an amplitude equation approach to contextual effects in visual

LETTER Communicated by Bard Ermentrout

An Amplitude Equation Approach to Contextual Effects inVisual Cortex

Paul C BressloffbressloffmathutaheduDepartment of Mathematics University of Utah Salt Lake City Utah 84112 USA

Jack D CowancowanmathuchicagoeduMathematics Department University of Chicago Chicago IL 60637 USA

A mathematical theory of interacting hypercolumns in primary visual cor-tex (V1) is presented that incorporates details concerning the anisotropicnature of long-range lateral connections Each hypercolumn is modeledas a ring of interacting excitatory and inhibitory neural populations withorientation preferences over the range 0 to 180 degrees Analytical meth-ods from bifurcation theory are used to derive nonlinear equations forthe amplitude and phase of the population tuning curves in which the ef-fective lateral interactions are linear in the amplitudes These amplitudeequations describe how mutual interactions between hypercolumns vialateral connections modify the response of each hypercolumn to modu-lated inputs from the lateral geniculate nucleus such interactions formthe basis of contextual effects The coupled ring model is shown to re-produce a number of orientation-dependent and contrast-dependent fea-tures observed in center-surround experiments A major prediction of themodel is that the anisotropy in lateral connections results in a nonuniformmodulatory effect of the surround that is correlated with the orientationof the center

1 Introduction

The discovery that a majority of neurons in the visual or striate cortex ofcats and primates (usually referred to as V1) respond selectively to thelocal orientation of visual contrast patterns (Hubel amp Wiesel 1962) initi-ated many studies of the precise circuitry underlying this property Twocortical circuits have been fairly well characterized There is a local circuitoperating at subhypercolumn dimensions (lt07 mm) in monkeys compris-ing strong orientation-specic recurrent excitation and weaker intrahyper-columnar inhibition (Michalski Gerstein Czarkowska amp Tarnecki 1983Hata Tsumoto Sato Hagihara amp Tamura 1988 Douglas Koch MahowaldMartin amp Suarez 1995) The other circuit operates between hypercolumns

Neural Computation 14 493ndash525 (2002) cdeg 2002 Massachusetts Institute of Technology

494 Paul C Bressloff and Jack D Cowan

connecting cells with similar orientation preferences separated by up to sev-eral millimeters of cortical tissue (Gilbert amp Wiesel 1989 Hirsch amp Gilbert1992 Malach Amir Harel amp Grinvald 1993 Fitzpatrick Zhang Schoeldamp Muly 1993 Grinvald Lieke Frostig amp Hildesheim 1994 Yoshioka Blas-del Levitt amp Lund 1996) The lateral connections that mediate this circuitarise almost exclusively from excitatory neurons (Rockland amp Lund 1983Gilbert amp Wiesel 1983) although 20 terminate on inhibitory cells and canthus have signicant inhibitory effects (McGuire Gilbert Rivlin amp Wiesel1991)

Stimulation of a hypercolumn via lateral connections modulates ratherthan initiates spiking activity (Hirsch amp Gilbert 1992 Toth Rao Kim Som-ers amp Sur 1996) Thus the lateral connectivity is ideally structured to pro-vide local cortical processes with contextual information about the globalnature of stimuli As a consequence these lateral connections have beeninvoked to explain a wide variety of context-dependent visual processingphenomena (Gilbert Das Ito Kapadia amp Westheimer 1996 Fitzpatrick2000) (Interestingly contextual processing also occurs in extrastriate visualareas and feedback from these areas to V1 has been invoked to explain as-pects of contextual processingWenderoth amp Johnstone 1988) One commonexperimental paradigm is to investigate the response to stimuli consisting ofcircular (center) and annular (surround) gratings of differing contrasts ori-entations and diameters When both center and surround are stimulatedwith strong suprathreshold inputs one typically nds that the surroundsuppresses the center unitrsquos tuning response for surround stimulation closeto the orientation of the unitrsquos peak tuning response and facilitates the re-sponse when the surround is stimulated at orientations sufciently dissim-ilar to the preferred orientation of the center (Blakemore amp Tobin 1972 Liamp Li 1994 Sillito Grieve Jones Cudeiro amp Davis 1995)

The center response to surround stimulation depends signicantly onthe contrast of center stimulation (Toth et al 1996 Levitt amp Lund 1997Polat Mizobe Pettet Kasamatsu amp Norcia 1998) For example a xedsurround stimulus tends to facilitate responses to stimuli at a preferred ori-entation when the center contrast is low but suppresses responses whenit is high Both effects are strongest when center and surround stimuliare iso-oriented The response to variations in the size of a stimulus isalso contrast dependent For example responses to a high-contrast circu-lar grating decline beyond a characteristic preferred stimulus size due toactivation of an inhibitory surround In addition the effective size of theexcitatory receptive eld tends to increase with decreasing contrast andat low contrasts becomes a monotonically increasing function of stimu-lus size (Jagadeesh amp Ferster 1990 Sceniak Ringach Hawken amp Shapley1999)

One common approach to modeling the role of lateral connections incenter-surround modulation is to consider a reduced local cortical circuitcomposed of excitatory and inhibitory populations receiving feedforward

Amplitude Equation Approach 495

inputs from the lateral geniculate nucleus (LGN) together with longndashrangeintracortical inputs some of which could provide feedback from higher cor-tical areas (Somers Nelson amp Sur 1994 Mundel 1996 Mundel Dimitrovamp Cowan 1997 Somers et al 1998 Li 1999 Dragoi amp Sur 2000 StetterBartsch amp Obermayer 2000 Grossberg amp Raizada 2000) Inclusion of someform of contrast-related asymmetry between local excitatory and inhibitoryneurons is then sufcient to account for the switch between low-contrastfacilitation and high-contrast suppression (Somers et al 1998 Stetter etal 2000) Although these (mainly) computational studies have providedsome insights into the possible role of lateral connections in mediatingcenter-surround effects a more general analytical frameworkhas so far beenlacking

In this article we develop such a framework by considering the non-linear dynamics of interacting hypercolumns In particular previous workon the dynamics of sharp orientation tuning in recurrent models of a cor-tical hypercolumn (Ben-Yishai Bar-Or amp Sompolinsky 1995 Somers Nel-son amp Sur 1995 Vidyasagar Pei amp Volgushev 1996 Ben-Yishai Hanselamp Sompolinsky 1997 Mundel et al 1997 Bressloff Bressloff amp Cowan2000 Pugh Ringach Shapley amp Shelley 2000) is extended to incorporatethe effects of lateral connections between hypercolumns A related com-putational approach has been initiated by McLaughlin Shapley Shelleyand Wielaard (2000) Our basic assumptions are as follows (1) each activehypercolumn is close to a bifurcation point signaling the onset of sharp ori-entation tuning and (2) the interactions between hypercolumns are weakWe rst use analytical methods from bifurcation theory to derive nonlin-ear equations for the amplitude and phase of the tuning curves in whichthe effective interactions are linear in the amplitudes (Amplitude equa-tions with linear interaction terms also arise in studies of weakly interact-ing neurons Hoppensteadt amp Izhikevich 1997) These amplitude equationsdescribe how mutual interactions betweenhypercolumns via lateral connec-tions modify the response of each hypercolumn to modulated inputs fromthe lateral geniculate nucleus such interactions form the basis of contextualeffects We then show how our model reproduces a number of orientation-dependent and contrast-dependent features observed in center-surroundexperiments

One novel aspect of our model is that we incorporate the explicit aniso-tropy of lateral connections as observed in a number of optical imaging ex-periments (Yoshioka et al 1996 Bosking Zhang Schoeld amp Fitzpatrick1997) In recent studies of the global dynamics of V1 idealized as a con-tinuous twondashdimensional sheet of interacting hypercolumns we showedthat the patterns of connection exhibit a very interesting symmetry (Wiener1994Cowan 1997Bressloff Cowan Golubitsky Thomas amp Wiener 2001a)they are invariant under the action of the planar Euclidean group E(2)mdashthe group of rigid motions in the planemdashrotations reections and trans-lations By virtue of the anisotropy of the lateral connections they are also


invariant with respect to certain shifts and twists of the plane such that anew E(2) group action is needed to represent its properties This shiftndashtwistsymmetry plays an important role in determining which cortical activitypatterns form through spontaneous symmetry breaking Our previous re-sults on the global dynamics of V1 can also be obtained by considering acontinuum version of the amplitude equations for weakly interacting hy-percolumns derived in this paper This establishes that both the feedforwardresponse properties of V1 and its intrinisic dynamical properties can be un-derstood within the same analytical framework

2 A Dynamical Model of V1 and Its Intrinsic Circuitry

Recent work using optical imaging (Blasdel amp Salama 1986 Blasdel 1992)augments the early work of Hubel and Wiesel (1962) concerning the large-scale organization of isondashorientation patches in V1 It shows that approxi-mately every 07 mm or so in monkey V1 there is an isondashorientation patchof a given preference w We therefore view the tangential structure of thecortex as a lattice of hypercolumns where each hypercolumn comprisesa continuum of iso-orientation patches that are designated by their pre-ferred orientation w 2 [iexcl90plusmn 90plusmn] This simplied anatomy is most easilymotivated (though the analysis does not depend on this assumption in anycrucial manner) if we assume a pinwheel twondashdimensional architecturefor V1 (Obermayer amp Blasdel 1993 Bonhoeffer Kim Malonek Shoham ampGrinvald 1995) The sharply orientation-tuned elements of a hypercolumnthen form a ring of interacting neural populations (A lattice model is alsoconsistent with the observation that each location in the visual eld is repre-sented in V1 roughly speaking by a hypercolumn-sized region containingall orientations) In addition to this two-dimensional tangential architec-ture V1 has a complex laminar vertical structure Although this verticalorganization has been studied in some detail its functional characteristicswith regard to orientation selectivity are largely unknown The solution weadopt here is to collapse this vertical organization into the dynamical inter-action between two neuronal components one excitatory (E) representingpyramidal cells and the other inhibitory (I) The latter lumps together anumber of different types of interneuron for example interneurons withhorizontally distributed axonal elds of the basket cell subtype and in-hibitory interneurons that have predominantly vertically aligned axonalelds (see Figure 1)

Suppose that there are N hypercolumns labeled i D 1 N Let al (ri w t)denote the average membrane potential or activity at time t of a populationof cells of type l D E I belonging to an isondashorientation patch of the ithhypercolumn i D 1 N where ri denotes the cortical position of thehypercolumn (on some coarsely grained spatial scale) and w 2 [iexcl90plusmn 90plusmn]is the orientation preference of the patch The activity variables al evolve to


Figure 1 Local interacting neuronal populations Pymdashexcitatory populationof pyramidal cells Bamdashspatially extended basket cell inhibitory populationMamdashlocal inhibitory population Martinotti chandelier and double bouquetpopulations

the set of equations

fal (ri w t)

tD iexclal (ri w t) C hl (ri w )

