universality in the evolution of orientation columns in...

www.sciencemag.org/cgi/content/full/science.1194869/DC1

Supporting Online Material for

Universality in the Evolution of Orientation Columns in the Visual Cortex

Matthias Kaschube, Michael Schnabel, Siegrid Löwel, David M. Coppola, Leonard E. White, Fred Wolf*

*To whom correspondence should be addressed. E-mail: [email protected] (M.K.); fred-

[email protected] (F.W.)

Published 4 November 2010 on Science Express DOI: 10.1126/science.1194869

This PDF file includes:

Materials and Methods

SOM Text

Figs. S1 to S24

Tables S1 to S5

References

Supporting Online Material

Universality in the Evolution of Orientation Columns in the Visual Cortex

Matthias Kaschube, Michael Schnabel, Siegrid Löwel, David M. Coppola, Leonard E. White &Fred Wolf

Contents

1 Supporting information on materials and methods: Analysis of pinwheel statistics 31.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Optical imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Column spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Pinwheel density estimation - overview . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Pinwheel density estimation - methods . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Pinwheel density variability in subregions . . . . . . . . . . . . . . . . . . . . . . . . 91.8 Pinwheel nearest neighbor distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.9 Confidence intervals and significance tests . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Supporting information on materials and methods: Model 132.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The long-range interaction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Symmetry properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Universal solution set near threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Derivation from long-range orientation selective interactions . . . . . . . . . . . . . 162.6 Energy functional of the full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Scaling argument for long-range interactions . . . . . . . . . . . . . . . . . . . . . . . 212.8 Biological interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.9 Mathematical idealizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Supporting information on materials and methods: Numerical integration and pinwheelidentification in model maps 253.1 Numerical solution of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Synthesizing model maps and pinwheel identification . . . . . . . . . . . . . . . . . 27

4 Statistical characterization of the common design 294.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 The common design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Comparison to phase randomized maps . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Comparison to the long-range interaction model . . . . . . . . . . . . . . . . . . . . . 334.5 Average pinwheel density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.6 Pinwheel density variability in subregions . . . . . . . . . . . . . . . . . . . . . . . . . 36

1

4.7 Pinwheel nearest neighbor distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Model orientation map designs and their pinwheel statistics 425.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Pinwheel statistics in the long-range interaction model . . . . . . . . . . . . . . . . . 425.3 Essentially complex planforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4 Long-range interaction model: Numerical solutions . . . . . . . . . . . . . . . . . . . 495.5 Maps shaped by pinwheel annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . 525.6 Pinwheel crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.7 Statistics of random pinwheel positions . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Supporting discussion of potential evolutionary origins of the common design 63

2

1 Supporting information on materials and methods: Analysis of pin-wheel statistics

1.1 Overview

In the following sections we present a complete description and assessment of the methods usedfor the analysis of pinwheel organization and statistics in the analyzed optical imaging data. Firstwe briefly characterize optical imaging and image pre-processing procedures. We then describethe calculation of maps of local column spacing from the optical imaging data and assess its con-sistency with Fourier methods. We then describe the methods used for the estimation of pinwheeldensity in orientation preference maps, the methods used for characterizing pinwheel densityvariability in subregions of orientation preference maps and for the assessment of pinwheel near-est neighbor distances. Finally, we describe the bootstrap resampling methods used to calculateconfidence intervals for all measured quantities and to perform statistical significance tests.

1.2 Optical imaging

Optical imaging of intrinsic signals was performed with an enhanced video acquisition system(Optical Imaging), as described previously (S 1, 2). The cortex was illuminated with orange(605nm) or red light (700nm) and visualized with a tandem lens macroscope attached to a lownoise video camera. The visual stimuli were moving, high-contrast rectangular wave gratings(0.1-0.5 cycle per degree) oriented at 0, 45, 90 or 135◦. Difference images were generated by sub-tracting the optical responses to presentation of each member of a pair of orthogonal gratings.

1.3 Preprocessing

For each hemisphere a region of interest (ROI) was defined containing the imaged part of area V1(and parts of area V2 in the case of ferrets). From each of the two difference maps I′(x) (0◦ − 90◦,45◦ − 135◦) inside the ROI we calculated a high-pass filtered map by

I(x) = I′(x)− J(x) (1)

whereJ(x) =

1W(x)

F−1 {K(k) I′(k)}

(2)

is a smoothed version of I′. F denotes Fourier transform and the Fermi-function

Khp(k) =1

1 + e−(khp−|k|)/βhp(3)

in Fourier space is parameterized by the high-pass cutoff frequency khp and the steepness β. Nor-malizing by W(x) =

∫ROI Khp(x′ − x) d2x′ in real space accounts for the boundary of the ROI. We

used βhp = 0.05 khp and khp = 2π/λhp with cutoff wavelength λhp = 1.4mm for the ferret and

3

λhp = 1.2mm for tree shrew and galago, substantially larger than the typical spacing of columnsin each species. Robustness of pinwheel densities against variation of high-pass cutoff wavelengthλhp is shown for the example of galago in Fig. S5, Section 1.6.

1.4 Column spacing

For each hemisphere, a 2-dimensional (2D) map of the local spacing of columns was calculatedusing a wavelet method introduced previously (S 3, 4). In addition to the high-pass filtering de-scribed above, difference maps I(x) were low-pass filtered in the Fourier domain by multiplicationwith the Fermi-kernel

Klp(k) =1

1 + e−(klp−|k|)/βlp, (4)

where βlp = 0.05klp, and the low-pass cutoff frequency klp = 2π/λlp with cutoff wavelength λlp =0.3mm for the ferret and λlp = 0.2mm for the galago and tree shrew. Morlet wavelets (kψ = 7, σy =1) were sampled with a spatial resolution of 0.05mm on n scales li equally spaced in an interval d(ferret, n = 19, d = [0.4, 1.3]mm; galago and tree shrew, n = 17, d = [0.3, 1.1]mm) using 16 waveletorientations θi = {0π/16, . . . , 15π/16}. The scale l∗ maximizing the angle averaged modulusof wavelet coefficients (estimated by cubic splines interpolation) defines the column spacing ΛIat location x. The 2D map of local column spacing Λ(x) was calculated by averaging columnspacings ΛI(x) obtained from the two difference maps.

Mean column spacings Λ = 〈Λ(x)〉x were cross checked by using an alternative estimationbased on Fourier analysis. From each preprocessed difference map I(x) we calculated its twodimensional power spectrum

P(k) =∣∣ I(k)

∣∣2 (5)

as the squared modulus of the Fourier transform of the map I(x). Averaging over the angle,

P(k) =∫ 2π

0dθP(k, θ) , (6)

yielded the one dimensional spectrum P. The dominant frequency q of the spectrum P determinesthe typical spacing through Λ = 2π/q. To estimate the frequency q we smoothed the spectrum Pwith a Gaussian kernel of width σ = 1.7mm−1 and fitted it with the function

f (k) = a0 exp

(− (k− a1)

2

2a22

)+ a3 + a4k + a5k2 (7)

where the ai are fitting parameters. The peak position q = a1 was obtained by a nonlinear least-squares fit based on a gradient-expansion algorithm. Finally, the values of typical column spac-ings obtained for the two difference maps were averaged yielding an estimation of mean columnspacing Λ for a given hemisphere based on Fourier analysis.

Fig. S1 shows mean column spacings Λ for the three analyzed species calculated by boththe wavelet and Fourier method. Values were generally very consistent. In tree shrews, e.g.,the correlation coefficient between the two methods was r = 0.99. Relative distances of column

4

1 40 800.5

0.6

0.7

0.8

0.9

1.0

1.1

1 5 100.40

0.50

0.60

0.70

0.80

0.90

1 10 200.4

0.5

0.6

0.7

0.8C

olum

n sp

acin

g

Hemispheres

Tree shrews FerretsGalagos

CBA

Figure S1. Method independence of mean column spacing estimation. (A to C) Column spacingsΛ as calculated by either wavelet analysis (blue crosses) or Fourier analysis (orange diamonds).Differences between both methods were very small (average relative difference in Λ: Tree shrew,1.5%, galago, 0.7%, ferret, 2.3%; relative distance between averages 〈Λ〉: 0.7%, 0.01%, 0.7%).

spacings Λ were on average 1.5%, and the distance of the average column spacing 〈Λ〉 was 0.7%.In galagos, the correlation was r = 0.99 with average distances 0.7% and distance of averages0.01%. In ferrets, we found r = 0.95, 2.3% and 0.7%, respectively (based on the N=73 mapssufficiently large to be analyzed by Fourier methods). Deviations between the Fourier and waveletmethod were largely due to a variation of column spacings across cortex. While generally smallin the galago, in both tree shrews and ferrets this spatial variation exceeded > 10% in some ofthe cases (see the examples in Fig. S2). The deviation between the two methods may be takenas an estimate of a systematic error of column spacing measurements. The potential systematicerror of the average pinwheel density 〈ρ〉 that may result would then amount to < 0.04 in treeshrews, < 0.001 in galagos, and < 0.05 in ferrets. We conclude that Fourier analysis confirms theestimations of column spacings obtained by the wavelet analysis.

1.5 Pinwheel density estimation - overview

The abundance of pinwheels in an orientation map (Fig. S2, A to C) can be quantified by the pin-wheel density which we define as the average number of pinwheels per orientation hypercolumn.Denoting the typical column spacing, i.e. the average spacing of adjacent iso-orientation columns,by Λ, and the frequency of occurrence of pinwheels per mm2 by ρ, the pinwheel density is givenby

ρ = ρΛ2 (8)

where Λ2 is the area of an orientation hypercolumn (S 5, 6). Defined in this way the pinwheel den-sity ρ is a dimensionless number that characterizes the layout of orientation columns independentof their absolute size. Because orientation columns exhibit large interindividual and systematicinterspecies differences in absolute column spacing (Fig. 2 A and (S 7, 5, 8, 3, 9)) the insensitivityof the pinwheel density to column spacing makes this dimensionless quantity a useful measurefor inter-individual and inter-species comparisons of orientation map design.

Within the visual cortex, even directly adjacent regions frequently exhibit strongly differing

5

0 1 2 30

1

0.2 0.4 0.6 0.8 1.0

2

3

4

Wavevector

Ferret

Tree shrew

Fermi

Map

Gauss

Pinwheel density via plateau

λ=0.5λ=0.0

Pow

er (

norm

aliz

ed)

Pin

whe

el d

ensi

ty

Low−pass filter cutoff wavelength

GalagoBA C

D E

L

R

R

L

(1/Λ)

1mm

R

L

(Λ)

Figure S2. (A to C) Optically recorded orientation maps in ferret (A), galago (B), and tree shrew(C) visual cortex (Section 1.2). Maps were high-pass filtered (Subection 1.3). Four pinwheels aremarked by white arrows in the magnified region in (A) (black frame). The upper left and lowerright pinwheels have the same topological charge, so does the pair lower left/upper right. In(C), two exemplary high (blue frame) and low (orange frame) pinwheel density regions are high-lighted. (D) Fermi- (as opposed to conventional Gaussian-) filtering efficiently eliminates highfrequency noise without deteriorating the signal. This is illustrated by comparing the average ra-dial component of the power spectrum of the galago map from (B) (high-pass filtered; normalized)with Gaussian and Fermi filter kernels in frequency representation. Fermi filtering effectively im-plements a cutoff in the frequency domain. (E) Operational definition of pinwheel density ρ: Valueof the ’plateau’ region where the spatially averaged pinwheel density ρλ = 〈ρλ(x)〉x is indepen-dent of low-pass cutoff wavelength λ. Left: ρλ for the two maps from C (orange: fit, short dashes,ρ =2.48; blue: fit, long dashes, ρ =3.78). Right: Choosing λ within the plateau (e.g. λ = 0.5)provided an estimate of the spatial location of pinwheels (marked by white points).

pinwheel arrangements (Fig. S2 C) and can vary substantially in the typical spacing of orienta-tion columns (S 3). To reliably quantify the layout of orientation columns in the visual cortex wetherefore developed an analysis method that respects this spatial heterogeneity. The pinwheeldensity ρ was calculated for geometrically defined subregions of each map from independent es-timates of local column spacing (Section 1.4) and of the frequency of pinwheel occurrence (Section1.6). To precisely estimate the typical local column spacing of a region we used a wavelet-basedmethod exhibiting relative errors in the range of only a few percent (S 3, 4). Pinwheel centers wereidentified and localized by calculating the zeros of band-pass filtered orientation polar maps. To

6

reliably determine the local pinwheel density we used a Fermi filter (Fig. S2 D) and consideredthe dependence of pinwheel density estimates using Eq. ((2)) on the filter cutoff wavelength (Fig.S2 E). Graphed against this wavelength, the estimated pinwheel density generally exhibited largeplateaus, indicating that in a wide range of filter parameters high frequency noise in the imagingdata was effectively eliminated while the pinwheel arrangement was essentially preserved. Wethus operationally defined the pinwheel density ρ of a region by the plateau value of the graph ofpinwheel density vs. filter cutoff-wavelength.

In contradistinction to the Fermi-filtering method, conventional filtering based on a Gaussianfilter kernel made it virtually impossible to effectively eliminate high spatial frequency measure-ment noise without substantially deteriorating the optical signal itself (Fig. S 2 D). In particular, itwas impossible to unambiguously estimate pinwheel densities as shown in Fig. S3 for ferret orien-tation maps. With conventional filtering the pinwheel density ρ was a monotonically decreasingfunction of the width (i.e. SD) of the Gaussian filter kernel.

0.0 0.2 0.4 0.6 0.80

2

4

6

8

Pin

whe

el d

ensi

ty

Filter kernel width (mm)

Figure S3. For conventional filter techniques us-ing Gaussian filter kernels, pinwheel density es-timates were ambiguous. In most analyzed ferretmaps, pinwheel density ρ vs. SD of the Gaus-sian filter kernel was a monotonically decreasingfunction. Thus, pinwheel densities cannot be de-fined objectively when orientation maps are pre-processed using Gaussian filters.

1.6 Pinwheel density estimation - methods

Pinwheel centers were identified as the zeros of Fermi-filtered orientation polar maps z(x) =I0◦−90◦(x) + iI45◦−135◦(x) obtained from the two difference maps. They were calculated by search-ing crossings of zero contours of the two difference maps (S 10).

To obtain an objective estimation of pinwheel density ρ, we varied the low-pass cutoff wave-lengths λlp over wide ranges (ferret, 90 values λlp equally spaced in [0.1, 0.99]mm; galago andtree shrew, 70 values in [0.1-0.79]mm; βlp = 0.05). We identified putative pinwheel centers foreach λlp and calculated smooth maps of local frequency of occurrence of pinwheels per mm2,ρ(x, λlp), by superimposing normalized Gaussians (with standard deviation σ0 = 0.25Λ) centeredat the identified pinwheel locations. Rescaling λ(x) = λlp/Λ(x), we defined the filter dependentlocal pinwheel density ρ(x, λ) = ρ(x, λ)Λ(x)2, the number of pinwheels in an area of size Λ(x)2

surrounding x as a function of the normalized low-pass cutoff parameter λ.For further evaluation we used two independent methods, the first based on visual inspection,

the second fully automated. For the first method, we divided orientation maps into rectangular

7

2.0 2.5 3.0 3.5 4.02.0

2.5

3.0

3.5

4.0

2.0 2.5 3.0 3.5 4.02.0

2.5

3.0

3.5

4.0

2.0 2.5 3.0 3.5 4.02.0

2.5

3.0

3.5

4.0

FerretsTree shrews GalagosPin

whe

el d

ensi

ty

Pinwheel density, fully automated method

CBA

Figure S4. Comparison of pinwheel densities by the fully-automated with those by the semi-automated method. (A to C) Pinwheel densities calculated by the two different methods werehighly correlated (tree shrew, r = 0.91, galago; r = 0.93; ferrets, r = 0.82).

regions of size . 5Λ2 defined on a grid (Fig. S2 C). For each region we calculated the filter depen-dent pinwheel density by averaging over locations, ρ(λ) = 〈ρ(x, λ)〉x. Generally, ρ(λ) was fairlyconstant over a broad range of cutoff wavelengths λ (Fig. S2 E). We operationally defined the pin-wheel density ρ of the region by the value of the plateau and estimated it by a constant fit basedon visual inspection. To do this unbiased, the ρ-axis was suppressed. Regions for which a plateaucould not be identified unambiguously (≈ 5%) were excluded from further analysis. Values fromdifferent regions were averaged weighted with their size in units of Λ2. The average density ρ ofa map was robust against variations, e. g. shifts or rotations, of the grid. In a set of cases for whichthe level of noise was relatively weak and spatially homogeneous, we calculated ρ(λ) and thusρ from a single region encompassing the entire map (ferret, 14 out of 82; galago, 7 out of 9; treeshrew, 8 out of 26). Estimations based on grids were consistent for these cases.

The second method applied an automated fitting scheme. At every location x, a piecewiselinear function

h(λ) = c0 + c1[−λ + λ0]+ + c2[λ− (λ0 + ∆λ)]+

was fitted to ρ(x, λ) (calculated for σ0 = 1Λ) in the interval 0.2 < λ < 1 (least-squares fit;[]+denotes rectification). The function h exhibits a plateau of value c0 in the interval [λ0, λ0 + ∆λ].The slopes c1 and c2 account for the λ-dependence of ρ(x, λ) outside of the plateau region (com-pare Fig. S2 E). To ensure identification of the plateau, we estimated the fit under variation ofλ0 and ∆λ (under constraints λ0 ≥ 0.2, ∆λ ≥ 0.4, and 0.2 + λ0 + ∆λ ≤ 1), and defined the localpinwheel density ρ(x) by c0 obtained from the best fit. Averaging over space yielded the pinwheeldensity ρ of a given region or map.

Finally, we estimated locations of pinwheel centers from ρ(x, λc(x)) (calculated for σ0 = 0.01Λ)where λc(x) is the normalized cutoff wavelength λ at the plateau center, estimated independentlyfor each x.

The fully- and semi automated method for estimating pinwheel densities ρ provided highlyconsistent results. This is demonstrated in Fig. S4 showing scatter plots of pinwheel densities ρobtained by the two methods. For each species, values were highly correlated: r = 0.91 for treeshrews, r = 0.93 for galagos and r = 0.82 for ferrets. For individual maps, differences in pinwheel

8

density ρ were on average 2.4% for the tree shrew, 2.7% for the galago, and 4.3% for the ferret.Average pinwheel densities obtained by the fully automated method were 〈ρ〉 = 3.13 [3.05, 3.21](mean [2.5%, 97.5%] standard confidence interval) for tree shrew, 〈ρ〉 = 3.19 [3.01, 3.37] for galago,〈ρ〉 = 3.19 [3.11, 3.27] for ferret, which, compared to the semi automated method, deviatiated by0.3%, 1.3%, 1.3%, respectively. For the grand average we obtained〈ρ〉 = 3.16 [3.10, 3.22], whichdeviated by 0.64% from the semi-automated result. The average pinwheel density for the dark-reared ferrets, 〈ρ〉 = 3.28 [3.14, 3.42] , deviated 1.8% from the semi automated value. For therandomized maps the fully automated method yielded 〈ρ〉 = 3.54 [3.46, 3.62], which differed by1.1% from the semi automated value. Thus, all values obtained with the fully automated methodwere close to those obtained with the semi automated method. Comparing fully automated val-ues, species averages deviated by less then 1% from the grand average, dark-reared ferrets byonly 3.7% (although this deviation was statistically significant) and random maps by 11.3%. Thus,the fully automated method confirmed our results that pinwheel densities are highly similar inthe three species and also similar in dark-reared animals whereas they deviate considerably inrandomized maps.

Thus, the starting point for both methods was the filter dependent pinwheel density ρ(x, λ).Using the semi-automated method the density ρ(x, λ) is first averaged over regions . 5Λ2 definedby an arbitrarily arranged regular grid superimposed on the map. Then the pinwheel density ρof the region was estimated by identifying the plateau region of the averaged filter dependentdensity ρ(λ). Using the fully automated method, the pinwheel density was estimated from theplateau of ρ(x, λ) by an automated fitting procedure before averaging the result over space. Theadvantage of the semi automated method is that the estimation of the plateau can be checked atevery step. In some regions, the plateau is more irregular with values fluctuating around a con-stant value, plateaus may be relatively short, or there may be multiple plateaus at different rangesof cutoff wavelength λ. Regions for which a plateau could not be determined unambiguously(regions with poor signal to noise ratio) were identified by visual inspection and excluded fromfurther analysis.

