Self-organization and the selection of pinwheel density in visual cortical development

0.52 MB·English

by  Matthias Kaschube

Self-Organization and the Selection of Pinwheel Density in
Visual Cortical Development
8 Matthias Kaschube1−4, Michael Schnabel1,2 and Fred Wolf1,2
0
0
2
n 1MaxPlanckInstituteforDynamicsandSelf-Organizationand2BernsteinCenterfor
a ComputationalNeuroscience,Goettingen,Germany,3Lewis-SiglerInstituteforIntegrative
J
Genomicsand4JosephHenryLaboratoriesofPhysics,PrincetonUniversity,PrincetonNJ,USA
3
2
Self-organization of neural circuitry is an appealing framework for understanding cortical

]
C development, yet its applicability remains unconfirmed. Models for the self-organization of
N neural circuits have been proposed, but experimentally testable predictions of these models
. have been less clear. The visual cortex contains a large number of topological point defects,
o
called pinwheels, which are detectable in experiments and therefore in principle well suited
i
b
fortestingpredictionsofself-organization empirically. Here,weanalyticallycalculatetheden-
-
q sity of pinwheels predicted by a pattern formation model of visual cortical development. An
[
important factor controlling the density of pinwheels in this model appears to be the pres-

1 enceofnon-locallong-rangeinteractions, apropertywhichdistinguishescortical circuits from
v
manynonlivingsystemsinwhichself-organizationhasbeenstudied. Weshowthatinthelimit
1
5 where the range of these interactions is infinite, the average pinwheel density converges to π.
6 Moreover, an average pinwheel density close to this value is robustly selected even for inter-
3
mediate interaction ranges, a regime arguably covering interaction-ranges in a wide range of
.
1
different species. In conclusion, our paper provides the first direct theoretical demonstration
0
8 andanalysisofpinwheeldensityselectioninmodelsofcorticalself-organization andsuggests
0 toquantitativelyprobethistypeofpredictioninfuturehigh-precisionexperiments.
:
v
i
X
1 Introduction
r
a
Neuronal circuits in the mammalian cerebral cortex are among the most complex systems in na-
ture. The biological mechanisms that contribute to their formation in early brain development
remain poorlyunderstood. However,it is unlikely thattheprecisearchitecture ofmature cortical
circuits can be attributed to genetic prespecification, since the number of genes in the genome is
insufficient[1]. Instead,dynamicalself-organization presumablyplaysamajorroleinshapingthe
architecture of neuronal circuits in the cerebral cortex. Dynamical self-organization is most thor-
oughlydescribedinnon-livingphysicalsystemsdrivenoutsideofthermodynamicequilibriumby
external forcing. Whereas the emergence of structure is externally driven, the structures formed
are primarily determined through interactions within the system itself[2, 3]. Neural circuits en-
compass various positive and negative feedback loops, and these could well form the basis for
cortical self-organization. However, evidence supporting this presumption is derived from theo-
reticalconsiderations[4,5,6]ratherthanempiricalobservation. Modelsfortheself-organizationof
1
neuralcircuits have been proposed,but experimentallytestable predictions ofthesemodelshave
beenlacking.
The system of orientation columns in the visual cortex is a paradigmatic systemfor studying
cortical developmentand the role of self-organization in this process. Most neuronsin the visual
cortexrespondselectivelyto a particular orientationof an elongatedvisual stimulus. Whereas in
columns perpendicular to the cortical surface, neurons prefer similar stimulus orientations, the
preferredorientationvaries mostlysmoothly[7]and repetitivelyacross thecortical surface giving
rise to a complex two-dimensional pattern called the map of orientation preference (Fig. 1a, b).
