F. Wolf and T. Geisel- Universality in visual cortical pattern formation

8/3/2019 F. Wolf and T. Geisel- Universality in visual cortical pattern formation

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Universality in visual cortical pattern formation

F. Wolf *, T. Geisel

Department of Nonlinear Dynamics, Max-Planck-Institut fur Stromungsforschung and Institute for Nonlinear Dynamics,

Fakultat fur Physik, Universitat Gottingen, D-37073 Gottingen, Germany

Abstract

During ontogenetic development, the visual cortical circuitry is remodeled by activity-dependent mechanisms of synaptic plas-

ticity. From a dynamical systems perspective this is a process of dynamic pattern formation. The emerging cortical network supports

functional activity patterns that are used to guide the further improvement of the networks structure. In this picture, spontaneous

symmetry breaking in the developmental dynamics of the cortical network underlies the emergence of cortical selectivities such asorientation preference. Here universal properties of this process depending only on basic biological symmetries of the cortical

network are analyzed. In particular, we discuss the description of the development of orientation preference columns in terms of a

dynamics of abstract order parameter fields, connect this description to the theory of Gaussian random fields, and show how the

theory of Gaussian random fields can be used to obtain quantitative information on the generation and motion of pinwheels, in the

two dimensional pattern of visual cortical orientation columns.

2003 Elsevier Ltd. All rights reserved.

Keywords: Area 17; Development; Experience-dependence; Cortical maps; Self-organization

1. Introduction

Universality, the phenomenon that collective prop-

erties of very different systems exhibit identical quanti-

tative laws, is of great importance for the mathematical

modeling of complex systems. Originally, the phenom-

enon of universality gained widespread recognition

when it was realized that the quantitative laws of phase

transitions in physically widely different equilibrium

thermodynamic systems were determined only by their

dimensionalities and symmetries and were otherwise

insensitive to the precise nature of physical interactions

(for an introduction see [26]). Subsequent research in

nonlinear dynamics and statistical physics has uncov-ered that universal behavior extends far beyond equi-

librium thermodynamics and is found for instance in

pattern forming systems far from equilibrium (see e.g.

[16]), in chaotic dynamics (see e.g. [40]), and in turbu-

lence (see e.g. [20]). It is for two reasons that universal

behavior is particularly important for the mathematical

modeling of complex systems such as the brain. First, in

order to understand the universal properties of a system

it is sufficient to study fairly simplified models as long asthey are in the right universality class. Second, predic-

tions for experiments that are derived from universal

model properties are critical: because universal proper-

ties are insensitive to changing microscopic interactions

and numerical parameters or refining the level of detail

in a model, verification or falsification of universal

predictions can determine whether a certain modeling

approach is appropriate or not. This is particularly

important in theoretical neuroscience because for the

neuronal networks of the brain a complete microscopic

characterization of all interactions cannot be achieved

experimentally and even if available would preclude

comprehensive mathematical analysis.In this chapter we will discuss in detail the universal

properties of a paradigmatic process in brain develop-

ment: the formation of so called orientation pinwheels

and the development of orientation columns in the vi-

sual cortex. In the visual cortex as in most areas of the

cerebral cortex information is processed in a 2-dimen-

sional (2D) array of functional modules, called cortical

columns [15,29]. Individual columns are groups of

neurons extending vertically throughout the entire cor-

tical thickness that share many functional properties.

Orientation columns in the visual cortex are composed* Corresponding author.

0928-4257/$ - see front matter 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jphysparis.2003.09.018

Journal of Physiology - Paris 97 (2003) 253264

www.elsevier.com/locate/jphysparis


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of neurons preferentially responding to visual contours

of a particular stimulus orientation [24]. In a plane

parallel to the cortical surface, neuronal selectivities

vary systematically, so that columns of similar func-

tional properties form highly organized 2D patterns,

known as functional cortical maps. In the case of ori-

entation columns, this 2D organization is characterized

by so called pinwheels, regions in which columns pre-

ferring all possible orientations are organized around

a common center in a radial fashion [8,43] (see Fig. 1).

It is a very attractive but still controversial hypothesis

that in the ontogenetic development of the brain the

emerging cortical organization is constructed by learn-

ing mechanisms which are similar to those that enable us

to acquire skills and knowledge in later life [28,41,42].

