Receptive Fields and Maps in

the Mo dels of

Ocular Dominance and

Orientation Columns

Kenneth D Miller

Published in Mo dels of Neural Networks IIIEDomanyJLvan Hemmen

and K Schulten Eds SpringerVerlag NY pp

Anearlier and briefer version of this article appeared in The Handbook

of Neural Networks MAArbib Ed The MIT Press under the ti

tle Models of Ocular Dominance and Orientation Columns Reused by

permission

ABSTRACT The formation of o cular dominance and orientation columns

in the mammalian visual cortex is briey reviewed Correlationbased mo d

els for their development are then discussed b eginning with the mo dels of

Von der Malsburg For the case of semilinear mo dels mo del b ehavior is

well understo o d correlations determine receptive eld structure intracor

tical interactions determine pro jective eld structure and the knitting

together of the two determines the cortical map This provides a ba

sis for simple but powerful mo dels of o cular dominance and orientation

column formation o cular dominance columns form through a correlation

based comp etition b etween left and righteye inputs while orientation

columns can form through a comp etition between ONcenter and OFF

center inputs These mo dels accountwell for receptive eld structure but

are not completely adequate to account for the details of cortical map struc

ture Alternative approaches to map structure including the selforganizing

feature map of Kohonen are discussed Finally theories of the computa

tional function of correlationbased and selforganizing rules are discussed

Depts of Physiology and Otolaryngology WM Keck Center for Integra

tive Neuroscience and Sloan Center for Theoretical Neurobiology Universityof

California San Francisco CA

ii Receptive Fields and Maps in the Visual Cortex

Postscript version of this gure

not available hence gure is

omitted from preprint of pap er

FIGURE Schematic of the mature visual system

Retinal ganglion cells from the two pro ject to separate layers of the lat

eral geniculate nucleus LGN from these twolayers pro ject to separate

patches or strip es within layer of visual cortex V Bino cular regions receiv

ing input from b oth eyes are depicted at the b orders b etween the eyesp ecic

patches The cortex is depicted in crosssection so that layers are ab oveand

layers b elow the LGNrecipientlayer Reprinted bypermissionfrom

c

by the AAAS

INTRODUCTION

The brain is a learning machine An animals exp erience shap es the neu

ral activity of its brain this activity in turn mo dies the brain so that

the animal learns from its exp erience This selforganization the brains

reshaping of itself through its own activity reviewed in has

long fascinated neuroscientists and mo delers

The classic example of activitydep endent neural development is the for

mation of o cular dominance columns in the or monkey primary visual

cortex reviewed in The is the uniquely mammalian

part of the brain It is thought to form the complex asso ciative represen

tations that characterize mammalian and intelligence The primary

visual cortex V is the rst cortical area to receive visual information It

receives signals from the lateral geniculate nucleus of the thalamus LGN

which in turn receives input from the retinae of the twoeyes Fig

To describ e o cular dominance columns several terms must b e dened

First the receptive eld of a cortical refers to the area on the retinae in

which appropriate light stimulation evokes a resp onse in the cell and also

to the pattern of light stimulation that evokes such a resp onse Second a

Kenneth D Miller iii

Postscript version of this gure

not available hence gure is

omitted from preprint of pap er

FIGURE Ocular dominance columns from cat V

A horizontal cut through the layerofVisshown Terminals serving a single

eyeare lab eled white Dark regions at edges are out of plane containing LGN

terminals Region shown is x mm Photograph generously supplied byDr

Y Hata

column is dened as follows V extends many millimeters in eachoftwo

horizontal dimensions Receptive eld p ositions vary continuously along

these dimensions forming a retinotopic map a continuous map of the visual

world In the third vertical dimension the cortex is ab out mm in depth

and consists of six layers Receptive eld p ositions do not signicantly

vary through this depth Such organization in which cortical prop erties

are invariant through the vertical depth of cortex but vary horizontallyis

called columnar organization and is a basic feature of cerebral cortex

Third ocular dominance must b e dened Cells in the LGN are monoc

ular resp onding exclusively to stimulation of a single eye Fig LGN

cells pro ject to layer of V where they terminate in alternating strip es

or patches of terminals representing a single eye Figs Most or

in some sp ecies all layer V cells are mono cular Cells in other layers

of V resp ond best to the eye that dominates layer resp onses at that

horizontal lo cation Thus V cells can be characterized by their ocular

dominance or eye preference The strip es or patches of cortex that are

dominated throughout the cortical depth by a single eye are known as

ocular dominancecolumns

The segregated pattern of termination of the LGN inputs to V arises