Throughoutthecorticalmap,therearepoint-likeorientationsingularities[8,9,10]calledpinwheel
centers[11]atwhichallstimulusorientationsarerepresentedincircular fashion. Numerousstud-
ies are consistent with the hypothesis that orientation maps develop through activity-dependent
self-organization. Theyformindarkrearedanimals[12],undersubstantialmanipulationofvisual
input[13, 14], and even in auditorycortex when rewired to be driven by visual inputs[15]. More-
over, an analogy between cortical developmentand pattern formation appears plausible. Like in
othersystemswherepatternformation has been observed[2, 3], the orientation map arises prob-
ablyfromaninitially non-selectivestate,itexhibitsatypicalperiodicityandaspatialextensionat
leastanorderofmagnitudelargerthanthebasicperiodicitylength.
The conditions under which orientation maps can arise through self-organization have been
thoroughlyinvestigated theoretically[16, 17, 18]. A recent and highly promising approach stress-
ing the analogy to pattern forming systems showed that pinwheels can be stabilized by activity-
dependentlong-range interactions [19]. A phenomenological order parameter field model based
ontheSwift-Hohenbergequation[20]wasproposedinwhichorientationmapsarisefromasuper-
criticalbifurcationofTuring-type.Inthisclassofmodels,thestabilizingnonlinearityincludesonly
keyfeaturesofvisualcorticalorganizationandisconstraintbybiologicallyplausiblesymmetryas-
sumptions. The model exhibits multiple structurally distinct quasiperiodic attractors resembling
orientationmapsinthevisualcortex.
The qualitative similarity of solutions of this model to orientation maps in the visual cortex
appears promising. However, it is unclear at present weather the model accounts for aspects of
corticalorganizationalsoquantitativelyandwhetherthisresemblanceisnotjustsuperficial. Com-
paringdirectlyorientationmapsin themodeland thevisualcortexwouldbedifficult, becauseof
the large number of possible map structures. Instead, it will be virtually unavoidable to take a
statisticalapproachtothisquestions. Asdiscreteentities,pinwheelscanbedetected,countedand
localizedwithhighaccuracy. Here,wecalculate inthemodeltheaveragedensityofpinwheelsin
thelimitofinfiniteinteractionrange. Weshowthatthisdensityisrepresentativeforalargeregime
of intermediate interaction ranges covering the estimated ranges in various mammalian species.
Therefore,thequantitypinwheeldensityappearsparticularlywellsuitedfortestingforsignatures
oflong-rangedominatedself-organizationinexperiment. Inthefollowing,webrieflydescribethe
systemoforientationpreferencesinthevisualcortexandpresentthelong-rangeself-organization
modelforitsactivity-dependentdevelopment.
1.1 Orientation preferencemaps
Inthevisualcortex,asinmostareasofthecerebralcortex,informationisprocessedina2-dimensio-
nal (2D) array of functional modules, called cortical columns [21, 22]. Individual columns are
groups of neurons extending vertically throughout the entire cortical thickness that share many
functionalproperties. Orientationcolumnsinthevisualcortexarecomposedofneuronspreferen-
2
Figure1: Patternsoforientationcolumnsandlong-rangehorizontalconnectionsintheprimaryvi-
sualcortexoftreeshrewvisualizedusingopticalimagingofintrinsicsignals(modifiedfrom[23]).
a,Activity patternsresultingfromstimulation withvertically andobliquely orientedgratings,re-
spectively. Whitebarsdepicttheorientationofthevisualstimulus. Activatedcolumnsarelabeled
dark gray. The used stimuli activate only columns in the primary visual area V1. The patterns
thusendattheboundarybetweenareasV1and V2. b, Thepatternoforientationpreferencescal-
culatedfromsuchactivitypatterns. Theorientationpreferencesofthecolumnsarecolorcodedas
indicatedbythebars. Apartofthepatternoforientationpreferencesisshownathighermagnifi-
cation. Twopinwheelcentersofoppositetopologicalchargearemarkedbyarrows. c,Long-range
horizontalconnectionsextendoverseveralmillimeters parallel tothecortical surface (treeshrew,
superimposedon the orientation preference map). White symbols indicate locations of cells that
werefilledbyatracer(biocytin);labeledaxonsareindicatedbyblacksymbols.
tially respondingto visual contoursof a particular stimulus orientation [7]. In a plane parallel to
thecorticalsurface,neuronalselectivitiesvarysystematically,sothatcolumnsofsimilarfunctional
propertiesformhighlyorganized2Dpatterns,knownasfunctionalcorticalmaps(Fig. 1a,b).