Several lines of evidence strongly suggest that the brain

in a very fundamental sense learns to see. First, visual

experience is very important for the normal develop-

ment of sight. If the use of the visual sense is prevented

early in life vision becomes irreversibly impaired [17].Since this is not due to a malformation of the eye or of

peripheral stages of the visual pathway, it suggests that

in development visual input it used to improve the

processing capabilities of the visual cortical networks

[17]. In addition, the performance of the developing vi-

sual system responds very sensitively to visual experi-

ence. In human babies, for instance, already a few hours

of visual experience lead to a marked improvement of

visual acuity [32]. Second, the synaptic organization of

the visual cortex is highly plastic and responds with

profound and fast functional and structural reorgani-

zation to appropriate experimental manipulations of

visual experience [3,49]. These and similar observations

suggest that the main origin of perceptual improvement

in early development is due to an activity-dependent and

thus use-dependent refinement of the cortical network,

in which neuronal activity patterns that arise in the

processing of visual information in turn guide the

refinement of the cortical network. Whereas, theoreti-

cally, this hypothesis is very attractive, it is, experi-

mentally, still controversial, whether neural activity

actually plays such an instructive role (for discussion see

[14,27,34]). In 1998, we discovered that experimentally

accessible signatures of such a activity refinement of the

cortical network are predicted by universal properties ofa very general class of models for the development of

visual cortical orientation preference maps [52]. We

could demonstrate that if the pattern of orientation

preferences is set up by learning mechanisms, then the

number of pinwheels generated early in development

exhibits a universal minimal value that depends only

on general symmetry properties of the cortical network.

This implies that in species exhibiting a lower number of

pinwheels in the adult pinwheels must move and anni-

hilate in pairs during the refinement of the cortical cir-

cuitry. Verification of this intriguing prediction would

provide striking evidence for the activity-dependent

generation of the basic visual cortical processing archi-

tecture. In the following, we will present a self-contained

treatment of the mathematical origin of this kind of

universal behavior.

The presentation is organized as follows. In Section 2,

we introduce the mathematical language used to de-

scribe the spatial layout of orientation preference col-

umns in the visual cortex and briefly describe their main

features as experimentally observed. In Section 3, the

description of the development of the orientation map

by an abstract dynamics of order parameter fields is

introduced and elementary consequences of its basic

symmetry properties are discussed. In Section 4, we

discuss how such a dynamics is derived from models of

cortical learning processes that explicitly describe how

activity patterns restructure the cortical architecture. In

Section 5, we present qualitative arguments demon-

strating that spontaneous breaking of symmetry in such

models will lead to the formation of pinwheels anddiscuss how the further motion of pinwheels is con-

strained by topological principles. By the very nature of

spontaneous symmetry breaking, the initial pattern of

orientation preferences is, mathematically, a random

variable. The pattern that will be generated by the

breaking of symmetry is undetermined, owing to a lack

of knowledge of the microscopic parameters and initial

conditions. In spite of this, it may be assumed that the

pattern will be one instance from a well-defined

ensemble of possible patterns. The ensemble can be

characterized by symmetry assumptions. This is the

starting point for the examination of the statistically

expected initial density of the pinwheels in Section 6. We

demonstrate that for every Gaussian ensemble this

density has a lower bound that depends only on sym-

metry properties. In Section 7, we will then show that

the assumption of Gaussian statistics for the random

pattern is fulfilled for a very general class of dynamical

models. In Section 8, we present additional symmetry

arguments indicating that the range of validity of our

theory also includes models in which the pattern of

orientation preferences interacts with the pattern of

ocular dominance and the pattern of phases of receptive

fields. Because the initial pinwheel density has a lower

bound, quantitative limits are placed on the dynamics ofthe pinwheels in the investigated classes of models.

Pinwheel densities that are less than the minimum initial

density can occur only as a result of pinwheel annihi-

lation during a phase of development subsequent to the

breaking of symmetry. In the discussion (Section 9), we

therefore compare the experimentally observed density

with the theoretically calculated bounds which identifies

species for which annihilation of pinwheels is predicted.

We conclude with an outlook on possible extensions of

the symmetry based analysis of universal properties in

cortical pattern formation.

254 F. Wolf, T. Geisel / Journal of Physiology - Paris 97 (2003) 253264


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2. The pattern of orientation preference columns

In the following, we will briefly introduce the math-

ematical description of the spatial layout of orientation

columns in the visual cortex in terms of complex valued

order parameter fields. Experimentally, the pattern of

orientation preferences can be visualized using the

optical imaging method [6,8]. In such an experiment, the

activity patterns Ekx produced by stimulation with agrating of orientations hk are recorded. Here x repre-

sents the location of a column in the cortex. Using the

activity patterns Ekx, a field of complex numbers zxcan be constructed that completely describes the pattern

of orientation columns:

zx X

k

ei2hkEkx: 1

The pattern of orientation preferences #x is then ob-tained from z

x

as follows:

#x 12

argz: 2

Typical examples of such activity patterns Ekx and thepatterns of orientation preferences derived from them

are shown in Fig. 1. Numerous studies confirmed that

the orientation preference of columns is a almost every-

where continuous function of their position in the cor-

tex. Columns with similar orientation preferences occur

next to each other in iso-orientation domains [46].