early in development Initially LGN inputs pro ject to layer of V in an

overlapping manner without apparent distinction byeye represented The

terminal arb ors of individual LGN inputs extend horizontally in layer

iv Receptive Fields and Maps in the Visual Cortex

for distances as large as mm for comparison a typical spacing b etween

cortical cells is p erhaps m Subsequently b eginning either prenatally

or shortly after birth dep ending on the sp ecies the inputs representing

each eye b ecome horizontally conned to the alternating approximately

mm wide o cular dominance patches

This segregation results from an activitydep endent comp etition b etween

the geniculate terminals serving the twoeyes see discussion in The

signal indicating that dierent terminals represent the same eye app ears to

be the correlations in their neural activities These correlations exist

due b oth to sp ontaneous activity which is lo cally correlated within each

and to visuallyinduced activity which correlates

the activities of retinotopically nearby neurons within each eyeandtoa

lesser extent b etween the eyes The segregation pro cess is comp etitive

If one eye is caused to have less activity than the other during a critical

p erio d in which the columns are forming the more active eye takes over

most of the cortical territory but the eye with reduced activity

suers no loss of pro jection strength in retinotopic regions in whichitlacks

comp etition from the other eye In summary o cular dominance

column formation is a simple system in which correlated patterns of neural

activity sculpt the patterns of neural connectivity

Orientation columns are another striking feature of visual cortical orga

nization Most V cells are orientation selective resp onding selectively to

lightdark edges over a narrow range of orientations The preferred orienta

tion of cortical cells varies regularly and p erio dically across the horizontal

dimension of cortex and is invariantinthevertical dimension The matura

tion of orientation selectivity is activitydep endent eg However

it has not yet b een p ossible to test whether the initial developmentofori

entation selectivity is activitydep endent This is b ecause some orien tation

selectivity already exists at the earliest developmental times at which vi

sual cortical resp onses can be recorded and it has not

b een p ossible to blo ck visual system activity immediately b efore this time

Nonetheless it has long b een a p opular notion that the initial development

of orientation selectivity like that of o cular dominance may o ccur through

a pro cess of activitydep endent synaptic comp etition

The inputs from LGN to V serving eacheye are of twotyp es ONcenter

and OFFcenter Both kinds of cells have circularly symmetric orientation

insensitive receptive elds and resp ond to contrast rather than uniform

luminance ONcenter cells resp ond to light against a dark background or

to light onset OFFcenter cells resp ond to dark against a lightbackground

or to light oset In the cat the orientationselectiveVcellsinlayer are

simple cel ls cells with receptive elds consisting of alternating oriented

subregions that receive exclusively ONcenter or exclusively OFFcenter

input Fig As shall b e discussed one theory for the developmentof

orientation selectivityis that like o cular dominance it develops through

a comp etition b etween two input p opulations in this case a comp etition

Kenneth D Miller v

FIGURE Two examples of receptive elds RFs

Regions of the visual eld from which a simple cell receives ONcenter white or

OFFcenter dark input are shown Note Ocular dominance columns Fig

represent an alternation across cortex in the typ e of input left or righteye

received by dierent cortical cells while a simple cell RF this gure represents

an alternation across visual space in the typ e of input ON or OFFcenter

received bya single cortical cell

between the ONcenter and the OFFcenter inputs

CORRELATIONBASED MODELS

To understand o cular dominance and formation two

pro cesses must b e understo o d the developmentofreceptive eld struc

ture under what conditions do receptive elds b ecome mono cular drivable

only by a single eye or orientation selective the developmentof periodic

cortical maps of receptive eld prop erties what leads o cular dominance or

preferred orientation to vary p erio dically across the horizontal dimensions

of cortex and what determines the p erio dic length scales of these maps

the problem is simplied by consideration of a twodimensional Typically

mo del cortex ignoring the third dimension in which prop erties such as

o cular dominance and orientation are invariant

One approach to addressing these problems is to b egin with a hyp othe

sized mechanism of synaptic plasticity and to study the outcome of cortical

development under such a mechanism Most commonly theorists havecon

sidered a Hebbian synapse a synapse whose strength is increased when

pre and p ostsynaptic ring are correlated and p ossibly decreased when

they are anticorrelated Other mechanisms such as activitydep endentre

vi Receptive Fields and Maps in the Visual Cortex

lease and uptake of a diusible mo dication factor can lead to similar dy

namics in which synaptic plasticity dep ends on the correlations among