CX

mDEI

NX

jD1

Z p 2

iexclp 2wlm (ri w |rj w 0 ) pound sm (am (rj w 0 t))

dw 0

p (21)

where f is a decay time constant (set equal to unity) hl (ri w ) is the input tothe ith hypercolumn (in this study presumed to be from the LGN) and sl istaken to be a smooth output function1

sl (x) D1

1 C eiexclgl (xiexclk l) (22)

1 If sl is interpreted as the ring rate of a single neuron with al its membrane potentialthen sl typically has a hard threshold that is sl (al ) D 0 for al lt gl However at thepopulation level such a ring-rate function can be smoothed out by either spontaneousactivity or dispersion of membrane properties within the population (Wilson amp Cowan1972) From a mathematical perspective the assumption of smoothness is needed in orderto carry out the bifurcation analysis of Section 3


where gl determines the slope or sensitivity of the inputndashoutput charac-teristics of the population and kl is a threshold The kernel wlm (ri w |rj w 0 )represents the strength or weight of connections from the mth populationof the iso-orientation patch w 0 at cortical position rj to the lth populationof the orientation patch w at position ri Note that in this article we do notspecify how visual stimuli impinging on the retina are mapped to the cor-tex via the LGN We simply assume that when there is an oriented stimulusin the (aggregate) classical receptive eld of the ith hypercolumn then theinput h(ri w ) is maximal at w D Wi where Wi is the stimulus orientation Wealso assume that the hypercolumns have nonoverlapping aggregate eldsso that each probes a distinct region of visual space

Optical imaging combined with labeling techniques has generated con-siderable information concerning the pattern of connections both withinand between hypercolumns (Blasdel amp Salama 1986 Blasdel 1992 Malachet al 1993 Yoshioka et al 1996 Bosking et al 1997) A particularly strikingresult concerns the intrinsic lateral connections in layers II III and (to someextent) V of V1 The axons of these connections make terminal arbors onlyevery 07 mm or so along their tracks (Rockland amp Lund 1983 Gilbert ampWiesel 1983) and they seem to connect mainly to cells with similar orien-tation preferences (Malach et al 1993 Yoshioka et al 1996 Bosking et al1997) In addition there is a pronounced anisotropy of the pattern of suchconnections its long axis runs parallel to a patchrsquos preferred orientation(Gilbert amp Wiesel 1983 Bosking et al 1997) Thus differing iso-orientationpatches connect to patches in neighboring hypercolumns in differing di-rections This contrasts with the pattern of connectivity within any onehypercolumn that is much more isotropic any given iso-orientation patchconnects locally in all directions to all neighboring patches within a radiusof less than 07 mm

Motivated by these observations concerning the intrinsic circuitry of V1we decompose w in terms of local connections from elements within thesame hypercolumn and patchy (excitatory) lateral connections from ele-ments in other hypercolumns

wlm (ri w |rj w 0 ) D wlm (w iexcl w 0 )di j C 2 bldmE Ow (ri w |rj w 0 ) (1 iexcl di j) (23)

where 2 is a parameter that measures the weight of lateral relative to localconnections Observations by Hirsch and Gilbert (1992) suggest that 2 issmall and therefore that the lateral connections modulate rather than driveV1 activity The relative strengths of the lateral inputs into local excitatoryand inhibitory populations are represented by the factors bl (Recall thatalthough the lateral connections are excitatorymdashRockland amp Lund 1983Gilbert amp Wiesel 1983mdash20 of the connections in layers II and III of V1end on inhibitory interneurons so the overall action of the lateral connec-tions can become inhibitory especially at high levels of activity Hirsch ampGilbert 1992) The contrast dependence of the factors bl will be considered


in Section 4 Substituting equation 23 into equation 21 gives

al (ri w t)t

D iexclal (ri w t) CX

mDEI

Z p 2

iexclp 2wlm (w iexcl w 0 )sm (am (ri w 0 t))

dw 0

pC hl (ri w ) C 2 bl

X

j 6Di

Z p2

iexclp 2Ow(ri w |rj w 0 )sE (aE (rj w 0 t))

dw 0

p(24)

for i D 1 NIn order to incorporate information concerning the anisotropic nature of

lateral connections we further decompose Ow according to2

Ow(ri w |rj w 0 ) D Ow(rij)p0(w 0 iexcl hij)p1 (w iexcl w 0 ) (25)

where rj D ri C rij (cos(hij) sin(hij)) In the special case pk (w ) D d (w ) thedistribution Ow(rij ) determines the weight of lateral connections betweeniso-orientation patches separated by a cortical distance rij along a visuotopicaxis whose direction hij is parallel to their (common) orientation preference(see Figure 2) The p -periodic functions pk (w ) then determine the spreadof the lateral connections with respect to the visuotopic axis for k D 0 (seeFigure 3a) and orientation preference for k D 1 (see Figure 3b) We takepk (w ) to be an even monotonically decreasing function of w over the domainiexclp 2 lt w lt p 2 such that

R p 2iexclp 2 pk (w )dw p D 1

3 Amplitude Equation for Interacting Hypercolumns

31 Sharp Orientation Tuning in a LocalCortical Circuit In the absenceof lateral connections (2 D 0) each hypercolumn is independently describedby the ring model of orientation tuning (Somers et al 1995 Ben-Yishai etal 1995 1997 Mundel et al 1997 Bressloff et al 2000) That is equation 24reduces to the pair of equations

aE

tD iexclaE C wEE curren sE (aE) iexcl wEI curren sI (aI) C hE (31)

aI

tD iexclaI C wIE curren sE (aE) iexcl wII curren sI (aI ) C hI (32)

where curren indicates a convolution operation

w curren f (ri w ) DZ p 2

iexclp 2w (w iexcl w 0 ) f (ri w 0 )

dw 0

p (33)

2 The distribution of lateral connections given by equation 25 differs from the oneused by Li (1999) in that there is no explicit crossndashorientation inhibition


Figure 2 Anisotropic lateral interactions connect iso-orientation patches lo-cated along a visuotopic axis parallel to their (common) orientation preferenceHorizontal patches are connected along axis A whereas oblique patches areconnected along axis B

for a hypercolumn at a xed cortical position ri In the case of w -independentexternal inputs hl (ri w ) D Nhl (ri) i D 1 N there exists at least one xed-point solution al (ri w ) D Nal (ri) of equations 31 and 32 which satises thealgebraic equations

NaE D WEE (0)sE (NaE) iexcl WEI (0)sI (NaI ) C NhE (34)

NaI D WIE (0)sE (NaE) iexcl WII (0)sI (NaI ) C NhI (35)

Figure 3 Angular spread in the anisotropic lateral connections with respect tothe (a) visuotopic axis and (b) orientation preference Shaded orientation patchesare connected


where Wlm (0) DR p2

iexclp2 wlm (w )dw p If Nhl is sufciently small relative to thethresholdkl then this xed point is unique and stable for each ri Under thechange of coordinates al al iexcl Nhl it can be seen that the effect of Nhl is to shiftthe threshold by the amount iexclNhl Thus there are two ways to increase theexcitability of the network and thus destabilize the xed point increasingthe external input Nhl or reducing the threshold kl In the case of externalvisual stimuli adaptation mechanisms tend to lter out the uniform part ofthe stimulus so that the contribution to Nhl from the LGN is expected to besmall We assume however that various arousal and attentional mechanismact so that neurons sit close to threshold and can respond to small spatial(and temporal) variations of Nhl (Tsodyks amp Sejnowski 1995) An alternativemechanism for raising the excitability of the cortex is through the action ofdrugs on certain brain stem nuclei which can induce the experience of see-ing geometric visual hallucinations (Ermentrout amp Cowan 1979 Bressloffet al 2001a Bressloff Cowan Golubitsky Thomas amp Wiener 2001b)

The local stability of (NaE NaI ) is found by linearization

bE

tD iexclbE C

hs 0

E (NaE)wEE curren bE iexcl s 0I (NbI )wEI curren bI

i(36)

bI

tD iexclbI C

pounds 0

E (NaE)wIE curren bE iexcl s 0I (NaI)wII curren bI

curren (37)

where bl (rj w t) D al (rj w t) iexcl Nal (rj ) and s 0l (x) D dsl (x)dx Equations 36

and 37 have solutions of the form

bl (rj w t) D Blelthz(rj)e2inw C z(rj )eiexcl2inw

i (38)

where z(rj ) is an arbitrary (complex) amplitude For each positive integer nthe eigenvalues l and corresponding eigenvectors B D (BE BI)T satisfy thematrix equation

(1 C l)B D

sup3s 0

E (NaE)WEE (n) iexcls 0I (NaI )WEI (n)

s 0E (NaE)WIE (n) iexcls 0

I (NaI)WII (n)

acuteB (39)

where Wlm (n) is the nth Fourier coefcient in the expansion of thep -periodicweights kernels wlm (w )

wlm (w ) D Wlm (0) C 21X

nD1

Wlm (n) cos(2nw ) l m D E I (310)

and we assume that wlm (w ) D wlm (iexclw )For the sake of illustration suppose that s 0

m (Nam) D m so that an increasein the excitability of the network can be modeled as an increase in m Equa-tion 39 then has solutions of the form

lsectn D iexcl1 C m Wsect

n (311)


for integer n where

Wsectn D

12

[WEE (n) iexcl WII (n) sect S (n)] (312)

are the eigenvalues of the weight matrix with

S (n) Dp

[WEE (n) C WII (n)]2 iexcl 4WEI (n)WIE (n) (313)

The corresponding eigenvectors (up to an arbitrary normalization) are

Bsectn D

sup3WEI (n)

12 [WEE (n) C WII (n) uml S (n)]

acute (314)

Suppose that Re WCn has a unique maximum as a function of n cedil 0 at n D 1

with Im WC1 D 0 and dene m c D 1WC

1 For m lt m c we have lsectn lt 0 for all

n and thus the homogeneous resting state is stable However as m is steadilyincreased the homogeneous state will destabilize at the critical pointm D m cdue to excitation of eigenmodes of the form BC

1

poundz(rj)e2iw C z(rj)eiexcl2iw

curren where

z(rj) is an arbitrary complex function of r It can be shown that the satu-rating nonlinearities of the system stabilize the tuning curves beyond thecritical point m c (Ermentrout 1998 Bressloff et al 2000) Using the po-lar representation z(r) D Z(r)e2ipw (r) the eigenmode can be rewritten asBC

p Z(rj) cos(2p[w iexclw (rj)]) Thus the maximum (linear) response of the jth hy-percolumn occurs at the orientation w (rj) C kp p k D 0 1 p iexcl1 for p gt 0

