link.springer.com · RESEARCH Open Access Coverage, continuity, and visual cortical architecture Wolfgang Keil1,2,3,4* and Fred Wolf1,2,3,4 Abstract Background: The primary visual

Coverage, continuity, and visual corticalarchitectureKeil and Wolf

Keil and Wolf Neural Systems & Circuits 2011, 1:17http://www.neuralsystemsandcircuits.com/content/1/1/17 (29 December 2011)

RESEARCH Open Access

Coverage, continuity, and visual corticalarchitectureWolfgang Keil1,2,3,4* and Fred Wolf1,2,3,4

Abstract

Background: The primary visual cortex of many mammals contains a continuous representation of visual space,with a roughly repetitive aperiodic map of orientation preferences superimposed. It was recently found thatorientation preference maps (OPMs) obey statistical laws which are apparently invariant among species widelyseparated in eutherian evolution. Here, we examine whether one of the most prominent models for theoptimization of cortical maps, the elastic net (EN) model, can reproduce this common design. The EN modelgenerates representations which optimally trade of stimulus space coverage and map continuity. While this modelhas been used in numerous studies, no analytical results about the precise layout of the predicted OPMs havebeen obtained so far.

Results: We present a mathematical approach to analytically calculate the cortical representations predicted by theEN model for the joint mapping of stimulus position and orientation. We find that in all the previously studiedregimes, predicted OPM layouts are perfectly periodic. An unbiased search through the EN parameter spaceidentifies a novel regime of aperiodic OPMs with pinwheel densities lower than found in experiments. In anextreme limit, aperiodic OPMs quantitatively resembling experimental observations emerge. Stabilization of theselayouts results from strong nonlocal interactions rather than from a coverage-continuity-compromise.

Conclusions: Our results demonstrate that optimization models for stimulus representations dominated bynonlocal suppressive interactions are in principle capable of correctly predicting the common OPM design. Theyquestion that visual cortical feature representations can be explained by a coverage-continuity-compromise.

IntroductionThe pattern of orientation columns in the primary visualcortex (V1) of carnivores, primates, and their close rela-tives are among the most intensely studied structures inthe cerebral cortex and a large body of experimental (e.g., [1-13]) and theoretical work (e.g., [14-39]) has beendedicated to uncovering its organization principles andthe circuit level mechanisms that underlie its develop-ment and operation. Orientation preference maps(OPMs) exhibit a roughly repetitive arrangement of pre-ferred orientations in which adjacent columns preferringthe same orientation are separated by a typical distancein the millimeter range [2-5,10]. This range seems to beset by cortical mechanisms both intrinsic to a particulararea [40] but potentially also involving interactions

between different cortical regions [41]. The pattern oforientation columns is however not strictly periodicbecause the precise local arrangement of preferredorientation never exactly repeats. Instead, OPMs appearas organized by a spatially complex aperiodic array ofpinwheel centers, around which columns activated bydifferent stimulus orientations are radially arranged likethe spokes of a wheel [2-5,10]. The arrangement ofthese pinwheel centers, although spatially irregular, isstatistically distinct from a pattern of randomly posi-tioned points [38] as well as from patterns of phase sin-gularities in a random pattern of preferred orientations[32,36,38,42] with spatial correlations identical to experi-mental observations [38,42]. This suggests that the lay-out of orientation columns and pinwheels althoughspatially aperiodic follows a definite system of layoutrules. Cortical columns can in principle exhibit almostperfectly repetitive order as exemplified by ocular domi-nance (OD) bands in the macaque monkey primary

* Correspondence: [email protected] for Dynamics and Self-organization, Am Fassberg 17, D-37077 Göttingen, GermanyFull list of author information is available at the end of the article

Keil and Wolf Neural Systems & Circuits 2011, 1:17http://www.neuralsystemsandcircuits.com/content/1/1/17

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visual cortex [43,44]. It is thus a fundamental questionfor understanding visual cortical architecture, whetherthere are layout principles that prohibit a spatiallyexactly periodic organization of orientation columns andinstead enforce complex arrangements of these columns.Recent comparative data have raised the urgency of

answering this question and of dissecting what is consti-tutive of such complex layout principles. Kaschube et al.[38] quantitatively compared pinwheel arrangements ina large dataset from three species widely separated inthe evolution of eutherian mammals. These authorsfound that the spatial statistics of pinwheels are surpris-ingly invariant. In particular, the overall pinwheel den-sity and the variability of pinwheel densities in regionsfrom the scale of a single hypercolumn to substantialfractions of the entire primary visual cortex were foundto be virtually identical. Characterizing pinwheel layouton the scale of individual hypercolumns, they found thedistributions of nearest-neighbor pinwheel distances tobe almost indistinguishable. Further supporting commonlayout rules for orientation columns in carnivores andprimates, the spatial configuration of the superficialpatch system [45] and the responses to drifting gratingstimuli were recently found to be very similar in cat andmacaque monkey primary visual cortex [46].From an evolutionary perspective, the occurrence of

quantitatively similar layouts for OPMs in primate treeshrews and carnivorous species appears highly informa-tive. The evolutionary lineages of these taxa divergedmore than 65 million years ago during the basal radia-tion of eutherian mammals [47-49]. According to thefossil record and cladistic reconstructions, their lastcommon ancestors (called the boreo-eutherial ancestors)were small-brained, nocturnal, squirrel-like animals ofreduced visual abilities with a telencephalon containingonly a minor neocortical fraction [47,50]. For instance,endocast analysis of a representative stem eutherianfrom the late cretaceous indicates a total anterior-pos-terior extent of 4 mm for its entire neocortex [47,50].Similarly, the tenrec (Echinops telfari), one of the closestliving relatives of the boreoeutherian ancestor [51,52],has a neocortex of essentially the same size and a visualcortex that totals only 2 mm2 [47]. Since the neocortexof early mammals was subdivided into several corticalareas [47] and orientation hypercolumns measurebetween 0.4 and 1.4 mm2 [38], it is difficult to envisionancestral eutherians with a system of orientation col-umns. In fact, no extant mammal with a visual cortex ofsuch size is known to possess orientation columns [53].It is therefore conceivable that systems of orientationcolumns independently evolved in laurasiatheria (suchas carnivores) and in euarchonta (such as tree shrewsand primates). Because galagos, tree-shrews, and ferretsstrongly differ in habitat and ecologically relevant visual

behaviors, it is not obvious that the quantitative similar-ity of pinwheel layout rules in their lineages evolved dri-ven by specific functional selection pressures (see [54]for an extended discussion). Kaschube et al. insteaddemonstrated that an independent emergence of identi-cal layout rules for pinwheels and orientation columnscan be explained by mathematically universal propertiesof a wide class of models for neural circuit self-organization.According to the self-organization scenario, the com-

mon design would result from developmental con-straints robustly imposed by adopting a particular kindof self-organization mechanism for constructing visualcortical circuitry. Even if this scenario is correct, onequestion still remains: What drove the different lineagesto adopt a similar self-organization mechanism? Aspointed out above, it is not easy to conceive that thisadoption was favored by the specific demands of theirparticular visual habitats. It is, however, conceivable thatgeneral requirements for a versatile and representation-ally powerful cortical circuit architecture are realized bythe common design. If this was true, the evolutionarybenefit of meeting these requirements might have driventhe adoption of large-scale self-organization and theemergence of the common design over evolutionarytimes.The most prominent candidate for such a general

requirement is the hypothesis of a coverage-continuity-compromise (e.g., [19,21,55,56]). It states that thecolumnar organization is shaped to achieve an optimaltradeoff between the coverage of the space of visual sti-mulus features and the continuity of their cortical repre-sentation. On the one hand, each combination ofstimulus features should be well represented in a corti-cal map to avoid ‘blindness’ to stimuli with particularfeature combinations. On the other hand, the wiringcost to establish connections within the map of orienta-tion preference should be kept low. This can beachieved if neurons that are physically close in the cor-tex tend to have similar stimulus preferences. These twodesign goals generally compete with each other. Thebetter a cortical representation covers the stimulusspace, the more discontinuous it has to be. The tradeoffbetween the two aspects can be modeled in what iscalled a dimension reduction framework in which corti-cal maps are viewed as two-dimensional sheets whichfold and twist in a higher-dimensional stimulus space(see Figure 1) to cover it as uniformly as possible whileminimizing some measure of continuity [21,57,58].From prior work, the coverage-continuity-compromise

appears to be a promising candidate for a principle toexplain visual cortical functional architecture. First,many studies have reported good qualitative agreementbetween the layout of numerically obtained dimension


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reducing maps and experimental observations[19,21,42,55-57,59-71]. Second, geometric relationshipsbetween the representations of different visual featuressuch as orientation, spatial frequency, and OD havebeen reproduced by dimension reduction models[25,56,63-65,67,68,72].Mathematically, the dimension reduction hypothesis

implies that the layouts of cortical maps can be under-stood as optima or near optima (global or local minima)of a free energy functional which penalizes both ‘stimu-lus scotomas’ and map discontinuity. Unfortunately,there is currently no dimension reduction model forwhich the precise layouts of optimal or nearly optimalsolutions have analytically been established. To justifythe conclusion that the tradeoff between coverage andcontinuity favors the common rules of OPM designfound in experiment, knowledge of optimal dimension-reducing mappings however appears essential.Precise knowledge of the spatial organization of opti-

mal and nearly optimal mappings is also critical for dis-tinguishing between optimization theories and frozennoise scenarios of visual cortical development. In a fro-zen noise scenario, essentially random factors such ashaphazard wiring [73], the impact of spontaneous activ-ity patterns [74], or an idiosyncratic set of visual experi-ences [75] determine the emerging pattern of preferredorientations. This pattern is then assumed to be ‘frozen’by an unknown mechanism, capable of preventingfurther modification of preferred orientations byongoing synaptic turnover and activity-dependent plasti-city. Conceptually, a frozen noise scenario is diametri-cally opposed to any kind of optimization theory. Evenif the reorganization of the pattern prior to freezing was

to follow a gradient descent with respect to some costfunction, the early stopping implies that the layout isneither a local nor a global minimum of this functional.Importantly, the layout of transient states is known toexhibit universal properties that can be completely inde-pendent of model details [25,32]. As a consequence, aninfinite set of distinct optimization principles will gener-ate the same spatial structure of transient states. Thisimplies in turn that the frozen transient layout is notspecifically shaped by any particular optimization princi-ple. Map layouts will thus in principle only be informa-tive about design or optimization principles of corticalprocessing architectures if maps are not just frozentransients.In practice, however, the predictions of frozen noise

and optimization theories might be hard to distinguish.Ambiguity between these mutually exclusive theorieswould result in particular, if the energy landscape of theoptimization principle is so ‘rugged’ that there is essen-tially a local energy minimum next to any relevant ran-dom arrangement. Dimension reduction models areconceptually related to combinatorial optimization pro-blems like the traveling salesman problem (TSP) andmany such problems are believed to exhibit ruggedenergy landscapes [76-78]. It is therefore essential toclarify whether paradigmatic dimension reduction mod-els are characterized by a rugged or a smooth energylandscape. If their energy landscapes were smooth witha small number of well-separated local minima, theirpredictions would be easy to distinguish from those of afrozen noise scenario.In this study, we examine the classical example of a

dimension reduction model, the elastic network (EN)