the activities of comp eting inputs Mo dels based on suchmechanisms are

referred to as correlationbased mo dels

The Von Der Malsburg Model Of V

Development

Von der Malsburg rst formulated a correlationbased mo del for the

development of visual cortical receptive elds and maps His mo del had two

basic elements First synapses of LGN inputs onto cortical neurons were

mo died by a Hebbian rule that is competitive so that some synapses were

strengthened only at the exp ense of others He enforced the comp etition by

holding constant the total strength of synapses converging on each cortical

cell conservation rule Second cortical cells tended to be activated in

clusters due to intrinsic cortical connectivity eg shortrange horizontal

excitatory connections and longerrange horizontal inhibitory connections

The conservation rule leads to comp etition among the inputs to a single

target cell Inputs that tend to be coactivated that is that have cor

related activities are mutually reinforcing working together to activate

the p ostsynaptic cells and thus to strengthen their own synapses Dierent

patterns that are mutually un or anticorrelated comp ete since strength

ening of some synapses means weakening of others Cortical cells eventually

develop receptive elds resp onsive to a correlated pattern of inputs

The clustered cortical activity patterns lead to comp etition b etween dif

ferent groups of cortical cells Each input pattern comes to b e asso ciated

with a cortical cluster of activity Overlapping cortical clusters contain

many coactivated cortical cells and thus b ecome resp onsive to overlap

ping correlated input patterns Adjacent nonoverlapping clusters contain

manyanticorrelated cortical cells and thus b ecome resp onsiveto un or

anticorrelated input patterns Thus over distances on the scale of an ac

tivity cluster cortical cells will have similar resp onse prop erties while on

the scale of the distance between nonoverlapping clusters cortical cells

will prefer un or anticorrelated input patterns This com bination of lo

cal continuity and largerscale heterogeneity leads to continuous p erio dic

cortical maps of receptive eld prop erties

In computer simulations this mo del was applied to the developmentof

orientation columns and ocular dominance columns For orien

tation columns inputs were activated in oriented patterns of all p ossible

orientations Individual cortical cells then develop ed selective resp onses

preferring one suchoriented pattern with nearby cortical cells preferring

nearby orientations For o cular dominance columns inputs were activated

in mono cular patterns consisting of a lo calized set of inputs from a single

eye Individual cortical cells came to b e driven exclusively by a single eye

Kenneth D Miller vii

and clusters of cortical cells came to b e driven by the same eye The nal

cortical pattern consisted of alternating strip es of cortical cells preferring

a single eye with the width of a strip e approximately set by the diameter

of an intrinsic cluster of cortical activity

In summary a comp etitive Hebbian rule leads individual receptive elds

to b ecome selective for a correlated pattern of inputs Combined with the

idea that the cortex is activated in intrinsic clusters this suggests an origin

for cortical maps coactivated cells in a cortical cluster tend to become

selective for similar coactivated patterns of inputs These basic ideas are

used in most subsequent mo dels

Mathematical Formulation

Atypical correlationbased mo del is mathematically formulated as follows

Let x y represent retinotopic p ositions in V and let

represent retinotopic p ositions in the LGN Let S x be the

synaptic strength of the connection from to x of the LGN pro jection

of typ e where may signify lefteye righteye ONcenter OFFcen ter

etc Let B x y represent the synaptic strength and sign of connection

from the cortical cell at y to that at xFor simplicity B x y is assumed

to take dierent signs for a xed y as x varies but alternatively sepa

rate excitatorypro jecting and inhibitorypro jecting cortical neurons may

be used Let ax and a represent the activity of a cortical or LGN

cell resp ectively

The activity ax of a cortical is assumed to dep end on a linear

combination of its inputs

X X

axf S x a B x y ay

y

Here f is some monotonic function such as a sigmoid or linear threshold

A Hebbian rule for the change in feedforward synapses can b e expressed

S x A x f ax f a

Here Ax is an arb or function expressing the numb er of synapses of

eachtyp e from to x a minimal form is Ax if there is a connection

from to x Ax otherwise A typical form for the functions f and

ver input patterns f is f aa hai where h ai indicates an average of a o

This yields a covariance rule synaptic change dep ends on the covariance

of p ostsynaptic and presynaptic activity

Next the Hebbian rule must b e made competitive This can b e accom

plished by conserving total synaptic strength over the p ostsynaptic cell

which in turn may be done either subtractively or multiplicatively

viii Receptive Fields and Maps in the Visual Cortex

The corresp onding equations are

d

S x S x xAx Subtractive

dt

d

S x S x xS x Multiplicative

dt

P P

S x S x

 

P P

and x Either form of con where x

Ax S x

 