In terms of the orientation tuning properties observed in real neuronsthe most relevant cases are p D 0 and p D 1 The former corresponds to abulk instability within a hypercolumn in which the new steady state exhibitsno orientation preference We call such a response the HubelndashWiesel modesince any orientation tuning must be extrinsic to V1 generated for exampleby local anisotropies of the geniculondashcortical map (Hubel amp Wiesel 1962) Abulk instability will occur when the local inhibition is sufciently weak Inthe second case each hypercolumn supports an activity prole consistingof a solitary peak centered about the angle w (rj) that is the population re-sponse is characterized by an orientation tuning curve One mechanism thatgenerates a p gt 0 mode is if the local connections comprise short-range exci-tation and longer-range inhibition which would be the case if the inhibitoryneurons are identied with basket cells (I D Ba) whose lateral axonal spreadhas a space constant of about 250 m m However it is also possible withina two-population model to generate orientation tuning in the presence ofshortndashrange inhibition as would occur if the inhibitory neurons are identi-ed with local interneurons (I D Ma) instead whose lateral axonal spreadis about 20 m m (see Figure 1) Example spectra WC

n for both of these casesare plotted in Figure 4 for gaussian coefcients

Wlm (n) Dp

2pjlmalmeiexcln2j 2lm 2 (315)


Figure 4 (a) Distribution of eigenvalues WCn as a function of n in the case of

long-range inhibition Parameter values are jEE D jIE D 15plusmn jII D jEI D 60plusmn aEE D 1 aEIaIE D 03 aII D 0 (b) Distribution of eigenvalues WC

n as a function ofn in the case of short-range inhibition Parameter values are jEE D 35plusmn jIE D 40plusmn jII D jEI D 5plusmn aEE D 05 aEIaIE D 05 aII D 1 Solid line shows real part anddashed line shows the imaginary part

where jlm determine the range of the axonal elds of the excitatory and in-hibitory populations For both examples it can be seen that ReWC

1 gt ReWCn

for all n 6D 1 so that the n D 1 mode becomes marginally stable rst lead-ing to the spontaneous formation of sharp orientation tuning curves (Notethat if ImWC

1 6D 0 at the critical point then the homogeneous resting statebifurcates to an orientation tuning curve whose peak spontaneously eitherrotates as a traveling wave or pulsates as a standing wave at a frequency de-termined by Iml

C1 (Ben-Yishai et al 1997 Bressloff et al 2000) We consider

the timendashperiodic case in Section 33The location of the peak w curren (rj) of the tuning curve at rj is arbitrary in the

presence of w-independent inputs hl (rj w ) D Nhl (rj) However the inclusionof an additional small-amplitude input Dhl (rj w ) raquo cos[2(w iexcl Wj)] breaksthe rotational invariance of the system and locks the location of the tuningcurve to the orientation corresponding to the peak of the stimulus thatis w curren (rj) D Wj This is illustrated in Figure 5 where the input and output


Figure 5 Sharp orientation tuning curve in a single hypercolumn Local re-current excitation and inhibition amplies a weakly modulated input from theLGN Dotted line is the baseline output without orientation tuning

of the excitatory population of a single hypercolumn are shown (see alsoSection 41) Thus the local intracortical connections within a hypercolumnserve to amplify a weakly oriented input signal from the LGN (Somers etal 1995 Ben-Yishai et al 1997)

The proportion of thalamocortical versus intracortical input in vivo andthe tuning of the thalamocortical input are matters of ongoing debate Itis still not known for certain whether orientation selectivity is introducedin a feedforward manner to subsequent layers or arises from the intrinsicrecurrent architecture of the upper layers of layer 4 or higher or from somecombination of these mechanisms Recent workappears to conrm the orig-inal proposal of Hubel and Wiesel (1962) that geniculate input is sharplytuned in the cat (Ferster Chung amp Wheat 1997) On the other hand aconsiderable body of work indicates that intracortical processes includingrecurrent excitation and intracortical inhibition are important in determin-ing orientation selectivity (Blakemore amp Tobin 1972 Sillito 1975 Douglaset al 1995 Somers et al 1995) In primates such as the macaque most layerIV C cells receiving direct thalamocortical input are unoriented and thusthe introduction of orientation asymmetry is clearly a cortical process Iforientation information is to be represented cortically and kept ldquoonndashlinerdquofor any period of time then specic intracortical circuitry not too dissimi-lar from the tuning circuitry of the primate (eg local recurrent excitationand inhibition) is necessary to stably represent the sharply tuned input Thework of Phleger and Bonds (1995) demonstrating a loss of stable orientationtuning in cats with blocking of intracortical inhibition supports this view

32 Cubic Amplitude EquationStationary Case So far we have estab-lished that in the absence of lateral connections each hypercolumn (labeled


by cortical position rj) can exhibit sharp orientation tuning when in a suf-ciently excited state The peak of the tuning curve is xed by inputs fromthe LGN signaling the presence of a local oriented bar in the classical re-ceptive eld (CRF) of the given hypercolumn We wish to investigate howsuch activity is further modulated by stimuli outside the CRF due to thepresence of anisotropic lateral connections between hypercolumns leadingto contextual effects

Our approach will be to exploit the fact that the lateral connections areweak relative to local circuit connections and to use bifurcation analysis toderive dynamical equations for the amplitudes of the excited modes closeto the bifurcation point These equations then allow us to explore the effectsof local and lateral inputs on sharp orientation tuning For simplicity weassume that each hypercolumn can be in one of two states of activationdistinguished by the index Acirc (ri) D 0 1 where ri is the cortical position ofthe ith hypercolumn If Acirc (ri) D 0 then the neurons within the hypercolumnare well below threshold so that there is only spontaneous activity sl frac140 On the other hand if Acirc (ri) D 1 then the hypercolumn exhibits sharporientation tuning with mean levels of output activity sl (Nal (ri)) where Nal (ri)is a xed-point solution of equations 34 and 35 close to the bifurcationpoint Any hypercolumn in the active state is taken to have the same meanoutput activity That is if we denote the set of active hypercolumns byJ D fi 2 1 NI Acirc (ri) D 1g then Nal (ri) D Nal for all i 2 J and we setsl (Nal ) D Nsl

First perform a Taylor expansion of equation 24 with respect to bl (ri w t)D al (ri w t) iexcl Nal i 2 J

bl

tD iexclbl C

X

mDEI

wlm currenhm bm C c mb2

m C c 0mb3

m C i

C Dhl

C 2 bl Ow plusmniexcl[ NsE C m bE C ] Acirc

cent(316)

where Dhl D hl iexcl Nhl and m D s 0l (Nal) c l D s 00 (Nal)2 c 0

l D s 000 (Nal)6 Theconvolution operation curren is dened by equation 33 and

[ Ow plusmn f ] (ri w ) DX

j2J j 6Di

Ow(rij)Z p 2

iexclp 2p0 (w 0 iexclhij)p1 (w iexclw 0 ) f (rj w 0 )

dw 0

2p 0 (317)

for an arbitrary function f (r w ) and Ow given by equation 25 Suppose thatthe system is 2 -close to the point of marginal stability of the homogeneousxed point associated with excitation of the modes esect2iw That is take m Dm c C 2 Dm where m c D 1WC

1 (see equation 311) Substitute into equation 316


the perturbation expansion

bm D 2 12b(1)m C 2 b (2)

m C 2 32b (3)m C cent cent cent (318)

Finally introduce a slow timescale t D 2 t and collect terms with equalpowers of 2 This leads to a hierarchy of equations of the form (up to O (2 32))

[ L b(1)]l D 0 (319)

[ L b(2)]l D v(2)l (320)

acuteX

mDEIc mwlm curren [b(1)

m ]2 C bl NsE Ow plusmn Acirc

[ L b(3)]l D v(3)l

acute iexclb (1)

l

tC

X

mDEI

wlm currenhDm b (1)

m C c 0m[b(1)

m ]3 C 2c mb(1)m b(2)

m

i

C Dhl C m cbl Ow plusmnplusmnb(1)

E Acircsup2

(321)

with the linear operator L dened according to

[ L b]l D bl iexcl m c

X

mDEI

wlm curren bm (322)

We have also assumed that the modulatory external input is O (2 32) andrescaled Dhl 2 32Dhl

Recall that each active hypercolumn is assumed to exhibit sharp orien-tation tuning so that the local connections are such that the rst equationin the hierarchy equation 319 has solutions of the form

b (1) (ri w t ) Dplusmn

z(ri t )e2iw C z(ri t ) eiexcl2iwsup2

B (323)

for all i 2 J with B acute BC1 dened in equation 314 and z denotes the

complex conjugate of z We obtain a dynamical equation for the complexamplitude z(ri t ) by deriving solvability conditions for the higher-orderequations We proceed by taking the inner product of equations 320 and321 with the dual eigenmode eb(w ) D e2iweB where

eB D

sup3WIE (1)

iexcl12 [WEE (1) C WII (1) iexcl S (1)]

acute(324)

so that

[ L Teb]l acute ebl iexcl m c

X

mDEI

wml curren ebm D 0


The inner product of any two vector-valued functions of w is dened as

hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)

With respect to this inner product the linear operator L satises heb| L bi Dh L Teb|bi D 0 forany b Since L b(p) D v(p) we obtain a hierarchy of solvabilityconditions heb|v(p)i D 0 for p D 2 3

It can be shown from equations 317 320 and 323 that the rst solvabilitycondition requires that

bl NsEP(1)0 P(1)

1

X

j2JOw(rij )eiexcl2ihij middot O (2 12) (326)

where

P(n)k D

Z p 2

iexclp 2eiexcl2niw pk (w )

dw

p

(327)

This is analogous to the so-called adaptation condition in the bifurcationtheory of weakly connected neurons (Hoppensteadt amp Izhikevich 1997) Ifit is violated then at this level of approximation the interactions betweenhypercolumns become trivial There are two ways in which equation 326can be satised the conguration of surrounding hypercolumns is suchthat

Pj Ow (rij)eiexcl2ihij middot O (2 12) or the mean ring rate NsE middot O (2 12) Here we

assume that NsE D O (2 12)The solvability condition heb|v(3)i D 0 generates a cubic amplitude equa-

tion for z(ri t ) As a further simplication we set c m D 0 since this doesnot alter the basic structure of the amplitude equation (Note however thatthe coefcients of the amplitude equation are c m dependent and hence thestability properties of a pattern may change ifc m 6D 0) Using equations 317321 and 323 we then nd that (after rescaling t )

z(ri t )t

D z(ri t )(Dm iexcl A|z(ri t )|2) C f (ri) C bX

j2JOw (rij)

poundhz(rj t ) C z(rj t )P(2)

0 eiexcl4ihij C sP(1)0 eiexcl2ihij

i (328)

for all i 2 J where s D NsE p

2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p

0eiexcl2iwDhl (ri w )

dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)

Equation 328 is our reduced model of weakly interacting hypercolumns Itdescribes the effects of anisotropic lateral connections and modulatory in-puts from the LGN on the dynamics of the (complex) amplitude z(ri t ) Thelatter determines the response properties of the orientation tuning curveassociated with the hypercolumn at cortical position ri The coupling pa-rameter b is a linear combination of the relative strengths of the lateral con-nectionsrsquo innervating excitatory neurons and those innervating inhibitoryneurons with DE DI determined by the local weight distribution SinceDE gt 0 and DI lt 0 we see that the effective interactions between hyper-columns have both an excitatory and an inhibitory component (The factorP(1)