Experimentally,thepatternoforientationpreferencescan be visualized usingoptical imaging
methods[8, 9]. Optical imaging ofintrinsic signals is based on thefact that the optical properties
differ in active vs. lessactive partsofthecortex[24]. Thisis utilized torecordpatternsofactivity
fromlightreflectance. Inatypicalexperiment,theactivitypatternsE (x)producedbystimulation
k
withagratingoforientationθ arerecorded. Herex= (x,y)representsthelocationofacolumnin
k
thecortex. Usingthe activity patternsE (x), a field ofcomplex numbers z(x) can be constructed
k
thatcompletelydescribesthepatternoforientationcolumns:
z(x) = ei2θk E (x). (1)
k
k
X
Thepatternoforientationpreferencesϑ(x)isthenobtainedfromz(x)asfollows:
1
ϑ(x)= arg(z). (2)
2
3
Typical examples of such activity patterns E (x) and the patterns of orientation preferences de-
k
rived from them are shown in Fig. 1a, b. Numerous studies confirmed that the orientation pref-
erence of columns is an almost everywhere continuous function of their position in the cortex.
The domains formed by neighboring columns with similar orientation preference are called iso-
orientationdomains[25].
1.2 Pinwheels
At many locations the iso-orientation domains are arranged radially around a common center
[9,10]. Aroundthesepinwheel[11]centers,stimulusorientationsarerepresentedincircular fash-
ion. Such an arrangement had been previously hypothesizedon the basis of electrophysiological
experiments[26,27]andtheoreticalconsiderations[28]. Theregionsexhibitingthiskindofradial
arrangementweretermedpinwheels[11]. Thecentersofpinwheelsarepointdiscontinuitiesofthe
field ϑ(x) where the mean orientation preference of nearby columns changes abruptly. They can
becharacterizedbyatopologicalcharge
1
q = ϑ(x)ds (3)
i
2π ∇
ICj
which indicates in particular whether the orientation preference increases clockwise around the
center of the pinwheel or counterclockwise. Here, C is a closed curve around a single pinwheel
j
center at x . Since ϑ is a cyclic variable within the interval [0,π) and up to isolated points is a
i
smoothfunctionofx,q canonlyhavethevalues
i
n
q = (4)
i
2
where n is an integer number [29]. If its absolute value q is 1/2, each orientation is represented
i
| |
once in the vicinity of a pinwheel center. Pinwheel centers with a topological charge of 1/2
±
are simple zeros of z(x). In experiments only pinwheels that had the lowest possible topological
charge q = 1/2 are observed. This means there are only two types of pinwheels: those whose
i
±
orientationpreferenceincreasesclockwiseandthosewhoseorientationpreferenceincreasescoun-
terclockwise. This organization has been confirmed in a large number of species and is therefore
believedtobeageneralfeatureofvisualcorticalorientationmaps[30,31,32,33,23,34].
1.3 Hypercolumn andpinwheeldensity
The pattern of preferred orientations is roughly repetitive [7, 21]. The column spacing Λ, i.e. the
spacingbetweenadjacentiso-orientationdomainspreferringthesamestimulusorientation,istyp-
ically in therange of 1mm. The column spacing Λ determinesthe size of thecortical hypercol-
∼
umn,whichisconsideredtobethebasicprocessingunitofthevisualcortex[7,22,35]. Wedefine
the size of a hyper column by Λ2. The pinwheel density is defined as the number of pinwheels
per unit area Λ2. Thus, by this definition, the pinwheel density is independent of the spacing of
columnsanddimension-less.