P

d

straint ensures that S x Alternative metho ds have b een

dt

develop ed to force Hebbian rules to b e comp etitive see Miller and MacKay

Finally synaptic weights may b e limited to a nite range s Ax

min

S x s Ax Typically s and s is some p ositive

max min max

constant

SemiLinear Models

In semilinear mo dels the f s in are chosen to be linear Then

after substituting for ax from and averaging over input patterns

assuming that all inputs have identical mean activit y and that changes

in synaptic weights are negligibly small over the averaging time

b ecomes

X

S x Ax I x y C k S y k

y

Here I x y is an elementoftheintracortical interaction matrix

I B B B

where the matrix B is dened in This summarizes intracortical synap

tic inuences including con tributions via synapses The covari

ance matrix

C h a a a a i

expresses the covariation of input activities The factors k and k are

constants Translation invariance has b een assumed in b oth cortex and

LGN

When there are two comp eting input p opulations can b e further

S

simplied by transforming to sum and dierence variables S S S

D

o p opulations so that S S S Assuming equivalence of the tw

C C C C b ecomes

o n

P

S

S S

S y k I x y C k S x Ax

y

P

D

D D

S x Ax I x y C S y y

Kenneth D Miller ix

S D

Here C C C C C C Subtractive renormalization

S

alters only for S by subtraction of xAx while leaving

D

for S unaltered Multiplicative renormalization alters b oth and

S D

by subtraction of xS x and xS x resp ectively

How SemiLinear Models Behave

Linear equations like and can be understo o d by nding the

eigenvectors or mo des of the op erators on the right side of the equation

The eigenvectors are the synaptic weight patterns that grow indep endently

and exp onentially each at its own rate The fastestgrowing eigenvectors

typically dominate development and determine basic features of the nal

pattern although the nal pattern ultimately is stabilized by nonlinearities

such as the limits on the range of synaptic weights or the nonlinearity

involved in multiplicative renormalization

D

I will fo cus on the behavior of for S for analysis of see

D

S describ es the dierence in the strength of two comp eting

input p opulations Thus it is the key variable describing the developmentof

o cular dominance segregation or development under an ONcenterOFF

center comp etition In many circumstances can be derived directly

D

from by linearization ab out S without need to assume a

D

semilinear mo del The condition S corresp onds to an initial condition

in which the pro jections of the twoinput typ es are approximately equal

Thus study of can lend insightinto early in more

general nonlinear correlationbased mo dels

Equation can b e simply solved in the case of full connectivity from

the LGN to the cortex when Ax for all x and Then mo des

D ik x il

of S x of the form e e grow exp onentially and indep endently

D D

with rate prop ortional to I k C l where I and C denote the Fourier

D

transforms of I and C resp ectively for a description of the mo des as real

wavenumber k determines rather than complex functions see The

D

the wavelength jk j of an oscillation of S across cortical cells while

D

the wavenumber l determines the wavelength jl j of an oscillation of S

across geniculate cells The fastestgrowing mo des which will dominate

early development are determined bythek and l that maximize I k and

D

C l resp ectively The p eak of a functions Fourier transform corresp onds

to the cosine wave that b est matches the function and thus represents the

principal oscillation in the function

To understand these mo des Fig consider rst the set of inputs

receiv ed by a single cortical cell that is the shap e of the mo de for a xed

cortical p osition x This can be regarded as the receptive eld of the

cortical cell Each receptive eld oscillates with wavenumber l This oscil

D

lation of S S S is an oscillation b etween receptive eld subregions

dominated by S inputs and subregions dominated by S inputs Thus in

o cular dominance comp etition mono cular cells cells whose entire recep

x Receptive Fields and Maps in the Visual Cortex

tive elds are dominated by a single eye are formed only by mo des with

l no oscillation Mono cular cells thus dominate development if the

D

p eak of the Fourier transform of the C governing leftright comp etition

is at l Now instead consider an ONOFF comp etition S and S rep

resent ON and OFFcenter inputs from a single eye Then the receptive

elds of mo des with nonzero l resemble simple cells they receive predom

inantly ONcenter and predominantly OFFcenter inputs from successive

alternating subregions of the visual world Thus simple cells can form if

D

the C governing ONOFF comp etition has its p eak at a nonzero l

Now consider the arb orizations or pro jective elds pro jecting from a

single geniculate p oint that is the shap e of the mo de for a xed genicu

late p osition These oscillate with wavenumber k In o cular dominance

comp etition this means that left and righteye cells from pro ject to

alternating patches of cortex When mono cular cells form l these

alternating patches of cortex are the o cular dominance columns alternat

ing patches of cortex receiving exclusively lefteye or exclusively righteye

input resp ectively Thus the width of o cular dominance columns the

wavelength of alternation between righteye and lefteye dominated cor

tical cells is determined by the peak of the Fourier transform of the

intracortical interaction function I In ONOFF comp etition with l