We tested the impact of low frequency noise on the pinwheel density by varying the high-passcutoff wavelength λhp underlying the analysis. As a representative example we analyzed galagomaps. Fig. S5 shows the galago average pinwheel density 〈ρ〉 for various cutoff wavelengths λhp.For each value of λhp, a completely independent analysis was performed including the calculationof the map of local column spacing Λ(x) and the filter dependent pinwheel density ρ(x, λ). Thefully automated method was used to estimate pinwheel densities ρ. Apart from the high-pass cut-off wavelength λhp, all parameters were identical for each data point. We found that the averagepinwheel density 〈ρ〉 was nearly constant between λhp = 1.0 and λhp = 2.0 demonstrating therobustness of the pinwheel density estimation against low frequency noise.

1.7 Pinwheel density variability in subregions

To calculate pinwheel density variability in subregions of size A (Fig. 2, C and E, Fig. 2, C and D),we sampled for each map in a given set (e.g. tree shrews, long-range interaction model, etc.) 105

circular shaped regions of various size and placed their centers at random locations of the map.

9

0.8 1.0 1.2 1.4 1.6 1.8 2.02.0

2.5

3.0

3.5

4.0P

inw

heel

den

sity

High−pass cutoff wavelength

Figure S5. Robustness of pinwheel density es-timation against variation of high-pass filtering.Galago average pinwheel density 〈ρ〉 for varioushigh-pass cutoff wavelengths λhp (in mm). Val-ues are (from left to right) 〈ρ〉 = 3.27, 3.21 3.19,3.17, 3.13, 3.13, 3.11. Over a broad range of cutoffwavelengths, the average pinwheel density 〈ρ〉 islargely filter independent.

Sizes of circular regions were uniformly distributed. Circular regions did, in general, only overlappartly with the ROI of a given map and pinwheel densities and area sizes were calculated forthe overlapping part only. Consequently, regions exhibited various sizes and regions with similarsize varied considerably in shape. To calculate pinwheel density variability for a given area Ai,we randomly selected from all regions in the set up to M=1000 regions with size in the interval[Ai, Ai + dA], where dA = min{|Ai+1 − Ai|, 0.1Λ2}, and calculated the SD of pinwheel densitiesand the number variance of pinwheels in these regions.

To assess statistical significance, we used bootstrap resampling (N=10000; see Section 1.9). Tothis end we obtained for every Ai a bootstrap distribution of SDs and number variances. Theaverages of these distributions for the three species are shown in Fig. 2 D and for the three speciesand the long-range interaction model in Fig. 3 F. For all animals, the model and the randomizedmaps, confidence intervals are shown in Fig. S10, Section 4.

We checked for accuracy and the influence of the ROI by analyzing “maps” of Poisson dis-tributed pinwheels with density ρ = π. Theoretically, an infinite 2D Poisson process is knownto exhibit a pinwheel density variability of SD(A) = √ρA−1/2 (see Section 5.7). We consideredtwo samples of Poisson maps each containing N=24 randomly generated maps: The first set con-sisted of maps on a rectangular area and with periodic boundary conditions (i.e. a torus) suchthat the circular regions with which we sampled pinwheel densities overlapped completely withthe maps. The second set consisted of Poisson maps identical to those in the first, but restrictedto the tree shrew ROIs. Subregions in the second set were sampled by the same procedure as forthe experimental maps. To facilitate comparison with the experimental data, the spatial scale inboth sets was chosen to be equal to column spacings from three shrew hemispheres. Since the ROIcontained only a subset of pinwheels, the total number of pinwheel was smaller in these maps (afactor of ∼ 1

6 ).Fig. S6 A shows means and 2.5 and 97.5 percentiles of bootstrap distributions of pinwheel

density variabilities for the two sets. In Fig. S6 B, curves were divided by√

πA−1/2. Both meansfollowed closely the theoretically expected curve up to area sizes of A ∼ 10 hypercolumns. Forlarger A, the set of maps confined to the ROIs started to deviate slightly. However, even up toarea sizes A ∼ 25 the deviation was smaller than ∼ 10% and the theoretically expected curvewas within 2.5% and 97.5% percentiles. Repeating the simulation for different samples of Poissonmaps, we could not observe a systematic trend of deviation towards smaller or larger values (data

10

not shown). Thus, the effect of the ROI on estimates of pinwheel density variability was relativelysmall for areas up to 25 hypercolumns.

1 10

0.6

0.8

1.0

1.2

1 100.1

1.0

PoissonPoisson+ROIPoissonPoisson+ROI

A B

Size (hypercolumns)Size (hypercolumns)

SD

SD

nor

mal

ized

Figure S6. Estimating pinwheel density variabilities in subregions with area size A: Precision andeffect of ROI. (A) mean (solid) and 2.5 and 97.5 percentiles (dashed) of bootstrap distributions ofpinwheel density standard deviations SD as a function of area size A. Red curves, N=24 “maps”of Poisson distributed pinwheels with density π per hypercolumn and periodic boundary condi-tions; yellow curves, the identical Possion maps, but confined to tree shrew ROIs (i.e. the totalnumber of pinwheels is reduced to∼ 1

6 and the geometry of regions of size A being sampled isnot circular, in general); black dashed curve, exact solution for an infinite 2D Poisson process,SD(A) =

√πA−1/2. (B) SDs and percentiles from (A) normalized by

√πA−1/2. Note that even

for the maps confined to a ROI the estimation follows closely the theoretically expected curve upto region sizes of A ∼ 10 hypercolumns and deviations remain fairly small (∼ 10%) up to sizes∼ 25 hypercolumns.

1.8 Pinwheel nearest neighbor distances

To calculate distributions of nearest neighbor distances of pinwheels with arbitrary charge (Fig. 2E) we identified for each pinwheel i in a given map the distances dij to all neighbors j 6= i of arbi-trary topological charge. Nearest neighbors to pinwheel i were found by minimizing dij. Distanceswere expressed in units of column spacing at the location of pinwheel i. All identified pinwheelswere included in these statistics. For distances between pinwheels of same (opposite) charge, onlydistances dij of pinwheel i to all pinwheels j of same (opposite) charge were considered. Notethat for each of the three statistics, the total number of distances is equal to the total number ofpinwheels.

When evaluating significances of differences of distributions between species, it is importantto note that the distances sampled from the same map cannot be considered independent datapoints. Thus, a direct comparison of nearest neighbor distributions by means of a KS-test would

11

be inappropriate. Instead, we used bootstrap resampling based on the assumption that distancesobtained from different maps are statistically independent as outlined in the following Section 1.9.

1.9 Confidence intervals and significance tests

All assessments of statistically significant differences between data sets were based on bootstrapresampling. For each data set (e.g. tree shrews, randomized maps, etc.) we sampled N=10000 ran-dom sets of identical size, each of which was generated by randomly selecting (with replacement)maps from the original set. For each data set we calculated the mean pinwheel density 〈ρ〉. Thebootstrap distributions of mean pinwheel densities for the different sets are shown in Fig. S9, Sec-tion 4. For each set we further calculated the pinwheel density variability SD(A) as a function ofregion size A. Thus, for every region size A we obtained a bootstrap distributions over N=10000values of SD(A). The mean of this distribution as a funtion of A is shown in Fig. 2 D for the threespecies and in Fig. 3 F for the three species together with the long-range interaction model. Fittingeach of these curves to c(〈ρ〉 /A)γ we obtained bootstrap distributions for the constants c and γ,shown in Fig. S10, E and F. Finally, we calculated distributions of nearest neighbor distances forpinwheels of arbitrary, same, and opposite topological charge for each of these sets. Bootstrap dis-tributions of the averages of these quantities, d , d++, and d+−, respectively, are shown in Fig. S11,D to F, Section 4. Confidence intervals were defined by the 2.5 and 97.5 percentiles of the bootstrapdistributions. Significance values for differences between two given data sets i and j were definedas p = min{2qj

i , 2qij}, where 100× qj

i is the left percentile of the interval of the bootstrap distribu-tion for set i that excludes the mean for set j. Calculating this p-value one can test the hypothesisthat the average of each of the two data sets is contained in the bootstrap distribution of averagesfor the other data set. This hypothesis is correct if the two data sets are statistically indistinguish-able. If p < 0.05, at least one of the two averages is located outside the 95% confidence interval ofthe other data set. We then reject this hypothesis and conclude that there is a significant differencebetween the two data sets.

12

2 Supporting information on materials and methods: Model

2.1 Overview

In this section, we state and derive the long-range interaction model and discuss its mathematicalproperties and the biological assumptions made. After defining the model, we point out that itis a representative example for an entire universality class of models. This class is specified by aset of mathematical symmetries of the dynamical equation that describes the formation and re-finement of the orientation preference map. We present a universal set of aperiodic solutions thatnear the threshold of pattern formation occur in any model from this class independent of modeldetails. This is the rational for terming the pinwheel statistics of these solutions “universal pin-wheel statistics”. As shown in the manuscript and further elaborated in Sections 4 and 5 of thisSOM, these aperiodic pinwheel-rich solutions as well as numerically calculated solutions of thelong-range interactions model agree well with the observed common organization of pinwheelarrangements in the visual cortex of ferret, galago, and tree shrew. To connect the long-rangeinteraction model to known features of visual cortical connectivity we then proceed by derivingit from a non-local model of pattern formation, in which distant cortical regions are linked byeffective connections that preferentially couple neurons with similar preferred orientations. Thisbiological derivation is complemented from a mathematical point of view by a formal argumentthat shows why – for any model within the considered universality class - some form of long-range interactions are expected to be necessary for stabilizing aperiodic solutions exhibiting theuniversal pinwheel statistics. We conclude by discussing the evidence for candidate biologicalsubstrates of the assumed long-range interactions. The non-local interactions incorporated in thelong-range interaction model are found consistent with the known anatomical and functional or-ganization of interactions in the visual cortex in carnivores, tree shrews and primates and duringthe relevant stages of visual cortical development.

2.2 The long-range interaction model

The long-range interaction model is defined by a dynamical equation for an order parameter fieldz(x, t) that completely characterizes the spatial pattern of preferred orientations at time t (thet-dependence will be omitted in the following). As an idealization, it is thus assumed that the de-velopmental formation of the orientation preference map can be described neglecting interactionswith other receptive field features. In the following, the layout of orientation columns is describedby a complex valued field (S 5, 6)

z(x) = |z(x)| ei2ϑ(x) (9)

where ϑ(x) is the orientation preference and |z(x)| a measure of the population averaged selectiv-ity at position x. The long-range interaction model is defined by

∂tz = LSHz + N3[z, z, z], (10)

where the linear part is the Swift-Hohenberg operator (S 11, 12)

LSH = r−(k2

c +∇2)2(11)

13

and the nonlinear part reads

N3 = (1− g) |z(x)|2z(x)− (2− g)∫

d2y Kσ (y− x)(

z(x)|z(y)|2 +12

z(x)z(y)2)

. (12)

The non-local interactions are weighted by a Gaussian kernel

Kσ =1

2πσ2 e−|y−x|2/(2σ2) (13)

with interaction range σ ≥ 0. The parameter 0 ≤ g ≤ 2 controls the relative strength of local andnon-local influences.

The choice of a Gaussian for the nonlinear interaction kernel Eq. (13) is made for simplicity.As explained in the following section, choosing other positive rotation-invariant kernel-functionsthat monotonically decay for distances beyond a typical interaction range σ will lead to similarbehavior.

Similarly, the Swift-Hohenberg operator Eq. (11) represents the simplest translation and ro-tation invariant differential operator that leads to a finite wavelength Turing-type bifurcation ofspatial patterns. It is used in our model for simplicity and to facilitate comparison with otherpattern forming systems (S 12, 13). Replacing the Swift-Hohenberg operator in Eq. (10) with aMexican hat-type convolution operator does not change the attractor states or dynamical behav-ior of the model near the pattern formation threshold.

The spectrum of the Swift-Hohenberg operator reads

λ = r−(k2 − k2

c)2

which is maximal for |k| = kc. For positive bifurcation parameter r > 0, the nonselective statez(x) = 0 is unstable and a pattern z with average column spacing close to Λ = 2π/kc emerges.The dynamics Eqs. (10-12) can be written as the gradient descent

∂tz(x) = − δ

δz(x)E[z] (14)

of the energy functional

E[z] = −∫

d2x z(x)(

r−(k2

c +∇2)2)

z(x) − 12

(1− g)∫

d2x |z(x)|4

+12

(2− g)∫

d2x∫

d2y Kσ (y− x)(|z(x)|2|z(y)|2 +

12

z(x)2z(y)2)

. (15)

Because of its variational structure Eq. (14), the long-range interaction model is an optimizationmodel. Its solutions are therefore guaranteed to converge to stable stationary states as time pro-ceeds.

14

2.3 Symmetry properties

Many properties of the long-range interaction model are insensitive to quantitative details of themodel equation. This results from a set of symmetry properties of this model that are sufficient tofix many of its stationary states and their stability properties. The long-range interaction model issymmetric with respect to translations,

F[Ty z] = Ty F[z] with Ty z(x) = z(x + y) , (16)

and rotations

F[Rβ z] = Rβ F[z] with Rβ z(x) = z([

cos(β) sin(β)− sin(β) cos(β)

]x)

(17)

of the cortical sheet. This means that patterns that can be converted into one another by translationor rotation of the cortical layers belong to equivalent solutions of the model, Eqs. (10-12), byconstruction. The dynamics is furthermore symmetric with respect to shifts in orientation

F[eiφ z] = eiφ F[z] . (18)

Thus, two patterns are also equivalent solutions of the model, if their layout of orientation do-mains is identical, but the preferred orientations differ everywhere by the same constant angle.Finally, the model is symmetric with respect to permutations

N3[u, v, w] = N3[w, u, v] . (19)

Here the non-linear operator is written in a trilinear form. This additional symmetry implies thatall two-orientation solutions, for instance real-valued solutions of the model, are unstable, whichin turn guarantees that all stimulus orientations are represented (S 14, 15).

2.4 Universal solution set near threshold

All models with the symmetry properties Eqs. (16-19) that exhibit a supercritical finite wave-length bifurcation exhibit a common set of stationary solution (referred to as planforms) near theinstability threshold (r � 1). These are superpositions of plane waves

z(x) = An−1

∑j=0

ei(ljkjx+φj) (20)

of n wavevectors

kj = kc

(cos

(jπn

), sin

(jπn

))(21)

distributed equidistantly on the upper half of the critical circle |k| = kc and real valued ampli-tude A. The binary variables lj = ±1 determine whether the mode with wave vector kj or withwavevector −kj is active. Because for any pair of anti-parallel wavevectors only one can be ac-tive in such a planform, there is no choice of variables lj for which the resulting pattern would

15

be real-valued. For this reason, such planforms are called essentially complex planforms (ECPs).Note that the layout of the resulting orientation map is independent of the pattern amplitude A.The value of A is determined by model parameters (see (S 15)). As a consequence of permutationsymmetry, all planforms of a given n have degenerate energy and stability properties (S 14, 15).In the large n limit, the pinwheel density averaged over all ECPs with a given n converges to π asshown in (S 16). The pinwheel statistics of this solution set for large n are largely insensitive to nand are called universal pinwheel statistics. They are characterized in Fig. 3, and in Section 4 andSection 5 of the SOM.

In the long-range interaction model, Eqs. (10-12), with g < 1 the order of stable planformsgrows for large σ as

n ≈ 2πσ/Λ (22)

and thus approximately linearly with the interaction range σ. This can be derived from the angle-dependent interaction function (S 15, 14)

g(α) = g + (2− g)2 cosh(k2

cσ2 cos α)

exp(−σ2k2c) , (23)

which describes the interaction between active modes with wavevectors k = kc(cos β, sin β) andh = kc(cos((β + α), sin(β + α)) and depends on the form of the nonlinear part Eq. (12). Oneshould note that for fixed σ and g < 1 in Eq. (12), ECPs with a range of n values are simultaneouslystable (S 14). Eq. (22) states the approximate center location of this range. For any model in thisuniversality class, the set of stable planforms is completely determined by the angle-dependentinteraction function such that models with identical interaction functions have an identical setof such attractors. Changing the exact shape of the kernel K will modify the precise form of theangle-dependent interaction function g(α), but preserve its essential properties because the shapeof g(α) is determined by the Fourier-transform of the kernel K. The Gaussian chosen for K inthe long-range interaction model is representative for unimodal kernels of range σ and allows theexplicit calculation of the interaction function g(α).

2.5 Derivation from long-range orientation selective interactions

In this section, we construct a pattern formation model, in which the orientation selectivity of long-range interactions are obtained from biologically plausible properties of neuronal connectivity.For this model we show that, near threshold, its dynamics are equivalent to the dynamics of thesimpler model, Eqs. (10-12) (S 15). Interactions between orientation preferences and selectivitiesof neurons at positions x and y are assumed to be proportional to a coupling strength functionW(x, y|z) that models the effective connectivity between y and x. Choosing

W(x, y| z) =1

2πσ2 e−|y−x|2

2σ2 e− |z(x)−z(y)|2

σ2z , (24)

these interactions will decay with the distance |y − x| and preferentially link neurons of similarpreferences. Here, the first part describes the distance dependent part of the coupling strength.The parameter σ is the range over which interactions decay. The z-dependent part is maximal forpairs of locations x and y at which the preferred angle is identical and minimal if the preferred

16

angle differs by 90◦. This coupling strength function mimics the patchy orientation-selective long-range connectivity of visual cortical neurons in cat (S 17, 18, 19, 20), ferret (S 21, 2), tree shrew (S1) and primate (S 22) and the structure of top-down interactions between V2 and V1 in primates(S 22, 23), while neglecting the effects of axial selectivity (S 1). The simplest term that couples theorientation preference of neurons at different positions is

Nlr = glr

∫d2y[z(y)− z(x)]W(x, y| z) . (25)

For glr > 0 the interaction drives coupled positions towards similar preferred orientations and forglr < 0 towards opposite (orthogonal) preferred orientations. Thus, for glr > 0, the interaction ofneurons is effectively fascilitatory and for glr < 0 it is effectively suppressive. We will see belowthat interactions need to be effectively suppressive to achieve equivalence of this structural modelwith Eq. (10). This is perhaps expected given that the coefficient of the nonlocal term in Eq. (12) isnegative for g < 2.

One should note that the non-local term Eqs. (24-25) satisfies the symmetries Eqs. (16-18).The dynamical equation for z is constructed by assuming that it can be decomposed into a localdynamics and the long-range coupling Eq. (25)

∂tz(x) = Llocz(x) + gsr|z(x)|2z(x) + glr

∫d2y[z(y)− z(x)]W(x, y| z) . (26)

Here, one should note that |z|2z is the only local cubic term that satisfies the symmetries Eqs.(16-18), Lloc is a local linear operator, and gsr and glr are the coefficients of local and long-rangeinteractions, respectively. For small z or relatively weak selectivity σz ∼ O(z), we can write

W(x, y| z) ≈ 12πσ2 e−

|y−x|22σ2

(1− |z(x)− z(y)|2

σ2z

)(27)

and

Nlr ≈ glr

∫d2y[z(y)− z(x)]

12πσ2 e−

|y−x|22σ2

(1− |z(x)− z(y)|2

σ2z

). (28)

= glrL[z] + glr N3[z, z, z] (29)

Thus, we obtain a non-local linear operator

L[z] = −z(x) +∫

d2y K(y− x)z(y) (30)

and a non-local cubic operator

N3[z, z, z] =1σ2

z|z(x)|2z(x) + 2

z(x)σ2

z

∫d2yK(y− x)|z(y)|2 − 2

|z(x)|2σ2

z

∫d2yK(y− x)z(y)

− z(x)2

σ2z

∫d2yK(y− x)z(y)− 1

σ2z

∫d2yK(y− x)z(y)|z(y)|2

+z(x)σ2

z

∫d2yK(y− x)z(y)z(y) . (31)

17

The linear term has a spectrum of eigenvalues

λL(k) = −1 + e−12 k2σ2

(32)

which adds to the spectrum of the operator Lloc. The first term in Eq. (32) is just a shift of the entirespectrum. The second term depends on k and thus changes the form of the spectrum. To describethe formation of a repetitive orientation map, the complete linear operator

Lloc[z] + glrL[z]

must exhibit a single finite wavelength maximum in its spectrum. Near pattern formation onset,the precise shape of the spectrum does not impact on the solutions and their stability.