Neighboring iso-orientation domains preferring the

same stimulus orientation exhibit a typical lateral

spacing K in the range of 1 mm, rendering the pattern ofpreferred orientations roughly repetitive. Furthermore,

it was found experimentally that the iso-orientation

domains are often arranged radially around a common

center. Such an arrangement had been previously hypo-

thesized on the basis of electrophysiological experi-

ments [2,43]. The regions exhibiting this kind of radial

arrangement were termed pinwheels (see Fig. 1). The

centers of pinwheels are point discontinuities of the field

#x where the mean orientation preference of near-by columns changes by 90. They can be characterized

by a topological charge which indicates in particular

whether the orientation preference increases clockwise

around the center of the pinwheel or counterclock-

wise:

qi 12p

ICj

r#xds; 3

where Cj is a closed curve around a single pinwheel

center at xi. Since # is a cyclic variable within theinterval 0;p and up to isolated points is a continuousfunction of x, qi can in principle only have the values

qi n2; 4

where n is an integer number [33]. If its absolute value

jqij is 1/2, each orientation is represented exactly once inthe vicinity of a pinwheel center. Pinwheel centers with a

topological charge of 1/2 are simple zeros of zx. Inexperiments only pinwheels that had the lowest possible

topological charge qi 1=2 are observed. This meansthere are only two types of pinwheels: those whose ori-

entation preference increases clockwise and those whose

orientation preference increases counterclockwise. This

organization has been confirmed in a large number of

species and is therefore believed to be a general feature

of visual cortical orientation maps [4,5,7,9,10,38,50].

3. Symmetries in the development of orientation columns

Owing to the large number of degrees of freedom of a

microscopic model of visual cortical development, the

description of the development of the pattern of col-

umns by equations for the synaptic connections betweenthe LGN and cortex is very complicated. On the order

of 106 synaptic strengths would be required to realisti-

cally describe, for example, the pattern of orientation

preference in a 4 4 mm2 piece of the visual cortex. This

complexity and the presently very incomplete knowledge

about the nature of realistic equations for the dynamics

of visual cortical development demand that theoretical

analyzes concentrate on aspects that are relatively

independent of the exact form of the equations and are

representative for a large class of models.

Because on a phenomenological level the pattern of

orientation columns can be represented by a simpleorder parameter field zx, models with the followingform provide a suitable framework:

o

otzx Fz gx; t: 5

Here Fz is a nonlinear operator and the randomterm gx; t describes intrinsic, e.g., activity-dependentfluctuations. In Eq. (5), it is assumed that, except for

random effects, changes in the pattern of orientation

columns during development can be predicted on the

basis of a knowledge of the current pattern. Swindale

was the first to study models of this type, with the intentto show that roughly periodical patterns of columns can

develop from a homogeneous initial state [44,45,47].

It is biologically plausible to assume that Eq. (5)

exhibits various symmetries. Considered anatomically,

the cortical tissue appears rather homogeneous. If we

look at the arrangement of the cortical neurons and

their patterns of connections, there is no region of the

cortical layers and no direction parallel to the layers that

is distinguishable from other regions or directions [11].

If the development of the pattern of orientation prefer-

ences can be described by an equation in the form of Eq.

F. Wolf, T. Geisel / Journal of Physiology - Paris 97 (2003) 253264 255


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(5), it is thus very plausible to require that it is sym-

metric with respect to translations,

FbTyz bTyFz with bTyzx zx y 6and rotations,

F

bRbz bRbFz withbRbzx z cosb sinb sinb cosb !x 7of the cortical layers. This means that patterns that can

be converted to one another by translation or rotation

of the cortical layers are equivalent solutions of Eq. (5).

If the orientation preference of a column is determined

by the afferent connections from the lateral geniculate

nucleus (LGN), it is also plausible to require that the

arrangement of iso-orientation domains contains no

information about the orientation preferences of the

columns. This is guaranteed by a further symmetry. If

Eq. (5) is symmetric with respect to shifts in orientation,

Fei/z ei/Fz 8then patterns whose arrangement of iso-orientation

domains is the same but whose orientation preference

values differ by a given amount, are equivalent solutions

of Eq. (5).

The three symmetries Eqs. (6)(8) imply a number of

basic properties of Eq. (5). Owing to the symmetry

under orientation shifts, zx 0 is a stationary solutionof Eq. (5) because

F

0

ei/F

0

)F

0

0:

9

Near this state, zx develops approximately accordingto a linear equation:

o

otzx bLzx gx; t; 10

where bL is a linear operator. Like F, the operator bLmust also commute with rotation and translation of the

cortical layer and with global shifts of orientation. Thus,

the Fourier representation of bL is diagonal and its ei-genvalues kk are only a function of the absolute valueof the wave vector k jkj.

A qualitative requirement placed in Eq. (5) is that it

be able to describe the spontaneous generation of aroughly repetitive pattern of orientation preferences

from an initially homogeneous state zx % 0. Thisrequirement further constrains the class of dynamic

equations that are to be considered. Eq. (10), and thus

also Eq. (5), describes the generation of a repetitive

pattern of orientation preferences when the spectrum

kk has positive eigenvalues, and exhibits them only inone interval of wave numbers kl; kh with 0 < kl < kh.Mathematically such a system is said to exhibit a Tur-

ing-type instability of the homogenous state zx % 0(see e.g. [16]).