the identity of the cortical cells receiving the ONcenter or OFFcenter part

of the pro jection varies as varies so individual cortical cells receiv e b oth

ON and OFFcenter input but from distinct subregions of the receptive

eld

In summary there is an oscillation within receptive elds with wavenum

D

ber l determined by the p eak of C and an oscillation within arb ors with

wavenumber k determined by the peak of I Fig These two oscil

lations are knit together to determine the overall pattern of synaptic

connectivity The receptive eld oscillation which matches the receptive

eld to the correlations quantitatively describ es von der Malsburgs nd

ing that individual receptive elds b ecome selective for a correlated pattern

of inputs Similarly the arb or oscillation matches pro jectiveelds to the

intracortical interactions and thus to the patterns of cortical activityclus

ters This quantitatively describ es the relationship b etween activity clusters

ik x

and maps Note that the factor e can b e regarded as inducing a phase

shift for varying x in the structure of receptiveeldsThus cortical cells

that are nearby on the scale of the arb or oscillation have similar receptive

elds while cells wavelength apart have opp osite receptive elds

An alternative viewp ointonthe same pattern is obtained by rewriting

ik lx ilx

the mo des as e e The argument l x represents the os

cillation with wavenumber l within the receptive eld now expressed in

co ordinates relativetothecenter of the receptive eld rather than in abso

lute p osition across the geniculate The argumentk l x represents a shift

for varying x in the phase of the receptive eld relativ eto the receptive

eld center For the case of o cular dominance with l this is just the

Kenneth D Miller xi

CORRELATION FUNCTION

LATERAL INTERACTION PROJECTIVE FIELD

FUNCTION

FIGURE Schematic of the outcome of semilinear correlationbased develop

ment

D

TOP The correlation function C determines the structure of receptiveelds

D

RFs White RF subregions indicate p ositive values of S dark subregions

D

negativevalues When C do es not oscillate individual cortical cells receiveonly

D

a single typ e of input as in o cular dominance segregation If C oscillates there

is a corresp onding oscillation in the typ e of input received by individual cortical

cells as in simple cell RFs Alternative RF structures could form as in the center

surround structure shown but oriented simplecelllike outcomes predominate for

reasonable parameters Simple cells then develop with various numbers of

subregions and various spatial phases only a single example of a cell with two

subregions and o dd spatial symmetry is pictured BOTTOM The intracortical

interactions I similarly determine the structure of pro jective elds Here solid

D

lines indicate p ositivevalues of S dotted lines indicate negativevalues Adapted from

xii Receptive Fields and Maps in the Visual Cortex

shift with wavenumber k between lefteye dominance and righteye domi

nance of cortical cells For ONOFF comp etition with l this represents

a p erio dic shifting with movement across cortex as to which subregions of

the receptive eld are dominated byONcenter inputs and which subre

gions are dominated by OFFcenter inputs Thus we can view the results

as an oscillation within receptive elds with wavenumber l combined with

a shift with cortical p osition in the spatial phase of receptive elds this

shift o ccurring with wavenumber k l the vector sum of the pro jective

eld or arb or oscillation and the receptive eld oscillation

The comp etitive renormalizing terms do not substantially al

ter these pictures except that multiplicative renormalization can suppress

o cular dominance development in some circumstances These results

hold also for lo calized connectivity nite arb ors and thus generally char

acterize the b ehavior of semilinear mo dels The ma jor dierence

in the case of lo calized connectivity is that if k or l corresp onds to a wave

length larger than the diameter of connectivity from or to a single cell

then it is equivalenttok or l resp ectively A go o d approximation

to the leading eigenvectors in the case of nite connectivityisgiven simply

ik x il

by Ax e e wherek and l are determined as ab oveby the p eaks

D

of I k andC l unpublished results

Understanding Ocular Dominance And

Orientation Columns With Semilinear

Models

This understanding of semilinear mo dels leads to simple mo dels for the de

velopment of b oth o cular dominance columns and orientation columns

as follows Fig

Mono cular cells develop through a comp etition of left and righteye

D

inputs in a regime in which C l is p eaked at l Thewavelength of

alternation is then determined by the peak of

I k

Orientationselective simple cells develop through a comp etition of ON

D

center and OFFcenter inputs in a regime in which C l is peaked at

 D

Subtractive renormalization has no eect on the developmentof S

Multiplicative renormalization lowers the growth rates of all mo des of b oth

D S S

S and S by the factor x which dep ends only on S Theresult is that

D

in order for S to grow at all its mo des must have larger unconstrained growth

S D

rates than those of S that is the p eak of the Fourier transform of C must

S

be larger than that of C In practice this condition is met only if there are

 

anticorrelations b etween S and S that is if C is signicantly negative When

D

this condition is met then the mo des that dominate S are just as describ ed

ab ove they are not altered by the constraint term in These and other eects of renormalizing terms are discussed in detail in

