, t) (blue) and v(x, t) (purple) for (a) t = 0, (b)t = 1, (c) t = 2, (d) t = 3. The continuous lines are the functions U0(x − ct)and V0(x − ct) which were found by solving equation (3.29). The functionsU0 and V0 were also used for the initial condition. The parameter values wereai = 1, ae = 0.4, σi = 1, σe = 2, β = 5, κ = 0.05, I = 0.24, ε = 0.006 with spatialand temporal grid sizes both being 0.01.

4. Numerical results

In order to proceed analytically, it was convenient to make an adiabatic approximationwhere we assumed that the depression variables qu and qv do not change with time.We also made the further approximation that the fronts move as a composite waveand never drift with respect to each other. In this section we verify that our analyticalassumptions are reasonable by considering numerical simulations of the full system givenby equations (3.17), (3.18), (3.2) and (3.4) with Qu,v → qu,v, though we retain theHeaviside firing rate function to allow us to compare directly the simulations and theanalysis. Simulations were performed on a constant lattice spaced grid of length ∆x withconstant time step ∆t such that the size of the composite waves is far smaller than thesimulation region. Figure 9 shows that our predicted mean front shape develops into astochastic traveling wave moving at the speed resulting from our analysis. It can be seenthat there is very good agreement between the mean profile and stochastic wavefrontposition. Moreover, given that the wave speed is approximately constant helps to verifyour assumption that qu and qv do not change significantly enough to affect the wavemotion.

In figure 10 we plot the mean and variance of the wave position as a function oftime, starting from initial conditions which are solutions to equations equation (3.28)and (3.29). In order to numerically calculate the mean location of the front as a function

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Figure 10. (a) Mean and (b) variance of the wave position for an ensemble of128 stochastic simulations. The parameters were the same as for figure 9.

Figure 11. Difference ∆X in the position of the left-eye and right-eye fronts asa function of time t.

of time, we carry out a large number of level set position measurements of the frontassociated with the left eye, say. That is, we determine the positions Xa(t) such thatU(Xa(t), t) = a, for various level set values a ∈ (0.5κ, 1.3κ), and then define the meanlocation to be X(t) = E[Xa(t)], where the expectation is first taken with respect to thesampled values a and then averaged over N trials. The corresponding variance is given byσ2X(t) = E[(Xa(t) − X(t))2]. It can be seen that X(t) varies linearly with t, consistently

with the assumption that there is constant speed wave, X(t) ∼ cεt. The variance initiallyincreases rapidly with respect to t, but then follows linear behavior consistent with adiffusive-like displacement of the wave from its uniformly translating position at longtime scales, σ2

X(t) ∼ 2D(ε)t. The initial sharp increase in the variance results from thefact that the left and right fronts do move slightly with respect to each other, resulting inhigher order behavior. However, we find numerically that this component of the varianceis small and bounded, so it becomes negligible as time increases. In order to show this,we carry out corresponding level set measurements of the position of the right front byintroducing Ya(t) such that V (Ya(t), t) = a. We then define the difference in front positionsaccording to ∆Xa(t) = Xa(t)−Ya(t) and set ∆X(t) = E[Xa(t)−Ya(t)]. Figure 11 shows atime plot of ∆X(t) for a fully developed stochastic wave, which verifies that this differencein motion is small (of the same order as the grid spacing for this particular simulation)and bounded.

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Figure 12. (a) Snapshot of a stochastic traveling wave (only left-eye activity isshown). (b) Distribution of the first-passage times needed for the wave to travel adistance L = 1, starting from the wave profile in (a). The parameter values wereai = 1, ae = 0.4, σi = 1, σe = 2, β = 5, κ = 0.05, I = 0.24, ε = 0.006 with spatialand temporal grid sizes both being 0.01. The best fit inverse Gaussian F(T ;µ, λ)for the histogram gives the parameters µ = 0.62, λ = 2200 which is in very goodagreement with the theoretical predictions of µ = L/c = 0.6, λ = L2/D = 2100.

Since our analysis predicts that the wave position will follow a Brownian motion, thetime taken for a wave to travel a distance L > 0 has a distribution given by the standardfirst-passage-time formula for Brownian motion with drift c. That is, let TL denote thefirst-passage time for the wave to travel a distance L: cTL + ∆(TL) = L given ∆(0) = 0.Then the first-passage-time density is given by an inverse Gaussian or Wald distribution:

f(TL) = F(TL;

L

c,L2

D

), (4.1)

where

F(T ;µ, λ) =

[λ

2πT 3

]1/2

exp

(−λ(T − µ)2

2µ2T

). (4.2)

Figure 12 shows the first-passage-time distribution generated over a large number ofsimulations for an initial condition given by a fully developed stochastic traveling wave.We find good agreement with our analysis, namely, that the distribution can be fittedby the predicted inverse Gaussian. If we choose to consider more experimentally realisticinitial conditions, which represent a sudden change of contrast in a small region of thedepressed stimulus, we find that the wave takes a substantial time to develop, whichresults in a slightly modified first-passage distribution, although it is still well fitted byan inverse Gaussian, as can be seen in figure 13. The corresponding development of thewave is illustrated in figure 14.