Figure 1 The dimension reduction framework. In the dimension reduction framework, the visual cortex is modeled as a two-dimensionalsheet that twists in a higher-dimensional stimulus (or feature) space to cover it as uniformly as possible while minimizing some measure ofcontinuity (left). In this way, it represents a mapping from the cortical surface to the manifold of visual stimulus features such as orientation andretinotopy (right).


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model. Since the seminal work of Durbin and Mitchison[21], the EN model has widely been used to study visualcortical representations [25,42,62-65,69-72,79]. The ENmodel possesses an explicit energy functional whichtrades off a matching constraint which matches corticalcells to particular stimulus features via Hebbian learn-ing, with a continuity constraint that minimizes Eucli-dean differences in feature space between neighboringpoints in the cortex [63]. In two ways, the EN model’sexplicit variational structure is very appealing. First, cov-erage and continuity appear as separate terms in thefree energy which facilitates the dissection of their rela-tive influences. Second, the free energy allows for theformulation of a gradient descent dynamics. The emer-gence of cortical selectivity patterns and their conver-gence toward a minimal energy state in this dynamicsmight serve as a model for an optimization process tak-ing place in postnatal development.Following Durbin and Mitchison, we consider the EN

model for the joint mapping of two visual features: (i)position in visual space, represented in a retinotopicmap (RM) and (ii) line orientation, represented in anOPM. To compute optimal dimension-reducing map-pings, we first develop an analytical framework forderiving closed-form expressions for fixed points, localminima, and optima of arbitrary optimization modelsfor the spatial layout of OPMs and RMs in which pre-dicted maps emerge by a supercritical bifurcation aswell as for analyzing their stability properties. By apply-ing this framework to different instantiations of the ENmodel, we systematically disentangle the effects of indi-vidual model features on the repertoire of optimal solu-tions. We start with the simplest possible model version,a fixed uniform retinotopy and an orientation stimulusensemble with only a single orientation energy and thenrelax the uniform retinotopy assumption incorporatingretinotopic distortions. An analysis for a second widelyused orientation stimulus ensemble including also unor-iented stimuli is given in Appendix 1. Surprisingly, in allcases, our analysis yields pinwheel-free orientationstripes (OSs) or stereotypical square arrays of pinwheelsas local minima or optimal orientation maps of the ENmodel. Numerical simulations of the EN confirm thesefindings. They indicate that more complex spatiallyaperiodic solutions are not dominant and that theenergy landscape of the EN model is rather smooth.Our results demonstrate that while aperiodic stationarystates exist, they are generally unstable in the consideredmodel versions.To test whether the EN model is in principle capable

of generating complex spatially aperiodic optimal orien-tation maps, we then perform a comprehensive unbiasedsearch of the EN optima for arbitrary orientation stimu-lus distributions. We identify two key parameters

determining pattern selection: (i) the intracortical inter-action range and (ii) the fourth moment of the orienta-tion stimulus distribution function. We derive completephase diagrams summarizing pattern selection in the ENmodel for fixed as well as variable retinotopy. Smallinteraction ranges together with low to intermediatefourth moment values lead to pinwheel-free OSs, rhom-bic, or hexagonal crystalline orientation map layouts asoptimal states. In the regime of large interaction rangestogether with higher fourth moment, we find irregularaperiodic OPM layouts with low pinwheel densities asoptima. Only in an extreme and previously unconsid-ered parameter regime of very large interaction rangesand stimulus ensemble distributions with an infinitefourth moment, optimal OPM layouts in the EN modelresemble experimentally observed aperiodic pinwheel-rich layouts and quantitatively reproduce the recentlydescribed species-invariant pinwheel statistics. Unex-pectedly, we find that the stabilization of such layouts isnot achieved by an optimal tradeoff between coverageand continuity of a localized population encoding by themaps but results from effectively suppressive long-rangeintracortical interactions in a spatially distributed repre-sentation of localized stimuli.We conclude our reexamination of the EN model with

a comparison between different numerical schemes forthe determination of optimal or nearly optimal map-pings. For large numbers of stimuli, numerically deter-mined solutions match our analytical predictions,irrespectively of the computational method used.

Results and discussionModel definition and model symmetriesWe analyze the EN model for the joint optimization ofposition and orientation selectivity as originally intro-duced by Durbin and Mitchison [21]. In this model, theRM is represented by a mapping R(x) = (R1(x), R2(x))which describes the receptive field center position of aneuron at cortical position x. Any RM can be decom-posed into an affine transformation x ↦ X from corticalto visual field coordinates, on which a vector-field ofretinotopic distortions r(x) is superimposed, i.e.:

R(x) = X + r(x)

with appropriately chosen units for x and R.The OPM is represented by a second complex-valued

scalar field z(x). The pattern of orientation preferences ϑ(x) is encoded by the phase of z(x) via

ϑ(x) =12arg(z(x)).

The absolute value |z(x)| is a measure of the averagecortical selectivity at position x. Solving the EN model


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requires to find pairs of maps {r(x), z(x)} that representan optimal compromise between stimulus coverage andmap continuity. This is achieved by minimizing a freeenergy functional

F = σ 2C +R (1)in which the functional C measures the coverage of a

stimulus space and the functional R the continuity ofits cortical representation. The stimulus space is definedby an ensemble {S} of idealized point-like stimuli, eachdescribed by two features: sz = |sz|e

2iθ and sr = (sx,sy)which specify the orientation θ of the stimulus and itsposition in visual space sr (Figure 2b). C and R aregiven by

C[z, r] = −〈ln

∫d2ye−(|sz−z(y)|

2+|sr−X−r(y)|2)/2σ 2〉S

R[z, r] = 12

∫d2yη||∇z(y)||2 + ηr

2∑j=1

||∇rj(y)||2 ,

with ∇ = (∂x, ∂y)T, and h Î [0, 1]. The ratios s 2/h ands 2/hr control the relative strength of the coverage termversus the continuity term for OPM and RM, respec-tively. 〈...〉S denotes the average over the ensemble ofstimuli.Minima of the energy functional F are stable fixed

points of the gradient descent dynamics

∂tz(x) = −2δF[z, r]δz̄(x)

≡ Fz[z, r](x)

∂tr(x) = −δF[z, r]δr(x)

≡ Fr[z, r](x)(2)

called the EN dynamics in the following. Thesedynamics read

∂tz(x) =〈[sz − z(x)

]e(x,S, z, r)

〉S + η�z(x) (3)

∂tr(x) =〈[sr − X − r(x)

]e(x,S, z, r)

〉S + ηr�r(x),(4)

where e(x, S, z, r) is the activity-pattern, evoked by astimulus S = (sr, sz) in a model cortex with retinotopicdistortions r(x) and OPM z(x). It is given by

e(x, . . .) =e−(|sr−X−r(x)|

2)/2σ 2e−(|sz−z(x)|2)/2σ 2∫

d2ye−(|sr−X−r(y)|2)/2σ 2e−(|sz−z(y)|2)/2σ 2.

Figure 2 illustrates the general features of the ENdynamics using the example of a single stimulus. Figure2a shows a model orientation map with a superimposeduniform representation of visual space. A single point-like, oriented stimulus S = (sr, sz) with position sr = (sx,sy) and orientation θ = 1/2 arg(sz) (Figure 2b) evokes a

cortical activity pattern e(x, S, z, r) (Figure 2c). Theactivity-pattern in this example is of roughly Gaussianshape and is centered at the point, where the location srof the stimulus is represented in cortical space. How-ever, depending on the model parameters and the sti-mulus, the cortical activity pattern may assume as well amore complex form (see also ‘Discussion’ section).According to Equations (3, 4), each stimulus and theevoked activity pattern induce a modification of theorientation map and RM, shown in Figure 2d. Orienta-tion preference in the activated regions is shifted towardthe orientation of the stimulus. The representation ofvisual space in the activated regions is locally contractedtoward the position of the stimulus. Modifications ofcortical selectivities occur due to randomly chosen sti-muli and are set proportional to a very small learningrate. Substantial changes of cortical representationsoccur slowly through the cumulative effect of a largenumber of activity patterns and stimuli. These effectivechanges are described by the two deterministic equa-tions for the rearrangement of cortical selectivities equa-tions (3, 4) which are obtained by stimulus-averagingthe modifications due to single activity patterns in thediscrete stimulus model [25]. One thus expects that theoptimal selectivity patterns and also the way in whichcortical selectivities change over time are determined bythe statistical properties of the stimulus ensemble. Inthe following, we assume that the stimulus ensemblesatisfies three properties: (i) The stimulus locations srare uniformly distributed across visual space. (ii) For thedistribution of stimulus orientations, |sz| and θ are inde-pendent. (iii) Orientations θ are distributed uniformly in[0, π].These conditions are fulfilled by stimulus ensembles

used in virtually all prior studies of dimension reductionmodels for visual cortical architecture (e.g.,[19,21,25,64,65,71,72,80,81]). They imply several symme-tries of the model dynamics equations (3, 4). Due to thefirst property, the EN dynamics are equivariant undertranslations