1 appearing in equation 329 which arises from the spread in lateral con-nections with respect to orientationmdashsee Figure 3bmdashgenerates a positiverescaling of the lateral interactions and can be absorbed into the parametersDE and DI) Note that in the case of isotropic lateral connections P (n)

k D 0for all n gt 1 so that z decouples from z in the amplitude equation 328

33 Cubic Amplitude Equation Oscillatory Case In our derivationof the amplitude equation 328 we assumed that the local cortical cir-cuit generates a stationary orientation tuning curve However as shownin Section 31 it is possible for a time-periodic tuning curve to occur whenImWC

1 6D 0 Taylor expanding 24 as before leads to the hierarchy of equa-tions 319 through 321 except that the linear operator L L t D L C tThe lowest-order solution equation 323 now takes the form

b (1) (ri w t t ) DhzL (ri t )ei(V0 tiexcl2w ) C zR (ri t )ei(V0tC2w )

iBC cc (332)

where zL and zR represent the complex amplitudes for anticlockwise (L) andclockwise (R) rotating waves (around the ring of a single hypercolumn) and

V0 D m c

p4WEI (1)WIE (1) iexcl [WEE (1) C WII (1)]2 (333)

Introduce the generalized inner product

hu | vi D limT1

1T

Z T2

iexclT2

Z p

0[uE (w t)vE (w t) C uI (w t)vI (w t)]

dw

pdt (334)

and the dual vectors ebL D eBei(V0tiexcl2w ) ebR D eBei(V0 tC2w ) Using the fact thathebL | L tbi D hebR | L tbi D 0 for arbitrary b we obtain the pair of solvabilityconditions hebL |v(p)i D hebR |v(p)i D 0 for each p cedil 2


The p D 2 solvability conditions are identically satised The p D 3solvability conditions then generate cubic amplitude equations for zL zR ofthe form

zL (ri t )t

D (1 C iV0)zL (ri t )hDm iexclA|zL (ri t ) |2 iexcl2A|zR (ri t ) |2

i

C fiexcl (ri) C bX

j2JOw(rij )

hzL (rj t ) C zR (rj t )P (2)

0 e4ihij

i(335)

and

zR (ri t )t

D (1 C iV0)zR (ri t )hDm iexclA|zR (ri t ) |2 iexcl2A|zL (ri t ) |2

i

C fC (ri) C bX

j2JOw (rij)

hzR (rj t ) C zL (rj t )P(2)

0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p

0eiexcli(V0tsect2w )

X

lDEI

eBlDhl (ri w t)dw

pdt (337)

It can be seen that the amplitudes couple only to timendashdependent inputsfrom the LGN

4 Contextual Effects

In the previous section we used perturbation techniques to reduce the orig-inal innite-dimensional system of Wilson-Cowan equations (24) to a cor-responding nite-dimensional system of amplitude equations (328) This isillustrated in Figure 6 The complex amplitude z(ri) D Zieiexcl2iw i determinesthe linear response function Ri (w ) of the ith hypercolumn according to

Ri (w ) D z(ri)e2iw C Nzi (ri)eiexcl2iw D 2Zi cos(2[w iexcl w i]) (41)

and this in turn determines the population tuning curves of the excitatoryand inhibitory populations In this section we use the amplitude equation328 to investigate (at least qualitatively) how contextual stimuli fallingoutside the classical receptive eld of a hypercolumn modify their responseto stimuli within the classical receptive eld such contextual effects aremediated by the lateral interactions between hypercolumns

41 A Single Driven Hypercolumn We begin by determining the re-sponse of a single hypercolumn to an LGN input in the absence of lateralinteractions (b D 0) and under the assumption that the orientation tuning


Figure 6 Reduction from a dynamical model of interacting hypercolumns to adynamical model of interacting population tuning curves

curves and external stimuli are stationary Let the input from the LGN tothe ith active hypercolumn be of the form

Dhl (ri w ) D glCi cos(2[w iexcl Wi]) (42)

where Ci Ci gt 0 is proportional to the contrast of the center stimulusWi is its orientation and gl cedil 0 determines the relative strengths of thefeedforward input to the local excitatory and inhibitory populations Sucha stimulus represents an oriented edge or bar in the aggregate receptiveeld of the hypercolumn It then follows from equation 330 that

f (ri) D Cieiexcl2iWi (43)

where we have set m cP

lDEIQBlgl D 1 for convenience This term is positive

if gE Agrave gI Setting b D 0 in equation 328 and writing z(ri ) D Zieiexcl2iwi withZi real we obtain the following pair of equations

dZi

dtD Zi (Dm iexcl AZ2

i ) C Ci cos(2[w i iexcl Wi]) (44)

dw i

dtD iexcl

Ci

2Zisin(2[w i iexcl Wi]) (45)

Assuming that the coefcients Dm gt 0 A gt 0 these have a stable xed-point solution (w curren

i Zcurreni ) with w curren

i D Wi (independent of the contrast) and Zcurreni


is thepositive rootof the cubic Zi (Dm iexclAZ2i )CCi D 0 Thus the hypercolumn

encodes the orientation Wi of a local bar in its aggregate receptive eld bylocking the peak of its tuning curve to Wi (see Figure 5) The amplitudeof the tuning curve is determined by Zcurren

i which is an increasing functionof contrast Note that in the absence of an external stimulus (Ci D 0) thetuning curve is marginally stable since wcurren

i is arbitrary This means that thephase will be susceptible to noise-induced diffusion Recall from Section 3that the amplitude of the LGN input was assumed to be O (2 32) whereasthe amplitude of the response is O (2 12) Since 2 iquest 1 the intrinsic circuitryof the hypercolumn amplies the LGN input

Now consider the oscillatory case In order that the amplitudes zL andzR of equations 335 and 336 couple to an LGN input the latter must betime dependent with a Fourier component at the natural frequency of os-cillation V0 of the hypercolumn It is certainly reasonable to assume thatexternal stimuli have a time-dependent part For example neurophysiolog-ical experiments often use ashing or drifting oriented gratings as externalstimuli (These are chosen to compensate for the spike frequency adaptationof neurons in the cortex) Time-dependent signals might also be generated inthe LGN Suppose that Ci is the contrast of the relevant temporal frequencycomponent and Wi is the orientation of the external stimulus Neglectinglateral inputs the amplitude equations 335 and 336 become

zL (ri t )t

D (1 C iV0)zL (ri t )

poundhDm iexcl A|zL (ri t )|2 iexcl 2A|zR (ri t ) |2

iC Cieiexcl2iWi (46)

and

zR (ri t )t

D (1 C iV0)zR (ri t )

poundhDm iexcl A|zR (ri t ) |2 iexcl 2A|zL (ri t )|2

iC Cie2iWi (47)

If Ci D 0 Dm gt 0 and A gt 0 there are three types of xed-point solution(1) anticlockwise rotating waves |zL |2 D Dm A zR D 0 (2) clockwise rotatingwaves |zR |2 D Dm A zL D 0 and (3) standing waves |zR |2 D |zL |2 D Dm 3AThe standing waves are unstable and the traveling waves are marginallystable due to the existence of arbitrary phases (Kath 1981 Aronson Ermen-trout amp Kopell 1990) (Note however that it is possible to obtain stablestanding waves when c m 6D 0 in equation 321) If Ci gt 0 the travelingwave solutions no longer exist but standing wave solutions do of the formzL D Zieiexcl2iWi eiy zR D Zie2iW ieiy with tan y D iexclV0 and Zi a positive root ofthe cubic

Zi[Dm iexcl 3AZ2i ] C Ci cos(y ) D 0 (48)


Equation 332 then implies that the tuning curves are of the form

4Zi cos(V0t C y ) cos(2[w iexcl Wi])

It can be shown that these solutions are unstable for sufciently low con-trasts but stable for high contrasts

The existence of a stable timendashperiodic tuning curve in the case of a singleisolated hypercolumn means that it effectively acts as a single giant oscilla-tor One could then investigate for example the synchronization propertiesof a network of hypercolumns coupled via anisotropic lateral interactionsHowever it is not clear how to interpret these oscillations biologically Onepossibility is that they correspond to the 40 Hz c frequency oscillations thatare thought to play a role in the synchronization of cell assemblies (GrayKonig Engel amp Singer 1989 Singer amp Gray 1995) One difculty with thisinterpretation is that the c oscillations appear in the spikes of individualneurons and thus the mechanism for generating them is likely to be washedout in any mean-eld theory analysis used to derive the rate models con-sidered in this article On the other hand recent work on integratendashandndashrenetworks has shown that timendashdependent ringrates can be viewed as mod-ulations of single neuron spikes (Bressloff et al 2000) Given the difcultiesconcerning the signicance of oscillatory tuning curves we assume thatthe normal operating regime of each hypercolumn is to generate stationarytuning curves in response to slowly changing LGN inputs

42 CenterndashSurround Interactions Consider the following experimen-tal situation (Blakemore amp Tobin 1972 Li amp Li 1994 Sillito et al 1995) aparticular hypercolumn designated the ldquocenterrdquo is stimulated with a gratingat orientation Wc while outside the receptive area of this hypercolumnmdashinthe ldquosurroundrdquomdashthere is a grating at some uniform orientation Ws as shownin Figure 7a In order to analyze this problem we introduce a mean-eldapproximation That is the active region of cortex responding to the center-suround stimulus of Figure 7a is taken to have the conguration shown inFigure 7b This consists of a center hypercolumn at rc D 0 interacting witha ring of N identical surround hypercolumns at relative positions rj with|rj | D R j D 1 N Note that the total input from surround to centeris strong relative to that from center to surround Therefore as a furtherapproximation we can neglect the latter and treat the system as a single hy-percolumn receiving a mixture of inputs from the LGN and lateral inputsfrom the surround (see Figure 7c)

Suppose that the surround hypercolumns are sharply tuned to the stim-ulus orientation Ws that is they have steady-state amplitudes of the formz(rj) D Zseiexcl2iWs where Zs is positive and real It follows from equation 328that the complex amplitude zc (t ) characterizing the response of the center


Figure 7 (a) Circular centerndashsurround stimulus conguration (b) Cortical con-guration of a center hypercolumn interacting with a ring of surround hy-percolumns (c) Effective single hypercolumn circuit in which there are directLGN inputs from the center and lateral inputs from the surround The relativestrengths of the lateral inputs to the excitatory (E) and inhibitory (I) popula-tions are determined by bE and bI respectively The corresponding LGN inputstrengths are denoted by gE and gI

satises the equation

dzc (t )dt

D zc (t ) (Dm iexcl A|zc (t ) |2)

C bws

plusmneiexcl2iWs C Le2iWs C QL

sup2C Cceiexcl2iWc (49)

where ws D NZs Ow (R) gt 0 Cc is the contrast of the center stimulus and

L DP(2)

0

N

NX

jD1

eiexcl4ihj QL DsP(1)

0

NZs

NX

jD1

eiexcl2ihj (410)

with hj the direction of the line from the center to the jth surround hy-percolumn (see Figure 7b) We now make the simplifying assumption thatL D QL D 0 due to the rotational symmetry of the mean-eld congurationWriting zc D Zceiexcl2iwc we then obtain the following pair of equations

dZc

dtD Zc (Dm iexcl AZ2

c ) C Cc cos(2[wc iexcl Wc])