The cubic term consists of a local and a non-local part. The following calculation shows thatits angle-dependent interaction function is identical to that of the simpler long-range interactionmodel, Eqs. (10-12). The angle-dependent interaction function for an arbitrary nonlinearity isgiven by

gj(α) = −e−ik0x[

N j3(eik0x, eih(α)x, e−ih(α)x)

+N j3(eih(α)x, eik0x, e−ih(α)x)

],

where k0 = kc(1, 0) and h(α) = kc(cos α, sin α). The six terms of the cubic part (Eq. (31))

N13 [z, z, z] = |z(x)|2z(x)

1σ2

z

N23 [z, z, z] = 2

z(x)σ2

z

∫d2yK(y− x)|z(y)|2

N33 [z, z, z] = −2

|z(x)|2σ2

z

∫d2yK(y− x)z(y)

N43 [z, z, z] = − z(x)2

σ2z

∫d2yK(y− x)z(y)

N53 [z, z, z] = − 1

σ2z

∫d2yK(y− x)z(y)|z(y)|2

N63 [z, z, z] = +

z(x)σ2

z

∫d2yK(y− x)z2(y)

18

yield

g1(α) = − 2σ2

z

g2(α) = − 2σ2

z

(ek2

c σ2(cos α−1) + 1)

g3(α) =4σ2

ze−

12 k2

c σ2

g4(α) =2σ2

ze−

12 k2

c σ2

g5(α) =2σ2

ze−

12 k2

c σ2

g6(α) = − 2σ2

ze−k2

c σ2(cos α+1) .

Only two of these functions, g2(α) and g6(α), depend on α. The other functions provide an offsetto the interaction function and have an effect equivalent to the local term in Eq. (26). The fullangle-dependent interaction function of Eq. (26) then reads

g(α) = −2gsr − glr4σ2

z

(1− 2e−

12 k2

c σ2)− glr

4σ2

zexp

(−k2

cσ2) cosh(k2

cσ2 cos α)

= g1 + 2g2 exp(−k2

cσ2) cosh(k2

cσ2 cos α)

= g2

(g1

g2+ 2 exp

(−k2

cσ2) cosh(k2

cσ2 cos α))

,

with effective coefficients

g2 = −2glr

σ2z

g1 = −2gsr − glr4σ2

z

(1− 2e−

12 k2

c σ2)

.

The shape of g(α) thus depends only on the ratio of g1 and g2 an is identical to that of the long-range interaction model, Eq. (10-12). Only the shape and not the absolute value of g(α) is relevantto pattern selection, i. e. the determination of stable solutions near threshold (S 12, 24, 15). Further-more, one of the effective coefficients g1, g2 can always be absorbed by a rescaling of the field z.One can thus, without loss of generality, reparametrize the model by introducing a single effectivecoupling parameter g. The substitution

2− g = g2

g = g1 (33)

identifies the angle-dependent interaction function of the structural model, Eq. (26), with thatof the long-range interaction model, Eq. (10). With this choice of parametrization, the coupling

19

parameter g is the minimal value of the angle-dependent interaction function for large σ. That theabove identification requires effectively suppressive long-range interactions glr < 0 and that thestability of ECPs with large n requires strongly suppressive long-range interactions is revealed bysolving the mappings Eq. (33) for glr and gsr:

glr = −σ2z

2(2− g)

gsr =32

(43− g)− 2(2− g)e−

12 kcσ2

The coefficient glr of the non-local term in the structural model is thus always negative for g < 2.For large σ, the coefficient of the local term gsr becomes positive if g < 4/3. Hence, for g < 4/3and large σ the dynamics is exclusively stabilized by the non-local suppressive interactions.

These results establish that the effective long-range interactions implemented in Eq. (10) canbe mediated by a system of patchy long-range connections as present in the primary visual cortex.In particular, they show that near threshold, the model based on long-range orientation selectiveinteractions, Eq. (26), and the long-range interaction model, Eqs. (10-12), exhibit the same setof attractor states and are predicted to show similar behavior. In the parameter regime in whichlarge n-ECPs with universal pinwheel statistics are stable (g < 1, σ > Λ), the effective interactionsmediated by the patchy long-range connections are strongly suppressive and solely responsiblefor pattern stabilization.

2.6 Energy functional of the full model

The energy functional of the model, Eq. (26), is given by

F [z] = −∫

d2x z(x)Llocz(x) − 12

gsr

∫d2x |z(x)|4

+ glrσ2

z4πσ2

∫∫d2w d2y e−

|y−w|22σ2 e

− |z(w)−z(y)|2

2σ2z . (34)

The first term denotes the energy of the linear operator while the second that of the local non-linearpart. Computing the functional derivative of the third term, FNonlocal , results in

δFNonlocal [z]δz(x)

=σ2

z2πσ2

∫∫d2w d2y δ(x−w)e−

|y−w|22σ2 e

− |z(w)−z(y)|2

2σ2z (z(w)− z(y))

(1

2σ2z

)σ2

z2πσ2

∫∫d2w d2y δ(x− y)e−

|y−w|22σ2 e

− |z(w)−z(y)|2

2σ2z (z(w)− z(y))

(− 1

2σ2z

)=

2σ2z

2πσ22σ2z

∫d2y e−

|y−x|22σ2 e

− |z(x)−z(y)|2

2σ2z (z(x)− z(y))

=1

2πσ2

∫d2y e−

|y−x|22σ2 e

− |z(x)−z(y)|2

2σ2z (z(x)− z(y)) ,

showing that Eq. (34) is indeed the energy functional of the model, Eq. (26).

20

2.7 Scaling argument for long-range interactions

In the long-range interaction model, Eqs. (10-12), aperiodic pinwheel-rich solutions exhibiting theuniversal pinwheel statistics are attractor states if non-local interactions are suppressive (g < 1)and sufficiently long-ranging. Conversely, if the interaction range σ is smaller than a critical value,only periodic patterns with crystalline pinwheel arrangements or pinwheel-free orientation mapsare stable (S 14, 15). Quantitatively, this is reflected by the fact that the number of active modesof stable solutions in this model is exactly proportional to the interaction range σ. Is this a specialproperty of Eqs. (10-12) or is it rather a general property of models in the universality class definedby the symmetries Eqs. (16-19)? A simple scaling argument is sufficient to demonstrate that in anysuch model, in which nonlinear interactions have a typical spatial range σ, there will be a criticalinteraction range below which aperiodic patterns, and in particular patterns with a large numberof active modes, cannot be stable solutions near the instability threshold.

Key to this argument are two general results that can be mathematically demonstrated withoutreference to model details in this model class (S 14). First, essentially complex planforms Eq. (20)with n > 1 are in general unstable, when the angle-dependent interaction function g(α) satisfiesthe inequality g(α) > g(0)/2 for all angles α and second, all such planforms with n > nc areunstable if g(α) > g(0)/2 for α < π/nc. The second property implies that a model can only havestable ECP solutions with n active modes if its angle-dependent interaction function g(α) exhibitsa peak of width less than π/n centered on α = 0. The amplitude of this peak must be such thatg(α) drops below half its peak value g(0) at α = π/nc. Note that for model, Eqs. (10-12), thefunction g(α) has peaks of width ∼ σ/Λ centered at α = 0 and α = π and the peak amplitude isindependent of σ for fixed coupling parameter g and large σ.

Dimensional analysis implies that the width of such a peak will in general be inversely propor-tional to the spatial range of nonlinear interactions in the model. One can see from the definitionof the interaction function g(α) in Eq. (23) that the width Δα of a peak of g(α) corresponds to acharacteristic scale Δk of the nonlinear interactions in Fourier space. Because distances in Fourierspace are inversely proportional to the corresponding distances in real space it is expected in gen-eral, thatΔα ∼ Δk ∼ 1/σv . In general, the number of active modes should therefore decrease withdecreasing distance σv and there will be a critical range below which no aperiodic planforms Eq.(20) are stable. In this sense, the stability of aperiodic pinwheel rich planforms Eq. (20) exhibit-ing universal statistics requires the presence of an effective spatial interaction range longer than aminimum critical value. This argument demonstrates that the influence of the range of non-localinteraction on the formation of aperiodic pinwheel rich patterns exhibiting universal pinwheelstatistics found in the model, Eqs. (10-12), is representative for the entire model class defined bythe symmetries, Eqs. (16-19), near the pattern formation threshold.

One should note that the presence of long-ranging interactions is a necessary but not a suffi-cient condition for the stabilization of large n-ECP solutions from the universal solution set. Theabove argument does not use a sufficient condition for the stability of such solutions. As shown in(S 14, 15), the range of stable n-ECPs is bounded from above and from below by two different in-stability mechanisms: (i) an intrinsic instability by which solutions with n modes decay into oneswith lower n, if n is too large (ii) an extrinsic instability mechanism by which solutions with a toolow number n of modes are unstable to the growth of additional active modes. These instabilities

21

constrain the range of stable n to a small finite set around a typical n that increases with σ. Thereis a simple picture for both instabilities: Each active mode can suppress the growth of additionalFourier components in a vicinity ∼ 1/σ. If two active modes are located closer than this criticaldistance on the critical circle they will strongly compete and one of them will extinguish the other.If, on the other hand, two neighboring active modes are separated substantially farther than thisdistance they cannot prevent the growth of an interspersed additional mode and the pattern willdecay into one with a larger n. In qualitative terms, these are the two instabilities bounding thestability region of an n-ECP. The condition used for the above argument represents only an upperbound to the onset of intrinsic instability.

2.8 Biological interpretation

There are well-known anatomical substrates for orientation-selective long-ranging interactions inthe visual cortex of carnivores (S 25, 18), scandentia (S 26, 1, 27), and primates (S 22, 28, 23, 29, 30).Anatomically, long-ranging interactions among neurons located in different hypercolumns in V1may be mediated by long-range tangential connections within V1 (S 26, 17, 31, 22, 28, 1) and byreciprocal systems of connections between V1 and V2 (S 23, 29, 30, 32, 33) or by callosal connectionsbetween the two hemispheres (S 18, 27, 34). It is worth mentioning that inter-areal and inter-hemispheric connections in fact appear to be involved in shaping the spatial layout of orientationcolumns within the primary visual cortex (S 35). The most obvious candidate circuit however areintra-areal connections within V1 that span at least several hypercolumns (S 31, 1, 36).

These intra-areal connections develop concurrent with the orientation preference map and arealready present in some form early in development, presumably in register with the emerging ori-entation columns (S 37, 38, 39, 40, 41). They are highly plastic and shaped by activity-dependentmechanisms (S 37, 31, 2). Interactions are thus able to stay in register with the pattern of orienta-tion columns even if the orientation preference map undergoes slow gradual changes (S 40, 2, 35).Although tangential long-range connections represent potentially only a limited fraction of lo-cal neuropil (see however (S 42)), they can substantially impact the level of impulse activity (S43, 36, 44). For instance, in cat visual cortex, silencing neurons in an individual orientation col-umn can change stimulus driven firing rates of neurons located several millimeters from the in-activation site by more than 100% (S 45). Long-range interactions are presumably frequently ofsuppressive nature (S 46, 47, 44, 48). In response to natural scene stimuli, for instance, visualcortical responses are substantially sparser when large regions of the visual field are stimulatedengaging long-range intracortical interactions than when only the classical receptive field of neu-rons is stimulated (S 48). Under such conditions local inhibitory neurons are strongly activated.Such effects are consistent with the fact that long-range intracortical connections originate largelyfrom principal excitatory neurons because their synapses target both excitatory and inhibitory lo-cal neuron populations (S 49, 30). As a result the net effect on principal neuron firing rates dependson the activation state of local networks and will generally become suppressive for high contraststimuli. Thus the horizontal range and orientation specificity as well as the supressive nature ofthe assumed interactions in the model appear fully consistent with the anatomy and physiologyof visual cortical interactions.

22

2.9 Mathematical idealizations

Both the long-range interaction model Eqs. (10-12) and the structural model Eqs. (24-25) aremathematically highly idealized descriptions of the formation of a system of orientation columnsby network self-organization. In particular, they do not provide an explicit characterization ofactivity patterns that drive the self-organization process; they neglect interactions that would co-ordinate the layout of orientation preference columns with the layout of other columnar systemssuch as ocular dominance, spatial frequency or direction preference columns (S 50, 7, 51, 52, 10), orwith the retinotopic representation of stimulus position (S 53, 54, 55). Furthermore, they assumea perfect homogeneity of column spacing that is not observed in the visual cortex (S 3, 35). It istherefore worthwhile to briefly discuss whether our model predictions are expected to be sensitiveor robust with respect to relaxing these idealizations.

Concerning interactions with other columnar systems (S 50, 7, 51, 52, 10), theoretical analysesindicate that our models can be easily generalized to include such interactions. We have recentlyshown for the example of a system of orientation columns interacting with a system of oculardominance columns that it is relatively easy to construct models in which interactions becomeeffectively hierarchical although each system can in principle influence the behavior of the other(S 56). Under such conditions, there is one dominant map that shapes the other while its own lay-out is primarily determined by intrinsic interactions. If inter-map coupling is of this type and theorientation preference map is the dominant structure, all layout properties of orientation columnswill be identical to those found in an uncoupled model. The other systems will nevertheless ex-hibit systematic geometric relationships with the system of orientation columns because the non-dominant map is nevertheless matched to the structure of the dominant map. One should alsonote that previous studies indicate that models including intermap interactions do not providea simple explanation of the common design. Simulations of models for the coordinated devel-opment of orientation preference and ocular dominance columns, for instance, exhibit pinwheeldensities that substantially deviate from the common design (S 6, 56).

Compared to such interactions between different column systems, interactions that couple thedevelopment of orientation preference maps to the retinotopic map are mathematically of a dif-ferent kind. Inclusion of such interactions can influence the symmetry properties of the resultingdynamics of the orientation preference map, reducing the separate rotation and orientation shiftsymmetries, Eqs. (17) and (18), to a single symmetry under joint rotations of space and of preferredorientation (S 57, 58), called shift-twist symmetry (S 59, 60, 61, 62, 63). Previous studies indicatethat in the absence of long-range interactions this reduced symmetry favors the crystallization ofpinwheels into periodic arrays over the formation of pinwheel sparse patterns of iso-orientationstripes by pinwheel annihilation (S 60, 61, 63, 64). The long-range interaction model used in ourstudy can be extended to the reduced shift-twist symmetry such that the pinwheel statistics in thelong-range dominated regime can remain essentially unaffected (S 64). This shows that the the-ory developed above is robust to the inclusion of interactions between the emerging orientationpreference map and the retinotopic organization.

The long-range interaction model favours the same wavelength of orientation columns at ev-ery position of the model cortex and its solutions are characterized by the dominance of a singletypical column spacing. In the visual cortex, however, orientation column spacing is itself a space

23

variant property that can vary substantially across an individual area (S 3, 35). As a consequencethe Fourier components of cortical orientation maps typically exhibit power over frequency bandsof finite width and show a quasi-continuous distribution of power within an annulus in Fourierspace (e.g. (S 7)). The stationary solutions of the long-range interaction model thus differ fromthe data in being composed of a discrete set of Fourier components of a single dominant wave-length. This property is expected to change towards a more realistic distribution of Fourier compo-nents when perfect translation invariance in the model is broken by the introduction of a realisticmap of local column spacing. The models considered in our study furthermore lack biologicaldetail in that they do not provide an explicit description of the activity patterns that drive theself-organization process. Theoretically, it is well understood that such activity-patterns enter thedevelopmental dynamics only via averages over statistical ensembles of activity patterns and thenare captured by an autonomous dynamics as in the models considered here (see e.g. (S 65, 6, 14)).As our analysis shows that an entire symmetry class of equations will essentially behave as ourrepresentative model Eqs. (10-12), one also expects a wide multiplicity of activity ensembles andmicroscopically detailed learning rules to lead to models that behave as predicted by the long-range interaction model. It is, however, not easy to conceive constructing such models from activ-ity patterns that are restricted to individual hypercolumns and driven by simple stimuli coveringonly the classical receptive field of a model cell (see e.g. (S 66)). Rather, it is much more natural toexpect microscopically detailed models to behave as the long-range interaction model if corticalactivity patterns cover larger regions of the visual cortex and if driven by complex scene stimuliinstead of simplified stimuli covering only the classical receptive field of a small set of neurons(S 67, 68). Analyzing models including such additional biological detail constitutes an interestingdirection for further studies.

24

3 Supporting information on materials and methods: Numerical inte-gration and pinwheel identification in model maps

3.1 Numerical solution of the model

Numerical solutions of Eq. (10) were obtained using a spectral method satisfying periodic bound-ary conditions. Both the spatial derivatives of the linear part (11) and the computationally de-manding convolutions of the nonlinear part (12) can be treated accurately and efficiently in theFourier domain. Spatial derivatives become multiplications avoiding phase errors commonly in-troduced when calculating derivatives with finite difference methods. To solve Eq. (10) numeri-cally in a rectangular region, the field z(x, y, t) was sampled on a regular grid

xi =(i− 1)Lx

Nx, i = 1, . . . Nx

yj =(j− 1)Ly

Ny, i = 1, . . . Ny (35)

with Lx and Ly the dimensions of the region and Nx and Ny the number of sample points in the xand y directions, respectively. The amplitudes in the frequency domain z((qx)k, (qy)l , t) are relatedto the sampled field z(xi, yj, t) through the discrete Fourier transform

z =Nx

∑i=1

Ny

∑j=1

z(xi, yj, t) e−i(qx)kxi e−i(qy)lyj (36)

where the wave numbers are confined to the set (qx)k = 2πkLx

with −Nx2 + 1 ≤ k ≤ Nx

2 and to

(qy)l = 2πlLy

with −Ny2 + 1 ≤ l ≤ Ny

2 , respectively. By the backward transform

z =1

Nx Ny

Nx/2

∑k=−Nx/2−1

Ny/2

∑l=−Ny/2−1

z((qx)i, (qy)i, t) ei(qx)kxi ei(qy)lyj (37)

the field z can be recovered from the modes z. In Fourier representation, the linear part (11) isdiagonal with eigenvalues

λ(qx, qy) = r− (q2c − (q2

x + q2y))

2 , (38)

that are positive around the critical wavenumber qc = 2πΛ only. The Fourier transform (36) maps

Eq. (10) to a system of nonlinear ordinary differential equations

˙z = λz− N(z, t) (39)

in which the coupling among the components of z is mediated through the nonlinearity

N(z, t) = (1− g)z ∗ z ∗ ˜z + (g− 2)(

z ∗ (Kσ z ∗ ˜z) +12

˜z ∗ (Kσ z ∗ z))

, (40)

25

the Fourier representation of the nonlinear part (12). The nonlinear term (40) was evaluated bypseudospectral methods where convolutions in real space are calculated by products in Fourierspace while convolutions in Fourier space are calculated by products in real space, respectively.For a finite collection of amplitudes this method leads to aliasing of modes which can cause nu-merical instabilities. However, the strong linear damping by the biharmonic term suppressescontaminated modes provided there is sufficient resolution in real space. For a cubic nonlinear-ity, modes with wavenumbers qalias ≥ qmax

2 = nπ2L are subject to aliasing. Therefore, the maximal

wavenumber qmax must be chosen large enough to ensure qmax > 2(qc + δ/2) where δ is the widthof the range of wavenumbers around qc not efficiently suppressed by the linear part (38). Numer-ical tests revealed that for near threshold a ratio of qc

qmax≤ 2

5 provided sufficient suppression of alleffects due to aliasing.