T̂yz(x) = z(x + y)

T̂yr(x) = r(x + y),

rotations

R̂βz(x) = e2iβz(−βx)

R̂βr(x) = β r(−βx)

with 2×2 rotation matrix

β =(cos β − sin βsin β cos β

),


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and reflections

P̂z(x) = z̄(x)

P̂r(x) = r(x),

where Ψ = diag(-1, 1) is the 2×2 reflection matrix.Equivariance means that

T̂yFz[z, r] = Fz[T̂yz, T̂yr] (5)

R̂βFz[z, r] = Fz[R̂βz, R̂βr] (6)

P̂Fz[z, r] = Fz[P̂z, P̂r], (7)

x

y

x

y

x

y

x

y

a b

c d

sy

sx

Figure 2 The EN model. (a) Example OPM (color code) together with a uniform map of visual space (RM) (grid lines). (b) Position sr = (sx, sy)and orientation θ of a ‘pointlike’ stimulus. (c) Cortical activity, evoked by the stimulus in b for the model maps in a. Dark regions are activated.Note, that in contrast to SOFM models, the activity pattern does not exhibit a stereotypical Gaussian shape. (d) Modification of orientationpreference and retinotopy, caused by the stimulus in b. Orientation preferences prior to stimulus presentation are indicated with grey bars, afterstimulus presentation with black bars. Most strongly modified preferences correspond to thick black bars. Modifications of orientationpreferences and retinotopy are displayed on an exaggerated scale for illustration purposes.


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with mutatis mutandis the same equations fulfilled bythe vector-field Fr[z, r].As a consequence, patterns that can be converted into

one another by translation, rotation, or reflection of thecortical layers represent equivalent solutions of themodel equations (3, 4), by construction. Due to the sec-ond assumption, the dynamics is also equivariant withrespect to shifts in orientation Sjz(x) = e

ijz(x), i.e.,

eiφFz[z, r] = Fz[eiφz, r] (8)

Fr[z, r] = Fr[eiφz, r]. (9)

Thus, two patterns are also equivalent solutions of themodel, if their layout of orientation domains and retino-topic distortions is identical, but the preferred orienta-tions differ everywhere by the same constant angle.Without loss of generality, we normalize the ensem-

bles of orientation stimuli such that 〈|sz|2〉S = 〈|sz|2〉 = 2throughout this article. This normalization can alwaysbe restored by a rescaling of z(x) (see [25,69]).Our formulation of the dimension-reduction problem

in the EN model utilizes a continuum description, bothfor cortical space and the set of visual stimuli. This facil-itates mathematical treatment and appears appropriate,given the high number of cortical neurons under onesquare millimeter of cortical surface (e.g., roughly 70000in cat V1 [82]). Even an hypothesized neuronal mono-layer would consist of more than 20 × 20 neurons perhypercolumn area Λ2, constituting a quite dense sam-pling of the spatial periodicity. Treating the featurespace as a continuum implements the concept that thecortical representation has to cover as good as possiblethe infinite multiplicity of conceivable stimulus featurecombinations.

The orientation unselective fixed pointTwo stationary solutions of the model can be estab-lished from symmetry. The simplest of these is theorientation unselective state with z(x) = 0 and uniformmapping of visual space r(x) = 0. First, by the shift sym-metry (Equation (8)), we find that z(x) = 0 is a fixedpoint of Equation (3). Second, by reflectional and rota-tional symmetry (Equations (5, 7)), we see that theright-hand side of Equation (4) has to vanish and hencethe orientation unselective state with uniform mappingof visual space is a fixed point of Equations (3, 4).This homogeneous unselective state thus minimizes

the EN energy functional if it is a stable solution of Equa-tions (3, 4). The stability can be determined by consider-ing the linearized dynamics of small deviations {r(x), z(x)} around this state. These linearized dynamics read

∂tr(x) � Lr[r] = 116πσ 4∫

d2y e−(x − y)24σ 2 Â r(y) + ηr � r(x) (10)

∂tz(x) � Lz[z] =(

1σ 2

− 1)z(x) + η�z(x) − 1

4πσ 4

∫d2ye−

(x−y)24σ 2 z(y), (11)

where (Â)ij = (xi - yi)(xj - yj) -2s2δij with δij being Kro-necker’s delta. We first note that the linearizeddynamics of retinotopic distortions and orientation pre-ference decouple. Thus, up to linear order and near thehomogeneous fixed point, both cortical representationevolve independently and the stability properties of theunselective state can be obtained by a separate examina-tion of the stability properties of both corticalrepresentations.The eigenfunctions of the linearized retinotopy

dynamics Lr[r] can be calculated by Fourier-transform-ing Equation (10):

∂t r̃i(k) = −2∑j=1

(σ 2e−k

2σ 2kikj + ηrk2δij)r̃j(k),

where k = |k| and i = 1, 2. A diagonalization of thismatrix equation yields the eigenvalues

λrL = −k2(ηr + e−σ2k2σ 2), λrT = −ηrk2

with corresponding eigenfunctions (in real space)

rL(x) = kφeikφx + c.c.

rT(x) = kφ+π/2eikφx + c.c.,

where kj = |k|(cos j, sin j)T. These eigenfunctionsare longitudinal (L) or transversal (T) wave patterns. Inthe longitudinal wave, the retinotopic distortion vector r(x) lies parallel to k which leads to a ‘compression wave’(Figure 3b, left). In the transversal wave pattern (Figure3b, right), the retinotopic distortion vector is orthogonalto k. We note that the linearized Kohonen model [61]was previously found to exhibit the same set of eigen-functions [80]. Because both spectra of eigenvalues λrT ,λrLare smaller than zero for every s >0, hr >0, and k >0(Figure 3a), the uniform retinotopy r(x) = 0 is a stablefixed point of Equation (4) irrespective of parameterchoice.The eigenfunctions of the linearized OPM dynamics

Lz[z] are Fourier modes ~ eikx by translational symme-

try. By rotational symmetry, their eigenvalues onlydepend on the wave number k and are given by

λz(k) = −1 + 1σ 2

(1 − e−k2σ 2

)− ηk2


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(see [25]). This spectrum of eigenvalues is depicted inFigure 3c. For h >0, lz(k) has a single maximum at

kc = 1σ

√ln

(1/η

). For

σ > σ ∗(η) =√1 + η ln η − η , (12)

this maximal eigenvalue r = lz(kc) is negative. Hence,the unselective state with uniform retinotopy is a stablefixed point of Equations (3, 4) and the only known solu-tion of the EN model in this parameter range. For s <s*(h), the maximal eigenvalue r is positive, and the non-selective state is unstable with respect to a band ofFourier modes ~ eikx with wave numbers around |k| ≈kc (see Figure 3c). This annulus of unstable Fouriermodes is called the critical circle. The finite wavelengthinstability [83-85] (or Turing instability [86]) leads tothe emergence of a pattern of orientation preferencewith characteristic spacing Λ = 2π/kc from the nonselec-tive state on a characteristic timescale τ = 1/r.

One should note that as in other models for the self-organization of orientation columns, e.g., [15,57], thecharacteristic spatial scale Λ arises from effective intra-cortical interactions of ‘Mexican-hat’ structure (short-range facilitation, longer-ranged suppression). Theshort-range facilitation in the linearized EN dynamics isrepresented by the first two terms on the right-handside of Equation (11). Since s

Figure 3d summarizes the result of the linear stabilityanalysis of the nonselective state. For s > s*(h), theorientation unselective state with uniform retinotopy isa minimum of the EN-free energy and also the globalminimum. For s < s*(h), this state represents a maxi-mum of the energy functional and the minima mustthus exhibit a space-dependent pattern of orientationselectivities.

Orientation stripesWithin the potentially infinite set of orientation selectivefixed points of the model, one class of solutions can beestablished from symmetry: {r(x) = 0, z(x) = A0e

ikx}. Inthese pinwheel-free states, orientation preference is con-stant along one axis in cortex (perpendicular to the vec-tor k), and each orientation is represented in equalproportion (see Figure 4a). Retinotopy is perfectly uni-form. Although this state may appear too simple to bebiologically relevant, we will see that it plays a funda-mental role in the state space of the EN model. It istherefore useful to establish its existence and basic char-acteristics. The existence of OS solutions follows directlyfrom the model’s symmetries (Equations (5) to (9)).Computing

Ty[Fz[eikx,0]] = Fz[Ty[eikx],Ty[0]] = Fz[eikyeikx,0] = eikyFz[eikx,0]

demonstrates that Fz[eikx, 0] is proportional to eikx.This establishes that the subspace of functions ~ eikx isinvariant under the dynamics given by Equation (3). ForA0 = 0, we recover the trivial fixed point of the ENdynamics by construction, as shown above. This meansthat within this subspace A0 = 0 is either a minimum ora maximum of the EN energy functional (Equation (1)).Furthermore, for A0 ® ∞ the EN energy tends to infi-nity. If the trivial fixed point is unstable, it correspondsto a maximum of the EN energy functional. Therefore,there must exist at least one minimum with A0 ≠ 0 inthe subspace of functions ~ eikx which then correspondsto a stationary state of the EN dynamics.Regarding the dynamics of retinotopic deviations, the

model’s symmetries equations can be invoked to showthat for the state {0, A0e

ikx}, the right-hand side ofEquation (4) has to be constant in space:

Ty[Fr[eikx,0]] = Fr[Ty[eikx],Ty[0]] = Fr[eikyeikx,0] = Fr[eikx,0]

If this constant was nonzero the RM would drift withconstant velocity. This, however, is impossible in a var-iational dynamics such that this constant must vanish.The OS solution (Figure 4a) is to the best of our knowl-edge the only exact nontrivial stationary solution ofEquations (3, 4) that can be established without anyapproximations.