4. From learning to dynamics

It is not difficult to construct models with the form of

Eq. (5) that represent the features of activity-dependent

plasticity in an idealized fashion [52]. One instructive

possibility is to start from an equation that describes

how the pattern of orientation preferences z

x

changes

under the influence of a sequence of patterns of afferent

activity Ai:

zix!Aizi1x: 11In a minimal model, the changes

dzx zi1x zix 12in the pattern must be dependent on both the current

pattern zix and the patterns of activity Ai:dzx fx;zi;Ai: 13If the maximum absolute value maxAjdzxj of amodification induced by a single activity pattern is much

smaller than the amplitude of the pattern and if the

patterns of afferent activity Ai are random variables

with a stationary probability distribution, then the

changes in zx on a long time scale are described by thefollowing equation:

o

otzx fx;z;A

A Fz; 14

where A

denotes averaging over the ensemble of

activity patterns [21]. This equation has the form of Eq.

(5).

A requirement that results directly from the activity-

dependent nature of synaptic plasticity is that only

the selectivity of the columns that are activated by

A is changed. For cortical activity patterns ex ex;z;A, this requirement is fulfilled by making themodification proportional to the cortical activity:

fx;z;A / ex;z;A: 15For simplicity, let us assume that a single activity pat-

tern A forces the activated cortical neuron to take on a

certain orientation preference h and a certain orientation

selectivity jsj, described by the complex numbersA jsAje2ihA; 16then the simplest modification rule has the form

fx;z;A / sA zxex;z;A: 17Several models have been proposed that can be inter-

preted in the just described way as an order parameter

dynamics [18,19,22,23,37]. These models are defined by

modification formulas with the form of Eq. (17). They

differ mainly in how the pattern ex of the activity ofthe cortex is modeled.



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5. Pinwheel generation and motion

It is easily shown that within this class of models

pinwheels will typically form during the initial symmetry

breaking phase of development. If the eigenvalues kkare real, which is expected when zx develops to astationary state, then beginning with a homogeneous

state zx % 0 the real and imaginary parts of zx ini-tially develop independently of each other. In particular,

the zero lines of the real and imaginary parts will de-

velop independently of each other and thus typically

intersect at points xi. These points are simple zeros of

zx and therefore are the centers of pinwheels with atopological charge qi 1=2.

It is important to note that the possible forms of a

change in the pinwheel configuration over time fxi; qigare already constrained by assuming an equation for the

developmental dynamics with the form of Eq. (5). Since

the field z

x

can only be a continuous function of time,

the entire topological charge of a given area A with aboundary A,

QA 12p

IA

r#xds Xxi2A

qi 18

is invariant as long as no pinwheel transgresses the

boundary of the area [33]. If the pattern contains only

pinwheels with qi 1=2, then only three qualitativelydifferent modifications of the pinwheel configuration are

possible. First, movement of the pinwheel within the

area; second, generation of a pair of pinwheels with

opposite topological charges; third, the annihilation of

two pinwheels with opposite topological charge whenthey collide. Only these transformations conserve the

value ofQA and are therefore permitted.

6. Random orientation maps

In order to estimate the pinwheel density which re-

sults from the initial breaking of symmetry, we will, in

this section, start by examining an ensemble of random

fields zx. Such an ensemble can be characterized by itsspatial correlation functions:

Cx; y hzxzyi; 19

Cx; y hzxzyi: 20Here angular brackets, h i, represent the expectationvalue for the ensemble. The form of these correlation

functions can be constrained by symmetry assumptions.

Because of the symmetries equations (6)(8), we assume

that the ensemble is statistically invariant with respect to

translations and rotations and that the patterns that can

be transformed into each other by a global orientation

shift zx ! ei/zx occur with the same probability. The

latter assumption implies that the expectation of zx isequal to zero:

hzxi 0: 21This means that any orientation preference can occur at

any location x in the cortex. Moreover, invariance under

orientation shifts implies that the correlation function

(20) is also equal to zero, since only in this case can the

following relation be fulfilled for any /:

hzxzy i hei/zxei/zyi: 22Re zx describes the patterns of columns that preferhorizontal and vertical stimuli. Imzx describes thepatterns of columns that prefer oblique stimuli. These

two patterns are not correlated and both have the same

correlation function because the correlation function

(20) is zero:

C

x; y

hz

x

z

y

i hRe zxRe zy ihIm zxIm zyi i hRe zxIm zyi hIm zxRe zyi 0: 23

Invariance with respect to translations and rotations

implies that the correlation function Cr is a functiononly of the distance r jx yj of the respective pair oflocations in the cortex:

Cx; y Cjx yj Cr: 24The correlation function Cr mainly provides infor-mation about the characteristic wavelength and the

correlation length of the pattern. The characteristicwavelength K of the pattern can be defined using the

Fourier transform ofCr:

Pjkj 12p

Zd2xCxeikx 25

which is called the power spectral density. It is of

advantage to use the mean wavenumber k to define the

characteristic wavelength:

K 2pk

2p

R1

0dkkPk : 26

Without loss of generality, the power spectral density isassumed to be normalized as follows:

R10

dkPk 1.There is an infinite number of ensembles of maps that

have the same two-point correlation function. In the

following considerations, we will assume that the ensem-

bles considered consists of Gaussian random fields. In

the next section, we will show that this assumption is

fulfilled for a large class of dynamic models, owing to

the central limit theorem.