Because of the stiffness of the biharmonic operator contained in the linear part (38), an implicittemporal integration scheme is demanded, since an explicit time-marching procedure would re-quire the use of an unacceptably small time step (S 69). To achieve this, the linear term wasintegrated using an exponential propagation procedure. The nonlinear term was treated explic-itly by a predictor corrector scheme, interpreting Eq. (39) formally as a system of linear equationsmodified by a nonlinear forcing term (following (S 70)). Multiplying both sides of the system (39)by the integrating factor exp(−λt) leads to

e−λt′ z(t′)∣∣∣t+∆t

t=∫ t+∆

tdt′ N(z(t′), t′)e−λt′ (41)

after integration over one time step ∆t. Assuming a sufficiently smooth solution z the nonlinearterm N can be approximated in the time interval t ≤ t′ ≤ t + ∆t by a linear function

N(z(t′), t′) ≈ N0 + (t′ − t)N1 (42)

where

N0 = N(z(t), t) ,

N1 =N(z(t + ∆t), t + ∆t)− N(z(t), t)

∆t(43)

are determined from the solution z at the endpoints of the time interval. Inserting the linearapproximation (42) with (43) into (41) one obtains

e−λ(t+∆t)z(t + ∆t) = e−λt z(t) + N0

∫ t+∆t

tdt′ e−λt′ + N1

∫ t+∆t

tdt′ (t′ − t)e−λt′ . (44)

After performing the integrals, this results in the time marching scheme

z(t + ∆t) = eλ∆t z(t) + N0eλ∆t − 1

λ+ N1

eλ∆t − (1 + λ∆t)λ2 (45)

for advancing z from t to t + ∆t. Note that expression (45) is still implicit since the term N1 dependsalso on z(t + ∆t). To evaluate z(t + ∆t), Eq. (45) was solved iteratively by a predictor-corrector

26

procedure. Setting in a first step N1 = 0, the obtained solution z(t + ∆t) was used to estimateN1, and to reevaluate z(t + ∆t) with this guess in a second step. This two step procedure alreadyprovided sufficient accuracy for our demands (see below). This temporal integration scheme isaccurate to 2nd order (S 70).

To ensure accuracy of the solution, the numerical error was controlled by means of an adap-tive step size procedure (S 69) adjusting the time step ∆t after every step in order to maintain apredefined accuracy ε. After each step the distance

∆z = |z∆t − z∆t/2| (46)

between solutions z(t + ∆t) obtained for step size ∆t and ∆t2 was calculated. If the maximal frac-

tional error defined by

∆ =max {∆z}

max {|z∆t/2|}(47)

exceeded the desired accuracy ε the step was reevaluated using the rescaled step size

∆′t = c∆t( ε

∆

)1/3(48)

where the exponent accounts for the scaling of the error ∆ ∼ ∆t3 in the used integration schemeand c is a safety factor. To ensure a maximal possible step size during integration, a scaling inverseto (48) was applied to the subsequent step whenever the error ∆ was below the desired accuracyε. For all simulations ε = 10−3 and c = 0.9 was used.

Solutions were obtained on a 128× 128 mesh using aspect ratios of Γ = 17− 25 (Γ = D/Λwhere D is the diameter of the system). Initial conditions z0 = z(t = 0) were band-pass filteredGaussian random fields with spectral width δk = kc and average amplitude A =r. Computerruntime increased approximately linear with the time t rescaled by the intrinsic time constant T =1/r of the linear dynamics (N3 = 0). The speed of marching was ≈ 1× 103/180s= 1× 106/50hon a Pentium 4 (2GHz). Consistency of the results was tested by single runs using an increasedspatial resolution of 256× 256.

3.2 Synthesizing model maps and pinwheel identification

Pinwheel centers were identified by the crossings of the zero contour lines of the real and imagi-nary part of the field z. To analyze pinwheels in numerical solutions of Eq. (10) (Fig. 4 D), solutionsz(k, t) to Eq. (39) were synthesized on a 512× 512 grid providing sufficient spatial resolution ofthe field z(x, t) in real space. Pinwheels were identified in each map in a quadratic subregion of3/4 of the total width spanning an area of 12Λ× 12Λ to 21Λ× 21Λ dependent on the aspect ratioΓ.

To calculate the expectation value of the pinwheel density 〈ρ〉 in an ensemble of planforms,Eq. (20), of order n (Fig. 4 C), planforms with randomly chosen sets of wavevector directions ljand phases φj were synthesized with aspect ratio Γ = 32 on a 2048× 2048 grid. In each planform,pinwheel locations were identified within a square subregion of size 8Λ× 8Λ. Realizations were

27

collected until sufficient precision of average pinwheel density estimate 〈ρ〉 was reached (mea-sured by the standard error measure (SEM) ∆ = s/

√N, were s is the standard deviation (SD) of

densities ρ and N the number realizations). The afforded precision of ∆ < 0.03 for 4 ≤ n ≤ 14 and∆ < 0.01 for 15 ≤ n ≤ 20 required between 100 and 1000 realizations per order n.

For calculating the pinwheel density variability and nearest neighbor statistics (Fig. 3, E toH), N = 26 ECPs (n = 20, Eq. (20)) were synthesized with average column spacings and ROIsmatched to the N = 26 tree shrew hemispheres.

28

4 Statistical characterization of the common design

4.1 Overview

The common design as presented in Fig. 2 of the main manuscript is characterized by the virtualidentity of i) pinwheel density, ii) pinwheel density variabilities as a function of subregion size andiii) three kinds of nearest neighbor (NN) distance distributions. To provide an objective quantita-tive assessment of all of these features and of their similarity in the different species, we definedsix distinct measures that comprehensively quantify properties (i-iii). The most important statisticis the pinwheel density 〈ρ〉 defined in Section 1. To quantify the dependence of pinwheel densityvariabilities on the size of cortical subregions, we defined a variability exponent, γ, and a vari-ability coefficient, c. These were estimated by fitting the SD(A) curves with a general power lawfunction. To assess the similarity of the NN distance distributions we used the average distancesof NN pinwheels, d, and the average distances of NN pairs of equal and opposite topologicalcharge, d++ and d+−, respectively. For all of these six parameters, we calculated means and boot-strap confidence intervals for the three investigated species and also for the dark-reared ferrets.Our data set enabled us to measure all of these quantities with a high degree of precision, oftenwith relative errors of only a few percent. To assess for all measures the similarities between thespecies we compared both the mean values in absolute terms and the overlap of bootstrap con-fidence intervals. These data as well as comparisons for phase randomized maps and solutionsof the long-range interaction model (n = 20, ECPs, Eq. (20)) are displayed in Figs. S7 and S8and Tabs. S1-3. In the following Sections 4.2-4.4, we first summarize the results of the differentcomparisons and then present all parameter estimates in full detail in Sections 4.5-4.7.

4.2 The common design

Our additional analyses confirmed the conclusion that the three investigated species as well asthe dark-reared ferrets exhibit a common design of orientation columns. Relative differences inthe mean values of the parameters between species were on average below 9% and with one ex-ception below 5% (Tab. S2). Even the maximal differences between species average values werealmost all below 7% (Tab. S2); thus all six quantities were very similar in absolute terms. In agree-ment with this overall similarity, almost all confidence intervals for the considered parametersoverlapped in the different species, indicating that pairwise differences of the mean were typi-cally not statistically significant (Fig. S7). In fact, no pairwise difference among ferrets, galagos,and dark-reared ferrets was statistically significant (Tab. S3). Tree shrew maps could not be sta-tistically distinguished from the other species by the variability exponent γ and the unsigned andopposite topological charge mean NN distances, d and d+− (Tab. S3). For the variability coeffi-cient c, a significant differences was only found between tree shrew and galago and for the sametopological charge mean NN distances, d++, between tree shrew and the other species (Tab. S3).In absolute terms, the differences were relatively small, 13.3% for the variability coefficient c andonly 4.7% for the same topological charge mean NN distance d++ (Tab. S3). The calculated sta-tistical significance of these two differences thus rather reflects the high precision with which thelayout parameters were estimated by our analyses than a substantial difference in layout.

29

Figure S7. Common design: Comparison of statistics for galago (G), ferret (F), tree shrew (TS),and dark-reared ferrets (D). Bars are centered around the population mean and span the 95%confidence intervals listed in Tab. S1. (A) Pinwheel density 〈ρ〉. (B) Pinwheel density variabilityexponent γ and coefficient c. (C) Pinwheel nearest neighbor statistics d, d++, and d+−.

Both of these small variations within the common design observed in the tree shrew can betraced to a feature of orientation column layout that is more pronounced in the tree shrew com-pared to other species: In tree shrew visual cortex, orientation columns near the border betweenareas V1 and V2 exhibit a tendency towards a more stripe-like organization than seen in the otherspecies. This feature leads to a slight increase of pinwheel density variability for large analyzedregions. In addition, this stripe-like region exhibits an increased fraction of closely space pin-wheel pairs of same charge, such that the mean NN distance for same charge pinwheels is slightlydecreased compared to the other two species. As already stated above, these species specific dif-

30

Table S1. Quantitative assessment of the common design in the three analyzed species, in themodel prediction (n = 20, ECPs, Eq. (20)) and in randomized maps. Listed are mean, and [2.5,97.5] percentile confidence intervals obtained by bootstrap resampling for the six parameters pin-wheel density 〈ρ〉, variability exponent γ, variability coefficient c, NN pinwheel distances of arbi-trary (d), equal (d++), and opposite (d+−) topological charge.

〈ρ〉 γ cTree shrew 3.12 [3.06, 3.18] 0.47 [0.45, 0.50] 0.85 [0.80, 0.89]

Galago 3.15 [2.97, 3.30] 0.49 [0.45, 0.55] 0.74 [0.68, 0.80]Ferret 3.15 [3.08, 3.22] 0.51 [0.46, 0.55] 0.79 [0.73, 0.86]

Dark-reared ferret 3.22 [3.13, 3.30] 0.52 [0.47, 0.58] 0.73 [0.65, 0.81]Model 3.17 [3.07, 3.26] 0.49 [0.44, 0.57] 0.71 [0.63, 0.77]

Randomized maps 3.50 [3.41, 3.59] 0.52 [0.47, 0.58] 0.71 [0.67, 0.76]

d d++ d+−

Tree shrew 0.376 [0.371, 0.382] 0.531 [0.525, 0.538] 0.416 [0.408, 0.424]Galago 0.379 [0.360, 0.401] 0.557 [0.542, 0.577] 0.414 [0.390, 0.442]Ferret 0.371 [0.363, 0.378] 0.554 [0.547, 0.562] 0.411 [0.401, 0.420]

Dark-reared ferret 0.374 [0.363, 0.387] 0.554 [0.546, 0.564] 0.414 [0.400, 0.430]Model 0.381 [0.374, 0.389] 0.573 [0.568, 0.579] 0.409 [0.399, 0.418]

Randomized maps 0.350 [0.343, 0.356] 0.534 [0.526, 0.543] 0.373 [0.365, 0.382]

Table S2. Summary: Quantitative assessment of the common design in the three species andits comparison to solutions of the long-range interaction model (n = 20, ECPs, Eq. (20)) and torandomized maps ensemble. Listed are the maximal and average relative pairwise distances.

〈ρ〉 γ c d d++ d+−

Species differences Maximum 1.0% 6.6% 13.3% 2.1% 4.7% 1.3%Average 0.7% 4.4% 8.9% 1.4% 3.2% 0.8%

Difference between Maximum 1.5% 3.6% 17.8% 2.8% 7.6% 2.8%model and species Average 0.8% 2.4% 11.0% 1.6% 4.6% 1.6%

Difference between Maximum 11.4% 9.4% 17.0% 8.0% 5.7% 10.9%randomized maps and species Average 10.8% 6.1% 10.0% 7.1% 2.8% 10.2%

ferences are, however, so small in quantitative terms that they are consistent with our concept ofan overall common design.

31

Figure S8. Agreement betweeen common design and universal pinwheel statistics. Comparisonof pinwheel statistics for galago (G), ferret (F), tree shrew (TS), the long-range interaction model(n = 20, ECPs, Eq. (20)) (M) and the randomized maps (R). Bars are centered around the pop-ulation mean and span the 95% confidence intervals listed in Tab. S1. (A) Pinwheel density 〈ρ〉.(B) Pinwheel density variability exponent γ and variability coefficient c. (C) Pinwheel nearestneighbor statistics d, d++, and d+−.

4.3 Comparison to phase randomized maps

Phase randomized maps provide an important comparison to the properties of the real maps: Byconstruction, phase randomized maps exhibit the same two point correlations as the experimen-tal maps but otherwise have a random arrangement of columns. Judged by their overall visualimpression they appear quite similar to real cortical orientation maps (Fig. 4 E) and they wouldbe expected to exhibit the same properties as real orientation maps if there were no further rules

32

of visual cortical design beyond local correlation and roughly repetitive spatial arrangement ofpreferred orientations. Such correlated random maps are predicted to arise immediately after theinitial formation of an orientation map by activity-dependent mechanisms but without furtheractivity-dependent refinement (S 6).

Our analyses, presented in detail in subsequent sections, show that phase randomized mapsdeviate from the experimentally observed common design in most but not all characteristics (Fig.S8, Tab. S2). This is very clear for the pinwheel density 〈ρ〉, which is significantly and substantiallylarger than the experimentally observed pinwheel density. The pinwheel density variability as afunction of subregion size observed in phase randomized maps nevertheless shared the mainfeatures of the common design. It exhibits a powerlaw decay of the pinwheel density SD forincreasing subregion size that has an exponent γ statistically indistinguishable from the valuesfound in all three species. This exponent of about 1/2 reflects the applicability of the statisticallaw of large numbers. This law is valid for the count statistics of randomized orientation maps,because fluctuations in pinwheel numbers arise homogeneously and independently in separateregions of cortical space (S 6, 58). As shown in Section 5.7 it also applies to the count statisticsof 2D Poisson processes as a model of randomly positioned pinwheels. It is important to notethat this behavior is nevertheless a nontrivial and essential property of the common design. InSections 5.5 and 5.6 we show models in which pinwheel count statistics substantially deviate fromthis behavior and have variability exponents clearly different from 1/2. The variability coefficientc in phase randomized maps was statistically indistinguishable from the value obtained in galagobut was significantly different from the values obtained in both ferret and tree shrew (Tab. S3).

With respect to the NN distances of pinwheels in phase randomized maps we found that theywere typically significantly different from the characteristics of the common design, although theirdistribution functions were qualitatively similar to those obtained for the measured maps (Tab.S3). Both the mean unsigned as well as the opposite sign NN distances were significantly smallerthan in the experimentally observed maps (Fig. S8). This is perhaps not unexpected, given thehigher pinwheel densities in the phase randomized maps. It nevertheless represents an inde-pendent indicator of a difference from the common design because random maps of fixed samepinwheel density can exhibit quite different pinwheel distance statistics. The mean same sign NNdistance of phase randomized maps is significantly smaller than the values obtained for ferretand galago (Tab. S3, Fig. S8). As stated above, in tree shrew V1, the mean same sign NN distancewas significantly smaller than in the other species and this smaller value was statistically indistin-guishable from the value obtained for randomized maps (Tab. S3). Thus, phase randomized mapsmatched only one of the three pinwheel NN distance parameters in only one of the three species.

4.4 Comparison to the long-range interaction model

Solutions of the long-range interaction model reproduced all essential features of the commondesign. This is shown in Fig. 3 of the main manuscript for the pinwheel density of numericallyobtained and exact solutions of the model for different values of the control parameters σ andr and illustrated for the pinwheel density variabilities and NN distance distributions of a set ofexact solutions of the model. To further assess the agreement of model solutions with all features

33

Tabl

eS3

.Qua

ntit

ativ

eas

sess

men

toft

heco

mm

onde

sign

inth

eth

ree

spec

ies

and

its

com

pari

son

toso

luti

ons

ofth

elo

ng-

rang

ein

tera

ctio

nm

odel

(n=

20,E

CPs

,Eq.

(20)

)an

dth

era

ndom

ized

map

sen

sem

ble.

List

edar

eth

ep-

valu

es(a

bove

diag

onal

)and

rela

tive

diff

eren

ces

(bel

owdi

agon

al).

The

arra

ngem

ento

fthe

valu

esfo

rth

esi

xpa

ram

eter

sin

each

entr

yis

indi

cate

din

the

uppe

rle

ftco

rner

ofth

em

atri

x.〈ρ〉

dγ

d++

cd+−

TSG

FD

MR

TS

0.65

0.86

0.51

10−

4

10−

30.

82

0.37

0.14

0.14

10−

4

0.11

0.28

0.31

0.20

0.71

10−

4

10−

40.

15

10−

410−

4

0.00

50.

4810−

410−

4

G1.

0%0.

6%3.

7%4.

7%13

.3%

0.5%

0.99

0.45

0.53

0.83

0.11

0.86

0.90

0.78

0.97

0.10

0.35

0.76

10−

410−

3

0.25

0.00

20.

3510−

3

F1.

0%1.

5%6.

6%4.

2%6.

5%1.

3%

0.1%

2.1%

2.8%

0.5%

6.9%

0.7%

0.14

0.59

0.54

0.99

0.11

0.68

0.68

0.00

60.

5010−

3

0.00

70.

73

10−

410−

4

0.50

10−

4

0.01

10−

4

D2.

0%0.

9%3.

6%0.

0%8.

6%0.

8%

M1.

5%1.

3%2.

9%7.

6%17

.8%

1.7%

0.5%

0.7%

0.8%

2.8%

4.5%

1.2%

0.5%

2.8%

3.6%

3.3%

11.3

%0.

4%

R11

.4%

7.3%

9.4%

0.6%

17.0

%10

.8%

10.4

%8.

0%5.

8%4.

2%3.

6%10

.3%

10.4

%5.

8%2.

9%3.

7%10

.5%

9.6%

34

of the common design we calculated all six quantitative parameters for model solutions represen-tative for the long range dominated regime and compared them to the values obtained from thebiological data. The set of model solutions consisted of N = 26 ECPs (n = 20, Eq. (20)) that weresynthesized with average column spacings and ROIs matched to the N = 26 tree shrew hemi-spheres. This analysis confirmed that essentially all features of the model maps agreed with thecommon design. Further quantification of model solutions obtained from numerical simulationsin different parameter regimes and from different sets of exact solutions of the long-range inter-action model are presented in Sections 5.3 and 5.4, respectively. These results further corroboratethe good agreement between the predictions of the long-range interaction model and the commondesign.

Almost all differences between the parameter values obtained from the model solutions andthose found in the experimental maps were of the same order of magnitude as the (often insignif-icant) interspecies differences (Tab. S2). The only quantities that exhibited somewhat larger dif-ferences between model parameters and experimental data were the variability coefficient c andthe mean same charge NN distance d++ (Tab. S2). Notably, these were the only parameters thatexhibited small but significant interspecies differences (Tab. S3). For this reason no single modelcan be expected to perfectly agree with all species in the predicted values for these parameters.For these two parameters as well as for all others the model was statistically indistinguishablefrom the galago (Fig. S8). As judged by its density variability coefficient c, this species had themost homogenous organization of pinwheels in our sample (Fig. S10 B). Since our model hasno source of spatial heterogeneity, it is plausible that it agrees best with maps from the most ho-mogenous species exhibiting the common design. All pinwheel statistics that were statisticallyindistinguishable between species such as pinwheel density, variability exponent, and the meantotal and opposite charge NN distances were matched very well by the model solutions (Tabs. S1and Fig. S8).

4.5 Average pinwheel density

We calculated average pinwheel densities 〈ρ〉 with confidence intervals, their interspecies differ-ences and tested for their statistical significance (compare Figs. 2, A and B, in the main text). Ouranalysis is based on bootstrap resampling (N=10000, see Section 1.9). Fig. S9 A shows bootstraphistograms of mean pinwheel densities 〈ρ〉 for tree shrew, galago and ferret. Here and in the fol-lowing we characterize such bootstrap distributions by mean and [2.5, 97.5] percentiles; we usesubscripts TS for tree shrew, G for galago, and F for ferret: 〈ρ〉TS = 3.12 [3.06, 3.18]; 〈ρ〉G = 3.15[2.97, 3.30]; 〈ρ〉F = 3.15 [3.08, 3.22]. Relative differences between the species averages, defined by| 〈ρ〉i−〈ρ〉j |/ 1

2 (〈ρ〉i + 〈ρ〉j), for species i and j, were everywhere smaller than 1.0% and on average0.7%. Significance values for difference between species were calculated as described in Section1.9. We found that average pinwheel densities 〈ρ〉were statistically indistinguishable between thethree analyzed species (p > 0.37).