Doubly periodic and quasi-periodic solutionsIn the EN model as considered in this study, the mapsof visual space and orientation preference are jointlyoptimized to trade off coverage and continuity leadingto mutual interactions between the two cortical repre-sentations. These mutual interactions vanish in the rigidretinotopy limit hr ® ∞ and the perfectly uniform reti-notopy becomes an optimal solution for arbitrary orien-tation column layout z(x). As it is not clear howessential the mutual interactions with position specificityare in shaping the optimal orientation column layout,we continue our investigation of solution classes by con-sidering global minima of optimization models withfixed uniform retinotopy. The mutual interactions willbe taken into account in a subsequent step.In the rigid retinotopy limit, minima of the energy

functional are stable stationary states of the dynamics ofthe OPM (Equation (3)) with r(x) = 0. To computeorientation selective stationary solutions of this OPMdynamics, we employ that in the vicinity of a supercriti-cal bifurcation where the nonselective fixed point z(x) =0 becomes unstable, the entire set of nontrivial fixedpoints is determined by the third-order terms of theVolterra series representation of the operator Fz[z, 0][35,84,85,87]. The symmetries given by Equations (5) to(9) restrict the general form of such a third-orderapproximation for any model of OPM optimization to

∂tz(x) ≈ Lz[z] +Nz3[z, z, z̄], (13)where the cubic operator Nz3 is written in trilinear

form, i.e.,

Nz3

⎡⎣∑

j

αjzj,∑k

βkzk,∑l

γlz̄l

⎤⎦ = ∑

j,k,l

αjβkγl Nz3[zj, zk, z̄l].

In particular, all even terms in the Volterra Seriesrepresentation of Fz[z, 0] vanish due to the Shift-Sym-metry (Equations (8, 9)). Explicit analytic computationof the cubic nonlinearities for the EN model is cumber-some but not difficult (see ‘Methods’ section) and yieldsa sum

Nz3[z, z, z̄] =11∑j=1

ajNj3[z, z, z̄]. (14)

The individual nonlinear operators Nj3 are with one

exception nonlocal convolution-type operators and aregiven in the ‘Methods’ section (Equation (38)), togetherwith a detailed description of their derivation.Only the coefficients aj depend on the properties of

the ensemble of oriented stimuli.


Page 9 of 55

n=1

n=2

n=3

n=5

n=15 ...

......

i=0

i=0

i=0 i=1

i=0 i=3

i=0 i=611

...

...

i=3

i=300

kx

ky

i=0

Figure 4 Exact and approximate orientation selective fixed points of OPM optimization models. (a) Pinwheel-free OS pattern. Diagramshows the position of the wave vector in Fourier space. (b) rPWC with four nonzero wave vectors. (c) Essentially complex planforms (ECPs). Theindex n indicates the number of nonzero wave vectors. The index i enumerates nonequivalent configurations of wave vectors with the same n,starting with i = 0 for the most anisotropic planform. For n = 3, 5, and 15, there are 2, 4, and 612 different ECPs, respectively. OPM layoutsbecome more irregular with increasing n.


Page 10 of 55

To calculate the fixed points of Equation (13), we usea perturbative method called weakly nonlinear analysisthat enables us to analytically examine the structure andstability of inhomogeneous stationary solutions in thevicinity of a finite-wavelength instability. Here, we exam-ine the stability of so-called planforms [83-85]. Plan-forms are patterns that are composed of a finite numberof Fourier components, such as

z(x) =∑j

Aj(t)eikjx

for a pattern of orientation columns. With the aboveplanform ansatz, we neglect any spatial dependency ofthe amplitudes Aj(t) for example due to long-wavedeformations for the sake of simplicity and analyticaltractability. When the dynamics is close to a finiitewavelength instability, the essential Fourier componentsof the emerging pattern are located on the critical circle|kj| = kc. The dynamic equations for the amplitudes ofthese Fourier components are called amplitude equa-tions. For a discrete number of N Fourier componentsof z(x) whose wave vectors lie equally spaced on the cri-tical circle, the most general system of amplitude equa-tions compatible with the model’s symmetries(Equations (5) to (9)) has the form [35,87]

Ȧi = rAi − AiN∑j=1

gij|Aj|2 − Āi−N∑j=1

fijAjAj− , (15)

with r >0. Here, gij and fij are the real-valued couplingcoefficients between the amplitudes Ai and Aj. Theydepend on the differences between indices |i - j| and areentirely determined by the nonlinearity Nz3[z, z, z̄] inEquation (13). If the wave vectors ki = (cos ai, sin ai)kcare parameterized by the angles ai, then the coefficientsgij and fij are functions only of the angle a = |ai - aj|between the wave vectors ki and kj. One can thus obtainthe coupling coefficients from two continuous functionsg(a) and f(a) that can be obtained from the nonlinearityNz3[z, z, z̄] (see ‘Methods’ section for details). In the fol-lowing, these functions are called angle-dependent inter-action functions. The amplitude equations arevariational if and only if gij and fij are real-valued. In thiscase they can be derived through

Ȧj(t) = −∂UA∂Āj

from an energy

UA = −rN∑i=1

|Ai|2 + 12N∑

i, j=1

gij|Ai|2|Aj|2 + 12N∑

i, j=1

fijĀiĀi−AjAj− . (16)

If the coefficients gij and fij are derived from Equation(1), the energy UA for a given planform solution corre-sponds to the energy density of the EN energy func-tional considering only terms up to fourth-order in z(x).The amplitude equations (15) enable to calculate an

infinite set of orientation selective fixed points. For theabove OS solution with one nonzero wave vector z(x) =A0e

ikx, the amplitude equations predict the so far unde-termined amplitude

|A0|2 = rgii (17)

and its energy

UOS = − r2gii . (18)

Since gii >0, this shows that OS stationary solutionsonly exist for r >0, i.e., in the symmetry breakingregime. As for all following fixed-points, UOS specifiesthe energy difference to the homogeneous unselectivestate z(x) = 0.A second class of stationary solutions can be found

with the ansatz

z(x) = A1eik1x + A2eik2x + A3e−ik1x + A4e−ik2x

with amplitudes Aj = |Aj|eijj and ∠(k1, k2) = a >0. By

inserting this ansatz into Equation (15) and assuminguniform amplitude |A1| = |A2| = |A2| = |A4| = A ,we obtain

A2 = rg00 + g0π + g0α + g0π−α − 2f0α . (19)

The phase relations of the four amplitudes are givenby

φ1 + φ3 = φ0φ2 + φ4 = φ0 + π .

These solutions describe a regular rhombic lattice ofpinwheels and are therefore called rhombic pinwheelcrystals (rPWCs) in the following. Three phases can bechosen arbitrarily according to the two above condi-tions, e.g., j0, Δ0 = j1 - j3 and Δ1 = j2 - j4. For anrPWC parameterized by these phases, Δ0 shifts theabsolute positions of the pinwheels in x-direction, Δ1shifts the absolute positions of the pinwheels in y-direc-tion, and j0 shifts all the preferred orientations by aconstant angle. The energy of an rPWC solution is

UrPWC = − 2rg00 + g0π + g0α + g0π−α − 2f0α . (20)


Page 11 of 55

An example of such a solution is depicted in Figure4b. We note that rPWCs have been previously found inseveral other models for OPM development[27,31,37,39,88]. The pinwheel density r of an rPWC, i.e., the number of pinwheels in an area of size Λ2, isequal to r = 4 sin a [54]. The angle a which minimizesthe energy UrPWC can be computed by maximizing thefunction

s(α) = g0α + g0π−α − 2f0α (21)in the denominator of Equation (20).The two solution classes discussed so far, namely OS

and rPWCs, exhibit one prominent feature, absent inexperimentally observed cortical OPMs, namely perfectspatial periodicity. Many cortical maps including OPMsdo not resemble a crystal-like grid of repeating units.Rather the maps are characterized by roughly repetitivebut aperiodic spatial arrangement of feature preferences(e.g., [5,10]). This does not imply that the precise layoutof columns is arbitrary. It rather means that the rules ofcolumn design cannot be exhaustively characterized bymapping a ‘representative’ hypercolumn.Previous studies of abstract models of OPM develop-

ment introduced the family of so-called essentially com-plex planforms (ECPs) as stationary solutions ofEquation (15). This solution class encompasses a largevariety of realistic quasi-periodic OPM layouts and istherefore a good candidate solution class for models ofOPM layouts. In addition, Kaschube et al. [38] demon-strated that models in which these are optimal solutionscan reproduce all essential features of the commonOPM design in ferret, tree-shrew, and galago. An n-ECPsolution can be written as

z(x) =n∑j=1

Ajeiljkjx,

with n = N/2 wave vectors kj = kc(cos(πj/n), sin(πj/n))distributed equidistantly on the upper half of the criticalcircle, complex amplitudes Aj and binary variables lj =±1 determining whether the mode with wave vector kjor -kj is active (nonzero). Because these planforms can-not realize a real-valued function they are called essen-tially complex [35]. For an n-ECP, the third term on theright-hand side of Equation (15) vanishes and the ampli-tude equations for the active modes Ai reduce to a sys-tem of Landau equations

Ȧi = rAi − Ain∑j=1

gij|Aj|2,

where gij is the n × n-coupling matrix for the activemodes. Consequently, the stationary amplitudes obey

|Ai|2 = rn∑j=1

(g−1

)ij. (22)

The energy of an n-ECP is given by

UECP = − r2∑i,j

(g−1

)ij. (23)

We note that this energy in general depends on theconfiguration of active modes, given by the lj’s, andtherefore planforms with the same number of activemodes may not be energetically degenerate.Families of n-ECP solutions are depicted in Figure 4c.