The centers of the pinwheels are the zeros of the field

zx. The pinwheel density is therefore obtained fromthe number of these zeros in an area A,



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N ZA

d2xdzx oRezx; Imzxox1;x2

; 27where

oRezx; Imzxox1;x2

oRezxox1

oImzxox2

oRezxox2

oImzxox1

28

is the Jacobian of zx. The expectation value of thenumber of zeros in an ensemble is

hNi ZA

d2x dzx oRezx; Imzxox1;x2

( ) 29from which follows that

q dzx oRezx; Imzxox1;x2

( )30

is the density of the pinwheels. This expectation value

will now be evaluated. Superficially, q as defined in Eq.

(30) is the expectation value for an ensemble of fields

zx and thus is a functional integral. It is, however,important to note that this expectation value depends on

locally defined parameters only, namely the values of the

field zx and its derivatives rzx at a given location x.To integrate Eq. (30) it is, therefore, sufficient to know

the joint probability density pzx;rzx. Since this isalso Gaussian, it is given by the cross-correlations and

autocorrelations of zx and rzx. Most of these cor-relations are equal to zero owing to the symmetry

requirements:

hzxrzxi 0; 31hzxzxi 0; 32hrzxrzxi 0; 33hzxrzxi 0; 34hzxzxi ca: 35hrzxrzxi cg: 36Eqs. (31)(33) result from the statistical invariance of

the ensemble under an orientation shift. Eq. (34) results

from the rotation invariance of the ensemble. Eq. (35)

defines the scale on which the order parameter is mea-sured. Thus, ca and cg are the only nontrivial correla-

tions of the distribution pzx;rzx. Because there isno correlation between zx and rzx, the distributionis composed of two factors:

pz;rz 1p3c2gca

exp

2 rzrz

cg

exp

zz

ca

: 37

Because this is true for any location x, the argument of

the field zx is omitted here and in the followingequations. Because Eq. (37) is factored, the expectation

in Eq. (30) is also factored. Thus,

q 1p3c2gca

Zd4rz exp

2 rzrz

cg

rz rz

rz rz

Zd2zdz exp

zz

ca

; 38

where jrz rzrz rzj is the Jacobian of zx.This integral is easily evaluated by converting to thespherical coordinates g2 0;1, h 2 0;p and /1/2 20; 2p in gradient space with the volume elementd4rz g3 dgj cosh sinhjdhd/1 d/2: 39Integration then yields

q 1p3c2gca

Z10

dgg5 exp

2g

2

cg

Zp

0

dhj cosh sinhj2Z2p0

d/1 d/2 cos/1 sin/2j cos/2 sin/1j

40

1

4p

cg

ca: 41

Since in the following discussion the dependence of this

expression on the power spectral density Pk is ofinterest, we express ca and cg as functionals of Pk:

ca Z

d2kPk; 42

cg Z

d2kjkj2Pk: 43

The pinwheel density is then given by

q 14p

Rd2kjkj2PkRd2kPk : 44

The exact form of the correlation function Cr and thestructure function Pk at the beginning of developmentis not known. In particular, it is to be expected that these

functions vary from species to species and from indi-

vidual to individual. In spite of this, the following

argument shows that Eq. (44) implies a quantitative

estimate of the initial pinwheel density. Since it may be

assumed that the pinwheel density is inversely propor-

tional to the square of the characteristic wavelength

q / K2

, we shall rewrite the expression for the density:

q k2

4p

R10

dkk3PkR10

dkkPk 3 pK2R1

0dkk3PkR1

0dkkPk 3 : 45

Owing to Jensens inequality [39],Z10

dkk3PkPZ1

0

dkkPk 3

46

it follows that q has a lower bound:

q pK2

1 a; 47



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where a > 0. Thus, the exact form of the power spectraldensity influences the expected pinwheel density only via

the positive-definite functional

a 3Z1

0

dkk k2

k2Pk

Z10

dkk k3

k3Pk; 48

where a is zero only when the power spectral density is

the Dirac delta distribution Pk dk k, i.e., whenthe correlation length of the pattern diverges. Thus, it

can be seen that a Gaussian random pattern of orien-

tation preferences has a minimum pinwheel density,

qm p

K2; 49

independently of the exact form of its spatial correla-

tions. Because two-dimensional Gaussian random fields

are ergodic [1], this lower bound is also valid for the

pinwheel density in an individual realization of such a

field.