C bws cos(2[wc iexcl Ws]) (411)

dwc

dtD iexcl

Cc

2Zcsin(2[wc iexcl Wc]) iexcl bws

2Zcsin(2[wc iexcl Ws]) (412)


Figure 8 Response of center population as a function of the relative orientationof the surround stimulus DW D Ws iexclWc Parameters in amplitude equations 411and 412 are Dm D 01 |b | Ows D 08 Cc D 1 and A D 1 (a) Solid line totalresponse dZc at preferred orientation as a fraction of the response in the absenceof surround modulation Dashed line amplitude of maximum response Zcurren

c (b )(b) Plot of the shift in the peak of the orientation tuning curve dwc D Wc iexcl wcurren

c (b )

Recall from equation 329 that the effective interaction parameter b hasboth an excitatory and an inhibitory component As noted previously ex-perimental data suggest that the effect of lateral inputs on the responseof a hypercolumn is contrast sensitive tending to be suppressive at highcontrasts and facilitatory at low contrasts (Toth et al 1996 Levitt amp Lund1997 Polat et al 1998) In terms of the amplitude equation 49 this impliesthat b D b (Cc) For the moment we assume that the contrast of the centerstimulus is large enough so that b lt 0

Let (Zcurrenc (b ) w curren

c (b )) be the unique stable steady-state solution of equa-tions 411 and 412 for a given b We dene two important quantities theshift dwc in the peak of the tuning curve and the fractional changedZc in thelinear response at the preferred orientation Wi

dwc D Wc iexcl w currenc (b ) dZc D

Zcurrenc (b ) cos(2dwc)

Zcurrenc (0)

(413)

If dZc lt 1 the effect of the lateral inputs is supressive whereas if dZc gt1 then it is facilitatory In Figure 8 we plot the resulting quantities dZcand dwc as a function of DW D Ws iexcl Wc It can be seen that surroundmodulation changes from suppression to facilitation as the difference inthe orientation of the surround and the center increases beyond around60plusmn This is consistent with experimental results on center-surround sup-pression and facilitation at high contrasts (Blakemore amp Tobin 1972 Liamp Li 1994 Sillito et al 1995) An important contribution to the asym-metry between suppression and facilitation arises from the shift in thepeak of the effective tuning curve in the presence of the surround as mea-sured by dwc Since this is positive there is an apparent increase in thedifference between the center and surround tuning curve peaks whichis analogous to the direct tilt effect observed in psychophysical experi-


ments (Wenderoth amp Johnstone 1988) although the effect is much largerhere

We now give a simple argument for the switch from suppression to fa-cilitation as the relative angle DW D Ws iexcl Wc is increased If the surroundis stimulated by a grating parallel to that of the center stimulus (Ws D Wc)then wcurren

c D Wc and equation 411 becomes

dZc


c ) C Cc C bws (414)

Similarly in the case of an orthogonal surround (Ws D Wc C 90plusmn)

dZc


c ) C Cc iexcl bws (415)

The steady-state amplitude Zcurrenc for b D 0 is an increasing function of Cc (see

Section 41) Hence if b lt 0 then Cc Cc iexcl |bwc | for a collinear surroundresulting in a suppression of the center response whereas Cc Cc C |bwc |for an orthogonal surround leading to facilitation

The occurrence of a facilitatory response to an orthogonal surround stim-ulus has been observed experimentally by a number of groups (Blakemore ampTobin 1972 Sillito et al 1995 Levitt amp Lund 1997 Polat et al 1998) At rstsight however this conicts with the consistent experimental nding thatstimulating a hypercolumn with an orthogonal stimulus suppresses the re-sponse to the original stimulus In particular DeAngelis Robson Ohzawaand Freeman (1992) show that crossndashorientation suppression (with orthog-onal gratings) originates within the receptive eld of most cat neurons ex-amined and is a consistent nding in both complex and simple cells Thedegree of suppression depends linearly on the size of the orthogonal gratingup to a critical dimension which is smaller than the classical receptive elddimension Such observations are compatible with the above ndings Inthe case of orthogonal inputs to the same hypercolumn we have the simplelinear summation Cc cos(2w ) C C0

c cos(2[w iexclp 2]) D (Cc iexclC0c) cos(2w ) where

Cc gt C0c gt 0 Thus the orthogonal input of amplitude C0

c reduces the ampli-tude of the original input and hence gives rise to a smaller response On theother hand the effect of an orthogonal surround stimulus is mediated bythe lateral connections and (in the suppressive setting) is input primarily tothe orthogonal inhibitory population which can lead to disinhibition andconsequently facilitation Similar arguments were used by Mundel et al(1997) and more recently by Dragoi and Sur (2000) in their numerical studyof interacting hypercolumns

The center response to surround stimulation depends signicantly onthe contrast of the center stimulation (Toth et al 1996 Levitt amp Lund 1997Polat et al 1998) For example a xed surround stimulus tends to facilitateresponses to preferred orientation stimuli when the center contrast is low


Figure 9 Variation in the response Rc of a center hypercolumn at its preferredorientation as a function of (log) contrast Cc for collinear high-contrast surround(solid curve) and no surround (dashed curve) Parameters in the amplitudeequations 411 and 412 are Dm D 01 A D 1 and Ows D 08 The contrast-dependent coupling is taken to be of the form b D (05 iexcl Cc)

but suppresses responses when it is high Both effects are strongest when thecenter and surround stimuli are at the same orientation It has recently beenshown that inclusion of some form of contrast-related asymmetry betweenlocal excitatory and inhibitory neurons is sufcient to account for the switchbetween low-contrast facilitation and high-contrast suppression (Somers etal 1998)

One way to model the thresholding properties of the lateral interac-tions is to distinguish between the differing classes of inhibitory inter-neurons For example certain types of local interneuron are well placedto provide a source of feedforward inhibition from lateral connections asillustrated in Figure 1 This feedforward inhibition will have its own thresh-old and gain that will be determined by the contrast of the LGN inputs andthere will thus be a contrast-dependent disynaptic inhibition arising fromthe lateral connections We incorporate such an effect into our cortical modelby including a contrast-dependent contribution to the lateral inputs of theexcitatory population in equation 24 bE bE iexcl bcurren where bcurren is contrastdependent In the case of high contrasts we expect bcurren to be sufciently largeso that the effective coupling b lt 0 This has been assumed in our analysisof high-contrast center-surround stimulation Now suppose that we con-sider the response of the center in the presence of a high-contrast surroundand varying center contrast For the sake of illustration suppose that bcurren

varies linearly with the contrast The response of the center at its preferredorientation Rc is plotted as a function of contrast in Figure 9 both withand without a collinear surround It can be seen that there is facilitation at


low contrasts and suppression at high contrasts which is consistent withexperimental data (Polat et al 1998)

43 Anisotropy in Surround Suppression So far we have considereda surround conguration in which the the anisotropic nature of the lateralconnections is effectively averaged away This is no longer the case if onlya subregion of the surround is stimulated For concreteness suppose thatonly a single hypercolumn in the surround is active and take its locationrelative to the center to be r D r (cosh sinh ) (see Figure 7b) Since the effectsof feedback from the center hypercolumn to the surround cannot now beignored we have to solve the pair of equations

dzc (t )dt

D zc (t ) (Dm iexcl A|zc (t ) |2) C Ceiexcl2iWc

C b Ow (r)hzs C NzsP

(2)0 eiexcl4ih C sP(1)

0 eiexcl2ihi

(416)

dzs (t )dt

D zs (t )(Dm iexcl A|zs (t ) |2) C Ceiexcl2iWs

C b Ow (r)hzc C NzcP


0 eiexcl2ihi

(417)

where for simplicity the contrast of the center and surround stimuli is takento be equal It is clear that if the lateral connections are isotropic then p0(w ) D1 for all w 2 [iexclp 2 p 2) so that P(2)

0 D 0 D P(1)0 and the effective interaction

between the center and surround is independent of their relative angularlocation h However if P(1)

0 and P(2)0 are nonzero then there will be some

dependence on the relative angle h which is a signature of the anisotropyof the lateral connections

For the sake of illustration suppose that Wc D Ws D 0 (iso-oriented centerand surround) Then by symmetry a xed-point solution exists for whichzc D zs D Zeiexcl2iw where

Z(Dm iexcl AZ2) C C cos(2w ) C b Ow(r)Z

poundh1 C P(2)

0 cos(4[w iexclh ]) C sP(1)0 cos(2[w iexclh])

iD 0 (418)

C sin(2w ) C b Ow(r)hP(2)

0 sin(4[w iexclh]) C sP(1)0 sin(2[w iexclh])

iD 0 (419)

Some solutions of these equations are plotted in Figure 10 with dZc DZ(b ) cos(w )Z(0) It can be seen that the degree of suppression of the center(at high-contrast) depends on both the relative angular position of the sur-round hypercolumn and the degree of spread in the lateral connections ascharacterized by the parameters P(1)

0 and P(2)0 For certain parameter values


Figure 10 Variation in the degree of suppression of a center hypercolumn asa function of the relative angular position h of a surround hypercolumn Bothcenter and surround are stimulated by a high-contrast horizontal bar such thath D 0plusmn corresponds to a collinear or end-ank conguration and h D 90plusmn corre-sponds to an orthogonal or side-ank conguration as shown The two curvesdiffer in the choice of spread parameters (i) sP (1)

0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)

0 D 04 Other parameters in equations 416 and 417 are m D 01A D 1 C D 10 and |b |w(r) D 05

(see curve ii in Figure 10) the suppressive effect of the surround hypercol-umn is maximal when located close to the end-ank position and minimalwhen located around the side-ank position However it is also possiblefor the suppressive effect to be minimal at some oblique conguration (seecurve i in Figure 10)

It is interesting that recent experimental data concerning the spatial orga-nization of surrounds in primary visual cortex of the cat imply that in manycases suppression originates from a localized region in the surround ratherthan being distributed uniformly in the region encircling the centerrsquos classi-cal receptive eld that is the surrounds are spatially asymmetric (WalkerOhzawa amp Freeman 1999) The suppressive portion of the surround canarise at any location with a bias toward the ends of the centerrsquos classicalreceptive Our analysis suggests that the effect of the surround depends ona subtle combination of the spread in the anisotropic lateral connections andthe spatial organization of the surround

5 Discussion

The mathematical model introduced here assumes that V1 dynamics is closeto threshold so that even weakly modulated signals from the LGN can trig-ger a tuned response One way to achieve this is if there is a balance betweenintracortical excitation and inhibition This and related ideas have been ex-plored in a number of recent studies of the roleof intracortical interactions onthe observed properties of cortical neurons (Douglas et al 1995 Carandiniet al 1997 Tsodyks amp Sejnowski 1995 Chance amp Abbott 2000 WielaardShelley McLaughlin amp Shapley 2001)