Fig. S9 B shows bootstrap histograms of mean pinwheel densities for the dark-reared ferrets,the long-range interaction model (n = 20, ECPs, Eq. (20)) and the randomized maps, in the follow-ing denoted by subscripts D, M, and R, respectively. For comparison we include the histograms

35

2.6 2.8 3.0 3.2 3.4 3.60.0

0.5

1.0Tree shrewGalagoFerretRandomModelDark

2.8 3.0 3.2 3.40.0

0.5

1.0Tree shrewGalagoFerret

Fre

q. (

norm

aliz

ed)

Pinwheel density Pinwheel density

A B

Figure S9. Bootstrap analysis of average pinwheel densities 〈ρ〉. (A) Average pinwheel densityfor the three species (N=10000 bootstrap samples). Mean, [2.5, and 97.5] percentiles: 3.12, [3.06,3.18] (tree shrew); 3.15, [2.97, 3.30] (galago); 3.15, [3.08, 3.22] (ferret). Relative difference betweenmeans were below 1%. (B) Same analysis including dark-reared ferrets (Dark), the long-rangeinteraction model (n = 20, ECPs, Eq. (20)) (Model), and the randomized maps (Random): 3.22,[3.13, 3.30] (dark-reard ferrets); 3.17, [3.07, 3.26] (model); 3.50, [3.41, 3.59] (randomized maps).Relative differences between means of normally and dark-reared ferrets were 2%, below 1.5%between the model and the three species, but larger than 10% between randomized maps and thethree species. Note that average pinwheel densities for all three species, dark-reared animals andthe model were indistinguishable (p > 0.14) while randomized maps deviated significantly fromthe species averages (p < 0.0001).

for the three species. Average pinwheel densities where 〈ρ〉D = 3.22 [3.13, 3.30], 〈ρ〉M = 3.17[3.07, 3.26], and 〈ρ〉R = 3.50 [3.41, 3.59]. Normally and dark-reared ferrets were statistically indis-tinguishable (p > 0.14) and species averages differed by 2.0%. The long-range interaction modelwas indistinguishable from the three species (p > 0.31) and differences were everywhere < 1.5%and 0.8% on average. In contrast, the randomized maps deviated > 10.4% from each of the threespecies and 10.8% on average. The bootstrap histogram for the randomized maps exhibited virtu-ally no overlap with any of the histograms obtained from the biological data. Differences betweenrandomized and real maps were highly significant (p < 0.0001).

4.6 Pinwheel density variability in subregions

In this section, we present a statistical characterization of the pinwheel density variability in sub-regions for the three species, the dark-reared ferrets, the long-range interaction model (n = 20,ECPs, Eq. (20)), and the randomized maps. Details of the calculation of pinwheel density vari-ability are described in Section 1.7. The statistical analysis is based on bootstrap resampling asdescribed in Section 1.9. To this end, we sampled for each data set N=10000 curves SD(A) ofpinwheel density standard deviation as a function of region size A.

36

1 100.5

0.6

0.7

0.8

0.9

1.0

0.5 0.6 0.7 0.8 0.9 1.00.0

0.5

1.0

1 100.5

0.6

0.7

0.8

0.9

1.0

1 100.1

1.0

Tree shrewGalagoFerret



0.3 0.4 0.5 0.6 0.7 0.80.0

0.5

1.0

Tree shrewGalagoFerretRandomModelDark

1 100.1

1.0


RandomModelDark


RandomModelDark


RandomModelDark


RandomModelDark


RandomModelDark


RandomModelDark

A B

E F

C D

SD

SD

nor

mal

ized

Fre

quen

cy (

norm

aliz

ed)

γ c

SD

Size (hypercolumns)Size (hypercolumns)

SD

nor

mal

ized

Figure S10. Pinwheel density variability in subregions. (A) Bootstrap analysis of pinwheel den-sity standard deviations (SD) in randomly selected regions of size A (N=10000 bootstrap samples).Shown are mean (solid) and 2.5 and 97.5 percentiles (dashed) for the three species. Plot ranges dif-fer in A due to differences in the size of the imaged regions. (B) SDs and percentiles normalized bySD(A) =

√πA−1/2, the density variability for a 2D random distribution of pinwheels (Poisson).

(C and D) as in (A and B), respectively, including dark-reared ferrets, the long-range interactionmodel (n = 20, ECPs, Eq. (20)) and the randomized maps. (E and F) Bootstrap distributions of fitconstants c and γ obtained by fitting SD(A) = c(〈ρ〉 /A)γ to the data in (C) (least square fit overintervals shown in (C)).

37

Fig. S10 A shows mean and 2.5 and 97.5 percentiles of the pinwheel density standard devi-ation curves. Distributions for the different species appeared very similar and relatively narrowover a large range of A. To obtain a more detailed view, we divided these curves in Fig. S10 B bySD(A) =

√πA−1/2 (i.e. by the density standard deviation of a 2D random (Poisson) pinwheel

array; see also Section 1.7, Fig. S6). Especially for small values, normalized SDs were roughlyconstant with A and differed only by a slight offset. Nevertheless, bootstrap distributions fromtree shrew and galago did not overlap strongly indicating a small but systematic difference be-tween these species. For intermediate areas (A > 3 hypercolumns) the tree shrew appeared todeviate towards larger density variability. This deviation partly reflects a tendency in tree shrewsfor pinwheel sparse regions to occur in particular regions of the cortex, such as near the V1/V2border. Fig. S10, C and D, include the curves for the dark-reared ferrets (see Fig. 4, A to C), thelong-range interaction model and the randomized maps (see Fig. 4, D to F). Dark-reared ferretshad distributions largely overlapping with the distributions from the three species. Also the long-range interaction model and the randomized maps exhibited distributions similar to those fromthe three species.

To characterize density variability as a function of area size in the different data sets and to testthe statistical significance of differences, we estimated the variability coefficient c and exponentγ by fitting the function SD(A) = c(〈ρ〉 /A)γ to the SD(A)-curves (〈ρ〉 is the average pinwheeldensity in a given set, e.g. 〈ρ〉TS for tree shrew). Fig. S10, E and F, shows the distribution forγ and c for the experimental data, the long-range interaction model and for randomized maps.The exponent γ was very similar for the three species and the dark-reared ferrets (Fig. S10 E):γTS = 0.47 [0.45, 0.50], γG = 0.49 [0.45, 0.55], γF = 0.51 [0.46, 0.55], γD = 0.52 [0.47, 0.58]. Relativedifferences between species means were < 6.6% (4.4% on average) and 3.6% between dark andnormally reared ferrets. Exponents γ were statistically indistinguishable p > 0.14. The long-rangeinteraction model prediction agreed well with experiments with γM = 0.49 [0.44, 0.57] and a rela-tive difference between model and species means of < 3.6% and 2.4% on average. The differencebetween model and experimental data was not significant (p > 0.50). Randomized maps deviatedslightly more from the species with γR = 0.52 [0.47, 0.58] and a the relative difference of < 9.4%and 6.1% on average. Randomized maps were indistinguishable from ferret and galago (p > 0.25),but statistically different from tree shrew (p < 0.006).

Compared with exponents γ, the distributions of variability coefficients c in the species werebroader and slightly offset from each other (Fig. S10 F): cTS = 0.85 [0.80, 0.89], cG = 0.74 [0.68,0.80], cF = 0.79 [0.73, 0.86], cD = 0.73 [0.65, 0.81]. Relative differences between species meanswere < 13.3% (8.9% on average) and 8.6% between dark and normally reared ferrets. The ferretwas statistically indistinguishable from tree shrew and galago (p > 0.11), but tree shrew andgalago differed significantly (p < 0.001). Thus, although pinwheel density variabilities were verysimilar in the three species, some small interspecies differences within the common design couldbe discerned.

As between experimental data sets, we also observed some degree of deviation in the coeffi-cient c for the long-range interaction model and the randomized maps. For the model we foundcM = 0.71 [0.63, 0.77] with a relative distance < 17.8% and 11% on average. It was distinguishablefrom tree shrew and ferret (p < 0.007) but indistinguishable from galago (p = 0.35). For random-

38

ized maps we obtained cR = 0.71 [0.67, 0.76] with a relative distance < 17.0% and 10% on average.Randomized maps differed from ferret and tree shrew (p < 0.01) but were indistinguishable fromgalago (p = 0.35).

Furthermore, all three species, the dark-reared ferrets, the model and the randomized mapswere well distinguishable from the properties of randomly positioned pinwheels for which thenormalized SD in Fig. S10, D and F, is constant and equal to 1 (see also Fig. S6).

Thus, pinwheel density variability with area size was very similar in tree shrew, galago, fer-ret and dark-reared ferrets. These statistics were reproduced well by the long-range interactionmodel and by the randomized maps while randomly placed pinwheels deviated substantially.Despite the basic similarity, we could discern slight differences among the three species and alsoto the long-range interaction model and the randomized maps. The ability to detect such subtledifferences is a consequence of the large size of the analyzed data sets and the resulting preciseestimation of parameters.

4.7 Pinwheel nearest neighbor distances

In this section, we compare the pinwheel NN statistics in the three species, tree shrew, galago,and ferret, the dark-reared ferrets, the long-range interaction model (n = 20, ECPs, Eq. (20)) andrandomized maps (compare Fig. 2 E, F in main text). Cumulative distributions of NN distancesappeared very similar in all animals, the dark-reared ferrets and in the long-range interactionmodel (Fig. S11, A to C). A relatively strong deviation was found for the randomized maps,especially for distances between pinwheels of arbitrary (Fig. S11 A) and opposite charge (Fig. S11F).

It is important to note that the distances observed in a given map are not statistically inde-pendent. Thus, a direct comparison of NN distributions by means of e.g. a KS-test would beinappropriate. Instead, we used a bootstrap resampling scheme as described in Section 1.9.

For pinwheels of arbitrary topological charge, mean distances d in the three species and indark-reared ferrets, were very similar (Fig. S11 D: dTS = 0.376 [0.371, 0.382], dG = 0.379 [0.360,0.401], dF = 0.371 [0.363, 0.378], dD = 0.374 [0.363, 0.387]. Relative differences between speciesmeans were < 2.1% (1.4% on average) and 0.9% between dark and normally reared ferrets. Specieswere statistically indistinguishable (p > 0.14) as were normally and dark-reared ferrets (p > 0.59).The solutions of the long-range interaction model appeared very similar to the data: dM = 0.381[0.374, 0.389] with a relative distance < 2.8% and 1.6% on average. This small difference wasnevertheless statistically significant for the ferret (p < 0.006), but not for galago or tree shrew(p > 0.20). Randomized maps, in contrast, differed considerably from each species with dR =0.350 [0.343, 0.356] and a relative difference > 5.8% and 7.1% on average. Randomized maps weresignificantly different from each of the species (p < 0.002).

For mean distances d++ between pinwheels of same charge (either both pinwheels of positiveor both of negative topological charge), deviations between animals were smaller than 5% (Fig.S11 F): d++

TS = 0.531 [0.525, 0.538], d++G = 0.557 [0.542, 0.577], d++

F = 0.554 [0.547, 0.562], d++D =

0.554 [0.546, 0.564]. Relative differences between species means were < 4.7% (3.2% on average)and < 0.1% between dark and normally reared ferrets. Differences between tree shrew and ferret

39

0.35 0.40 0.45 0.500.0

0.5

1.0

0.45 0.50 0.55 0.600.0

0.5

1.0

0.30 0.35 0.40 0.450.0

0.5

1.0 Tree shrewGalagoFerretRandomModelDark

0.0 0.5 1.00.0

0.5

1.0

Tree shrewGalagoFerretDarkRandomModel

0.0 0.5 1.00.00.0

0.5

1.0

0.0 0.5 1.00.00.0

0.5

1.0BA C

D FE

Fre

quen

cy

NN distance (same charge)Nearest neighbor distance

Cum

ulat

ive

freq

uenc

y

NN distance (opposite charge)

Mean NN distance Mean NN distance (opposite charge)Mean NN distance (same charge)

Figure S11. Statistical evaluation of pinwheel nearest neighbor (NN) distances by bootstrap re-sampling. (A to C) Cumulative distributions of nearest neighbor distances between pinwheelsof arbitrary (A), same (B) and opposite (C) topological charge. (D to F) Distributions of meandistances between pinwheels of arbitrary (d, D), same (d++, E) and opposite (d+−, F) charge bybootstrap resampling (N=10000). Especially for arbitrary and opposite charge, experimental dataand the long-range interaction model (n = 20, ECPs, Eq. (20)) were in very good agreementwhereas randomized maps deviated considerably.

or galago were significant (p < 0.0001), but not between ferret and galago (p > 0.83). The long-range interaction model solutions appeared very similar to the data: d++

M = 0.573 [0.568, 0.579]with a relative distance < 7.6% and 4.6% on average. The model was statistically indistinguishablefrom galago (p > 0.10), but distinguishable from the other species (p < 0.001). For the randomizedmaps we found d++

R = 0.534 [0.526, 0.543] and a relative difference < 5.7% and 2.8% on average.They were statistically indistinguishable from tree shrew (p > 0.48), but distinguishable fromgalago and ferret (p < 0.001).

For pinwheels of opposite charge, mean distances d+− in the three species and in dark-rearedferrets, were very similar (Fig. S11 E: d+−

TS = 0.416 [0.408, 0.424], d+−G = 0.414 [0.390, 0.442],

d+−F = 0.411 [0.401, 0.420], d+−

D = 0.414 [0.400, 0.430]. Relative differences between species meanswere < 1.3% (0.8% on average) and 0.8% between dark and normally reared ferrets. Specieswere statistically indistinguishable (p > 0.28) as were normally and dark-reared ferrets (p >0.68). The long-range interaction model appeared very similar to the data: d+−

M = 0.409 [0.399,0.418] with a relative distance to the biological data < 2.8% and 1.6% on average. The modelwas indistinguishable from each species (p > 0.15). In contrast, the randomized maps differed

40

considerably from each species with d+−R = 0.373 [0.365, 0.382] and a relative difference > 9.6%

and 10.2% on average. These differences were all significant (p < 0.001).We conclude that even considering a few subtle, but statistically significant differences, the

degree of similarity between the three species and also between the species and the long-range in-teraction model solutions was very high. Average distances between species were typically below2% and everywhere below 5%. Average distances between species and the long-range interactionmodel were typically below 3% and everywhere below 8%. In contrast, the randomized mapsdiffered stronger from the experimental data, sometimes by more than 10%. An even strongerdeviation was found for randomly (Poisson) distributed pinwheels (see Fig. S24, Section 5.7).

41

5 Model orientation map designs and their pinwheel statistics

5.1 Overview

In the main manuscript and in Section 4, we showed that the common design of pinwheel ar-rangements in ferret, tree shrew and galago is well reproduced by solutions of the long-rangeinteraction model Eqs. (10-12). There, we focused on essentially complex planforms (ECPs) whichare solutions of the long-range interaction model in a parameter regime near pattern formationonset (r � 1) and are known in closed form. We studied these solutions in a large interactionrange regime and, to ensure a fair comparison with the experimental results, confined to ROIs ofthe experimental data. In this section of the SOM, we provide additional and more detailed char-acterization of various analytical and numerical solutions of the long-range interaction model.We analyzed the pinwheel statistics in analytical solutions obtained for intermediate and largeinteraction ranges near the instability threshold (r � 1). Moreover, to assess the stability of thepinwheel statistics over time, and to study our model further away from threshold, we analyzednumerically obtained solutions for intermediate and short interaction ranges. We found that un-der most conditions, the long-range interaction model produces essentially invariant pinwheelstatistics. We begin this section with an overview of these results and by summarizing them inTabs. S5 and S4. A detailed description of these analyses is provided in subsequent Sections 5.3and 5.4.

In Sections 5.5 and 5.6 we then assess the pinwheel statistics in the absence of long-range in-teractions. In this regime, in which many alternative models operate, orientation maps are eithershaped by pinwheel annihilation (S 6, 71, 15, 72, 14, 73) or by the crystallization of pinwheels intoperfectly repetitive arrays (S 74, 71, 60, 61, 57, 63, 75, 56). We found that the pinwheel statistics de-viate substantially from the common design in this short-range interaction regime. As a reference,we also calculated all considered pinwheel statistics for random uncorrelated pinwheel positions(Section 5.7). Taken together, these results confirm that pinwheel arrangements predicted by thelong-range interaction model robustly and uniquely reproduce the common design observed inferret, tree shrew and galago visual cortex. Our analyses furthermore shows that all consideredstatistical features are selective markers of the common design, i.e. that none of them is con-strained to the experimentally observed regularity by general geometrical or statistical principles.

5.2 Pinwheel statistics in the long-range interaction model

In the main text, we demonstrate the good agreement between solutions of the long-range interac-tion model and the common design of orientation maps by presenting selected results from bothsimulations and analytical solutions (Fig. 3). In particular, we present the average pinwheel densi-ties for closed form ECP solutions, Eq. (20), with n = 4 to n = 20 active modes, average pinwheeldensities obtained from numerical simulations of the long-range interaction model for variousvalues of the growth rate r and for interaction ranges σ between 1 and 2Λ. In addition, we showpinwheel density fluctuations and NN statistics for ECP solutions with n = 20 modes. These re-sults and the quantitative comparison of the common design with the universal pinwheel statisticsexhibited by n = 20 ECP solutions (presented in Section 4), demonstrate that the common design

42

is reproduced very well by the long-range interaction model for r � 1 and σ � 1Λ. In addition,the numerical results for finite r and σ between 1 and 2Λ show that solutions of the long-rangeinteraction model can closely mimic all essential features of the common design even for sub-stantially shorter interaction ranges provided interactions reach beyond a single hypercolumn. InSections 5.3 and 5.4, these conclusions are further corroborated by a complete characterization ofthe pinwheel statistics for n = 8 and n = 20 ECP solutions and by numerical solutions for finiter and σ = 1Λ and σ = 1.7Λ. In this section, we provide an overview of these theoretical resultsand discuss their mutual relationships. Here and in the following sections we set, without loss ofgenerality, Λ = 1, such that the interaction range σ/Λ is simply given by σ.

Tab. S4 summarizes how closely the six quantitative parameters characterizing the pinwheelstatistics of different solutions, examined in Sections 5.3 and 5.4, and the species averaged param-eter values agree. Parameter values and confidence intervals for model estimates of all parame-ters of the pinwheel statistics are given in Tab. S5. Tab. S4 reveals that all considered solutionsagree well with the common design. Relative differences between model statistics and the corre-sponding species means are typically of the same magnitude and frequently even smaller than themaximal interspecies differences. Our data show that parameter values are very similar amongdifferent solutions of the long-range interaction model. Confidence intervals for all parameters areoverlapping for all examined solutions. Tab. S5 also states absolute values and confidence inter-vals of the parameters obtained in tree shrew, galago, and ferret. Almost all confidence intervalsfor the considered solutions overlap with those found in galago and ferret, and in tree shrew for allparameters that are statistically indistinguishable between species. Notably, the mean pinwheeldensities for n = 20 ECPs and for all characterized numerical solutions fall within the confidenceintervals for pinwheel density in all three species and also into the confidence interval of the grandaverage pinwheel density 3.14 [3.08, 3.20].

The ECPs that solve the long-range interaction model for r � 1 best match the experimentallyobserved pinwheel densities for relatively large n (Fig. 3C). These large n solutions are stable forσ � 1, i.e. for interactions extending substantially beyond the scale of a single hypercolumn.ECP solutions with n = 20 are representative for this regime. Tabs. S4 and S5 show that all otherfeatures of the pinwheel statistics, i.e. those describing pinwheel density fluctuations and NN dis-tances, are similar to the large n statistics for substantially smaller n (n = 8). This already indicatesthat an interaction range much larger than the column spacing Λ is not a strict requirement forreproducing the common design. In fact, the finite time, finite growth rate numerical solutionsanalyzed in Section 5.4 agree with all features of the common design already for σ = 1.0 and 1.7,i.e. for interactions that essentially just reach beyond a single hypercolumn.

To further elucidate these results it is useful to relate the pinwheel densities of numerical andanalytical solutions in this parameter regime in more detail. The mean pinwheel densities of ECPsolutions are shown in Fig. 3C for numbers of active modes increasing from n = 4, the smallestnumber for which ECP orientation maps are aperiodic., to n = 20, representative for large n ECPs.The graph in Fig. 3C shows that the pinwheel density values do not approach the large n limit ina monotonous fashion. Instead, the mean pinwheel density slightly fluctuates as a function of n,exhibiting local maxima at n = 6, 9, 12, and 18, minima at 5, 7, 10, and 13 and assuming a maximalvalue larger than π for n = 4. Note that the precise mean pinwheel density for n = 8 (Fig. 3C) is

43

Table S4. Similarity of pinwheel statistics in the long-range interaction model solutions character-ized in Sections 5.3 and 5.4 to the common design. Shown are for the six parameters quantifyingthe common design their relative distances between model solutions and species mean values.Maximal relative deviations of parameter values between species are provided for comparison(reproduced from Tab. S2). Note that for almost all conditions relative deviations between modelpredictions and species means are of similar size as interspecies variations within the commondesign.