The 1-ECP corresponds to the pinwheel-free OS patterndiscussed above. For fixed n ≥ 3, there are multipleplanforms not related by symmetry operations whichconsiderably differ in their spatial layouts. For n ≥ 4, thepatterns are spatially quasi-periodic, and are a generali-zation of the so-called Newell-Pomeau turbulent crystal[89,90]. For n ≥ 10, their layouts resemble experimen-tally observed OPMs. Different n-ECPs however differconsiderably in their pinwheel density. Planforms whosenonzero wave vectors are distributed isotropically on thecritical circle typically have a high pinwheel density (seeFigure 4c, n = 15 lower right). Anisotropic planformsgenerally contain considerably fewer pinwheels (see Fig-ure 4c, n = 15 lower left). All large n-ECPs, however,exhibit a complex quasi-periodic spatial layout and anonzero density of pinwheels.In order to demonstrate that a certain planform is an

optimal solution of an optimization model for OPM lay-outs in which patterns emerge via a supercritical bifur-cation, we not only have to show that it is a stationarysolution of the amplitude equations but have to analyzeits stability properties with respect to the gradient des-cent dynamics as well as its energy compared to allother candidate solutions.Many stability properties can be characterized by

examining the amplitude equations (15). In principle,the stability range of an n-ECPs may be bounded by twodifferent instability mechanisms: (i) an intrinsic instabil-ity by which stationary solutions with n active modesdecay into ones with lower n. (ii) an extrinsic instabilityby which stationary solutions with a ‘too low’ number ofmodes are unstable to the growth of additional activemodes. These instabilities can constrain the range ofstable n to a small finite set around a typical n [35,87].A mathematical evaluation of both criteria leads to pre-cise conditions for extrinsic and intrinsic stability of aplanform (see ‘Methods’ section). In the following, aplanform is said to be stable, if it is both extrinsicallyand intrinsically stable. A planform is said to be an opti-mum (or optimal solution) if it is stable and possesses


Page 12 of 55

the minimal energy among all other stationary planformsolutions.Taken together, this amplitude equation approach

enables to analytically compute the fixed points andoptima of arbitrary optimization models for visual corti-cal map layout in which the functional architecture iscompletely specified by the pattern of orientation col-umns z(x) and emerges via a supercritical bifurcation.Via a third-order expansion of the energy functionaltogether with weakly nonlinear analysis, the otherwiseanalytically intractable partial integro-differential equa-tion for OPM layouts reduces to a much simpler systemof ordinary differential equations, the amplitude equa-tions. Using these, several families of solutions, OSs,rPWCs, and essentially complex planforms, can be sys-tematically evaluated and comprehensively compared toidentify sets of unstable, stable and optimal, i.e., lowestenergy fixed points. As already mentioned, the aboveapproach is suitable for arbitrary optimization modelsfor visual cortical map layout in which the functionalarchitecture is completely specified by the pattern oforientation columns z(x) which in the EN model is ful-filled in the rigid retinotopy limit. We now start by con-sidering the EN optimal solutions in this limit andsubsequently generalize this approach to models inwhich the visual cortical architecture is jointly specifiedby maps of orientation and position preference that arematched to one another.

Representing an ensemble of ‘bar’-stimuliWe start our investigation of optimal dimension-redu-cing mappings in the EN model using the simplest andmost frequently used orientation stimulus ensemble, thedistribution with sz-values uniformly arranged on a ringwith radius rsz =

√2 [57,64-66,91]. We call this stimulus

ensemble the circular stimulus ensemble in the follow-ing. According to the linear stability analysis of the non-selective fixed point, at the point of instability, wechoose s = s*(h) such that the linearization given inEquation (11) is completely characterized by the conti-nuity parameter h. Equivalent to specifying h is to fixthe ratio of activation range s and column spacing Λ

σ /� =12π

√log(1/η) (24)

as a more intuitive parameter. This ratio measures theeffective interaction-range relative to the expected spa-cing of the orientation preference pattern. In abstractoptimization models for OPM development a similarquantity has been demonstrated to be a crucial determi-nant of pattern selection [35,87]. We note, however, thatdue to the logarithmic dependence of s/Λ on h, a slightvariation of the effective interaction range may

correspond to a variation of the continuity parameter hover several orders of magnitude. In order to investigatethe stability of stationary planform solutions in the ENmodel with a circular orientation stimulus ensemble, wehave to determine the angle-dependent interaction func-tions g(a) and f(a). For the coefficients aj in Equation(14) we obtain

a1 = 14σ 6 − 1σ 4 + 12σ 2 a2 = 14πσ 6 − 18πσ 8 a3 = − 116πσ 8 + 18πσ 6a4 = − 18πσ 8 + 14πσ 6 − 18πσ 4 a5 = − 116πσ 8 a6 = 18πσ 6 − 116πσ 8a7 = 112π2σ 10 − 112π2σ 8 a8 = 124π2σ 10 a9 = − 364π3σ 12a10 = 112π2σ 10 − 112π2σ 8 a11 = 124π2σ 10 .

The angle-dependent interaction functions of the ENmodel with a circular orientation stimulus ensemble arethen given by

g(α) =1σ 4

(1 − 2e−k2c σ 2 − e2k2c σ 2(cosα−1)

(1 − 2e−k2c σ 2 cosα

))+

12σ 2

(e2k

2c σ

2(cosα−1) − 1)+

8σ 6

e−2k2c σ

2sinh4(1/2k2c σ

2 cos α)

f (α) =1σ 4

(1 − e−2k2c σ 2 (cosh(2k2c σ 2 cosα) + 2 cosh(k2c σ 2 cos α)) + 2e−k2c σ 2)

+1

2σ 2

(e−2k

2c σ

2cosh(2k2c σ

2 cos α) − 1)+

4σ 6

e−2k2c σ

2sinh4

(1/2k2c σ

2 cos α).

(25)

These functions are depicted in Figure 5 for two dif-ferent values of the interaction range s/Λ. We note thatboth functions are positive for all s/Λ which is a suffi-cient condition for a supercritical bifurcation from thehomogeneous nonselective state in the EN model.Finally, by minimizing the function s(a) in Equation

(21), we find that the angle a which minimizes theenergy of the rPWC fixed-point is a = π/2. This corre-sponds to a square array of pinwheels (sPWC). Due tothe orthogonal arrangement oblique and cardinal orien-tation columns and the maximized pinwheel density ofr = 4, the square array of pinwheels has the maximalcoverage among all rPWC solutions.Optimal solutions close to the pattern formation thresholdWe first tested for the stability of pinwheel-free OSsolutions and the sPWCs, by analytical evaluation of thecriteria for intrinsic and extrinsic stability (see ‘Methods’section). We found both, OSs and sPWCs, to be intrinsi-cally and extrinsically stable for all s/Λ. Next, we testedfor the stability of n-ECP solutions with 2 ≤ n ≤ 20. Wefound all n-ECP configurations with 2 ≤ n ≤ 20 to beintrinsically unstable for all s/Λ. Hence, none of theseplanforms represent optimal solutions of the EN modelwith a circular stimulus ensemble, while both OSs andsPWC are always local minima of the energy functional.By evaluating the energy assigned to the sPWC (Equa-

tion 20) and the OS pattern (Equation 18), we nextidentified two different regimes: (i) For short interactionrange s/Λ ≲ 0.122 the sPWC possesses minimal energyand is therefore the predicted global minimum. (ii) Fors/Λ ≳ 0.122 the OS pattern is optimal.Figure 6a shows the resulting simple phase diagram.

sPWCs and OSs are separated by a phase border at s/Λ


Page 13 of 55

Π� Π Π� ΠΠ� Π Π� Π

a b

Figure 5 Angle-dependent interaction functions for the EN model with fixed retinotopy and circular orientation stimulus ensemble. (a,b) g(a) and f(a) for s/Λ = 0.1 (a) and s/Λ = 0.2 (b).

Figure 6 Optimal solutions of the EN model with a circular orientation stimulus ensemble [57,64-66,91]and fixed representation ofvisual space. (a) At criticality, the phase space of this model is parameterized by either the continuity parameter h (blue labels) or theinteraction range s/Λ (red labels, see text). (b, c) OPMs (b) and their power spectra (c) in a simulation of Equation (3) with r(x) = 0 and r = 0.1,s/Λ = 0.1 (h = 0.67) and circular stimulus ensemble (see also Additional file 1). (d) Analytically predicted optimum for s/Λ ≲ 0.122 (quadraticpinwheel crystal). (e) Pinwheel density time courses for four different simulations (parameters as in b; gray traces, individual realizations; blacktrace, simulation in b; red trace, mean value). (f) Mean squared amplitude of the stationary pattern, obtained in simulations (parameters as in b)for different values of the control parameter r (black circles) and analytically predicted value (solid green line). (g, h) OPMs (g) and their powerspectra (h) in a simulation of Equation (3) with r(x) = 0 and s/Λ = 0.15 (h = 0.41) (other parameters as in b, see also Additional file 2). (i)Analytically predicted optimum for s/Λ ≳ 0.122 (orientation stripes). (j) Pinwheel density time courses for four different simulations (parametersas in g; gray traces, individual realizations; black trace, simulation in g; red trace, mean value). (k) Mean squared amplitude of the stationarypattern, obtained in simulations (parameters as in g) for different values of the control parameter r (black circles) and analytically predicted value(solid green line).