51 Balanced Excitation and Inhibition As discussed by Tsodyks andSejnowski (1995) when a network with balanced excitation and inhibitionis close to threshold several properties obtain (1) Fluctuations about thethreshold are large and remain so even in the continuum limit (2) Thenetwork switches discontinuously from threshold to a tuned excited stateindicating theexistenceof a subcritical bifurcation (3) Critical slowing downoccurs that is the network state changes more slowly from threshold totuned the closer the stimulus is to threshold (see also Cowan amp Ermentrout1978)mdashgiven realistic neural parameters it takes about 50 msec to reach thepeak of the tuned state However (4) if the input orientation bias is switchedinstantaneously to a new value the net switches to thenew tuned state in lessthan 50 msec particularly if the inhibition in the network is faster than theexcitation Finally (5) the orientation-tuned response is contrast invariantMost if not all of these properties are found in the mathematical modeldescribed above

On a more technical level we note that one consequence of having bal-anced excitation and inhibition is that to have any effect on the bifurcationprocess whereby new orientation tuned stable states emerge external stim-uli for example those from the LGN which we labeled as hl (rj w ) must beof the form

hl (rj w ) D glCjf1 iexcl 2 32 C 2 32 cos(2[w iexcl Wj])g

where gl measures the relative strengths of LGN inputs to excitatory andinhibitory populations in V1 and Cj is the stimulus contrast at the jth hyper-column Thus the nonoriented component (1 iexcl 2 32)glCj is 0(1) and couplesto the steady-state equations whereas the orientation-tuned component2 32glCj cos(2[w iexcl Wj]) is 0(2 32) and couples to the cubic amplitude equa-tions It follows that if V1 is not sitting close to a bifurcation point so thatmuch stronger modulations are needed to drive it to an orientation-tunedstate then a different mathematical analysis is needed In this respect itis clear that the noise uctuations described above can play an importantrole in keeping the network dynamics close to the bifurcation point Asrecently noted by Anderson Lampl Gillespie and Ferster (2000) noiseuctuations can also secure contrast invariance in the tuned response of apopulation of spiking neurons which is a property we found in the uncou-pled hypercolumn model (Bressloff et al 2000) using sigmoid populationequations that effectively take account of effects produced by stochasticresonance

52 Divisive Inhibition Another well-known set of studies concernsthe use of recurrent inhibition to model the nearly linear behavior of sim-ple cortical cells Thus Carandini et al (1997) have explored the use ofshunting or divisive inhibition provided by a pool of inhibitory neuronsto linearize neural responses and produce contrast invariance and satura-


tion properties consistent with observations As Majaj Smith and Movshon(2000) recently showed many contextual effects can be modeled withinthis framework if the inhibition acts to normalize the neural responseThus crossndashorientation suppression within the classical receptive eld iseasily obtained In general many suppression experiments can be mod-eled with such a mechanism but it is more difcult to account for effectssuch as crossndashorientation facilitation by stimuli in the nonclassical receptiveeld

The idea of using division rather than subtraction to model the effectsof inhibition has been around for a long time One early mathematical usewas by Grossberg (1973) who developed population equations similar tothose of Wilson and Cowan (1972 1973) but using divisive inhibition Animmediate problem is then to nd a biophysical basis for such a mechanismRecent work by Wielaard et al (2001) has provided a natural solution If oneuses conductance-based models of the evolution of changes in membranepotentials in a network of integratendashandndashre neurons with recurrent exci-tation and inhibition and if the conductances are sufciently large then thesteady-state voltages reached depend on both differences between excita-tion and inhibition and also on ratios of conductances in such a way as toimplement divisive inhibition One can then use such conductance modelsto see how linear simple cell properties can emerge from network inter-actions It is the balance between excitation and inhibition that produceseffective linearity

A similar but less biophysically directmodel was also recently introducedby Chance and Abbott (2000) However they went further in devising a cir-cuit to maintain rapid switching in the presence of critical slowing downby using a three-population model comprising one excitatory and two (di-visive) inhibitory populations This circuit is more elaborate than that ofTsodyks and Sejnowski (1995) and in fact is very close to the three-neuroncircuit we have studied in this article (see Figure 1) except that we have usedsubtractive inhibition But the mathematical analysis carried out in this ar-ticle proceeds by rst linearizing the basic equations about an equilibriumstate and then carrying out various perturbation expansions It follows thateither subtractive or divisive inhibition will lead to very similar equationsand conclusions The only difference lies in the particular parameter valuesneeded to obtain the various behaviors Thus our results are qualitativelysimilar to what can be expected from a three-neuron circuit employing di-visive rather than subtractive inhibition particularly as we deal only withsteady-state behaviors in this article

Given the framework described above we note again that the analysisoutlined in this article provides an account of both suppression and facili-tation in a single model We expect that the equivalent divisive inhibitorymodel will also exhibit such properties at least if stationary tuning curvesare analyzed It remains to be seen what differences if any exist in the caseof timendashdependent responses


53 Contrast Dependence One other benet is provided by the three-neuron circuit described here In various parts of the article we have im-posed contrast dependence of the lateral connectivity in a somewhat adhoc manner simply by assuming the effective coupling coefcient b of thelateral connections between hypercolumns to be of the form

b D bE iexcl bcurren (C)

where bE is the excitatory coupling between hypercolumns and C is thestimulus contrast Evidently if bcurren (C) increases monotonically with contrastthere exists a contrast threshold at which b lt 0 One relatively straight-forward way to achieve this is to suppose that basket cells which receiveabout 20 of the signal from excitatory lateral connections (McGuire etal 1991) have relatively high thresholds and are therefore not activatedby low-contrast stimuli In effect this implies that the effective excitatorypart of the aggregate hypercolumn receptive eld is larger at low contrast(Movshon pers comm 2001)

Acknowledgments

We thank Trevor Mundel for many helpful discussions This work was sup-ported in part by grant 96-24 from the James S McDonnell Foundation toJ D C The researchof PC Bwas supported by a grant from theLeverhulmeTrust P C B wishes to thank the Mathematics Department University ofChicago for its hospitality and support P C B and J D C also thank Geof-frey Hinton and the Gatsby Computational Neurosciences Unit UniversityCollege London for hospitality and support

References

Anderson S Lampl I Gillespie D C amp Ferster D (2000) The contribution ofnoise to contrast invariance of orientation tuning in cat visual cortex Science290 (5498) 1968ndash1972

Aronson D G Ermentrout G B amp Kopell N (1990) Amplitude response ofcoupled oscillators Physica D 41 403ndash449

Ben-Yishai R Bar-Or R L amp Sompolinsky H (1995) Theory of orientationtuning in visual cortex Proc Nat Acad Sci 92 3844ndash3848

Ben-Yishai R Hansel D amp Sompolinsky H (1997) Traveling waves and theprocessing of weakly tuned inputs in a cortical network module J ComputNeurosci 4 57ndash77

Blakemore C amp Tobin E (1972) Lateral inhibition between orientation detec-tors in the catrsquos visual cortex Exp Brain Res 15 439ndash440

Blasdel G G (1992) Orientation selectivity preference and continuity in mon-key striate cortex J Neurosci 12 3139ndash3161

Blasdel G G amp Salama G (1986) Voltage-sensitive dyes reveal a modularorganization in monkey striate cortex Nature 321 579ndash585


Bonhoeffer T Kim D Malonek D Shoham D amp Grinvald A (1995) Opticalimaging of the layout of functional domains in area 1718 border in cat visualcortex European J Neurosci 7(9) 1973ndash1988

Bosking W H Zhang Y Schoeld B amp Fitzpatrick D (1997) Orientationselectivity and the arrangementof horizontal connections in tree shrew striatecortex J Neurosci 17 2112ndash2127

Bressloff P C Bressloff N W amp Cowan J D (2000) Dynamical mechanismfor sharp orientation tuning in an integratendashandndashre model of a cortical hy-percolumn Neural Comput 12 2473ndash2511

Bressloff P C Cowan J D Golubitsky M Thomas P J amp Wiener M (2001a)Geometric visual hallucinations Euclidean symmetry and the functional ar-chitecture of striate cortex Phil Trans Roy Soc Lond B 356 299ndash330

Bressloff P C Cowan J D Golubitsky M Thomas P J amp Wiener M (2001b)What geometric visual hallucinations tell us about the visual cortex NeuralComput 13 1ndash19

Carandini M Heeger D J amp Movshon J A (1997) Linearity and normaliza-tion in simple cells of the macaque primary visual cortex J Neurosci 17(21)8621ndash8644

Chance F S amp Abbott L F (2000) Divisive inhibition in recurrent networksNetwork Comp Neural Sys 11 119ndash129

Cowan J D (1997) Neurodynamics and brain mechanisms In M ItoY Miyashita amp E T Rolls (Eds) Cognition computation and consciousness(pp 205ndash233) Oxford Oxford University Press

Cowan J D amp Ermentrout G B (1978) Some aspects of the ldquoeigenbehavior rdquo ofneural nets In S Levin (Ed) Studies in mathematical biology 15 (pp 67ndash117)Providence RI Mathematical Association of America

DeAngelis G Robson J Ohzawa I amp Freeman R (1992) Organization ofsuppression in receptive fields of neurons in cat visual cortex J Neurophysiol68(1) 144ndash163

Douglas R Koch C Mahowald M Martin K amp Suarez H (1995) Recurrentexcitation in neocortical circuits Science 269 981ndash985

Dragoi V amp Sur M (2000) Some properties of recurrent inhibition in primaryvisual cortex Contrast and orientation dependence on contextual effects JNeurophysiol 83 1019ndash1030

Ermentrout G B (1998) Neural networks as spatial pattern forming systemsRep Prog Phys 61 353ndash430

Ermentrout G B amp Cowan J D (1979) A mathematical theory of visual hal-lucination patterns Biol Cybernetics 34 137ndash150

Ferster D Chung S amp Wheat H (1997) Orientation selectivity of thalamicinput to simple cells of cat visual cortex Nature 380 249ndash281

Fitzpatrick D (2000) Seeing beyond the receptive eld in primary visual cortexCurr Op in Neurobiol 10 438ndash443

Fitzpatrick D Zhang Y Schoeld B amp Muly M (1993) Orientation selectivityand the topographic organization of horizontal connections in striate cortexSoc Neurosci Abstracts 19 424

Gilbert C Das A Ito M Kapadia M amp Westheimer G (1996) Spatial inte-gration and cortical dynamics Proc Nat Acad Sci 93 615ndash622


Gilbert C amp Wiesel T (1983) Clustered intrinsic connections in cat visualcortex J Neurosci 3 1116ndash1133

Gilbert C amp Wiesel T (1989) Columnar specicity of intrinsic horizontal andcorticocortical connections in cat visual cortex J Neurosci 9 2432ndash2442

Gray C M Konig P Engel A K amp Singer W (1989) Oscillatory responsesin cat visual cortex exhibit interndashcolumnar synchronization which reectsglobal stimulus properties Nature 338 334ndash337

Grinvald A Lieke E Frostig R amp Hildesheim R (1994) Cortical pointndashspread function and longndashrange lateral interactions revealed by realndashtimeoptical imaging of macaque monkey primary visual cortex J Neurosci 142545ndash2568