1.4 Intra-corticalconnectivity
Visual cortical neurons are embedded in densely connected networks [36]. Besides a strong con-
nectivityverticaltothecorticalsheetbetweenneuronsfromdifferentlayerswithinacolumn,neu-
4
ronsalsoformextensiveconnectionshorizontaltothecorticalsurfacelinkingdifferentorientation
columns. Theseconnections extendfor several millimeters parallel tothe cortical surface and are
thereforecalledlong-rangehorizontalconnections. AsshowninFig. 1bfortheexampleofthetree
shrew,theseconnectionsareclusteredprimarilyconnectingdomainsofsimilarorientationprefer-
ence. Theyhave beenobservedin various mammals [37, 23, 38, 12]and repeatedlyhypothesized
toplayanimportanttoroleinvisualprocessingtaskssuchascontourintegration.
1.5 Activity-dependentdevelopment
Innormaldevelopment,orientationcolumnsfirstformataboutthetimeofeyeopening[39,13,12]
whichfortheferretisapproximatelyatpostnatalday(PD)31. Thisisjustafewdaysafterneurons
firstrespondtovisualstimuli. Asubsetofneuronsshoworientationpreferencefromthattimeon,
buttheadultpatternisnotattaineduntilsevenweeksafterbirth[40]. Roughlyclusteredhorizontal
connectionsare presentby around PD 27 [41]. Many lines ofevidence suggestthat theformation
of orientation columns is a dynamical process dependent on neuronal activity and sensitive to
visual experience [17, 42]. This is suggested not only by the time line of normal development,
butalsoreceivessupportfromvariousexperimentsmanipulating thesensoryinputtothecortex.
Mostintriguingly,whenvisualinputsarerewiredtodrivewhatwouldnormallybecomeprimary
auditorycortex,orientation selectiveneuronsand a patternoforientationcolumns evenforms in
thisbrainregionthatwouldnormallynotatallbeinvolvedintheprocessingofvisualinformation
[15, 42]. This observation suggests that the capability to form a system of orientation columns
is intrinsic to the learning dynamics of the cerebral cortex given appropriate inputs. Moreover,
thecomparisonofdevelopmentunderconditionsofmodifiedvisualexperiencedemonstratesthat
adequatevisualexperienceisessentialforthecompletematurationoforientationcolumnsandthat
impairedvisualexperience,aswithexperimentallyclosedeye-lids[13,12]orbyrearingkittensin
astripedenvironmentconsistingofasingleorientation[14],cansuppressorimpairtheformation
oforientationcolumns(butseealso[43]).
Viewed from a dynamical systemsperspective, the activity-dependentremodeling of the cor-
ticalnetworkisaprocessofdynamical patternformation. Inthispicture,spontaneoussymmetry
breaking in the developmental dynamics of the cortical network underlies the emergence of cor-
tical selectivities such as orientation preference [16]. The subsequent convergence of the cortical
circuitry towards a mature pattern of selectivities can be viewed as the development towards an
attractorofthedevelopmentaldynamics[18]. Thisisconsistentwiththeinterpretationofcortical
development as an optimization process. In the following, we will briefly describe a model [19]
thatisbasedonthisview.
1.6 Modelingcortical self-organization
Self-organizationhasbeenobservedtorobustlyproducelargescalestructuresinvariouscomplex
systems. Often, the class of patterns emerging depends on fundamental system properties such
as symmetries rather than on system specific details. Pattern formation can therefore often be
describedbyabstractmodelsincorporatingthesepropertiesonly. ForsystemsundergoingaTuring
typeinstability,canonicalmodelequationsareoftheSwift-Hohenberg[20,2]type
∂ z = F[z]
t
= L z+N [z]+N [z]+ (5)
SH 2 3
···
5
wherethelinearpartis
L = r k2+ 2 2 (6)
SH − c ∇
and z(x,t) is a complex scalar field. If the bifurc(cid:0)ation pa(cid:1)rameter r < 0, the homogeneous state
z(x) = 0 is stable. For r > 0, a pattern with wavelength close to Λ = 2π/k emerges. The lowest
c
order nonlinearities N and N are quadratic and cubic in z, respectively. The form of these non-
2 3
linearities determines the class of the emerging pattern, i.e. whether hexagons, rotating stripes,
spiralwaves,oranothertypeofpatternemerges.