〈ρ〉 γ c d d++ d+−

n = 8 planforms 5.3% 16.1% 17.6% 1.7% 4.0% 4.3%n = 20 planforms 1.7% 9.1% 12.1% 0.6% 2.1% 3.3%

σ = 1.7, t = 1 0.8% 11.5% 15.4% 1.0% 1.5% 1.5%σ = 1.7, t = 10 1.2% 10.9% 14.4% 0.4% 1.7% 1.8%

σ = 1.7, t = 300 1.8% 4.3% 6.6% 1.9% 2.0% 4.8%σ = 1.0, t = 1 0.4% 10.2% 15.2% 1.5% 1.5% 1.0%σ = 1.0, t = 10 0.2% 9.9% 14.1% 0.5% 1.4% 2.6%

σ = 1.0, t = 300 1.1% 7.2% 3.6% 2.5% 1.9% 5.4%

Inter-species deviation 1.0% 6.6% 13.3% 2.1% 4.7% 1.3%

at the upper end of the confidence interval in Tab. S5. To compare the pinwheel densities of thenumerically obtained solutions for finite r and the ECP solutions valid for r � 1, we determinedwhich ECPs are attractors for σ = 1 and σ = 1.7 (g = 0.98, r � 1). For σ = 1, ECPs with n = 4− 6are stable, while for σ = 1.7, ECPs with n = 6− 11 are stable (all other ECPs are unstable). Thus theECP attractors for σ = 1.7 contain two sets (n = 7, 10) with particularly small pinwheel densitiesand for σ = 1 the attractors contain the ECPs, n = 4, that exhibit the largest mean pinwheeldensity of all aperiodic ECPs. Thus the properties of the relevant ECP attractors are consistentwith the observation that the numerically obtained mean pinwheel density for σ = 1 is slightlylarger than for σ = 1.7, in particular for small r. The fact that, at fixed σ, ECPs with an entire rangeof numbers of active modes are stable explains why the dependence of mean pinwheel density onσ for numerically obtained solutions is a relatively smooth function of σ (compare Fig. 3C and D).For fixed σ, the solutions are predicted to converge to a multiplicity of ECP attractor states with anentire range of active mode numbers. As a consequence, the mean pinwheel density for numericalsolutions is expected to represent an average over ECPs with different n. Fig. 3D also reveals aweak dependence of numerically obtained pinwheel densities on the growth rate r, such that thepinwheel density slightly increases with decreasing r. This weak dependence likely reflects smallnon-universal finite r corrections to the model solutions (S 24).

Finally, the data presented in Section 5.4 also indicate that σ = 1 is at the lower end of the pa-rameter range for which solutions of the long-range interaction model mimic the common design.While the parameters quantifying pinwheel statistics of numerically obtained finite time solutionsat σ = 1 are close to values characteristic of the common design, the detailed structure of pinwheel

44

Table S5. Quantitative assessment of the pinwheel statistics in the long-range interaction modelfor different sets of parameters and integration times. The two exact essentially complex planform(ECP) solutions provide examples close to the pattern formation threshold (r � 1) with interme-diate (n = 8) and relatively long-range (n = 20) interactions. The other solutions were obtainednumerically in a regime further away from threshold (r = 0.1) for intermediate (σ = 1.7) and rela-tively short-ranged (σ = 1.0) interactions and the time dependence of the pinwheel statistics wasassessed. Values for the three species are provided for comparison. All entries show mean and[2.5, 97.5] percentile confidence intervals obtained by bootstrap resampling. Note that for almostall conditions the pinwheel density for the model is within the confidence intervals for all threespecies.

〈ρ〉 γ cECP n = 8 2.97 [2.79, 3.13] 0.57 [0.56, 0.58] 0.65 [0.62, 0.69]ECP n = 20 3.09 [3.03, 3.14] 0.53 [0.52, 0.55] 0.70 [0.68, 0.72]

σ = 1.7, t = 1 3.12 [3.10, 3.13] 0.57 [0.54, 0.55] 0.67 [0.66, 0.68]σ = 1.7, t = 10 3.10 [3.08, 3.12] 0.54 [0.53, 0.55] 0.68 [0.67, 0.69]σ = 1.7, t = 300 3.08 [3.05, 3.11] 0.51 [0.50, 0.52] 0.74 [0.72, 0.76]σ = 1.0, t = 1 3.13 [3.11, 3.15] 0.54 [0.53, 0.55] 0.67 [0.66, 0.69]σ = 1.0, t = 10 3.14 [3.12, 3.16] 0.54 [0.53, 0.55] 0.68 [0.67, 0.69]σ = 1.0, t = 300 3.11 [3.06, 3.15] 0.45 [0.44, 0.47] 0.82 [0.79, 0.86]


d d++ d+−

ECP n = 8 0.369 [0.321, 0.457] 0.569 [0.534, 0.700] 0.396 [0.332, 0.525]ECP n = 20 0.373 [0.361, 0.389] 0.559 [0.548, 0.578] 0.400 [0.385, 0.422]

σ = 1.7, t = 1 0.379 [0.369, 0.387] 0.556 [0.549, 0.561] 0.407 [0.393, 0.419]σ = 1.7, t = 10 0.377 [0.369, 0.389] 0.557 [0.550, 0.569] 0.406 [0.396, 0.419]σ = 1.7, t = 300 0.368 [0.360, 0.378] 0.558 [0.543, 0.572] 0.394 [0.380, 0.402]σ = 1.0, t = 1 0.381 [0.374, 0.391] 0.555 [0.547, 0.561] 0.409 [0.401, 0.426]σ = 1.0, t = 10 0.373 [0.364, 0.381] 0.555 [0.547, 0.564] 0.402 [0.392, 0.412]σ = 1.0, t = 300 0.366 [0.351, 0.387] 0.558 [0.543, 0.572] 0.391 [0.371, 0.407]


45

density fluctuations for these solutions start to exhibit slight deviations from the biological obser-vations. As shown in Fig. S16, the SD(A) graph at t = 300 exhibits an increased density variabilityfor large regions. For A = 30, pinwheel density variability is larger than predicted for a randomarrangement of pinwheels and thus larger than in any of the considered species. For values of σbelow 1, pinwheel crystals (e.g. at σ = 0.82 for g = 0.98, r � 1) and pinwheel free maps (e.g.at σ = 0.34 for g = 0.98, r � 1) become attractors of the model (S 15) and lead to pinwheelstatistics qualitatively distinct from the common design. These types of pinwheel statistics arecharacterized in Sections 5.5 and 5.6.

5.3 Essentially complex planforms

0 0.5 10

0.5

1

Nearest neighbor distance

Fre

q.

(in

teg

rate

d)

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

A B C

FED

Figure S12. (A) Essentially complex planforms for n = 8 active modes as defined in Eq. (S49).(B and C) Nearest neighbor spacing distributions for pinwheels of arbitrary (B) and opposite andsame (C) topological charge, together with fits of f (x) = a xn/(1 + exp((x − x0)/b)) to thesedistributions (black) and to the experimental data (yellow, reproduced from Fig. 2, E and F). (Eand F) Cumulative distribution functions (CDFs) of the histograms in (B) and (C) together withCDFs of the fits. (D) Standard deviation SD of pinwheel densities in randomly selected circularregions of size A (in units of Λ2). The black dashed line represents c

√πA−γ the SD for a 2D

Poisson process of pinwheel density π. The inset shows the curve for a larger region togetherwith SD(A) = c (〈ρ〉/A)γ (red dashed line) with fit constants c and γ.

In this section, we describe the pinwheel statistics for two sets of model solutions, for interme-diate and for long interaction ranges. As shown in (S 15) stable asymptotic states of the model,

46

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

A B C

FED

Figure S13. Essentially complex planforms for n = 20 active modes as defined in Eq. (S49). (Band C) Nearest neighbor spacing distributions for pinwheels of arbitrary (B) and opposite andsame (C) topological charge, together with fits of f (x) = a xn/(1 + exp((x − x0)/b)) to thesedistributions (black) and to the experimental data (yellow, reproduced from Fig. 2, E and F). (Eand F) Cumulative distribution functions (CDFs) of the histograms in (B) and (C) together withCDFs of the fits. (D) Standard deviation SD of pinwheel densities in randomly selected circularregions of size A (in units of Λ2). The black dashed line represents c

√πA−γ the SD for a 2D

Poisson process of pinwheel density π. The inset shows the curve for a larger region togetherwith SD(A) = c (〈ρ〉/A)γ (red dashed line) with fit constants c and γ.

Eqs. (10-12), near the critical point are given by essentially complex planforms (ECPs)

z(x) =n

∑j=1

eiljkjxj+φj (49)

which are classified by their number of active modes n. We analyzed the average pinwheel density,the pinwheel density variability, and the NN distances for an ensemble of ECPs with n = 8, i.e.solutions for an intermediate range of interaction, and for an ensemble with n = 20, i.e. solutionsfor a relative long range of interactions. Modes kj were equally distributed on the half-circlekj = kc (cos(jπ/N), sin(jπ/N)) and signs lj were randomly chosen from lj ∈ {+1,−1}, suchthat either the mode kj or the antiparallel mode −kj was active. Phases φj were randomly chosenfrom the interval φj ∈ [0, 2π). We considered an ensemble of 30 realizations z(x), being eachrepresented on a discretized torus with 1024×1024 pixels. The critical wavenumber kc was chosensuch that each planform contained 484 hypercolumns.

47

We found that pinwheel densities are only slightly higher for the n = 20 ECPs comparedto n = 8. For the overall pinwheel densities we found 〈ρ〉 = 2.97 [2.79, 3.13] (mean [2.5, 97.5]percentiles of bootstrap distributions) for n = 8, and 〈ρ〉 = 3.09 [3.03, 3.14] for n = 20 activemodes.

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

Size (hypercolumns)

SD

100

101

10−1

100

SizeS

D10

−110

010

110

210

310

−210

−110

010

1

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1FE

t = 1 t = 10

BA C

t = 300

D

Figure S14. Pinwheel stabilization in the presence of long-range interactions for the model dy-namics Eqs. (10-12) for (g = 0.98, σ = 1.7, r = 0.1). (A to C) Snapshots of the dynamics attimes t = 1, t = 10 and t = 300 (measured in units of the intrinsic timescale τ = r−1). Patternsundergo some rearrangement before converging to a stable stationary state, while the average pin-wheel density 〈ρ〉 remains fairly constant over time. (D to F) Standard deviation SD of pinwheeldensities in randomly selected circular regions of size A (in units of Λ2). The black dashed linerepresents the SD for a 2D Poisson process of pinwheel density π. The inset shows the curve for alarger region together with SD(A) = c (〈ρ〉/A)γ (red dashed line) with fit constants c and γ.

Also the SD(A) differs only little for the two sets of ECPs (Figs. S12 D and S13 D). Fittingc(〈ρ〉/A)γ yields (γ = 0.57 [0.56, 0.58], c = 0.65 [0.62, 0.69]) for N = 8, and (γ = 0.53 [0.52, 0.55],c = 0.70 [0.68, 0.72]) for N = 20.

NN distances are also consistent between the two sets of ECPs (Fig. S12, B, C, E and F, andFig. S13, B, C, E and F). For arbitrary charge we obtained (n =1.22, x0 = 0.51, b = .038) for n = 8,(n =1.34, x0 = 0.52, b = 0.033) for n = 20. For same charge we obtained (n = 5.53, x0 = 0.57, b =0.04) for n = 8 and (n = 5.52, x0 = 0.58, b = 0.041) for n = 20. For opposite charge we obtained(n = 1.26, x0 = 0.49, b =0.073) for n = 8 and (n = 1.34, x0 = 0.51, b =0.066) for n = 20. For theaverage values of NN distances we obtained (d = 0.369 [0.321, 0.457], d++ = 0.569 [0.534, 0.700],d+− = 0.396 [0.332, 0.525]) for n = 8 and (d = 0.373 [0.361, 0.389], d++ = 0.559 [0.548, 0.578],d+− = 0.400 [0.385, 0.422]) for n = 20.

48

We conclude that in our model, the pinwheel statistics are essentially insensitive against vari-ation of interaction range.

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

0 0.5 10

1

2

3

4


Fre

q.

(no

rma

lize

d)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

FE

t = 1 t = 10

BA Ct = 300

D

Figure S15. Nearest neighbor distances for the solutions shown in Fig. S14 (g = 0.98, σ = 1.7, r =0.1). (A to C) Nearest neighbor spacing histograms for pinwheels of arbitrary topological charge(blue) together with f (x) = a xn/(1 + exp((x − x0)/b)) fitted to the histograms (orange) and tothe histograms of experimental maps (black, reproduced from Fig. 2 E). (D to F) Nearest neighbordistances for pinwheels of opposite and same topological charge.

5.4 Long-range interaction model: Numerical solutions

In this section, we analyze the pinwheel statistics in the long-range interaction model (Eqs. (10-12)) in a regime further away from the pattern formation threshold. Furthermore, we analyzethe temporal evolution of the pinwheel statistics and we perform this analysis for two interactionranges, σ = 1.7 and σ = 1.0. First, we assess the pinwheel statistics for intermediate-rangedlateral interactions by integrating the model dynamics numerically for the parameters r = 0.1,σ = 1.7, g = 0.98. Fig. S14, A to C, shows solutions at time points t = 0, t = 10 and t = 300(measured in units of the intrinsic time scale τ = r−1). As can be seen from the graphs in Fig.S14, orientation columns reorganize over time. The pinwheel statistics, however, remain virtuallyinvariant over time and are very similar to the statistics obtained for the ECPs in the previoussection. We calculated the pinwheel density, the pinwheel density variability and the pinwheelNN statistics at these time points. The calculation was carried out for a set of N = 30 solutions ofsize 484 Λ2 obtained for different random initial conditions. For the average pinwheel density wefound: 〈ρ〉 = 3.12 [3.10, 3.13] for t = 1, 〈ρ〉 = 3.10 [3.08, 3.12] for t = 10 and 〈ρ〉 = 3.08 [3.05, 3.11]for t = 300. Thus, in the case of intermediate-ranged lateral interactions, the average pinwheel

49

density 〈ρ〉 remains essentially constant over time.

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1FE

t = 1 t = 10

BA C

t = 300

D

Figure S16. Pinwheel stabilization in the presence of relatively short-range interactions for themodel dynamics Eqs. (10-12) for (g = 0.98, σ = 1.0, r = 0.1). (A to C) Snapshots of the dynamicsat times t = 1, t = 10 and t = 300 (measured in units of the intrinsic timescale τ = r−1). Patternsundergo some rearrangement before converging to a stable stationary state, while the average pin-wheel density 〈ρ〉 remains fairly constant over time. (D to F) Standard deviation SD of pinwheeldensities in randomly selected circular regions of size A (in units of Λ2). The black dashed linerepresents the SD for a 2D Poisson process of pinwheel density π. The inset shows the curve for alarger region together with SD(A) = c (〈ρ〉/A)γ (red dashed line) with fit constants c and γ.

Next, we calculated pinwheel variability in subregions of size A (Fig. S14, D to F). As before,we fitted c(〈ρ〉/A)γ to SD(A) with c and γ fit constants. For the parameter estimates for the differ-ent time points we found (γ = 0.57 [0.54, 0.55], c = 0.67 [0.66, 0.68]) for t = 1, (γ = 0.54 [0.53, 0.55],c = 0.68, [0.67, 0.69]) for t = 10 and (γ = 0.51 [0.50, 0.52], c = 0.74 [0.72, 0.76]) for t = 300. Theseparameters stay nearly constant for all three time points and the values of γ suggest that SD(A)decays ∼ A−1/2. Thus, also the pinwheel density variability exhibits a high degree of stabilityover time.

Next, we computed the NN spacing distributions and found that they are also virtually iden-tical for all three time points (Fig. S15). Following the procedure above, we fitted histograms tothe function

f (x) = a xn/(1 + exp((x− x0)/b)) , (50)

where a is determined by the normalization condition∫

dx f (x) = 1. For pinwheels of arbitrarycharge, fit parameters (n, x0, b) for the different times are similar: n =1.36, x0 = 0.53, b = 0.031)for t = 1, n =1.37, x0 = 0.53, b = 0.03 for t = 10, and n =1.29, x0 = 0.51, b = 0.036 for

50

t = 300. Also distance distribution between pinwheels of same charge change only little: n = 5.46,x0 = 0.57, b = 0.042 for t = 1, n = 5.44, x0 = 0.57, b = 0.042 for t = 10, and n = 5.42, x0 = 0.57,b = 0.046 for t = 300. In the model, distributions appear to be shifted towards slightly largervalues compared to experiment consistent with Fig. S11. For pinwheels of opposite charge wefound n = 1.33, x0 = 0.52, b =0.062 for t = 1, n = 1.37, x0 = 0.52, b =0.065 for t = 10, andn = 1.29, x0 = 0.49, b =0.073 for t = 300. For the average values of NN distances we obtained(see Tab. S5) (d = 0.379 [0.369, 0.387], d++ = 0.556 [0.549, 0.561], d+− = 0.407 [0.393, 0.419]) fort = 1, (0.377 [0.369, 0.389], d++ = 0.557 [0.550, 0.569], d+− = 0.406 [0.396, 0.419]) for t = 10, and(d = 0.368 [0.360, 0.378], d++ = 0.558 [0.543, 0.572], d+− = 0.394 [0.380, 0.402]) for t = 300. TheseNN statistics for the model in the regime of intermediate-ranged lateral interactions change onlylittle during development and at all times, model distributions are in good agreement with thedistributions obtained for the ECPs in the previous section.

Next, we assessed whether relatively short-ranged lateral interactions would still reproducethe observed common design. To this end, we integrated the model dynamics, Eqs. (10-12), nu-merically for the parameters r = 0.1, σ = 1.0, g = 0.98. Fig. S16, A to C, shows solutions at timepoints t = 0, t = 10 and t = 300 (measured in units of the intrinsic time scale τ = r−1). Wecalculated the pinwheel density, the pinwheel density variability and the pinwheel NN statisticsat these time points. The calculation was carried out for a set of N = 30 solutions of size 484Λ2 obtained for different random initial conditions. For the average pinwheel density we found:〈ρ〉 = 3.13 [3.11, 3.15] for t = 1, 〈ρ〉 = 3.14 [3.12, 3.16] for t = 10 and 〈ρ〉 = 3.11 [3.06, 3.15] fort = 300. Thus, the average pinwheel density 〈ρ〉 remains fairly constant over time, consistent withthe experimental observations (Fig. 4 C).

We also calculated the pinwheel variability in subregions of size A (Fig. S16, D to F) and fittedc(〈ρ〉/A)γ to SD(A) with c and γ fit constants. Parameter estimates for the different time pointsare (γ = 0.54 [0.53, 0.55], c = 0.67 [0.66, 0.69]) for t = 1, (γ = 0.54 [0.53, 0.55], c = 0.68 [0.67, 0.69])for t = 10 and (γ = 0.45 [0.44, 0.47], c = 0.82 [0.79, 0.86]) for t = 300. The parameters stay nearlyconstant for all three time points and the values of γ suggest that SD(A) decays ∼ A−1/2. Thevalue of γ obtained for very late times t = 300 is slightly decreased reflecting an increased vari-ability in large subregions (A > 10). Thus, apart from this effect occurring at very late times, thepinwheel density variability remains largely consistent with the variability observed in experi-ment even for the considered regime of relatively short-range interactions.

Also in this regime of relatively short-ranged interactions, NN spacing distributions are vir-tually identical for all three time points and similar to those observed in experiment (Fig. S17).For pinwheels of arbitrary charge, fit parameters (n, x0, b) for the different times are: (n =1.4,x0 = 0.53, b = 0.029) for t = 1, (n =1.32, 2, b = 0.03) for t = 10, and (n =1.27, x0 = 0.50, b = 0.04)for t = 300. Also distance distributions between pinwheels of same charge change only little:(n = 5.47, x0 = 0.58, b = 0.042) for t = 1, (n = 5.38, x0 = 0.57, b = 0.046) for t = 10, and (n = 5.36,x0 = 0.56, b = 0.048) for t = 300. Compared to experiment, distributions appear to be shiftedtowards slightly larger values in the model consistent with Fig. S11. For distances between pin-wheels of opposite charge we found (n = 1.37, x0 = 0.53, b =0.062) for t = 1, (n = 1.33, x0 = 0.51,b =0.072) for t = 10, and (n = 1.31, x0 = 0.48, b =0.076) for t = 300. For the average valuesof NN distances we obtained (see Tab. S5) (d = 0.381 [0.374, 0.391], d++ = 0.555 [0.547, 0.561],

51

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed

)

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed

)

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

0 0.5 10

1

2

3

4


Fre

q.