So far we have assumed that all the noise is in the activity variables. We nowstudy numerically how noise affects distributions of first-passage times when placed inthe depression variables. We assume once again that a traveling wave crosses the cortexsignificantly faster than the relaxation time of synaptic depression, so qu and qv can betaken to be constant with respect to time. However, they are no longer constant withrespect to space nor with respect to trials, that is, qu = Qu(x) and qv = Qv(x) withQu, Qv random variables of x. So although the wave itself will travel deterministicallyin a given trial, the functions Qu(x) and Qv(x) will be different across trials. In several

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Figure 13. Same as for figure 12 except that the initial condition is takento be a right-eye dominant homogeneous steady state perturbed by a stepfunction in left-eye activity. Although the wave now needs time to developinto a stochastic traveling front, the best fit inverse Gaussian still shows strongqualitative agreement.

Figure 14. Simulation of the activity variable U(x, t) evolving from a step inputin the left-eye activity for the times t = 0–3 with the same parameters as infigure 12.

experimental studies [44, 45] of binocular rivalry waves, a dominance switch was inducedabout halfway through a normal dominance cycle by locally changing the contrast in thedepressed stimulus. The latter can be represented by increasing the input strength I overa small region in our model. This suggests taking qu and qv to evolve according to

τs dqu(x, t) = [1− qu(x, t)] dt+ η dWqu(x, t) (4.3)

τs dqv(x, t) = [1− qv − βqv] dt+ η dWqv(x, t) (4.4)

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Figure 15. Effects of quenched noise in the depression variables. (a) Initialcondition given by a right-eye dominant homogeneous steady state perturbed by anarrow square pulse of left-eye activity. (b) Distribution of the first-passage timesneeded for a wave to travel a distance L = 6 starting from the initial conditionshown in (a). The parameter values were ai = 1, ae = 0.4, σi = 1, σe = 2, β = 5, κ =0.05, I = 0.24, η = 0.037〈Q〉, where 〈Q〉 is the spatial average of the quencheddepression variables (to make the results comparable with multiplicative noiseones). The spatial grid size ∆x = 0.1 and the temporal grid step is ∆t = 0.01.The solid curve in (b) is the best fit inverse Gaussian.

Figure 16. Same as for figure 15, except that the noise is now smaller, withη = 0.025〈Q〉.

over the time interval [t0, T ]. It is assumed that a switch from left-eye to right-eyedominance occurs at t = t0, so u(x, t) < κ and v(x, t) > κ for t ∈ (t0, T ). The time T is thenchosen such that the system is about 2/3 of the way through a rivalry oscillation, suchthat Qu(x) = qu(x, T ) and Qv(x) = qv(x, T ). (Similar results were obtained for differentcycle lengths and choices of T away from the beginning or end of a cycle.) Thus, thequenched random variables Qu(x) and Qv(x) are obtained by taking a snapshot of twolines of independent Ornstein–Uhlenbeck processes. Figures 15 and 16 show numericallythat the first-passage-time distribution is still well approximated by an inverse Gaussiandistribution. Figure 17 illustrates the development and propagation of a wave on top ofquenched random depression variables. As can be seen, there is still an approximatelyconstant overall shape, though the front’s position will no longer move at a constantspeed.

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Figure 17. Development of an activity variable wave on top of quenchedstochastic depression variables Qu(x), Qv(x). The parameters are the same asfor figure 16.

5. Discussion

In this paper we have analyzed the effects of noise on a neural field model of binocularrivalry waves. Formulating the problem in terms of continuum neural field equationsallowed us to study the short time behavior associated with the propagation of eyedominance from an analytical perspective, both for deterministic and for stochastic waves.In the deterministic case, we have previously shown that some form of slow adaptationsuch as synaptic depression is needed in order to provide a symmetry breaking mechanismthat allows propagation of a binocular rivalry wave [21]. That is, the equations for theleft-eye and right-eye networks have to be different on the time scales on which travelingwaves propagate. One particularly interesting implication of our previous work was thatpurely noise-driven switching between rivalrous states in the absence of adaptation couldnot by itself generate rivalry waves, since it would not produce bias in the appropriatedirection. However, as we have shown in this paper, noise does have a significant effecton binocular rivalry waves in the presence of slow adaptation. In particular, our analysispredicts that motion of the wave in the presence of noise in the activity variables has adiffusive component that leads to a distribution of first-passage times given by an inverseGaussian. We also found a similar distribution of first-passage times for quenched noisein the depression variables. Such results could be experimentally testable.

In future work it would be interesting to extend our neural field model to a morerealistic network topology. Instead of considering one line of neurons for each eye, wecould consider a circle of neurons for each point in the visual field—this would allow us totake orientation information into account using an extension of the coupled ring model [19,20]. We could then investigate the experimental observation that the speed of binocular

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rivalry waves depends on the orientation of the left- and right-eye grating stimuli [85].Moreover, experiments on waves crossing gaps in perceptual grating stimuli have foundthat gap crossing is a stochastic variable and we would hope that it is possible to extendour neural field framework to situations involving non-constant underlying orientationssuch as this one.

Acknowledgments

This publication was based on work supported in part by the National Science Foundation(DMS-1120327), the King Abdullah University of Science and Technology AwardNo. KUK-C1-013-04, and the Systems Biology Doctoral Training Centre, University ofOxford.

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