Followingthis paradigm, we adopteda modeloftheform Eq. (5)with nonlinearities derived
fromkeyfeaturesofthevisualcortex(following [19,44]). Asin experimentalrecordings,orienta-
tioncolumnsarerepresentedbyacomplexfield[28,18]
z(x) = z(x) ei2ϑ(x) (7)
| |
where ϑ is the orientation preference and z a measure of selectivity at location x = (x,y) in the
| |
map. The factor 2 in the exponent accounts for the π-periodicity of stimulus parameter orienta-
tion. Constructing the nonlinearity of the model relies on the following assumptions. The model
includestheeffectsoflong-rangeintracorticalconnectionsbetweencolumnswithsimilar orienta-
tion preference (Fig. 1c). Unlike many non-living systemsin which interactions are purely local,
long-range interactions are an important and distinctive feature of neuronal circuits in the cortex
andparticularlyintheprimaryvisualcortex. Further,basedonthespatialhomogeneityofcircuits
acrosscortex,itisassumedthatthedynamicsissymmetricwithrespecttotranslations,
F[Tˆ z] = Tˆ F[z] with Tˆ z(x) = z(x+y), (8)
y y y
androtations
cos(β) sin(β)
F[Rˆ z] = Rˆ F[z] with Rˆ z(x) = z x (9)
β β β sin(β) cos(β)
(cid:18)(cid:20) − (cid:21) (cid:19)
ofthecorticalsheet. Thismeansthatpatternsthatcanbeconvertedintooneanotherbytranslation
orrotationofthecorticallayersbelongtoequivalentsolutionsofthemodel,Eq. (5),byconstruc-
tion. Itisfurtherassumedthatthedynamicsissymmetricwithrespecttoshiftsinorientation
F[eiφz]= eiφF[z]. (10)
Thus,twopatternsarealsoequivalentsolutionsofthemodel,iftheirlayoutoforientationdomains
isidentical,butthepreferredorientationsdiffereverywherebythesameconstantangle. Solutions
shallcontain representationsofall stimulusorientations. Forsimplicity, couplingstoothervisual
cortical representations such as ocular dominance or retinotopy are neglected. Considering only
leadingordertermsuptocubicnonlinearitiesanonlinearityfulfillingtheserequirementsisgiven
by
N [z(x)] = 0
2
N [z(x)] = (1 g) z(x)2z(x)+
3
− | |
1
(g 2) d2yK (y x) z(x)z(y)2 + z¯(x)z(y)2 , (11)
σ
− − | | 2
Z (cid:18) (cid:19)
6
withlong-rangeinteractionsmediatedthroughconvolutionswithaGaussian
1 − x2
Kσ(x) = 2πσ2e 2σ2 (12)
with range σ. The second parameter 0 g 2 controls local and nonlocal interactions. The
≤ ≤
first term is the only strictly local term consistent with the required symmetries, the second non-
localtermrepresentsthesimplestnon-localtermthatissymmetricwithrespecttotheseandwith
respecttopermutations
N (u,v,w) = N (w,u,v). (13)
3 3
Here the non-linear operator is written in a trilinear form as introduced in [19, 44]. This addi-
tional symmetry implies that all two-orientation solutions, for instance real valued solutions, of
the model, are unstable, which in turn guarantees that all stimulus orientations are represented.
One should note that in this model the spatial range of the nonlinearity σ is a control parameter
independentof the wavelength Λ. The patterns selected for different ratios σ/Λ are displayed in
Fig. 2b. It can also be derived as an approximation to a model for the combined development of
orientationpreferenceandlong-rangehorizontalconnections[44]. ThemodelasdefinedinEq. (5-
13)isvariational. ItisconsistentwithsynapticmodelsbasedonHebbianplasticity,e.g. [17,45,46].