Page 14 of 55

≈ 0.122. We numerically confirmed these analytical pre-dictions by extensive simulations of Equation (3) with r(x) = 0 and the circular stimulus ensemble (see ‘Meth-ods’ section for details). Figure 6b,c shows snapshots ofa representative simulation with short interaction range(r = 0.1, s/Λ = 0.1 (h = 0.67)) (see also Additional file1). After the phase of initial pattern emergence (symme-try breaking), the OPM layout rapidly approaches asquare array of pinwheels, the analytically predictedoptimum (Figure 6d). Pinwheel density time courses(see ‘Methods’ section) display a rapid convergence to avalue close to the predicted density of 4 (Figure 6e). Fig-ure 6f shows the stationary mean squared amplitudes ofthe pattern obtained for different values of the controlparameter r (black circles). For small control para-meters, the pattern amplitude is perfectly predicted byEquation (19) (solid green line). Figure 6g,h shows snap-shots of a typical simulation with longer interactionrange (r = 0.1, s/Λ = 0.15 (h = 0.41)) (see also Addi-tional file 2). After the emergence of an OPM withnumerous pinwheels, pinwheels undergo pairwise anni-hilation as previously described for various models ofOPM development and optimization [25,27,35]. The OPpattern converges to a pinwheel-free stripe pattern,which is the analytically computed optimal solution inthis parameter regime (Figure 6i). Pinwheel densitiesdecay toward zero over the time course of the simula-tions (Figure 6j). Also in this parameter regime, themean squared amplitude of the pattern is well-predictedby Equation (17) for small r (Figure 6k).In summary, the phase diagram of the EN model with

a circular stimulus ensemble close to threshold isdivided into two regions: (i) for a small interactionrange (large continuity parameter) a square array of pin-wheels is the optimal dimension-reducing mapping and(ii) for a larger interaction range (small continuity para-meter) OSs are the optimal dimension-reducing map-ping. Both states are stable throughout the entireparameter range. All other planforms, in particularquasi-periodic n-ECPs are unstable. At first sight, thisstructure of the EN phase diagram may appear rathercounterintuitive. A solution with many pinwheel-defectsis energetically favored over a solution with no defectsin a regime with large continuity parameter where dis-continuity should be strongly penalized in the ENenergy term. However, a large continuity parameter atpattern formation threshold inevitably leads to a shortinteraction range s compared to the characteristic spa-cing Λ (see Equation (24)). In such a regime, the gain incoverage by representing many orientation stimuli in asmall area spanning the typical interaction range, e.g.,with a pinwheel, is very high. Our results show that thegain in coverage by a spatially regular positioning of

pinwheels outweighs the accompanied loss in continuityabove a certain value of the continuity parameter.EN dynamics far from pattern formation thresholdClose to pattern formation threshold, we found only twostable solutions, namely OSs and sPWCs. Neither of thetwo exhibits the characteristic aperiodic and pinwheel-rich organization of experimentally observed OPMs.Furthermore, the pinwheel densities of both solutions (r= 0 for OSs and r = 4 for sPWCs) differ considerablyfrom experimentally observed values [38] around 3.14.One way toward more realistic stable stationary statesmight be to increase the distance from pattern forma-tion threshold. In fact, further away from threshold, ourperturbative calculations may fail to correctly predictoptimal solutions of the model due to the increasinginfluence of higher order terms in the Volterra seriesexpansion of the right-hand side in Equation (3).To asses this possibility, we simulated Equation (3)

with r(x) = 0 and a circular stimulus ensemble for verylarge values of the control parameter r. Figure 7 dis-plays snapshots of such a simulation for r = 0.8 as wellas their pinwheel density time courses for two differentvalues of s/Λ. Pinwheel annihilation in the case oflarge s/Λ is less rapid than close to threshold (Figure7a,b). The OPM nevertheless converges toward a lay-out with rather low pinwheel density with pinwheel-free stripe-like domains of different directions joinedby domains with essentially rhombic crystalline pin-wheel arrangement. The linear zones increase theirsize over the time course of the simulations, eventuallyleading to stripe-patterns for large simulation times.For smaller interaction ranges s/Λ, the OPM layoutrapidly converges toward a crystal-like rhombicarrangement of pinwheels, however containing severaldislocations (Figure 24a in Appendix 1) [84]. Disloca-tions are defects of roll or square patterns, where tworolls or squares merge into one, thus increasing thelocal wavelength of the pattern [83,85]. Nevertheless,for all simulations, the pinwheel density rapidlyreaches a value close to 4 (Figure 7c) and the squarearrangement of pinwheels is readily recognizable. Bothfeatures, the dislocations in the rhombic patterns anddomain walls in the stripe patterns, have been fre-quently observed in pattern-forming systems far fromthreshold [84,85].In summary, the behavior of the EN dynamics with

circular stimulus ensemble far from pattern formationthreshold agrees very well with our analytical predic-tions close to threshold. Again, orientation stripes andsquare pinwheel crystals are identified as the only sta-tionary solutions. Aperiodic and pinwheel-rich patternswhich resemble experimentally OPM layouts were notobserved.


Page 15 of 55

Taking retinotopic distortions into accountSo far, we have examined the optimal solutions of theEN model for the simplest and most widely used orien-tation stimulus ensemble. Somewhat unexpected fromprevious reports, the optimal states in this case do notexhibit the irregular structure of experimentallyobserved orientation maps. Our treatment however dif-fers from previous approaches in that the mapping ofvisual space so far was assumed to be undistorted andfixed, i.e., r(x) = 0. We recall that in their seminal publi-cation, Durbin and Mitchison [21] in particular demon-strated interesting correlations between the map oforientation preference and the map of visual space.These correlations suggest a strong coupling betweenthe two that may completely alter the model’s dynamicsand optimal solutions.It is thus essential to clarify whether the behavior of

the EN model observed above changes or persists if werelax the simplifying assumption of undistorted retino-topy and allow for retinotopic distortions. By analyzingthe complete EN model dynamics (Equations (3, 4)), we

study the EN model exactly as originally introduced byDurbin and Mitchison [21].We again employ the fact that in the vicinity of a

supercritical bifurcation where the nonorientation selec-tive state becomes unstable, the entire set of nontrivialfixed points of Equations (3, 4) is determined by thethird-order terms of the Volterra series representationof the nonlinear operators Fz[z, r] and Fr[z, r]. Themodel symmetries equations (5) to (9) restrict the gen-eral form of the leading order terms for any model forthe joint optimization of OPM and RM to

∂tz(x) = Lz[z] +Qz[r, z] +Nz3[z, z, z̄] + · · · (26)

∂tr(x) = Lr[r] +Qr[z, z̄] + · · · . (27)Because the uniform retinotopy is linearly stable, reti-

notopic distortions are exclusively induced by a couplingof the RM to the OPM via the quadratic vector-valuedoperator Qr[z, z̄] . These retinotopic distortions will inturn alter the dynamics of the OPM via the quadratic

0 500 1000 1500 20000

1

2

3

4

time [τ]

Pin

whe

el d

ensi

ty ρ

0 200 400 6000

1

2

3

4

time [τ]

Pin

whe

el d

ensi

ty ρ

Figure 7 Numerical analysis of the EN dynamics with circular orientation stimulus ensemble and fixed representation of visual spacefar from pattern formation threshold. (a) OPMs and their power spectra in a simulation of Equation (3) with r(x) = 0, r = 0.8, s/Λ = 0.3 (h =0.028) and circular orientation stimulus ensemble. Pinwheel density time courses for four different simulations (parameters as in a; gray traces,individual realizations; black trace, simulation in a; red trace, mean value) (c, d) OPMs and their power spectra in a simulation of Equation (3)with r(x) = 0, r = 0.8, s/Λ = 0.12 (h = 0.57) and circular orientation stimulus ensemble. (d) Pinwheel density time courses for four differentsimulations (parameters as in c; gray traces, individual realizations; black trace, simulation in c; red trace, mean value).


Page 16 of 55

complex-valued operator Qz[r, z]. Close to the point ofpattern onset (r ≪ 1), the timescale of OPM develop-ment, τ = 1/r, becomes arbitrarily large and retinotopicdeviations evolve on a much shorter timescale. Thisseparation of timescales allows for an adiabatic elimina-tion of the variable r(x), assuming it to always be at theequilibrium point of Equation (27):

r(x) = −L−1r[Qr[z, z̄]

]. (28)

We remark that as λrT/L(k) < 0 for all finite wave

numbers k >0, the operator Lr[r] is indeed invertiblewhen excluding global translations in the set of possibleperturbations of the trivial fixed point. From Equation(28), the coupled dynamics of OPM and RM is thusreduced to a third-order effective dynamics of the OPM:

∂tz(x) ≈ Lz[z]−Qz[L−1r[Qr[z, z̄]

], z]︸︷︷︸

Nr3[z,z,z̄]

+Nz3[z, z, z̄]

= Lz[z] +Nr3[z, z, z̄] +Nz3[z, z, z̄].

(29)

The nonlinearity Nr3[z, z, z̄] accounts for the couplingbetween OPM and RM. Its explicit analytical calculationfor the EN model is rather involved and yields a sum

Nr3[z, z, z̄] =12∑j=1

ajrNjr[z, z, z̄].

The individual nonlinear operators Njr are nonlinearconvolution-type operators and are presented in the‘Methods’ section together with a detailed description oftheir derivation. Importantly, it turns out that the coeffi-cients ajr are completely independent of the orientationstimulus ensemble.The adiabatic elimination of the retinotopic distortions

results in an equation for the OPM (Equation (29))which has the same structure as Equation (13), the onlydifference being an additional cubic nonlinearity. Due tothis similarity, its stationary solutions can be determinedby the same methods as presented for the case of afixed retinotopy. Again, via weakly nonlinear analysis weobtain amplitude equations of the form Equation (15).The nonlinear coefficients gij and fij are determinedfrom the angle-dependent interaction functions g(a) andf(a). For the operator Nr3[z, z, z̄] , these functions aregiven by

gr(α) =

((1 − σ 2 − 2e−k2c σ 2

)e2k

2c σ

2(cos α−1) + e−k2c σ 2)2

2σ 4(ηr + σ 2e−2k

2c σ

2(cos α−1))fr(α) =

12

(gr(α) + gr(α + π)