Kenneth D Miller xiii

l The mean wavelength of alternation of ONcenter and OFFcenter

subregions in the simple cells receptive elds is determined by the p eak of

D

C l This wavelength corresp onds to a cells preferred spatial frequency

under stimulation by sinusoidal luminance gratings In individual mo des

all cortical cells have the same preferred orientation but their spatial phase

varies p erio dically with cortical p osition The mixing of such mo des of all

orientations leads to a periodic variation of preferred orientation across

cortex The p erio d with which preferred orientations change across cortex

is more complex to determine

This mo del of o cular dominance column formation is similar to that of

von der Malsburg The latter mo del assumed anticorrelation b etween

the twoeyes this was required due to the use of multiplicative renormaliza

tion With subtractive renormalization o cular dominance col

umn formation can o ccur even with partial correlation of the twoeyes

The mo del can b e compared to exp eriment particularly through the predic

tion of the relation b etween intracortical connectivity and o cular dominance

column width

The mo del of orientationselectivecelldevelopment is quite dierent from

that of von der Malsburg Von der Malsburg p ostulated that oriented

input patterns lead to the developmentof orientationselective cells The

ONOFF mo del instead p ostulates that ONOFF comp etition results in

oriented receptive elds in the absence of oriented input patterns the

circular symmetry of the input patterns is sp ontaneously broken This

symmetrybreaking p otential of Hebbian developmentwas rst discovered

by Linsker In all of these mo dels the continuityand periodic alter

nation of preferred orientation is due to the intracortical connectivityThe

ONOFF mo del can b e compared to exp eriment most simply by the mea

D

surementof C to determine whether it has the predicted oscillation

Related SemiLinear Models

Linsker prop osed a mo del that was highly inuential in two

resp ects First he pointed out the p otential of Hebbian rules to sp onta

neously break symmetry yielding orientationselective cells given approxi

mately circularly symmetric input patterns Second he demonstrated that

Hebbian rules could lead to segregation within receptive elds so that a cell

came to receive purely excitatory or purely inhibitory input in alternating

subregions of the receptive eld This mo del was thoroughly analyzed in

Linsker used a semilinear mo del with a single input typ e that could

have p ositive or negative synaptic strengths s s He largely

min max

restricted study to the case of a single p ostsynaptic cell Because the equa

tion for a single input typ e and a single p ostsynaptic cell Eq with

xiv Receptive Fields and Maps in the Visual Cortex

I x y x y is circularly symmetric its eigenfunctions are also

eigenfunctions of the rotation op erator Thus the eigenfunctions can be

written in p olar co ordinates r as cosn f r sinn f r where

nj nj

f r is a radial function and n and j are integers indexing the eigen

nj

functions In quantum mechanics atomic orbitals are named Nx where N

is a numb er representing one plus the total numb er of angular and radial

no des and x is a letter denoting the numb er of angular no des spdfg

corresp onding to n angular no des Thus s is a function with

zero no des s has one no de which is radial p has one no de whichisan

gular p has two no des one radial one angular etc This naming scheme

can b e applied to any rotationally symmetric system and in particular can

b e applied to the eigenfunctions of Linskers system a fact whic h

physicists have found amusing The nature of these eigenfunctions their

dep endence on parameters and their role in determining the outcomes

Linsker observed in simulations are describ ed in

For our present purp oses the essential results of this analysis are as

follows Two factors underlay Linskers results One factor was that oscilla

tions in a correlation function can induce oscillations in a receptive eld as

describ ed ab ove The other factor was a constraint in the mo del xing the

p ercentage of p ositive or negative synapses received by a cell this forced

an alternation of p ositive and negative subregions even when the correla

tion function did not oscillate These two causes were not disentangled in

Linskers simulations but only the rst app ears likely to be of biological

relevance

Tanaka has indep endently formulated mo dels of o cular domi

nance and orientation columns that are similar to those describ ed in sec

tion The ma jor dierence is that he works in a regime in whicheach

cortical cell comes to receive only a single LGN input Tanaka denes cor

tical receptive elds as the convolution of the input arrangement with the

intracortical interaction function This means that a cortical cells receptive

eld is due to its single input from the LGN plus its input from all other

cortical cells within reach of the intracortical interaction function Thus

orientation selectivity in this mo del arises from the breaking of circular

symmetry in the pattern of inputs to dierent cortical cells rather than to

individual cortical cells

THE PROBLEM OF MAP STRUCTURE

The ab ove mo dels accountwell for basic features of primary visual cortex

However many details of real cortical maps are not replicated by these



The assumption is made that the arb or and correlation functions dep end only on distance