Itis theonly modelknowntotheauthorsthat exhibits stable aperiodicsolutionsdominatedby a
singlespatialscaleΛ. Thesesolutionsresembleorientationmapsobservedinthevisualcortex. In
thefollowing,wediscussthestructureofthesesolutionsandthephasediagramofthemodel.
1.7 Weaklynonlinear analysis
Solutions of the model close to the bifurcation point r = 0 are known in closed form, derived by
means of a perturbation method called weakly nonlinear stability analysis[47, 2]. When the dy-
namicsisclosetoafinitewavelengthinstability,theessentialFouriercomponentsoftheemerging
patternarelocatedonthecriticalcircle. Solutionsareplanformpatterns
z(x) = A eikjx (14)
j
j
X
composedofafinitenumberofFouriercomponentswithwavevectorsonthecriticalcircle, k =
j
| |
k . Bysymmetry,thedynamics ofamplitudes A of aplanform are governedby amplitude equa-
c i
tions
A˙i = Ai gij Aj 2Ai fijAjAj−A¯i− (15)
− | | −
j j
X X
−
wherej denotestheindexofthemodeantiparalleltomodej. TheformofEq. (15)isuniversalfor
modelsofacomplexfieldz satisfyingsymmetryassumptions(8-10). Allmodeldependenciesare
included in the coupling coefficients g and f and may be obtained from F[z] by multiscale ex-
ij ij
pansion[47,2]. Denotingtheanglebetweenthewavevectorsk andk byαandδ theKronecker
i j ij
delta,thecoefficientsread[19,44]
1
g = 1 δ g(α)
ij ij
− 2
(cid:18) (cid:19)
fij = 1 δij δi−j f(α) (16)
− −
(cid:0) (cid:1)
7
a n = 1 b
g
1
n = 2
i = 0 i = 1
n = 3
n = 1 2 3 4 5
i = 0 i = 3
n = 5
σ/Λ
n = 15
i = 0 i = 200 i = 611
Figure2: a,Essentiallycomplexplanformswithdifferentnumbersn = 1,2,3,5,15ofactivemodes:
The patterns of orientation preferences θ(x) are shown. The diagrams to the left of each pattern
display the position of the wavevectors of active modes on the critical circle. For n = 3, there
are two patterns; for n = 5, there are four; and for n = 15, there are 612 different patterns. b,
Phasediagramofmodel. Ifnon-localinteractionsare dominant(g < 1)andlong-ranging(σ large
comparedtoΛ),quasiperiodicplanformsareselected. Reproducedfrom[44].
where
g(α) = g+2(2 g)exp σ2k2 cosh σ2k2cos(α)
− − c c
1
f(α) = g(α) (cid:0) (cid:1) (cid:0) (cid:1) (17)
2
arecalledangle-dependentinteractionfunctions.
StationarysolutionsofEq. (15)aregivenbyfamilies ofplanforms[19,44]
n−1
z(x) = A ei(ljkjx+φj) (18)
j
| |
j=0
X
ofordernwithwavevectors
jπ jπ
k = k cos ,sin (19)
j c
n n
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
distributed equidistantly on the upper half of the critical circle and binary values l = 1 deter-
j
±
miningwhetherthemodewithwavevectork orwithwavevector k isactive. Theseplanforms
j j
−
cannot realize a real valued function and are called essentially complex planforms (Fig. 2). For
theseplanforms the third term in Eq. (15) vanishes and the effective amplitude equations for the
activemodesreducetoasystemofLandauequations
A˙ = A g A 2A (20)
i i ij j i
− | |
j
X
8
withstationarysolutionsEq. (18)withamplitudesofequalmodulus
−1/2
A = g (21)
i ij
| |  
j
X
 
and an arbitrary phaseφ independentofthemode configurationl . Ifthedynamics is stabilized
i j
bylong-rangenonlocalinteractions(g < 1,σ > Λ),largenplanformsaretheonlystablesolutions.