7. Random orientation maps from a Turing instability

The assumption was made in the previous section

that the ensemble of possible initial patterns show

Gaussian statistical properties. We will now show that

this is actually the case for a large class of models. This

establishes that the lower bound for the pinwheel den-

sity calculated in the previous section places a quanti-

tative constraint on the dynamics of pinwheels in this

class of models. For this purpose, we will use the class of

models defined in Section 3 that can be represented by

an equation for the dynamics of the order parameter

field zx:o

otzx Fz nx; t: 50

The assumed symmetries of this equation imply that

zx 0 is a stationary solution. In the vicinity of thispoint, the dynamics of zx is approximately linear:o

otzx bLzx nx; t; 51

where

bL is a linear operator. This equation generates an

orientation map from an initially homogeneous state.

As discussed in Section 3, the class of operators bL isfurther constrained by the requirement that Eq. (50)

describe the spontaneous generation of a pattern of

orientation preferences starting with an initially homo-

geneous state. Equation (51), and thus also Eq. (50),

describes the generation of a repetitive pattern of ori-

entation preferences if the spectrum kk has positiveeigenvalues only within a single interval kl; kh of k-values. Using the Greens function

Gx; t 12p

Zd2keikxkjkjt; 52

Eq. (51) with the initial condition z0x % 0 at time t 0is solved to yield

zLx; t Z

d2y

Zt0

dt0Gy x; t t0 ny; t0 z0xdt0:53

Since zLx; t is the sum of linear transforms of therandom fields z0x and nx; t, it will always haveGaussian statistics when z0x and nx; t are alsoGaussian. This is independent of the form of their cor-

relation functions:

Cz0r hz0xz0x ri; 54Cnr; t hnx; t0nx r; t0 ti: 55In general, the statistical properties of zLx; t are alsoGaussian for a much larger class of random processes.

The field zLx; t is an integral over a number of randomvariables. When this integral consists of a large number

of independent terms then the central limit theorem [39]gives the conditions under which the statistical proper-

ties of zLx; t are Gaussian, even if z0x and nx; t arenot Gaussian. We will now briefly present these condi-

tions and discuss their biological significance. For sim-

plicity we will limit ourselves to sufficiently smooth

functions z0x and nx; t such that the right side of Eq.(53) can be expressed approximately by a sum:

zLx; t %Xi;j

Dy2i DtjGyi x; t tj nyi; tj

z0xidj;0: 56

The central limit theorem applies if the distributions of

z0x and nx; t fulfill the Lindeberg criterion:

limb!1

Zjwj>b

dww2Pw 0; 57

where the variable w represents either z0x or nx; t atan arbitrary location x at any time t and Pw is theprobability density. Eq. (57) is fulfilled when jz0xj andjnx; tj are bounded or have a finite variance. Bothappear to be plausible for the observed biological pro-

cess. There is very little orientation selectivity, which is

observed in the visual cortex considerably before the

eyes open [12]. Thus, it can be taken as certain that thejz0xj field, which describes this early selectivity, isbounded. A similar argument can be made for the

fluctuations nx; t. All available evidence indicates thatthe process in which the initial orientation map is gene-

rated is continuous [12]. This speaks for the assumption

that fluctuations in this process are bounded or at least

have a finite variance.

In order to determine the conditions under which

zLx; t consists of a large number of independent terms,it is necessary to compare the correlation times and

lengths ofz0x and nx; t with the temporal and spatial



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scales of the Greens function Gx; t. The characteristictime scale ofGx; t iss 1=kkmax; 58where kkmax is the largest eigenvalue of bL. The ampli-tude jG0;Dtj ofGDx;Dt increases with time with timeconstant s. The Greens function also has a character-

istic spatial scale L on which GDx;Dt decays as afunction of the distance jDxj for a given time differenceDt. In order to estimate L, let us assume that kk has aquadratic maximum between kl and kh. The spatial

Fourier transform eGk;Dt of the Greens functionGDx;Dt is then approximated in the vicinity of themaximum kmax by a Gaussian function:eGk;Dt ekjkjt 59

% exp kmax 1

jkj kmaxkh kl=2 2

t

: 60

The bandwidth Dk jkh klj=2 ffiffiffiffiffiffiffiffiffiffiDt=sp of this functionis a function of time. Its inverse provides an estimate of

L:

L % 2ffiffiffiffiffiffiffiffiffiffiDt=s

pjkh klj : 61

Thus, the field zLx; t Eq. (53) integrates fluctuationsnx; t Dt that occurred in the past at time t Dtwithin a distance L. The size of this volume as given by

Eq. (61) increases diffusively: L /ffiffiffiffiffiffiffiffiffiffiDt=s

p. If it is as-

sumed that the correlation function Cz0r decays on aspatial scale Lz0 and the same is assumed for Cz0r onthe spatial and temporal scales Ln and sn, so that

Cz0r6Az0 ejxj=Lz0 ; 62Cnr; t6An ejxj=Lnjtj=s 63are fulfilled with positive constants Az0 and An, then the

number of independent terms in zLx; t is large ifsn ( s 64and

maxfLz0 ;Lng (1

jkh klj : 65

If zL

x; t

is determined mainly by n

x; t

, then either

sn ( s or Ln ( 1jkhklj is a sufficient condition. IfzLx; t isdetermined by z0x, then Lz0 (

2ffiffiffiffi

t=sp

jkhklj is sufficient, where

t is the time at which the amplitude of zx; t saturates.Especially the first of the two cases is compatible with

the biological situation. Fluctuations caused by afferent

activity patterns, like the activity patterns themselves,

can be correlated only over time intervals of a few

hundred milliseconds. Since the first pattern of orien-

tation preferences is formed over a period of several

hours to several days, the number of independent

activity-induced fluctuations is surely large. Therefore,

the statistical properties of the initial pattern can be

assumed to be Gaussian.