),

verifying that, Nr3[z, z, z̄] is independent of the orienta-tion stimulus ensemble. Besides the interaction range s/Λ the continuity parameter hr Î [0, ∞] for the RMappears as an additional parameter in the angle-depen-dent interaction function. Hence, the phase diagram ofthe EN model will acquire one additional dimensionwhen retinotopic distortions are taken into account. Wenote, that in the limit hr ® ∞, the functions gr(a) and fr(a) tend to zero and as expected one recovers theresults presented above for fixed uniform retinotopy.The functions gr(a) and fr(a) are depicted in Figure 8for various interaction ranges s/Λ and retinotopic conti-nuity parameters hr.Coupled essentially complex n-planformsIn the previous section, we found that by an adiabaticelimination of the retinotopic distortions in thedynamics equations (26, 27) the system of partial inte-gro-differential equations can be reduced to a singleequation for the OPM. In this case, the stationary solu-tions of the OPM dynamics are again planforms com-posed of a discrete set of Fourier modes

z(x) =N∑j

Ajeikjx, (30)

with |k| = kc. However, each of these stationary plan-form OPM solutions induces a specific pattern of reti-notopic distortions by Equation (28). The joint mapping{X + r(x), z(x)} is then an approximate stationary solu-tion of Equations (26, 27) and will be termed coupledplanform solution in the following. In contrast to othermodels for the joint mapping of orientation and visualspace (e.g., [31,33,92]), the coupling between the repre-sentation of visual space and orientation in the ENmodel is not induced by model symmetries but a mereconsequence of the joint optimization of OPM and RMthat requires them to be matched to one another.For planforms given by Equation (30), it is possible to

analytically evaluate Equation (28) and compute theassociated retinotopic distortions r(x). After a somewhatlengthy calculation (see ‘Methods’ section), one obtains

r(x) = −n∑

k=1,j

This means that the OS solution does not induce anydeviations from the perfect retinotopy as shown pre-viously from symmetry. This is not the case for thesquare pinwheel crystal (sPWC)

zsPWC(x) ∝ sin(kcx1) + i sin(kcx2) ,the second important solution for undistorted retino-

topy. Inserting this ansatz into Equation (31) and

neglecting terms of order O((

e−k2c σ

2)2)

or higher, we

obtain

rsPWC(x) ∝ e−k2c σ 2

σ 2λr2(2kc)

(kc sin(2kcx1)kc sin(2kcx2)

).

These retinotopic distortions are a superposition ofone longitudinal mode in x-direction and one in y-direction, both with doubled wave number ~ 2kc. Thedoubled wave number implies that the form of retino-topic distortions is independent of the topologicalcharge of the pinwheels. Importantly, the gradient ofthe retinotopic mapping R(x) = X + rsPWC(x) isreduced at all pinwheel locations. The coupled sPWC

Π� Π Π� Π Π� Π Π� Π

Π� Π Π� Π

Π� Π Π� Π

Π� Π Π� Π

Π� Π Π� Π

a b

c d

e f

Figure 8 Angle-dependent interaction functions for the coupling between OPM and RM in the EN model. (a, b) gr(a) and fr(a) for hr =0.005 and s/Λ = 0.3 (a) and 0.1 (b). (c, d) gr(a) and fr(a) for hr = 0.05 and s/Λ = 0.3 (c) and 0.1 (d). (e, f) gr(a) and fr(a) for hr = 0.5 and s/Λ =0.3 (e) and 0.1 (f).


Page 18 of 55

is therefore in two ways a high coverage mapping asexpected. First, the representations of cardinal andoblique stimuli (real and imaginary part of z(x)) areorthogonal to each other. Second, the regions of high-est gradient in the orientation map correspond to lowgradient regions in the RM.In Figure 9, the family of coupled n-ECPs is displayed,

showing simultaneously the distortions of the RM andthe OPM. Retinotopic distortions are generally weakerfor anisotropic n-ECPs and stronger for isotropic n-ECPs. However, for all stationary solutions the regionsof high gradient in the orientation map coincide withlow gradient regions (the folds of the grid) in the RM.This is precisely what is generally expected from adimension-reducing mapping [21,62,63,91]. In the fol-lowing section, we will investigate which of these solu-tions become optimal depending on the two parameterss/Λ and hr that parameterize the model.

The impact of retinotopic distortionsAccording to our analysis, at criticality, the nontrivialstable fixed points of the EN dynamics are determinedby the continuity parameter h Î (0, 1) for the OPM or,equivalently, the ratio σ /� = 12π

√log(1/η) and the con-

tinuity parameter hr for the mapping of visual space.We first tested for the stability of pinwheel-free orienta-tion stripe (OS) solutions and rPWC solutions of Equa-tion (15), with coupling matrices gij and fij as obtainedfrom the nonlinearities in Equation (29). The anglewhich minimizes the energy UrPWC (Equation (20)) isnot affected by the coupling between retinotopic andOPM and is thus again a = π/4. By numerical evalua-tion of the criteria for intrinsic and extrinsic stability,we found both, OSs and sPWCs, to be intrinsically andextrinsically stable for all s/Λ and hr.Next, we tested for the stability of coupled n-ECP

solutions for 2 ≤ n ≤ 20. We found all coupled n-ECPconfigurations with n ≥ 2 to be intrinsically unstable forall s/Λ and hr. Evaluating the energy assigned tosPWCs and OSs, we identified two different regimes: (i)for shorter interaction range s/Λ the sPWC is the mini-mal energy state and (ii) for larger interaction range s/Λ the optimum is an OS pattern as indicated by thephase diagram in Figure 10a. The retinotopic continuityparameter has little influence on the energy of the twofixed points. The phase border separating stripes fromrhombs runs almost parallel to the hr-axis. We numeri-cally confirmed these analytical predictions by extensivesimulations of Equation (3, 4) (see ‘Methods’ section fordetails). Figure 10c shows snapshots of a representativesimulation with small interaction range (r = 0.1, s/Λ =0.1 (h = 0.67), hr = h). After the initial symmetry break-ing phase, the OPM layout rapidly converges toward a

crystalline array of pinwheels, the predicted optimum inthis parameter regime (Figure 10c). Retinotopic devia-tions are barely visible. Figure 10b displays pinwheeldensity time courses for four such simulations. Notethat in one simulation, the pinwheel density drops toalmost zero. In this simulation, the OP pattern con-verges to a stripe-like layout. This is in line with thefinding of bistability of rhombs and stripes in all para-meter regimes. Although the sPWC represents the glo-bal minimum in the simulated parameter regime, OSsare also a stable fixed point and, depending on theinitial conditions, may arise as the final state of a frac-tion of the simulations. In the two simulations with pin-wheel densities around 3.4, patterns at later simulationstages consist of different domains of rhombic pinwheellattices with a < π/2.Figure 10d,e shows the corresponding analysis with

parameters for larger interaction range r = 0.1, s/Λ =0.15 (h = 0.41), hr = h. Here after initial pinwheel crea-tion, pinwheels typically annihilate pairwisely and theOPM converges to an essentially pinwheel-free stripepattern, the predicted optimal solution in this parameterregime (Figure 10e). Retinotopic deviations are slightlylarger. The behavior of the EN model for the joint opti-mization of RM and OPM thus appears very similarcompared to the fixed retinotopy case. Perhaps surpris-ingly, the coupling of both feature maps has little effecton the stability properties of the fixed points and theresulting optimal solutions.As in the previous case, the structure of the phase dia-

gram in Figure 10a appears somewhat counterintuitive.A high coverage and pinwheel-rich solution is the opti-mum in a regime with large OPM continuity parameterwhere discontinuities in the OPM such as pinwheelsshould be strongly penalized. A pinwheel-free solutionwith low coverage and high continuity is the optimumin a regime with small continuity parameter. Asexplained above, a large OPM continuity parameter atpattern formation threshold implies a small interactionrange s/Λ (see Equation (24)). In such a regime, thegain in coverage by representing many orientation sti-muli in a small area spanning the typical interactionrange, e.g., with a pinwheel, is very high. Apparently thisgain in coverage by a regular positioning of pinwheelsoutweighs the accompanied loss in continuity for verylarge OPM continuity parameters. This counterintuitiveinterplay between coverage and continuity thus seems tobe almost independent of the choice of retinotopic con-tinuity parameters.The circular orientation stimulus ensemble contains

only stimuli with a fixed and finite ‘orientation energy’or elongation |sz|. This raises the question of whetherthe simple nature of the circular stimulus ensemblemight restrain the dynamics of the EN model. The EN


Page 19 of 55

kx

ky

n=1

n=3

n=5

i=0

n=8

i=1i=0

i=0

i=0

i=4

i=12

. . .

. . .

. . .

. . .

. . .

. . .

Figure 9 Coupled n-ECPs as dimension-reducing solutions of the EN model. Coupled n-ECP are displayed in visual space showingsimultaneously the distortion of the RM and the OPM (s/Λ = 0.3 (h = 0.028), hr = h, circular stimulus ensemble). The distorted grid represents athe cortical square array of cells. Each grid intersection is at the receptive field center of the corresponding cell. Preferred stimulus orientationsare color-coded as in Figure 2a. As in Figure 4, n and i enumerate the number of nonzero wave vectors and nonequivalent configurations ofwave vectors with the same n, respectively. The coupled 1-ECP is a pinwheel-free stripe pattern without retinotopic distortion. Only the mostanisotropic and the most isotropic coupled n-ECPs are shown for each n. Note that for all ECPs, high gradients within the orientation mappingcoincide with low gradients of the retinotopic mapping and vice versa. Retinotopic distortions are displayed on a fivefold magnified scale forvisualization purposes.


Page 20 of 55

dynamics are expected to depend on the characteristicsof the activity patterns evoked by the stimuli and thesewill be more diverse and complex with ensembles con-taining a greater diversity of stimuli. Therefore, werepeated the above analysis of the EN model for a richerstimulus ensemble where orientation stimuli are uni-formly distributed on the disk {sz, |sz| ≤ 2}, a choiceadopted by a subset of previous studies, e.g., [19,25,81].In particular, this ensemble contains unoriented stimuliwith |sz| = 0. Intuitively, the presence of these unor-iented stimuli might be expected to change the role ofpinwheels in the optimal OPM layout. Pinwheels’ popu-lation activity is untuned for orientation. Pinwheel cen-ters may therefore acquire a key role for therepresentation of unoriented stimuli. Nevertheless, wefound the behavior of the EN model when consideredwith this richer stimulus ensemble to be virtually indis-tinguishable from the circular stimulus ensemble. Details

of the derivations, phase diagrams and numericallyobtained solutions are given in Appendix 1.