Kenneth D Miller xv

mo dels One reason may b e the simplicity of the mo del of cor

tex the real cortex is dimensional rather than has cellsp ecic con

nectivity rather than connectivity that dep ends only on distance and has

plastic rather than xed intracortical connections Another reason is that

the details of map structure inherently involve nonlinearities bywhichthe

fastestgrowing mo des interact and comp ete whereas the semilinear frame

work only fo cuses on early pattern formation in which the fastestgrowing

mo des emerge and mix randomly without interacting

Some simple mo dels that fo cus on map development rather than re

ceptive eld development strikingly match the map structures observed

in monkeys One suchmodel uses the selforganizing feature map

SOFM of Kohonen In the SOFM only a single cluster of cortical

cells is activated in resp onse to a given input pattern This is an abstraction

of the idea that the cortex resp onds in lo calized activity clusters The single

activated cluster is centered on the cell whose weightvector b est matches

the direction of the input activation vector Hebbian learning then takes

place on the activated cells bringing their weightvector closer to the input

activation vector The size of an activity cluster is gradually decreased as

the mapping develops this is akin to annealing helping to ensure a nal

mapping that is optimal on b oth coarse and ne scales

Except for the restriction to a single activity cluster and the gradual

decrease in cluster size the SOFM is muchlike the correlationbased mo d

els However an abstract representation of the input is generally used In

correlationbased mo dels the input space mayhave thousands of dimen

sions one for each input cell In the SOFM mo del of visual cortex the input

space instead has ve dimensions two represent retinotopic p osition and

one represents eachof o cular dominance orientation selectivity and pre

ferred orientation Each cortical cell receives ve synapses corresp onding

to these ve inputs Assumptions are made as to the relative size of or

variance of the input ensemble along each dimension There is no obvious

biological interpretation for this comparison between dimensions Under

the assumptions that the o cular dominance and orientation dimensions are

short compared to the retinotopic dimensions and that only one input

p ointisactivated at a time Hebbian learning can lead to maps of orienta

tion and o cular dominance that are in detail remarkably like those seen

in macaque monkeys

The SOFM and other mo dels based on the elastic net algorithm

uous mappings in which a constant distance across lead to lo cally contin

the cortex corresp onds to a roughly constant distance in the reduced in

put space This means that when one input feature is changing rapidly

across cortex the others are changing slowly Thus the mo dels predict

that orientation changes rapidly where o cular dominance changes slowly

and vice versa It may b e this feature that is key to replicating the details

of macaque orientation and o cular dominance maps A mo del that forces

such a relationship to develop b etween o cular dominance and orientation

xvi Receptive Fields and Maps in the Visual Cortex

while assuring p erio dic representation of each also gives a go o d matchto

primate visual maps

The SOFM also replicates asp ects of the retinotopic maps seen in higher

areas of cat visual cortex For these studies the input and output

space are eachtaken to b e twodimensional representing retinotopic p osi

tion The input space is taken to b e a halfcircle representing a hemiretina

and the shap e of the output space is varied When this shap e is long and

narrow as in cat cortical areas and the retinotopic map develop ed by

the SOFM has a characteristic pattern of discontinuities closely resembling

those observed exp erimentally in those areas The SOFM achieves maps

in which nearbypoints in the output space corresp ond to nearbypoints in

the input space while each area of the input space receives approximately

equal representation provided each is equally activated see further

discussion of the SOFM b elow The success of the SOFM mo dels of retino

topic maps suggests that these are constraints that should b e satised by

any mo del of cortical maps One would like to determine more precisely

the constraints on a retinotopic mapping embodied by the SOFM that

are sucient to replicate these results

It has recently b een rep orted that input correlations can alter the spac

ing of o cular dominance columns in cat visual cortex by p erhaps

A smaller ocular dominance column spacing develops when the ac

tivities of the two eyes are correlated by normal vision than when the

twoeyes activities are decorrelated decorrelation is achieved by inducing

divergent strabismus which causes the two eyes to see dierent parts of

the visual world This eect was anticipated theoretically by Go o dhill

the two who argued essentially that correlation of the activities of

eyes brings them closer together and so the twoeyes should b e brought

closer together in their cortical representation by reduction of column size

This eect couldalsohave b een anticipated by the SOFM mo dels of oc

ular dominance b ecause decorrelation corresp onds to an increase in the

variance of o cular dominance and thus an increase in the size of the o c

ular dominance dimension which results in increased column size In

semilinear mo dels in contrast the column width do es not app ear to be

signicantly aected bybetweeneye correlations Rather as the degree of

betweeneye correlation is increased more bino cular cells form at the col

umn b orders until at some critical level of correlation o cular dominance



This story is still somewhat tentative Another worker K Murphyprivate

communication rep orts no dierence b etween the column spacings found in nor

mal and strabismic She rep orts that in b oth cases column p erio ds range

from m whereas Lowel rep orts that normal cats have p erio ds of

m and strabismic cats have p erio ds of m Neither study

used littermatched controls given natural variation in o cular dominance column

width such studies may b e needed to determine the strength and robustness of a strabismic eect