Grossberg S (1973) Contour enhancement short term memory and constan-cies in reverberating neural networks Studies in Applied Mathematics 52(3)213ndash257

Grossberg S amp Raizada R D S (2000) Contrast-sensitive perceptual groupingand objectndashbased attention in the laminar circuits of primary visual cortexVision Res 40 1413ndash1432

Hata Y Tsumoto T Sato H Hagihara K amp Tamura H (1988) Inhibition con-tributes to orientation selectivity in visual cortex of cat Nature 335 815ndash817

Hirsch J amp Gilbert C (1992) Synaptic physiology of horizontal connectionsin the catrsquos visual cortex J Physiol Lond 160 106ndash154

Hoppensteadt F amp Izhikevich E (1997) Weakly connected neural networks NewYork Springer

Hubel D amp Wiesel T (1962) Receptive elds binocular interaction and func-tional architecture in the catrsquos visual cortex J Neurosci 3 1116ndash1133

Jagadeesh B amp Ferster D (1990) Receptive eld lengths in cat striate cortex canincrease with decreasing stimulus contrast Soc Neurosci Abstr 16 13011

Kath W L (1981) Resonance in periodically perturbed Hopf bifurcation StudAppl Math 65 95ndash112

Levitt J B amp Lund J (1997) Contrast dependence of contextual effects in pri-mate visual cortex Nature 387 73ndash76

Li C amp Li W (1994) Extensive integration eld beyond the classical receptiveeld of catrsquos striate cortical neuronsmdashclassication and tuning propertiesVision Res 34(18) 2337ndash2355

Li Z (1999) Pre-attentive segmentation in the primary visual cortex SpatialVision 13 25ndash39

Majaj N Smith M A amp Movshon J A (2000) Contrast gain control inmacaque area MT Soc Neuroci Abstr

Malach R Amir Y Harel M amp Grinvald A (1993) Relationship between in-trinsic connections and functional architecture revealed by optical imagingand in vivo targeted biocytin injections in primate striate cortex Proc NatlAcad Sci 90 10469ndash10473

McGuire B Gilbert C Rivlin P amp Wiesel T (1991) Targets of horizontalconnections in macaque primary visual cortex J Comp Neurol 305 370ndash392

McLaughlin D Shapley R Shelley M amp Wielaard D J (2000) A neuronalnetwork model of macaque primary visual cortex (V1) Orientation tuningand dynamics in the input layer 4Ca Proc Natl Acad Sci 97 8087ndash8092


Michalski A Gerstein G Czarkowska J amp Tarnecki R (1983) Interactionsbetween cat striate cortex neurons Exp Brain Res 53 97ndash107

Mundel T (1996) A theory of cortical edge detection Unpublished doctoral dis-sertation University of Chicago

Mundel T Dimitrov A amp Cowan J D (1997) Visual cortex circuitry and orien-tation tuning InM C Mozer M I Jordanamp T Petsche (Eds)Advances inneu-ral information processing systems 9 (pp 886ndash893) Cambridge MA MIT Press

Obermayer K amp Blasdel G (1993) Geometry of orientation and ocular dom-inance columns in monkey striate cortex J Neurosci 13 4114ndash4129

Phleger B amp Bonds A B (1995) Dynamic differentiation of GABAAndashsensitiveinuences on orientation selectivity of complex cells in the cat striate cortexExp Brain Res 104 81ndash88

Polat U Mizobe K Pettet M W Kasamatsu T amp Norcia A M (1998)Collinear stimuli regulate visual responses depending on a cellrsquos contrastthreshold Nature 391 580ndash584

Pugh M C Ringach D L Shapley R amp Shelley M J (2000) Computationalmodeling of orientation tuning dynamics in monkey primary visual cortexJ Comput Neurosci 8 143ndash159

Rockland K S amp Lund J (1983) Intrinsic laminar lattice connections in primatevisual cortex J Comp Neurol 216 303ndash318

Sceniak M P Ringach D L Hawken M J amp Shapley R (1999) Contrastrsquoseffect on spatial summation by macaque V1 neurons Nature Neurosci 2733ndash739

Sillito A M (1975) The contribution of inhibitory mechanisms to the receptivendasheld properties of neurones in the striate cortex of the cat J Physiol Lond250 305ndash329

Sillito A M Grieve K L Jones H E Cudeiro J amp Davis J (1995) Visualcortical mechanisms detecting focal orientation discontinuities Nature 378492ndash496

Singer W amp Gray C M (1995) Visual feature integration and the temporalcorrelation hypothesis Ann Rev of Neurosci 18 555ndash586

Somers D Nelson S amp Sur M (1994) Effects of longndashrange connections ongain control in an emergent model of visual cortical orientation selectivitySoc Neurosci Abstr 20 6467

Somers D Nelson S amp Sur M (1995) An emergent model of orientationselectivity in cat visual cortical simple cells J Neurosci 15 5448ndash5465

Somers D Todorov E V Siapas A G Toth L J Kim D-S amp Sur M (1998)A local circuit approach to understanding integration of longndashrange inputsin primary visual cortex Cerebral Cortex 8 204ndash217

Stetter M Bartsch H amp Obermayer K (2000) A mean eld model for orienta-tion tuning contrast saturation and contextual effects in the primary visualcortex Biol Cybernetics 82 291ndash304

Toth L J Rao S C Kim D Somers D amp Sur M (1996) Subthreshold facil-itation and suppression in primary visual cortex revealed by intrinsic signalimaging Proc Natl Acad Sci 93 9869ndash9874

Tsodyks M amp Sejnowski T (1995) Rapid state switching in balanced corticalnetwork models Network Comp Neural Sys 6 1ndash14


Vidyasagar T R Pei X amp Volgushev M (1996) Multiple mechanisms underly-ing the orientation selectivity of visual cortical neurons TINS 19(7) 272ndash277

Walker G A Ohzawa I amp Freeman R D (1999) Asymmetric suppressionoutside the classical receptive eld of the visual cortex J Neurosci 19 10536ndash10553

Wenderoth P amp Johnstone S (1988) The different mechanisms of the directand indirect tilt illusions Vision Res 28 301ndash312

Wielaard D Shelley M McLaughlin D amp Shapley R (2001) How simplecells are made in a nonlinear network model of the visual cortex J Neurosci21 5203ndash5211

Wiener M C (1994) Hallucinations symmetry and the structure of primary vi-sual cortex A bifurcation theory approach Unpublished doctoral dissertationUniversity of Chicago

Wilson H R amp Cowan J D (1972) Excitatory and inhibitory interactions inlocalized populations of model neurons Biophys J 12 1ndash24

Wilson H R amp Cowan J D (1973) A mathematical theory of the functionaldynamics of cortical and thalamic nervous tissue Kybernetik 13 55ndash80

Yoshioka T Blasdel G G Levitt J B amp Lund J (1996) Relation betweenpatterns of intrinsic lateral connectivity ocular dominance and cytochromeoxidasemdashreactive regions in macaque monkey striate cortex Cerebral Cortex6 297ndash310

Received January 10 2001 accepted May 25 2001


connecting cells with similar orientation preferences separated by up to sev-eral millimeters of cortical tissue (Gilbert amp Wiesel 1989 Hirsch amp Gilbert1992 Malach Amir Harel amp Grinvald 1993 Fitzpatrick Zhang Schoeldamp Muly 1993 Grinvald Lieke Frostig amp Hildesheim 1994 Yoshioka Blas-del Levitt amp Lund 1996) The lateral connections that mediate this circuitarise almost exclusively from excitatory neurons (Rockland amp Lund 1983Gilbert amp Wiesel 1983) although 20 terminate on inhibitory cells and canthus have signicant inhibitory effects (McGuire Gilbert Rivlin amp Wiesel1991)

Stimulation of a hypercolumn via lateral connections modulates ratherthan initiates spiking activity (Hirsch amp Gilbert 1992 Toth Rao Kim Som-ers amp Sur 1996) Thus the lateral connectivity is ideally structured to pro-vide local cortical processes with contextual information about the globalnature of stimuli As a consequence these lateral connections have beeninvoked to explain a wide variety of context-dependent visual processingphenomena (Gilbert Das Ito Kapadia amp Westheimer 1996 Fitzpatrick2000) (Interestingly contextual processing also occurs in extrastriate visualareas and feedback from these areas to V1 has been invoked to explain as-pects of contextual processingWenderoth amp Johnstone 1988) One commonexperimental paradigm is to investigate the response to stimuli consisting ofcircular (center) and annular (surround) gratings of differing contrasts ori-entations and diameters When both center and surround are stimulatedwith strong suprathreshold inputs one typically nds that the surroundsuppresses the center unitrsquos tuning response for surround stimulation closeto the orientation of the unitrsquos peak tuning response and facilitates the re-sponse when the surround is stimulated at orientations sufciently dissim-ilar to the preferred orientation of the center (Blakemore amp Tobin 1972 Liamp Li 1994 Sillito Grieve Jones Cudeiro amp Davis 1995)

The center response to surround stimulation depends signicantly onthe contrast of center stimulation (Toth et al 1996 Levitt amp Lund 1997Polat Mizobe Pettet Kasamatsu amp Norcia 1998) For example a xedsurround stimulus tends to facilitate responses to stimuli at a preferred ori-entation when the center contrast is low but suppresses responses whenit is high Both effects are strongest when center and surround stimuliare iso-oriented The response to variations in the size of a stimulus isalso contrast dependent For example responses to a high-contrast circu-lar grating decline beyond a characteristic preferred stimulus size due toactivation of an inhibitory surround In addition the effective size of theexcitatory receptive eld tends to increase with decreasing contrast andat low contrasts becomes a monotonically increasing function of stimu-lus size (Jagadeesh amp Ferster 1990 Sceniak Ringach Hawken amp Shapley1999)

One common approach to modeling the role of lateral connections incenter-surround modulation is to consider a reduced local cortical circuitcomposed of excitatory and inhibitory populations receiving feedforward













fal (ri w t)


CX

mDEI

NX

jD1

Z p 2


dw 0

p (21)


sl (x) D1











al (ri w t)t


mDEI

Z p 2


dw 0


X

j 6Di

Z p2


dw 0

p(24)








aE


aI





dw 0

p (33)









where Wlm (0) DR p2



bE

tD iexclbE C

hs 0


i(36)

bI

tD iexclbI C

pounds 0


curren (37)




i (38)


(1 C l)B D

sup3s 0



I (NaI)WII (n)

acuteB (39)



nD1





n (311)


for integer n where

Wsectn D

12



S (n) Dp



Bsectn D

sup3WEI (n)


acute (314)





1


curren where





Wlm (n) Dp







1 gt ReWCn















bl

tD iexclbl C

X

mDEI


m C c 0mb3

m C i

C Dhl


cent(316)




j2J j 6Di

Ow(rij)Z p 2


dw 0

2p 0 (317)





bm D 2 12b(1)m C 2 b (2)



[ L b(1)]l D 0 (319)