Are there stimulus ensembles for which realistic,aperiodic maps are optimal?So far, we have presented a comprehensive analysis ofoptimal dimension-reducing mappings of the EN modelfor two widely used orientation stimulus distributions(previous sections and Appendix 1). In both cases,optima were either regular crystalline pinwheel latticesor pinwheel-free orientation stripes. These results mightindicate that the EN model for the joint optimization ofOPM and RM is per se incapable of reproducing thestructure of OPMs as found in the visual cortex. Draw-ing such a conclusion is suggested in view of the appar-ent insensitivity of the model’s optima to the choice ofstimulus ensemble. The two stimulus ensembles consid-ered so far however do not exhaust the infinite space of

Figure 10 Phase diagram of the EN model with variable retinotopy for a circular stimulus ensemble [57,64-66,91]. (a) Regions of the hr-s/Λ-plane in which n-ECPs or rPWCs have minimal energy. (b) Pinwheel density time courses for four different simulations of Equations (3, 4)with r = 0.1, s/Λ = 0.13 (h = 0.51), hr = h (grey traces, individual realizations; red trace, mean value; black trace, realization shown in c). (c) OPMs(upper row), their power spectra (middle row), and RMs (lower row) obtained in a simulation of Equations (3, 4); parameters as in b. (d) Pinwheeldensity time courses for four different simulations of Equations (3, 4) with r = 0.1, s/Λ = 0.3 (h = 0.03), hr = h (grey traces, individual realizations;red trace, mean value; black trace, realization shown in e). (e) OPMs (upper row), their power spectra (middle row), and RMs (lower row) in asimulation of Equations (3, 4); parameters as in d.


Page 21 of 55

stimulus distributions that are admissible in principle.From the viewpoint of ‘biological plausibility’ it is cer-tainly not obvious that one should strive to examine sti-mulus distributions very different from these, as long asthe guiding hypothesis is that the functional architectureof the primary visual cortex optimizes the joint repre-sentation of the classical elementary stimulus features.If, however, stimulus ensembles were to exist, for whichoptimal EN mappings truly resemble the biologicalarchitecture, their characteristics may reveal essentialingredients of alternative optimization models for visualcortical architecture.Adopting this perspective raises the technical question

of whether an unbiased search of the infinite space ofstimulus ensembles only constrained by the model’ssymmetries (Equations (5) to (9)) is possible. To answerthis question, we examined whether the amplitude equa-tions (15) can be obtained for an arbitrary orientationstimulus distribution. Fortunately, we found that thecoefficients of the amplitude equations are completelydetermined by the finite set of moments of order lessthan 5 of the distributions. The approach developed sofar can thus be used to comprehensively examine thenature of EN optima resulting for any stimulus distribu-tion with finite fourth-order moment. While such astudy does not completely exhaust the infinite space ofall eligible distributions, it appears to only excludeensembles with really exceptional properties. These areprobability distributions with diverging fourth moment,i.e., ensembles that exhibit a heavy tail of essentially‘infinite’ orientation energy stimuli.Since the coupling between OPM and RM did not

have a large impact in the case of the two classical sti-mulus ensembles, we start the search through thespace of orientation stimulus ensembles by consideringthe EN model with fixed retinotopy r(x) = 0. The coef-ficients ai for the nonlinear operators N3i [z, z, z̄] inEquation (14) for arbitrary stimulus ensembles aregiven by

a1 =〈|sz|4〉16σ 6 − 〈

|sz |2〉2σ 4 +

12σ 2 a2 =

〈|sz|2〉8πσ 6 − 〈

|sz|4〉32πσ 8 a3 = − 〈

|sz|4〉64πσ 8 +

〈|sz|2〉16πσ 6

a4 = − 〈|sz|4〉

32πσ 8 +〈|sz|2〉8πσ 6 − 18πσ 4 a5 = − 〈

|sz|4〉64πσ 8 a6 =

〈|sz|2〉16πσ 6 − 〈

|sz|4〉64πσ 8

a7 =〈|sz|4〉48π2σ 10 −

〈|sz|2〉24π2σ 8 a8 =

〈|sz|4〉96π2σ 10 a9 = −

3〈|sz|4〉256π3σ 12

a10 =〈|sz|4〉48π2σ 10 − 〈

|sz|2〉24π2σ 8 a11 =

〈|sz|4〉96π2σ 10 .

(32)

The corresponding angle-dependent interaction func-tions are given by (see ‘Methods’ section)

g(α) =

〈|sz|2〉2σ 4

(1 − 2e−k2c σ 2 − e2k2c σ 2(cosα−1)

(1 − 2e−k2c σ 2 cos α

))+

12σ 2

(e2k

2c σ

2(cosα−1) − 1)+2〈|sz|4〉σ 6

e−2k2c σ

2sinh4

(1/2k2c σ

2 cos α)

f (α) =

〈|sz|2〉2σ 4

(1 − e−2k2c σ 2 (cosh(2k2c σ 2 cos α) + 2 cosh(k2c σ 2 cos α)) + 2e−k2c σ 2)

+1

2σ 2

(e−2k

2c σ

2cosh(2k2c σ

2 cos α) − 1)+

〈|sz|4〉σ 6

e−2k2c σ

2sinh4

(1/2k2c σ

2 cos α).

(33)

Again, without loss of generality, we set 〈|sz|2〉 = 2. At

criticality, both functions are parameterized by the conti-nuity parameter h Î (0, 1) for the OPM or, equivalently,the interaction range σ /� = 12π

√log(1/η) and the fourth

moment 〈|sz|4〉 of the orientation stimulus ensemble. The

fourth moment, is a measure of the peakedness of a sti-mulus distribution. High values generally indicate astrongly peaked distribution with a large fraction of non-oriented stimuli (|sz|

4 ≈ 0), together with a large fractionof high orientation energy stimuli (|sz|

4 large).The dependence of g(a) on the fourth moment of the

orientation stimulus distribution and f(a) suggests thatdifferent stimulus distributions may indeed lead to dif-ferent optimal dimension-reducing mappings.The circular stimulus ensemble possesses the minimal

possible fourth moment, with 〈|sz|4〉 = (〈|sz|

2〉)2 = 4. Thefourth moment of the uniform stimulus ensemble is 〈|sz|

4〉 = 16/3. The angle-dependent interaction functionsfor both ensembles (Equation (25), Figure 22 in Appen-dix 1) are recovered, when inserting these values intoEquation (33).To simplify notation in the following, we define

s4 =〈|sz|4〉 − 〈|sz|2〉2 = 〈|sz|4〉 − 4

as the parameter characterizing an orientation stimu-lus distribution. This parameter ranges from zero forthe circular stimulus ensemble to infinity for ensembleswith diverging fourth moments. Figure 11 displays theangle-dependent interaction functions for differentvalues of s/Λ and s4. In all parameter regimes, g(a) andf(a) are larger than zero. The amplitude dynamics aretherefore guaranteed to converge to a stable stationaryfixed point and the bifurcation from the nonselectivefixed point in the EN model is predicted to be supercri-tical in general.By evaluating the energy assigned to the rPWC and n-

ECPs, we investigated the structure of the two-dimen-sional phase space of the EN model with an arbitraryorientation stimulus distribution. First, it is not difficultto show that the angle a which minimizes the energyUrPWC (Equation (20)) of an rPWC is a = π/4 for all s/Λ and s4. Hence, a square lattice of pinwheels (sPWC)is in all parameter regimes energetically favored overany other rhombic lattice configuration of pinwheels.Figure 12 displays the phase diagram of the EN modelwith an arbitrary orientation stimulus distribution. Fororientation stimulus distributions with small fourthmoments, optimal mappings consist of either parallelpinwheel-free stripes or quadratic pinwheel crystals.These distributions include the circular and the uniformstimulus ensembles with s4 = 0 and s4 = 4/3. Above acertain value of the fourth moment around s4 = 6, n-


Page 22 of 55

ECPs with n >2 become optimal mappings. For a shortinteraction range s/Λ, hexagonal pinwheel crystals dom-inate the phase diagram in a large region of parameterspace. With increasing interaction range, we observe asequence of phase transitions by which higher n-ECPsbecome optimal. For n >3, these optima are spatially

aperiodic. In all parameter regimes, we found that then-ECP with the most anisotropic mode configuration(Figure 4c, left column) is the energetically favored statefor n >3. Pinwheel densities of these planforms are indi-cated in Figure 12 and are typically smaller than 2.0.We note that this is well below experimentally observed


a b



c

e

d

f

Figure 11 Angle-dependent interaction functions for the EN model with fixed retinotopy for different fourth-moment values of theorientation stimulus distribution and effective interaction-widths. (a, b) g(a) and f(a) for s4 = 8 and s/Λ = 0.1 (a) and s/Λ = 0.3 (b). (c, d)g(a) and f(a) for s4 = 20 and s/Λ = 0.1 (c) and s/Λ = 0.3 (d). (e, f) g(a) and f(a) for s4 = 100 and s/Λ = 0.1 (e) and s/Λ = 0.3 (f).


Page 23 of 55

pinwheel density values [38]. Optimal mappings oforientation preference for finite fourth moment in theEN model are thus either orientation stripes, periodicarrays of pinwheels (hexagonal, square), or aperiodicpinwheel arrangements with low pinwheel density.

We numerically tested these analytical predictions bysimulations of Equation (3) (r(x) = 0) with two addi-tional stimulus ensembles with s4 = 6 and s4 = 8 (see‘Methods’ section). Figure 13a shows snapshots of asimulation with (r = 0.1, s/Λ = 0.2 (h = 0.2)) and s4 = 6

Figure 12 Stripe-like, crystalline, and quasi-crystalline cortical representations as optimal solutions to the mapping of orientationpreference with fixed uniform retinotopy in the EN model. The graph shows the regions of the s4-s/Λ-plane in which n-ECPs or sPWCshave minimal energy. For n ≥ 3, pinwheel densities of the energetically favored n-ECP configuration are indicated.


Page 24 of 55

(see also Additional file 3). After the initial phase of pat-tern emergence, the OPM layout converges toward anarrangement of fractured stripes whi

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link.springer.com · RESEARCH Open Access Coverage, continuity, and visual cortical architecture Wolfgang Keil1,2,3,4* and Fred Wolf1,2,3,4 Abstract Background: The primary visual