8. Hidden parameters

The pattern of orientation preferences does not de-

velop completely independently of the patterns of other

properties of the neurons in the visual cortex. In the

previous analyzes, it was assumed that the initial for-

mation of a orientation preference pattern can be de-

scribed by an autonomous system. Observations and

theoretical arguments, however, indicate that the pat-

tern of orientation preferences interacts with other

selectivity patterns. For example, iso-orientation lines

tend to cross the boundaries of ocular dominance col-

umns at right angles (Fig. 2). Moreover, oriented

receptive fields not only have an orientation preference,

but they also have a phase (Fig. 3). Because in micro-

scopic models the forms of the receptive fields governthe development of the afferent connections, these

models imply that the orientation preference pattern and

the pattern of the phases of receptive fields interact

[35,36]. In this section we show based on further sym-

metry arguments that the conclusions of the previous

section are also justified even if it is assumed that the

orientation preference pattern interacts with these two

patterns.

For the discussion of the effects of interaction be-

tween the ocular dominance patterns, the receptive field

phases, and the developing system of orientation col-

umns, we will represent the two patterns by additional

order parameter fields. The pattern of ocular dominance

columns can be described by a real field ox, whereox > 0 represents the columns that prefer the left eyeand ox < 0 represents the columns that prefer the righteye. The phase W pattern of the receptive fields can also

be represent by a field. The phase of a receptive field is a

cyclic variable. It can, therefore, be described by a

complex scalar. The locations of the cortical neurons are

given by xi and the neurons phases are Wi. The coarse

grained pattern of phases is then given by

p

x Xi hxi xeiWi ; 66

where hDx is an arbitrary window function. Thisfunction is to be chosen so that only neurons in the

vicinity of x contribute to the value of px. The mag-nitude of this field then indicates whether the phases

of these neurons are similar. It goes to zero when the

distribution of these phases is random and has a value

of one when the phases are identical. When the orien-

tation preference pattern interacts with the phase and

ocular dominance patterns, Eq. (50) must be replaced

by



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p0x % 0; 69ox % 0: 70With the initial condition p0x % 0, it is assumed thatthe phases are initially in a random state. With the initial

condition o0x % 0, it is assumed that the projectionsfrom the two eyes initially overlap to a large extent. In

general, z0

x

0;p0

x

0; o

x

0 is a stationary

state of the development dynamics:

F0; 0; 0 0: 71This is justified by the fact that the homogeneous state

should have no tendency to develop a specific pattern of

orientation preferences. Such a tendency would be the

case ifF0; 0; 0 6 0.In the vicinity of this stationary state, a linear equa-

tion describes the development:

o

otzx bLzx bLppx bLoox nx; t: 72

Two plausible symmetries cause the two operators bLpand bLo to be zero simultaneously. bLo is zero if exchanging the columns of the left and right eyes does

not influence the development of the ocular dominance

pattern, i.e., when

Fzx;px;ox Fzx;px; ox: 73All models that have been proposed for the coordinated

development of orientation preference columns and

ocular dominance columns fulfill this condition [48]. In

this case, Eq. (72) must also be invariant with respect to

the inversion ox ! ox. This is possible only if

bLo is

zero. Analogously,

bLp must also be zero when Eq. (67) is

invariant with respect to inversion of the receptive fields

px ! px:Fzx;px; ox Fzx;px; ox: 74Receptive fields of cortical neurons consist of subfields

that receive input from on- and off-center neurons. The

exchanging of the on- and off-center inputs is mathe-

matical equivalent to a phase shift by p, i.e., a change in

sign of the field px (Fig. 3). Eq. (67) can, therefore, beinvariant when the on- and off-center inputs are ex-

changed only when the conditions of Eq. (74) andbLp 0 are fulfilled. Thus, even with the assumption thatthe orientation preference pattern interacts with the

pattern of ocular dominance columns and the phases of

the receptive fields, Eq. (51) can describe the formation

of the first orientation preference pattern. The reason

for this is that the interaction with the other column

patterns is determined first by nonlinear terms when the

assumed symmetries are present. These interactions,

therefore, cannot influence the initial formation of pin-wheels.