[ L b(2)]l D v(2)l (320)

acuteX



[ L b(3)]l D v(3)l

acute iexclb (1)

l

tC

X

mDEI

wlm currenhDm b (1)

m C c 0m[b(1)

m ]3 C 2c mb(1)m b(2)

m

i


E Acircsup2

(321)



X

mDEI

wlm curren bm (322)





B (323)



eB D

sup3WIE (1)


acute(324)

so that


X

mDEI

wml curren ebm D 0



hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)



bl NsEP(1)0 P(1)

1

X


where

P(n)k D

Z p 2


dw

p

(327)





z(ri t )t


j2JOw (rij)



i (328)


2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p


dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References
























































































fal (ri w t)


CX

mDEI

NX

jD1

Z p 2


dw 0

p (21)


sl (x) D1











al (ri w t)t


mDEI

Z p 2


dw 0


X

j 6Di

Z p2


dw 0

p(24)








aE


aI





dw 0

p (33)









where Wlm (0) DR p2



bE

tD iexclbE C

hs 0


i(36)

bI

tD iexclbI C

pounds 0


curren (37)




i (38)


(1 C l)B D

sup3s 0



I (NaI)WII (n)

acuteB (39)



nD1





n (311)


for integer n where

Wsectn D

12



S (n) Dp



Bsectn D

sup3WEI (n)


acute (314)





1


curren where





Wlm (n) Dp







1 gt ReWCn















bl

tD iexclbl C

X

mDEI


m C c 0mb3

m C i

C Dhl


cent(316)




j2J j 6Di

Ow(rij)Z p 2


dw 0

2p 0 (317)





bm D 2 12b(1)m C 2 b (2)



[ L b(1)]l D 0 (319)

[ L b(2)]l D v(2)l (320)

acuteX



[ L b(3)]l D v(3)l

acute iexclb (1)

l

tC

X

mDEI

wlm currenhDm b (1)

m C c 0m[b(1)

m ]3 C 2c mb(1)m b(2)

m

i


E Acircsup2

(321)



X

mDEI

wlm curren bm (322)





B (323)



eB D

sup3WIE (1)


acute(324)

so that


X

mDEI

wml curren ebm D 0



hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)



bl NsEP(1)0 P(1)

1

X


where

P(n)k D

Z p 2


dw

p

(327)





z(ri t )t


j2JOw (rij)



i (328)


2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p


dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References




















































































al (ri w t)t


mDEI

Z p 2


dw 0


X

j 6Di

Z p2


dw 0

p(24)








aE


aI





dw 0

p (33)









where Wlm (0) DR p2



bE

tD iexclbE C

hs 0


i(36)

bI

tD iexclbI C

pounds 0


curren (37)




i (38)


(1 C l)B D

sup3s 0



I (NaI)WII (n)

acuteB (39)



nD1





n (311)


for integer n where

Wsectn D

12



S (n) Dp



Bsectn D

sup3WEI (n)


acute (314)





1


curren where





Wlm (n) Dp







1 gt ReWCn















bl

tD iexclbl C

X

mDEI


m C c 0mb3

m C i

C Dhl


cent(316)




j2J j 6Di

Ow(rij)Z p 2


dw 0

2p 0 (317)





bm D 2 12b(1)m C 2 b (2)



[ L b(1)]l D 0 (319)

[ L b(2)]l D v(2)l (320)

acuteX



[ L b(3)]l D v(3)l

acute iexclb (1)

l

tC

X

mDEI

wlm currenhDm b (1)

m C c 0m[b(1)

m ]3 C 2c mb(1)m b(2)

m

i


E Acircsup2

(321)



X

mDEI

wlm curren bm (322)





B (323)



eB D

sup3WIE (1)


acute(324)

so that


X

mDEI

wml curren ebm D 0



hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)



bl NsEP(1)0 P(1)

1

X


where

P(n)k D

Z p 2


dw

p

(327)





z(ri t )t


j2JOw (rij)



i (328)


2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p


dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References



















































































where Wlm (0) DR p2



bE

tD iexclbE C

hs 0


i(36)

bI

tD iexclbI C

pounds 0


curren (37)




i (38)


(1 C l)B D

sup3s 0



I (NaI)WII (n)

acuteB (39)



nD1





n (311)


for integer n where

Wsectn D

12



S (n) Dp



Bsectn D

sup3WEI (n)


acute (314)





1


curren where





Wlm (n) Dp







1 gt ReWCn















bl

tD iexclbl C

X

mDEI


m C c 0mb3

m C i

C Dhl


cent(316)




j2J j 6Di

Ow(rij)Z p 2


dw 0

2p 0 (317)





bm D 2 12b(1)m C 2 b (2)



[ L b(1)]l D 0 (319)

[ L b(2)]l D v(2)l (320)

acuteX



[ L b(3)]l D v(3)l

acute iexclb (1)

l

tC

X

mDEI

wlm currenhDm b (1)

m C c 0m[b(1)

m ]3 C 2c mb(1)m b(2)

m

i


E Acircsup2

(321)



X

mDEI

wlm curren bm (322)





B (323)



eB D

sup3WIE (1)


acute(324)

so that


X

mDEI

wml curren ebm D 0



hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)



bl NsEP(1)0 P(1)

1

X


where

P(n)k D

Z p 2


dw

p

(327)





z(ri t )t


j2JOw (rij)



i (328)


2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p


dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References













































































for integer n where

Wsectn D

12



S (n) Dp



Bsectn D

sup3WEI (n)


acute (314)





1


curren where





Wlm (n) Dp







1 gt ReWCn















bl

tD iexclbl C

X

mDEI


m C c 0mb3

m C i

C Dhl


cent(316)




j2J j 6Di

Ow(rij)Z p 2


dw 0

2p 0 (317)





bm D 2 12b(1)m C 2 b (2)



[ L b(1)]l D 0 (319)

[ L b(2)]l D v(2)l (320)

acuteX



[ L b(3)]l D v(3)l

acute iexclb (1)

l

tC

X

mDEI

wlm currenhDm b (1)

m C c 0m[b(1)

m ]3 C 2c mb(1)m b(2)

m

i


E Acircsup2

(321)



X

mDEI

wlm curren bm (322)





B (323)



eB D

sup3WIE (1)


acute(324)

so that


X

mDEI

wml curren ebm D 0



hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)



bl NsEP(1)0 P(1)

1

X


where

P(n)k D

Z p 2


dw

p

(327)





z(ri t )t


j2JOw (rij)



i (328)


2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p


dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References

















































































1 gt ReWCn















bl

tD iexclbl C

X

mDEI


m C c 0mb3

m C i

C Dhl


cent(316)




j2J j 6Di

Ow(rij)Z p 2


dw 0

2p 0 (317)





bm D 2 12b(1)m C 2 b (2)



[ L b(1)]l D 0 (319)

[ L b(2)]l D v(2)l (320)

acuteX



[ L b(3)]l D v(3)l

acute iexclb (1)

l

tC

X

mDEI

wlm currenhDm b (1)

m C c 0m[b(1)

m ]3 C 2c mb(1)m b(2)

m

i


E Acircsup2

(321)



X

mDEI

wlm curren bm (322)





B (323)



eB D

sup3WIE (1)


acute(324)

so that


X

mDEI

wml curren ebm D 0



hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)



bl NsEP(1)0 P(1)

1

X


where

P(n)k D

Z p 2


dw

p

(327)





z(ri t )t


j2JOw (rij)



i (328)


2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p


dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References





















































































bl

tD iexclbl C

X

mDEI


m C c 0mb3

m C i

C Dhl


cent(316)




j2J j 6Di

Ow(rij)Z p 2


dw 0

2p 0 (317)





bm D 2 12b(1)m C 2 b (2)



[ L b(1)]l D 0 (319)

[ L b(2)]l D v(2)l (320)

acuteX



[ L b(3)]l D v(3)l

acute iexclb (1)

l

tC

X

mDEI

wlm currenhDm b (1)

m C c 0m[b(1)

m ]3 C 2c mb(1)m b(2)

m

i


E Acircsup2

(321)



X

mDEI

wlm curren bm (322)





B (323)



eB D

sup3WIE (1)


acute(324)

so that


X

mDEI

wml curren ebm D 0



hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)



bl NsEP(1)0 P(1)

1

X


where

P(n)k D

Z p 2


dw

p

(327)





z(ri t )t


j2JOw (rij)



i (328)


2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p


dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References














































































bm D 2 12b(1)m C 2 b (2)



[ L b(1)]l D 0 (319)

[ L b(2)]l D v(2)l (320)

acuteX



[ L b(3)]l D v(3)l

acute iexclb (1)

l

tC

X

mDEI

wlm currenhDm b (1)

m C c 0m[b(1)

m ]3 C 2c mb(1)m b(2)

m

i


E Acircsup2

(321)



X

mDEI

wlm curren bm (322)





B (323)



eB D

sup3WIE (1)


acute(324)

so that


X

mDEI

wml curren ebm D 0



hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)



bl NsEP(1)0 P(1)

1

X


where

P(n)k D

Z p 2


dw

p

(327)





z(ri t )t


j2JOw (rij)



i (328)


2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p


dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References














































































hu|vi DZ p

0[uE (w )vE (w ) C uI (w )vI (w )]

dw

p (325)



bl NsEP(1)0 P(1)

1

X


where

P(n)k D

Z p 2


dw

p

(327)





z(ri t )t


j2JOw (rij)



i (328)


2 m cBE

b D P(1)1

X

lDEI

Dlbl (329)

f (ri ) D m cX

lDEI

eBl

Z p


dw

p(330)


and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References













































































and

Dl Dm 2

c BE

eBTBeBl A D iexcl 3

eBTB

X

lDEI

eBlc0l B3

l (331)







iBC cc (332)


V0 D m c



hu | vi D limT1

1T

Z T2

iexclT2

Z p


dw

pdt (334)




zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References














































































zL (ri t )t


i

C fiexcl (ri) C bX

j2JOw(rij )


0 e4ihij

i(335)

and

zR (ri t )t


i

C fC (ri) C bX

j2JOw (rij)


0 eiexcl4ihij

i (336)

where

fsect (ri) D limT1

m c

T

Z T2

iexclT2

Z p


X

lDEI

eBlDhl (ri w t)dw

pdt (337)
















dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References





















































































dZi



dw i

dtD iexcl

Ci











zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References


















































































zL (ri t )t




and

zR (ri t )t



iC Cie2iWi (47)













dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References






















































































dzc (t )dt


C bws




L DP(2)

0

N

NX

jD1


0

NZs

NX

jD1

eiexcl2ihj (410)


dZc




dwc

dtD iexcl

Cc






c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References















































































c (b )






Zcurrenc (0)

(413)






dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References
















































































dZc




dZc


















dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References




















































































dzc (t )dt




0 eiexcl2ihi

(416)

dzs (t )dt




0 eiexcl2ihi

(417)








poundh1 C P(2)


iD 0 (418)



iD 0 (419)





0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References














































































0 D 01 P (2)0 D 06 and (ii)

sP(1)0 D 08 P(2)




5 Discussion

















Acknowledgments


References



























































































Acknowledgments


References












































































an amplitude equation approach to contextual effects in visual

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