Inthislong-rangeregime,theorderofplanformsgrowsas
n 2πσ/Λ, (22)
∼
approximately linear with the interaction range. For a given order n, different planforms are de-
generatedinenergy. ThisisaconsequenceofthepermutationsymmetryEq. (13). Thissymmetry
also implies that the relevant stable solutions are essentially complex planforms which in turn
guaranteesthatallstimulusorientationsarerepresented.
2 Calculation of pinwheel density
2.1 Largerange limitof interactions: Planformanisotropy
First,wecalculate theaverage pinwheeldensityρ for an ensembleofplanforms, Eq. (18), with a
l
fixedsetofwavevector directionsl = (l0,l1,...,ln−1) butarbitrary phasesφj in thelimit n .
→ ∞
Here, and until notedotherwise, shall denoteaverage over phasesφ . In this limit, z and local
j
hi
linearfunctionalsofzhaveGaussianstatisticssuchthatthedensityofpinwheelsisdeterminedby
thesecondorderstatisticsofthefield. Second,weevaluatetheexpectationvalueofρ overallsets
l
ofl.
Forlargeσ,togoodapproximationgii 1andgij gandhence Ai 1/√ng. Planforms(18)
≈ ≈ | |≈
simplifyto
n−1
2
z(x) = ei(ljkjx+φj) (23)
n
r j=0
X
where for later convenience the constant √2g was absorbed into z(x). Pinwheels are the zeros of
thefieldz(x). ThenumberofpinwheelsinagivenareaAisobtainedby
N = d2xδ(z(x))J(z(x)), (24)
ZA
whereδ(x)denotesDirac’sdeltafunctionand
∂R(x)∂I(x) ∂R(x)∂I(x)
J(z(x)) = (25)
∂x ∂y − ∂y ∂x
(cid:12) (cid:12)
(cid:12) (cid:12)
istheJacobian ofthefield (cid:12) (cid:12)
(cid:12) (cid:12)
z(x) = R(x)+iI(x) (26)
split here for later convenience into its real and imaginary part. Averaging Eq. (24) over the
ensembleofphasesφ reads
j
N = d2x δ(z(x))J(z(x)) (27)
h i h i
ZA
9
implyingthat
ρ = δ(z(x))J(z(x)) (28)
l
h i
is the expectation value of the pinwheel density for a fixed set of l. This expectation value only
dependsonlocalquantities,namelyonthefield,Eq. (23),anditsspatialderivatives
n−1
2
z(x) = i l k ei(ljkjx+φj) (29)
j j
∇ n
r j=0
X
such that knowledge of the joint probability density p(z, z) is sufficient to evaluate Eq. (28).
∇
Owingtothecentrallimittheorem,thisprobabilitydensitybecomesGaussianinthelargenlimit.
Eq. (28)isthendeterminedbythefirstandsecondorderstatisticsofz and z. Furthermore,since
∇
theirstatisticsisthesameateachlocationx,itissufficienttoevaluateEq. (28)forz(0), z(0). The
∇
spatialdependencyisthusomittedinthefollowing. TheaverageinEq. (28)isgivenbyanintegral
overthejointprobabilitydensity
p(v) = 1 e−12vTC−1v (30)
(2π)3√detC
ofcomponents
v = (R,I,∂ R,∂ I,∂ R,∂ I) . (31)
x x y y
withcovariancematrixC whichshallbeanalyzedinthefollowing.
First,thediagonalelementsofC areevaluated. Using(18)and(26),theauto-correlationsofthe
fieldare
n−1
2
R2 = cosφjcosφj′
n
j,j′=0
(cid:10) (cid:11) X (cid:10) (cid:11)
= 1 (32)
fortherealpartand
n−1
2
I2 = sinφjsinφj′
n
j,j′=0
(cid:10) (cid:11) X (cid:10) (cid:11)
= 1 (33)
fortheimaginarypart. Forthespatialderivativesoneobtains
n−1
2
(∂xR)2 = ljlj′kxjkxj′ sinφjsinφj′
n
D E j,Xj′=0 (cid:10) (cid:11)
n−1
1
= k2 (34)
n xj
j=0
X
10