9. Discussion

The analyzes presented above demonstrate that the

density of the pinwheels in a Gaussian random pattern

of orientation preferences is always larger than

qm p=K2, where K is the characteristic wavelength ofthe pattern. This feature is universal, meaning that it

applies, no matter what the detailed structure of the

fields statistical correlations are. In addition, we showedthat orientation preference patterns generated by a dy-

namic instability have Gaussian statistical properties in

a large class of models owing to the central limit theo-

rem, and that this property is not affected by interaction

with two other patterns of neuronal selectivities. These

results imply that the dynamics of pinwheels is quanti-

tatively constrained in a large class of models for the

development of orientation preference patterns. Inde-

pendent of modeling details, pinwheel densities that are

less than the lower bound for the initial density

qm p=K2 can develop under the given conditions onlyby pairwise annihilation of pinwheels. This implies that

pinwheels must move during development and must bepairwise annihilated in those species in which low pin-

wheel densities are observed in adults (Fig. 4).

Because of this, it is of interest to compare densities in

adult animals with the calculated lower limit. The

characteristic wavelength K differs considerably from

species to species. Values of the relative pinwheel density

q qK2 that are less than p imply pinwheel annihilationduring development. Observed relative densities range

from 2.0 to 4.0 (see Fig. 4). Macaques have the highest

relative density. The lowest relative densities observed so

far are in the visual cortex of adult tree shrews (Tupaia).

Fig. 3. Orientation and phase of receptive fields. The receptive fields of

cortical neurons are made up of subfields that receive on- and off-

center input from the LGN. These fields usually have an orientation

and a spatial phase. The figure shows two examples with the same

orientation but different phases. In (a) and (b) the areas of the field of

view from which the cell receives on input are light and those from

which it receives off input are dark. The form of typical receptive

fields can be described by Gabor functions with the form gr cosKr WerRr=2 [25], where the vector r is the location in the field ofview relative to the center of the receptive field, the vectorK defines the

orientation and spatial frequency of the receptive field,R is a positive-

definite, symmetric matrix that governs the size of the field, and W is

the phase of the receptive field. (a) shows an arbitrary example; (b)

shows the receptive field that results by exchanging the on and offinputs. Exchanging the on and off inputs is equivalent to a phase shift

by p.



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The relative densities of both tree shrews and cats are

distinctly less than p. Thus, pinwheel movement and

annihilation is implied for both species by the theory

developed above. Pinwheel movement cannot be ex-

cluded in ferrets and squirrel monkeys. This would beespecially the case when the initial pinwheel density is

substantially larger than the minimum. The high pin-

wheel density observed in macaques can occur as a re-

sult of a dynamic instability without any subsequent

rearrangement of the pattern. Thus, tree shrews are the

species in which the most extensive rearrangement is

expected during development.

The ontogenetic development of the cerebral cortex is

a process of astonishing complexity. In every mm3 of

cortical tissue in the order of 106 neurons must be wired

appropriately for their respective functions such as the

analysis of sensory inputs, the storage of skills and

memory, or motor control [11]. In the brain of an adult

animal, each neuron receives input via about 104 syn-

apses from neighboring and remote cortical neurons and

from subcortical inputs [11]. At the outset of postnatal

development, the network is formed only rudimentarily:

In the cats visual cortex, most neurons have just fin-

ished the migration from their birth zone lining the

cerebral ventricle to the cortical plate at the day of birth

[31]. The number of synapses in the tissue is then only

10% and at the time of eye-opening, about two weeks

later, only 25% of its adult value [13]. In the following 2

3 months the cortical circuitry is substantially expanded

and reworked and the individual neurons acquire theirfinal specificities in the processing of visual information

[17]. It is intriguing, that the motion and annihilation of

pinwheels, a global aspect of such remodeling, can be

predicted without reference to the details of the bio-

logical mechanisms involved in this process.

It is natural to ask whether the symmetry based ap-

proach used above to derive this prediction can be ex-

tended to study further aspects of cortical pattern

formation. This is indeed possible, however, under more

restrictive conditions. Mathematically, we have used

primarily the linear equation, Eq. (10), which describes

the generation of a pattern of orientation columns from

an initially homogeneous state. Eq. (5), which models

the entire process of development must be nonlinear in

order to describe the saturation of the orientation

selectivity jzxj and a possible subsequent reorganiza-tion of the pattern. The symmetry with respect to shifts

of orientation means that when F

is represented by a

power series only terms with an odd power can occur. If

the parameter values in Equation (5) are assumed to be

in the vicinity of an instability of the homogeneous state

zx 0 and the transition to repetitive patterns is con-tinuous, then also the stationary patterns that bifurcate

from zx 0 can be studied perturbatively in a largelymodel independent fashion. Using such an approach

general conditions for the final stability of pinwheel

patterns can be obtained [51].

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Fig. 4. Constraints on the development of the pinwheel density by

symmetries. The range of permissible trajectories of the scaled pin-

wheel density qt is shaded gray. Two permissible trajectories areshown. The y-axis on the right shows the scaled pinwheel densities

observed in several species. Densities that are less than p can develop

only by annihilation of pinwheels (modified from [52]).



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F. Wolf and T. Geisel- Universality in visual cortical pattern formation

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