(no

rma

lize

d)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

FE

t = 1 t = 10

BA Ct = 300

D

Figure S17. Nearest neighbor distances for the solutions shown in Fig. S16 (g = 0.98, σ = 1.0, r =0.1). (A to C) Nearest neighbor spacing histograms for pinwheels of arbitrary topological charge(blue) together with f (x) = a xn/(1 + exp((x − x0)/b)) fitted to the histograms (orange) and tothe histograms of experimental maps (black, reproduced from Fig. 2 E). (D to F) Nearest neighbordistances for pinwheels of opposite and same topological charge.

d+− = 0.409 [0.401, 0.426]) for t = 1, (0.373 [0.364, 0.381], d++ = 0.555 [0.547, 0.564], d+− =0.402 [0.392, 0.412]) for t = 10, and (d = 0.366 [0.351, 0.387], d++ = 0.558 [0.543, 0.572], d+− =0.391 [0.371, 0.407]) for t = 300.

We conclude that the pinwheel statistics in the long-range interaction model remain in goodagreement with the observed common design even considerably far away from the pattern forma-tion threshold (r = 0.1) and for interaction-ranges as small as σ = 1. Furthermore, the pinwheelstatistics are established as early as t = 1 and remain fairly stable during development. This is ingood agreement with the observed stability of pinwheel densities in ferret visual cortex betweenpostnatal weeks 5 and 20 (Fig. 4 C).

5.5 Maps shaped by pinwheel annihilation

Here, we characterize pinwheel density variabilities and NN pinwheel distances in orientationmaps shaped by pinwheel annihilation. In models exhibiting pinwheel annihilation (S 6, 71, 15,72, 14, 73), not only pinwheel density but all other pinwheel statistics are time dependent andchange monotonically during the course of map reorganization. In such models, orientation mapsinitially mimic random maps and as a consequence exhibit variability exponents around 1/2. This,however, changes quickly with map rearrangement. Pinwheel annihilation leads to the emergenceof pinwheel sparse stripe-like regions in the orientation maps and the size of these regions sys-

52

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1FE

t = 1 t = 10

BA C

t = 300

D

Figure S18. Pinwheel statistics in the absence of long-range interactions for the model dynamicsEqs. (10-12) for (g = 2, r = 0.1). (A to C) Snapshots of the dynamics at times t = 1, t = 10and t = 300 (measured in units of the intrinsic timescale τ = r−1). Patterns show extensivereorganization and pinwheels annihilate. (D to F) Standard deviation SD of pinwheel densities inrandomly selected circular regions of size A (in units of Λ2). The black dashed line represents theSD for a 2D Poisson process of pinwheel density π. The inset shows the curve for a larger regiontogether with SD(A) = c (〈ρ〉/A)γ (red dashed line) with fit constants c and γ.

tematically increases over time. We find that pinwheel density variability is reduced for small sizesubregions and enhanced for large size subregions that cover boundaries of different stripe-likedomains exhibiting high local pinwheel densities (see Fig. S18). As a consequence, maps shapedby pinwheel annihilation exhibit reduced variability exponents (γ ≈ 0.35) that reflect a more shal-low decay of density variability with subregion size as compared to the common design. Thisis a direct demonstration that the applicability of the law of large numbers to count fluctuationsin orientation maps is a nontrivial property. We find that it can be violated in maps that have asubstantial visual resemblance to cortical orientation maps (see e.g. Fig. S18 B). In addition, mapsshaped by pinwheel annihilation also exhibit a long-term decay in the absolute size of densityvariabilities, reflecting the fact that the size of density variabilities decays together with the meandensity. As a consequence, the variability coefficient c drops with time to values substantiallybelow the values characteristic of the common design as pinwheel annihilation proceeds.

Pinwheel annihilation is also strongly affecting all types of NN distance distributions and leadsto the emergence of distributions qualitatively distinct from those characterizing the common de-sign. Pinwheel annihilation proceeds by the collision and pairwise annihilation of pinwheels ofopposite topological charge. It thus reduces the number of short distance pinwheel pairs and se-

53

lectively removes short distance pairs of opposite topological charge. This is reflected by boththe uncharged pinwheel NN distance distribution and the opposite charge pinwheel NN distancedistribution (Fig. S19, B and E). In the long term, all remaining NN pinwheel pairs are of sametopological charge leading to the convergence of the uncharged and same charged pinwheel NNdistance distributions. These remaining same charge pinwheel pairs are located at the bordersof different domains of stripe-like organization and exhibit a stereotyped spacing of about 1/2column spacings.

To obtain these results, we performed numerical integration of the model dynamics, Eqs. (10-12), for the parameter r = 0.1 and g = 2, i.e in the absence of long-range interactions, startingfrom random initial conditions (N = 30 realizations of size 484 Λ2). It was shown previously thatthe Eqs. (10-12) mimic the behavior of other Hebbian models of visual cortical development thatlead to pinwheel annihilation (S 6, 14). As for the long-range dominated regime (Section 5.4), wecalculated the pinwheel density, pinwheel density variability and NN statistics for three differenttime points: t = 1, t = 10 and t = 300.

Shortly after the initial emergence of the first orientation preference map, patterns exhibit ahigh pinwheel density, but pinwheels of opposite charge start to move towards each other andannihilate (Fig. S18, A to C). This leads to the formation of domains of orientation stripes withpinwheel positions largely confined to the domain boundaries. The average pinwheel density 〈ρ〉decays drastically over time. We found 〈ρ〉 = 3.11 for t = 1, 〈ρ〉 = 2.29 for t = 10 and 〈ρ〉 = 0.31for t = 300.

Also the pinwheel density variability is strongly affected by the progressive map rearrange-ment and annihilation of pinwheel pairs. Parameter estimates for the different time points are:(γ = 0.55, c = 0.68) for t = 1, (γ = 0.37, c = 1.07) for t = 10 and (γ = 0.35, c = 1.07) fort = 300 (Fig. S18, D to F). Thus, SD curves deviate from the law of large numbers that applies tothe common design. Late pinwheel sparse maps contain large pinwheel free regions and arraysof pinwheels clustered between them. The pinwheel density variability is sensitive to such inho-mogeneities in the maps. With time the density variability becomes more pronounced for largeregions but less so for smaller regions leading to a lower value of the variability exponent γ.

Finally, also the NN spacing distributions are strongly affected by the map rearrangement(Fig. S19). For pinwheels of arbitrary topological charge we found that distributions develop asingle prominent peak at approximately half a wavelength Λ at large times (Fig. S19, A to C).The reason for this was that pinwheels of same topological charge tend to assemble at a distanceof approximately half a wavelength Λ forming linear fracture zones between two stripe patternsdevoid of pinwheels (Fig. S19, D to F). In contrast, distances of opposite charge become very large,since pinwheel pairs of opposite charge, when sufficiently close to each other, are likely to decayby annihilation. Thus, the NN statistics changes drastically during development and intermediateand late model distributions are inconsistent with the distributions obtained from experiment.

5.6 Pinwheel crystals

Many dynamical models for the development of orientation preference maps lead to the emer-gence of crystalline arrays of pinwheels (see e.g. (S 74, 71, 60, 61, 57, 63, 75, 56)). The experimen-

54

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed

)

0 0.5 10

2

4

6

8


Fre

q. (n

orm

aliz

ed

)

0 0.5 10

5

10

15

20


Fre

q. (n

orm

aliz

ed)

0 0.5 10

5

10

15

20


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

0 0.5 10

2

4

6

8


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

FE

t = 1 t = 10

BA Ct = 300

D

Figure S19. Nearest neighbor distances for the solutions shown in Fig. S18 (g = 2, r = 0.1). (Ato C) Nearest neighbor spacing histograms for pinwheels of arbitrary topological charge at timest = 1, t = 10 and t = 300. (D to F) Nearest neighbor distances for pinwheels of opposite andsame charge. Distance is measured in units of Λ. Note that with time, the spacing distribution foroppositely charged pinwheels undergoes dispersion (due to pinwheel annihilation). In contrast,for pinwheels of the same charge, the distribution develops a peak at a distance s ≈ Λ/2.

tally observed organization of orientation preference maps is, however, clearly non-crystalline sothat these models may be rejected based on qualitative comparison. Nevertheless it is instructiveto examine their pinwheel statistics to assess the specificity of the parameters of pinwheel orga-nization that define the common design. If some of these parameters agreed with the statisticsobtained from the biological data, this would indicate a lack of selectivity of our characteriza-tion of the pinwheel statistics. Notably all considered pinwheel statistics of crystalline pinwheelarrangements generally deviate from the universal design.

For crystalline pinwheel arrangements, pinwheel densities and pinwheel NN distance distri-butions can be calculated analytically and reveal that a pinwheel density of π is not a genericprediction of crystalline pinwheel arrangements. Their neighbor distance distributions generallyconsist of a single discrete peak. Most interesting are the pinwheel density variabilities exhibitedby crystalline pinwheel arrangements. They reveal a substantial deviation from the predictionof the law of large numbers for which γ = 1/2. For crystalline pinwheel arrangements densityvariabilities are found to decay much faster with subregion size with γ = 3/4. This deviation isof the opposite sign as the one exhibited by maps shaped by pinwheel annihilation. This resultfurther corroborates the conclusion that the variability exponent is a nontrivial characteristic ofthe common design.

55

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

A B C

FED

Figure S20. (A) Rhombic pinwheel crystals with α = 90◦. (B and C) Nearest neighbor spacingdistributions for pinwheels of arbitrary (B) and opposite and same (C) topological charge. Dis-tance is measured in units of Λ. (E and F) Cumulative distribution functions of the histograms in(B) and (C), respectively. (D) Standard deviation SD of pinwheel densities in randomly selectedcircular regions of size A (area A measured in units of Λ2). The black dashed line represents theSD for a 2D Poisson process of pinwheel density ρ = π. The inset shows the curve for a largerregion together with the fit to the curve c(ρ/A)γ.

First, we consider rhombic pinwheel crystals(S 60, 61, 57, 56) as, for example, obtained in

z(x) = sin(k1x) + i sin(k2x) (51)

where real and imaginary parts are sine waves in direction k1 and k2, respectively. Without lossof generality we chose k1 = kc(1, 0) and k2 = kc(cos α, sin α). For α = 90◦ the resulting lattice isquadratic (Fig. S20 A), for α ∈ [0, π], pinwheel positions form a rhombic lattice. An example forα = 45◦ is shown in Fig. S21 A.

Different kinds of periodic pinwheel arrangements are obtained for patterns consisting of threeFourier modes (S 74, 57, 15, 75, 56). We consider planforms

z(x) =3

∑j=1

eiljkjxj+φj (52)

with the three modes kj are equally spaced on the half-circle kj = kc (cos(jπ/3), sin(jπ/3)) suchthat the relative angle of neigbouring modes amounts to 60◦, which is required to provide a peri-odic pattern z(x). We note that patterns Eq. (52) are solutions of Eqs. (10-12) and become stable

56

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

A B C

FED

Figure S21. (A) Rhombic pinwheel crystals with α = 45◦. (B to F) as in Fig. S20.

for σ ≈ 0.1. Phases φj are randomly chosen from the interval φj ∈ [0, 2π). Signs lj are randomlychosen from lj ∈ {+1,−1}, such that either the mode kj or the antiparallel mode−kj is a memberof the planform. Since the resulting set of 23 possibilities can be reduced to the configurationslA = (+1,−1 + 1) and lB = (+1, +1 + 1) by rotation and reflection it is sufficient to discuss thesetwo cases. Examples of patterns for lA and lB are shown in Figs. S22 A and S23 A. Pinwheels arelocated on two distinct hexagonal grids, depending on their topological charge ( + 1

2 or − 12 ). The

relative position of both grids is different for lA and lB which results in a high pinwheel densityplanform for lA and a low pinwheel density planform for lB.

Perhaps not unexpected, pinwheel crystals differ in almost all aspects from the statistics of thecommon design. In the following, we describe the pinwheel statistics in these different types ofpinwheel crystals. For the rhombic lattices the pinwheel density ρ depends on α and reads

ρ = 4 sin α, (53)

which implies 0 < ρ ≤ 4. For the two cases depicted in Figs. S20 and S21 the pinwheel den-sity equals ρ = 4 (α = 90◦) and ρ ≈ 2.83 (α = 45◦), respectively. For the hexgonal lattices thepinwheel density ρ depends on the class of active modes and can be calculated analytically. ForlA = (+1,−1 + 1) (Fig. S22 A) we find

ρ = 3√

3 ≈ 5.19. (54)

For lB = (+1, +1 + 1) (Fig. S23 A) the pinwheel density is three times lower,

ρ =√

3 ≈ 1.73. (55)

57

Thus, hexagonal lattices appear inconsistent with the observed common pinwheel density, whilerhombic lattices can reproduce a wide spectrum of densities.

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

A B C

FED

Figure S22. (A) Hexagonal pinwheel crystals with high pinwheel density. (B to F) as in Fig. S20.

Next we analyze the pinwheel density variability in the different crystal patterns. Analyticalresults for the number variance, even in the case of regular arrangements like in pinwheel crystals,are hard to obtain since, in general, pinwheel locations are correlated. Still, for large A � 1 it ispossible to derive the expected asymptotic behavior for A → ∞ from simple scaling arguments.For pinwheel crystals one expects a number variance

NV(A) ∼ A1/2 (56)

and thusSD(A) ∼ A−3/4 (57)

which is based on the following reasoning: Suppose the window is a circle of area A. Superim-posed on random locations of the pinwheel lattice it will contain a fluctuating number of pin-wheels. However, since pinwheels have orderly arrangement in the bulk, the fluctuations in thepinwheel count can only originate from the pinwheels located near the boundary. The number ofboundary pinwheels under consideration, NB, scales with the circumference of the circle. Assum-ing 〈N2

B〉 − 〈NB〉2 ∼ 〈NB〉 we obtain the estimate Eq. (S56).We numerically checked the validity of this prediction by calculating SD(A) for 200 different

window sizes centered at N = 105 random locations in a pinwheel crystal consisting of N =22× 22 hypercolumns. Fitting the variability to SD(A) = c (〈ρ〉/A)γ, as above, we find (γ =

58

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

A B C

FED

Figure S23. (A) Hexagonal pinwheel crystals with low pinwheel density. (B to F) as in Fig. S20.

0.73, c = 0.26) for rhombic lattices with α = 90◦ and (γ = 0.74, c = 0.32) for α = 45◦, in goodagreement with Eq. (S57). For hexagonal pinwheel crystals we find (γ = 0.74, c = 0.24) for thehigh pinwheel density planform and (γ = 0.73, c = 0.44) for the low density planform, in goodagreement with Eq. (S57).

Both the exponent and the prefactor deviate substantially from those obtained in experimentand in the long-range interaction model. Thus, the density variabilities in crystals do not followthe law of large numbers that is characteristic for the common design and expressed by an expo-nent of γ ≈ 1

2 . Instead, regular pinwheel lattices show an exponent close to γ = 34 .

Pinwheel NN distances for crystals are relatively simple and can be calculated analytically.For a rhombic pinwheel crystal the distance between pinwheels is fixed and depends on the angleα. As a consequence, the distribution of NN spacings between pinwheels of arbitrary topologicalcharge is a delta function (Fig. S20 B and Fig. S21 B)

P(s) = δ(s− smin) (58)

with

P(s) =

{δ(s− s1) : 0 < α ≤ π/3δ(s− s2) : π/3 < α ≤ π,

(59)

wheres1 = Λ/(2 cos(α/2)) (60)

ands2 = Λ/(2 sin(α/2)). (61)

59

Likewise,P+−(s) = δ(s− s1) (62)

for pinwheel pairs of opposite charge, and

P++(s) = δ(s− s2) (63)

for pinwheel pairs of same charge (shown in Figs. S20 C and S21 C). Note, that for 0 < α ≤ π/3nearest neighbors have opposite topological charge whereas for π/3 < α ≤ π they have thesame topological charge (compare Fig. S20, B and C, with Fig. S21, B and C). The correspondingcumulative distributions Σ(s) =

∫ s0 ds′P(s) are Heaviside step functions (Fig. S20, E and F, and

Fig. S21, E and F).For hexagonal pinwheel crystals the pinwheel distance statistics depends on the arrangement

of active modes. The distribution of NN spacings between pinwheels of arbitrary topologicalcharge is again a delta function

P(s) = δ(s− smin) (64)

withsmin =

23√

3(65)

for both cases (Figs. S22 B and S23 B).Likewise,

P(s) = δ(s− s1) (66)

for pinwheel pairs of opposite charge, and

P(s) = δ(s− s2) (67)

for pinwheel pairs of same charge. The values s1 and s2 depend on the class of active modes andare given by

s1 =2

3√

3s2 =

23

(68)

for the high pinwheel density planform (Fig. S22 C) and by

s1 =2

3√

3s2 =

2√3

(69)

for the low pinwheel density planform (shown in Fig. S23 C). The corresponding cumulativedistributions Σ(s) =

∫ s0 ds′P(s) are Heaviside step functions (Figs. S22, E and F, and S23, E and F).

5.7 Statistics of random pinwheel positions

A simple and analytically tractable model of irregular pinwheel positions is the superposition oftwo independent 2D Poisson processes for the positions of each of the two topological charges ofpinwheels. In this model, pinwheel density is a free parameter that fixes all pinwheels nearestneighbor distance statistics. The law of large numbers applies such that the variability exponent is

60

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

0 0.5 10

0.5

1


Fre

q.

(in

teg

rate

d)

OppositeSame charge

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

Size (hypercolumns)

SD

100

101

10−1

100

Size

SD

10−110

010

110

210

310

−210

−110

010

1

0 0.5 10

1

2

3

4


Fre

q. (n

orm

aliz

ed)

OppositeSame charge

A B C

FED

Figure S24. 24 Statistics for randomly positioned pinwheels in two dimensions (2D Poisson pro-cess with density ρ = π). (A) Pinwheel positions for two independent 2D Poisson processes, eachwith density π/2 (red referring to positive, blue to negative topological charge). (B and C) Nearestneighbor spacing distributions for pinwheels of arbitrary (B) and opposite and same (C) topo-logical charge. (E and F) Cumulative distribution functions of the histograms in (B) and (C) forpinwheels of arbitrary (E) and opposite and same (F) topological charge. (D) Standard deviationSD of pinwheel densities in randomly selected circular regions of size A (area A measured in unitsof Λ2). The black dashed line represents the SD for a 2D Poisson process of pinwheel density π. Theinset shows the curve for a larger region together with the fit to the curve c (ρ/A)γ. Statistics in (Bto F) were obtained by averaging over 1000 realizations of size 484 Λ2.

exactly 1/2. The variability coefficient is c = 1 independent of density, showing that the variabilitycoefficients characteristic of the common design are sub-Poisson.

Thus, we consider two independent Poisson processes for pinwheels of positive and negativetopological charges with densities ρ+ = ρ− = ρ/2 = π/2. The unit of space is denoted by Λ. LetN(A) denote the number of pinwheels within a window of area A (measured in units of Λ2) thefluctuations of ρ(A) are related to the number variance NV, defined as

NV(A) = 〈N(A)2〉 − 〈N(A)〉2, (70)

where the brackets 〈·〉 represent the average over the ensemble. The pinwheel density variability

SD(A) :=√〈ρ(A)2〉 − 〈ρ(A)〉2, (71)

expressed in terms of the number variance, is given by

61

SD(A) =√

NV(A)/A. (72)

Since for a Poisson process 〈N2〉 − 〈N〉2 = 〈N〉, the number variance is

NV(A) = ρA (73)

and the standard deviation of the pinwheel density is simply given by

SD(A) =√

ρ A−1/2. (74)

Generating N=1000 realizations of size 484 Λ2, a fit of SD(A) to the general form SD(A) =c (ρ/A)γ yields(γ = 0.5, c = 1.00), in perfect agreement with Eq. (S74).

The nearest neighbor spacing distribution of a 2D-Poisson process with point density ρ is (seee.g.(S 76))

P(s) = 2πρse−πρs2, (75)

which in our context represents the distribution for pinwheels of arbitrary charge (Fig. S24 B).Distributions for opposite and equally charged pinwheels are identical and obtained from Eq.(S75) by replacing ρ → ρ/2 (Fig. S24 C). The mean nearest neighbor distances are d = (4ρ)−1/2

and d++ = d+− = (2ρ)−1/2. For ρ = π this yields d ≈ 0.28 and d++ = d+− = 0.4. Thecorresponding cumulative distributions are given by

Σ(s) :=∫ s

0dt P(t) = 1− e−πρs2

(76)

and shown in Fig. S24, E and F.An exponent of γ = 1

2 in Poisson maps reflects the law of large numbers. That a similar ex-ponent is found also in experiment (Fig. 2 D and Fig. S10) suggests its applicability to pinwheelcounting in experimentally measured orientation maps. In other statistics, Poisson maps deviatesubstantially from the common design. The prefactor c is considerably larger than in orientationmaps (Fig. 2 D and Fig. S10). Poisson maps also deviate strongly in their NN statistics. For in-stance, mean distances of pinwheels of arbitrary charge and same charge are substantially shorterthan in the biological data.

62

6 Supporting discussion of potential evolutionary origins of the com-mon design

How can the strong similarity of orientation column layouts among species long separated inmammalian evolution be explained? One possibility is that the taxa sampled in this study mayhave retained a common ancestral orientation column system. According to the fossil record andcladistic reconstructions, early mammals were small-brained, nocturnal, squirrel-like animals ofreduced visual abilities with a telencephalon containing only a minor neocortical fraction (S 77,78). For instance, endocast analysis of Asioryctes nomegetensis, a representative stem eutherian fromthe late cretaceous, indicates a total anterior-posterior extent of 4mm for its entire neocortex (S 77,78). Similarly, the tenrec (Echinops telfari), one of the closest living relatives of the boreoeutherianancestor (S 79, 80) has a neocortex of essentially the same size and a visual cortex that totals only2mm2 (S 77). Since the neocortex of early mammals was subdivided into several cortical areas (S77) and orientation hypercolumns measure between 0.4mm2 and 1.4mm2 (see Fig. 2 B and (S 3)),it is difficult to conceive of ancestral eutherians with a system of orientation columns. In fact, noextant mammal with a visual cortex of such size is known to possess orientation columns (S 81).The phylogenetic relations depicted in Fig. l A then would rather imply that systems of orientationcolumns evolved independently in laurasiatheria (such as carnivores) and in euarchonta (such astree shrews and primates).

Alternatively, one might hypothesize that ancestral mammals in the cretaceous already pos-sessed a system of few or miniaturized orientation columns and pinwheels, implying that rodentsand lagomorphs subsequently lost this system. In this scenario, the orientation column systems indifferent mammalian lineages would be derived from a common ancestor. Regardless of whetherthe systems of orientation columns in the sampled taxa derived by homology or homoplasy, thequantitative similarity of their design is especially striking vis-à-vis the divergence of other fea-tures that characterize the functional architecture of the visual cortex. For example, the pattern ofmonocular input from the thalamus to the visual cortex varies widely among the sampled taxa,and even among different species in the same taxon (e.g., primates) (S 82, 81). Thus the com-mon design of orientation column systems reported here, even if it were derived from a commonancestor, apparently must have been precisely preserved throughout the evolutionary history ofmodern carnivores and primates, despite significant diversification in the organization of feedfor-ward inputs and intrinsic circuitry within the visual cortex.

Our results show that self-organization of visual cortical circuits dominated by suppressivelong-range interactions can robustly reproduce the common design. This implies that this type ofdevelopmental mechanism strongly constrains the range of possible OR-map designs and couldthus explain the independent emergence of the common design in separated mammalian lineages.Alternatively, one may consider the hypothesis that some unknown selective pressure drove thesedisparate lineages towards the common design. However, the three analyzed species evolved un-der quite distinct functional demands on their visual systems. Galagos are nocturnal, arboreal,frugivores from the African rainforest; tree shrews are diurnal, semiarboreal, insectivores from de-cidouous tropical forests of southeast Asia; and ferrets, although long domesticated, are presum-ably derived from crepuscular or nocturnal, terrestrial, rodent-predators from western Europe.

63

The diversity of niches inhabited by the three species thus essentially argues against attributingthe common design to common selective pressures shaping the evolution of visual cortical archi-tecture. Consistent with this view, American gray squirrels (Sciurus carolinensis), highly visualanimals with a niche and visual environment comparable to the tree shrew, lack OR-columns al-together (S 81).

References

(S1) W. H. Bosking, Y. Zhang, B. Schofield, and D. Fitzpatrick. Orientation selectivity and thearrangement of horizontal connections in tree shrew striate cortex. J. Neurosci., 17:2112–2127, 1997.

(S2) L. E. White, D. M. Coppola, and D. Fitzpatrick. The contribution of sensory experience tothe maturation of orientation selectivity in ferret visual cortex. Nature, 411:1049–1052, 2001.

(S3) M. Kaschube, F. Wolf, T. Geisel, and S. Löwel. Genetic influence on quantitative features ofneocortical architecture. J. Neurosci., 22:7206–7217, 2002.

(S4) M. Kaschube, F. Wolf, M. Puhlmann, S. Rathjen, K. F. Schmidt, T. Geisel, and S. Löwel. Thepattern of ocular dominance columns in cat primary visual cortex: intra- and interindividualvariability of column spacing and its dependence on genetic background. Eur. J. Neurosci.,22:7206–7217, 2003.

(S5) N. V. Swindale. The development of topology in the visual cortex: a review of models.Network: Computation in Neural Systems, 7:161–247, 1996.

(S6) F. Wolf and T. Geisel. Spontaneous pinwheel annihilation during visual development. Na-ture, 395:73–78, 1998.

(S7) K. Obermayer and G. G. Blasdel. Geometry of orientation and ocular dominance columnsin monkey striate cortex. J. Neurosci., 13:4114–4129, 1993.

(S8) T. Müller, M. Stetter, M. Hübener, F. Sengpiel, T. Bonhoeffer, I. Gödecke, B. Chapman,S. Löwel, and K. Obermayer. An analysis of orientation and ocular dominance patternsin the visual cortex of cats and ferrets. Neural Comp., 12:2573–2595, 2000.

(S9) X. Xu, W. H. Bosking, L. E. White, D. Fitpatrick, and V. A. Casagrande. Functional organi-zation of visual cortex in the prosimian bush baby revealed by optical imaging of intrinsicsignals. J. Neurophysiol., 94:2748–2762, 2005.

(S10) S. Löwel, K. E. Schmidt, D. S. Kim, F. Wolf, F. Hoffsümmer, W. Singer, and T. Bonhoeffer.The layout of orientation and ocular dominance domains in area 17 of strabismic cats. Eur.J. Neurosci., 10:2629–2643, 1998.

(S11) J. Swift and P. C. Hohenberg. Hydrodynamic fluctuations at the convective instability. Phys.Rev. A, 15:319–328, 1977.

64

(S12) M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium. Rev. Mod. Phys.,65:851–1112, 1993.

(S13) M. C. Cross and H. Greenside. Pattern Formation and Dynamics in Nonequilibrium Systems.Cambridge University Press, New York, USA, 2009.

(S14) F. Wolf. Methods and models in neurophysics, Les Houches, Session LXXX, chapter 12. Elsevier,Amsterdam, 2005.

(S15) F. Wolf. Symmetry, multistability, and long-range interactions in brain development. Phys.Rev. Lett., 95:208701, 2005.

(S16) M. Kaschube, M. Schnabel, and F. Wolf. Self-organization and the selection of pinwheeldensity in visual cortical development. New Journal of Physics, 10:015009–015029, 2008.

(S17) C. D. Gilbert and T. N. Wiesel. Columnar specificity of intrinsic horizontal and corticocorti-cal connections in cat visual cortex. J. Neurosci., 9:2432–2442, 1989.

(S18) K. E. Schmidt, D. S. Kim, W. Singer, T. Bonhoeffer, and S. Löwel. Functional specificity oflong-range intrinsic and interhemispheric connections in the visual cortex of strabismic cats.J. Neurosci., 17:5480–5492, 1997.

(S19) Z. F. Kisvarday, E. Toth, M. Rausch, and U. T. Eysel. Orientation-specific relationship be-tween populations of excitatory and inhibitory lateral connections in the visual cortex of thecat. Cereb. Cortex, 7:605–618, 1997.

(S20) T. Yousef, T. Bonhoeffer, D. S. Kim, U. T. Eysel, E. Toth, and Z. F. Kisvarday. Orientationtopography of layer 4 lateral networks revealed by optical imaging in cat visual cortex (area18). Eur. J. Neurosci., 11:4291–4308, 1999.

(S21) M. Weliky, K. Kandler, D. Fitzpatrick, and L. C. Katz. Patterns of excitation and inhibitionevoked by horizontal connections in visual cortex share a common relationship to orienta-tion columns. Neuron, 15:541–552, 1995.

(S22) R. Malach, Y. Amir, M. Harel, and A. Grinvald. Relationship between intrinsic connectionsand functional architecture revealed by optical imaging and in vivo targeted biocytin injec-tions in primate striate cortex. Proc. Natl. Acad. Sci. U.S.A., 90:10469–10473, 1993.

(S23) A. Angelucci, J. B. Levitt, E. J. Walton, J. M. Hupe, J. Bullier, and J. S. Lund. Circuits for localand global signal integration in primary visual cortex. J. Neurosci., 22:8633–8646, 2002.

(S24) P. Manneville. Dissipative Structures and Weak Turbulence. Academic Press, San Diego, 1990.

(S25) K. E. Schmidt, R. Goebel, S. Löwel, and W. Singer. The perceptual grouping criterion ofcolinearity is reflected by anisotropies of connections in the primary visual cortex. Eur. J.Neurosci., 9:1083–1089, 1997.

65

(S26) K. S. Rockland and J. S. Lund. Widespread periodic intrinsic connections in the tree shrewvisual cortex. Science, 215:1532–1534., 1982.

(S27) W. H. Bosking, R. Kretz, M. L. Pucak, and D. Fitzpatrick. Functional specificity of callosalconnections in tree shrew striate cortex. J. Neurosci., 20:2346–2359, 2000.

(S28) A. Burkhalter, K. L. Bernardo, and V. Charles. Development of local circuits in human visualcortex. J. Neurosci., 13:1916–1931, 1993.

(S29) A. Shmuel, M. Korman, A. Sterkin, M. Harel, S. Ullman, R. Malach, and A. Grinvald. Retino-topic axis specificity and selective clustering of feedback projections from v2 to v1 in the owlmonkey. J. Neurosci., 25:2117–2131, 2005.

(S30) A. Angelucci and P. C. Bressloff. Contribution of feedforward, lateral and feedback con-nections to the classical receptive field center and extra-classical receptive field surround ofprimate v1 neurons. Prog. Brain Res., 154:93–120., 2006.

(S31) S. Löwel and W. Singer. Selection of intrinsic horizontal connections in the visual cortex bycorrelated neuronal activity. Science, 255:209–12, 1992.

(S32) K. F. Schmidt and S. Lowel. Strabismus modifies intrinsic and inter-areal connections in catarea 18. Neuroscience, 152:128–137, 2008.

(S33) N. L. Rochefort, P. Buzas, N. Quenech’du, A. Koza, U. T. Eysel, C. Milleret, and Z. F. Kisvar-day. Functional selectivity of interhemispheric connections in cat visual cortex. Cereb. Cortex,19:2451–2465, 2009.

(S34) K. E. Schmidt, S. G. Lomber, and G. M. Innocenti. Specificity of neuronal responses inprimary visual cortex is modulated by interhemispheric corticocortical input. Cereb. Cortex,2010.

(S35) M. Kaschube, M. Schnabel, F. Wolf, and S. Löwel. Interareal coordination of columnar archi-tectures during visual cortical development. Proc. Natl. Acad. Sci. U S A, 106:17205–17210,2009.

(S36) H. J. Chisum, F. Mooser, and D. Fitzpatrick. Emergent properties of layer 2/3 neurons reflectthe collinear arrangement of horizontal connections in tree shrew visual cortex. J. Neurosci.,23:2947–2960, 2000.

(S37) E. M. Callaway and L. C. Katz. Emergence and refinement of clustered horizontal connec-tions in cat striate cortex. J. Neurosci., 10:1134–1153, 1990.

(S38) J. C. Durack and L. C. Katz. Development of horizontal projections in layer 2/3 of ferretvisual cortex. Cereb. Cortex, 6:178–183, 1996.

(S39) R. A. W. Galuske and W. Singer. The origin and topography of long-range intrinsic projec-tions in cat visual cortex: a developmental study. Cereb. Cortex, 6:417–430, 1996.

66

(S40) E. S. Ruthazer and M. P. Stryker. The role of activity in the development of long-rangehorizontal connections in area 17 of the ferret. J. Neurosci., 16:7253–7269, 1996.

(S41) L. E. White and D. Fitzpatrick. Vision and cortical map development. Neuron, 56:327–338,2007.

(S42) A. Stepanyants, L. M. Martinez, A. S. Ferecsko, and Z. F. Kisvarday. The fractions of short-and long-range connections in the visual cortex. Proc. Natl. Acad. Sci. U. S. A., 106:3555–60,2009.

(S43) J. B. Levitt and J. S. Lund. Contrast dependence of contextual effects in primate visual cortex.Nature, 387:73–76, 1997.

(S44) H. Adesnik and M. Scanziani. Lateral competition for cortical space by layer-specific hori-zontal circuits. Nature, 464:1155–1160, 2010.

(S45) J. M. Crook, R. Engelmann, and S. Löwel. GABA-inactivation attenuates collinear facilita-tion in cat primary visual cortex. Exp. Brain Res., 143:295–302, 2002.

(S46) C. D. Gilbert and T. N. Wiesel. The influence of contextual stimuli on the orientation selec-tivity of cells in primary visual cortex of the cat. Vision Res., 30:689–701, 1990.

(S47) J. A. Hirsch and C. D. Gilbert. Synaptic physiology of horizontal connections in the cat’svisual cortex. J. Neurosci., 11:1800–1809, 1991.

(S48) B. Haider, M. R. Krause, A. Duque, Y. Yu, J. Touryan, J. A. Mazer, and D. A. McCormick.Synaptic and network mechanisms of sparse and reliable visual cortical activity during non-classical receptive field stimulation. Neuron, 65:107–121, 2010.

(S49) M. Stemmler, M. Usher, and E. Niebur. Lateral interactions in primary visual cortex: Amodel bridging physiology and psychophysics science. Psychophysics Sci., 269:1877–1880,1995.

(S50) Eyal Bartfeld and Amiram Grinvald. Relationships between orientation–preference pin-wheels, cytochrome oxidase bolbs, and ocular–dominance columns in primate striate cortex.Proc. Natl. Acad. Sci. U S A, 89:11905–11909, 1992.

(S51) M. Hubener, D. Shoham, A. Grinvald, and T. Bonhoeffer. Spatial relationships among threecolumnar systems in cat area 17. J. Neurosci., 17:9270–9284, 1997.

(S52) M. C. Crair, E. S. Ruthazer, D. C. Gillespie, and M. P. Stryker. Ocular dominance peaks atpinwheel center singularities of the orientation map in cat visual cortex. J. Neurophysiol.,77:3381–3385, 1997.

(S53) A. Das and C. D. Gilbert. Distortions of visuotopic map match orientation singularities inprimary visual cortex. Nature, 387:594–598, 1997.

67

(S54) W. H. Bosking, J. C. Crowley, and D. Fitzpatrick. Spatial coding of position and orientationin primary visual cortex. Nat. Neurosci., 5:874–882, 2002.

(S55) P. Buzas, M. Volgushev, U. T. Eysel, and Z. F. Kisvarday. Independence of visuotopic rep-resentation and orientation map in the visual cortex of the cat. Eur. J. Neurosci., 18:957–968,2003.

(S56) L. Reichl, S. Lowel, and F. Wolf. Pinwheel stabilization by ocular dominance segregation.Phys. Rev. Lett., 102:208101, 2009.

(S57) P. J. Thomas and J. D. Cowan. Symmetry induced coupling of cortical feature maps. Phys.Rev. Lett., 92(18):188101, 2004.

(S58) M. Schnabel, M. Kaschube, S. Löwel, and F. Wolf. Random waves in the brain: Symmetriesand defect generation in the visual cortex. Eur. Phys. J. Special Topics, 145:137, 2007.

(S59) P. C. Bressloff. Geometric visual hallucinations, euclidean symmetry and the functionalarchitecture of striate cortex. Phil. Trans. R. Soc. B, 356:299, 2001.

(S60) N. Mayer, J. M. Herrmann, and T. Geisel. Curved feature metrics in models of visual cortex.Neurocomp., 44-46:533–539, 2002.

(S61) H. Y. Lee, M. Yahyanejad, and M. Kardar. Symmetry considerations and development ofpinwheels in visual maps. Proc. Natl. Acad. Sci. U.S.A., 100:16036–16040, 2003.

(S62) P. C. Bressloff. Euclidean shift-twist symmetry in population models of self-aligning objects.SIAM J. Appl. Math., 64:1668, 2004.

(S63) N. M. Mayer, M. Browne, J. M. Herrmann, and M. Asada. Symmetries, non-Euclidean met-rics, and patterns in a Swift-Hohenberg model of the visual cortex. Biol Cybern, 99:63–78,2008.

(S64) M. Schnabel, M. Kaschube, and F. Wolf. Pinwheel stability, pattern selection and the geom-etry of visual space. arXiv, pages 0801.3832v2 [q–bio.NC], 2008.

(S65) K. D. Miller. A model for the development of simple cell receptive fields and the orderedarrangement of orientation columns through activity–dependent competition between on–and off–center inputs. J. Neurosci., 14:409–441, 1994.

(S66) E. Erwin, K. Obermayer, and K. Schulten. Models of orientation and ocular dominancecolumns in the visual cortex: A critical comparison. Neural Comp., 7:425–468, 1995.

(S67) J. A. Bednar and R. Miikkulainen. Selforganization of spatiotemporal receptive fields andlaterally connected direction and orientation maps. Neurocomputing, 52Ð54:473Ð480, 2003.

(S68) J. A. Bednar, A. Kelkar, and R. Miikkulainen. Scaling self-organizing maps to model largecortical networks. Neuroinformatics, 2:275–302, 2004.

68

(S69) W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes. Cam-bridge University Press, Cambridge, 2nd edition, 1992.

(S70) M. C. Cross, D. Meiron, and Y. Tu. Chaotic domains: A numerical investigation. Chaos,4:607–619, 1994.

(S71) AA Koulakov and DB Chklovskii. Orientation preference patterns in mammalian visualcortex: a wire length minimization approach. Neuron, 29:519–527, 2001.

(S72) M. W. Cho and S. Kim. Understanding visual map formation through vortex dynamics ofspin Hamiltonian models. Phys. Rev. Lett., 92:018101, 2004.

(S73) M. W. Cho and S. Kim. Different ocular dominance map formation influenced by orientationpreference columns in visual cortices. Phys. Rev. Lett., 94:068701, 2005.

(S74) C. von der Malsburg. Self-organization of orientation sensitive cells in the striate cortex.Kybernetik, 14:85–100, 1973.

(S75) A. Grabska-Barwinska and C. von der Malsburg. Establishment of a scaffold for orientationmaps in primary visual cortex of higher mammals. J. Neurosci., 28(1):249–257, 2008.

(S76) F. Haake. Quantum Signatures of Chaos. Springer New York, Inc., Secaucus, NJ, USA, 2006.

(S77) J. H. Kaas. Evolution of Nervous Systems: A Comprehensive Reference, chapter 3.03 Reconstruct-ing the Organization of Neocortex of the First Mammals and Subsequent Modifications.Elsevier, Amsterdam, 2006.

(S78) Z. Kielian-Jaworowsk, R. L. Cifelli, and Z.-X. Luo. Mammals from the Age of Dinosaurs.Columbia University Press, New York, 2004.

(S79) O. R. Bininda-Emonds, M. Cardillo, K. E. Jones, R. D. MacPhee, R. M. Beck, R. Grenyer, S. A.Price, R. A. Vos, J. L. Gittleman, and A. Purvis. The delayed rise of present-day mammals.Nature, 446:507–512, 2007.

(S80) J. R. Wible, G. W. Rougier, M. J. Novacek, and R. J. Asher. Cretaceous eutherians andLaurasian origin for placental mammals near the K/T boundary. Nature, 447:1003–1006,2007.

(S81) S. D. Van Hooser. Similarity and diversity in visual cortex: is there a unifying theory ofcortical computation? Neuroscientist, 13:639–656, 2007.

(S82) J. C. Horton and D. L. Adams. The cortical column: a structure without a function. Philos.Trans. R. Soc. Lond., B, Biol. Sci., 360:837–862, 2005.

69

universality in the evolution of orientation columns in...

Documents