Kenneth D Miller xvii

segregation no longer o ccurs unpublished results That is the twoeyes

are brought closer together through alteration of receptive elds rather

than through alteration of the map One can anticipate several biological

mechanisms that might b e added to instead yield reduction in column size

such as nonlinearities that discourage formation of bino cular cells or non

linearities in cortical activation that cause the size of activity clusters to

dep end on the correlations of the inputs

Finally it has recently b een noted that cat orientation maps are signi

cantly smo other than could b e achieved by simple linear considerations

The analysis in suggests that these maps could result mathematically

from a lo cal diusion of preferred orientations It will b e interesting to

develop a biologically interpretable mo del of such a pro cess

THE COMPUTATIONAL SIGNIFICANCE

OF CORRELATIONBASED RULES

Efficient Representation of Information

A simple correlationbased rule for a single p ostsynaptic cell can if prop erly

designed lead to development of a receptive eld that corresp onds to the

principal comp onent of the input data that is to the principal eigenvector

of the covariance matrix of the inputs to the cell This recep

tive eld in turn maximizes the variance of the p ostsynaptic cells activity

given the ensemble of input patterns It has b een argued that correlation

based rules thus maximize the information carried in the p ostsynaptic cells

activity ab out the input patterns Intuitively byvarying as muchas

p ossible in its resp onse to dierent inputs the p ostsynaptic cell draws the

greatest p ossible distinction b etween the dierent input patterns

More generallyanumb er of closely related and in many circumstances

identical computational functions have b een prop osed for brain areas near

the sensory p eriphery These include maximization of information ab out

the inputs minimization of redundancy or correlation in the activities

of output cells statistical indep endence of the output activities or

enco ding of the input information as compactly as p ossible for example

requiring as little dynamic range as p ossible p er neuron These func



Go o dhills theory is identical to the mo del of except that the trans

lation from input activations to cortical activations has b een altered Miller et al

used linear activation rules whereas Go o dhill uses the SOFM rule in

which a single activity bubble of cells surrounds the b estactivated or winning

cortical cell also Go o dhill mo dies the rule so that the more often a cell has won

the less likely it is to win again Thus use of nonlinear intracortical interactions

can transform a semilinear mo del into one in whichintero cular correlation mo dulates column width

xviii Receptive Fields and Maps in the Visual Cortex

tions all involve representing the input information in an ecientwayin

the of information theory These measures of eciency takeinto ac

count the statistics of the input ensemble but disregard the semantics

the meaning or survival value to the animal of the inputs

The interpretation that the function of a correlationbased rule is to

yield such an ecient representation is inviting but it carries two ma

jor problems First the principal comp onent representation achieved by

correlationbased rules is optimally ecient only for a Gaussian distribu

tion of input patterns or in other words it reects only the second order

or twop oint statistics the covariance of the input data It is p ossible

that a great deal of information may reside in higherorder statistics but

a correlationbased rule as conceived ab ove will ignore this information

Intrator has suggested that a variant of standard Hebbian rules can in

stead maximize a thirdorder statistic of the output activity and argues

that this may b e a b etter statistic for distinguishing among the elements

of realworld ensembles While one statistic or the other may be

b est for characterizing a given set of data b oth approaches can suer from

the limitation that they are maximizing one particular statistic rather than

maximizing some measure of eciency

Second this interpretation applies only to a single isolated p ostsynaptic

cell Multiple cells viewing the same input ensemble will extract the same

information from it under a given correlationbased rule This do es not

add new information ab out the input but only redundantly rep eats the

same information Thus although a single cell mayhave a receptiveeld

that maximizes the information it could carry ab out the input ensemble

a group of such cells generally will not much impro ve on the p erformance

of a single cell and will not carry the maximal p ossible information ab out

the input ensemble

One way out of this dilemna is to intro duce couplings b etween the p ostsy

naptic cells that force them to learn indep endent parts of the input ensem

ble Unfortunately excitatory couplings tend to pro duce correlated cells

while inhibitory couplings pro duce anticorrelated cells The ostensible goal

however is to pro duce uncorrelated cells cells whose activities carry inde

p endent information ab out the input ensemble Thus biological couplings

will not work A theoretical way out involves using connections between

the p ostsynaptic cells that are mo died byantiHebbian rules if two cells

have correlated activity the connection b etween them b ecomes more nega

tive if two cells haveanticorrelated activity the connection b etween them



For simplicity in this discussion I am ignoring noise Dep ending on the signal

to noise ratio one will wish to strike a particular balance b etween variety carry

ing more indep endent comp onents of the input ensemble and redundancy eg

see However except in the extreme case of high noise where complete

redundancy is called for multiple comp onents will always b e needed to maximize

the information given multiple output cells

Kenneth D Miller xix

b ecomes more p ositive The result is that the cells b ecome uncorrelated

Many authors have indep endently prop osed rules that involve such anti

Hebbian learning on lateral connections eg or related ideas

However no biological sign of antiHebbian synaptic mo dication has

thus far b een observed

An alternativeway out of this dilemna stems from the observation that

biological receptive elds are lo calized Thus nearby cells see overlapping

but not identical sets of inputs Consider two extreme cases First when

each input cell is connected to a single output cell receptive elds are com

pletely lo calized In the limit of low noise the output layer replicates the

activity of the input layer so all information is preserved However when

noise is signicant some information is lost bythisidentity mapping and

alternative connectivityschemes may yield greater information ab out the

inputs Second when there is global connectivity so that all input cells are

connected to all output cells receptive elds are completely delo calized

Under a correlationbased rule each output cell learns the same receptive

eld Then in the low noise limit most information is b eing thrown away

only one dimension of the input pattern is b eing distinguished However

supp ose that this dimension is the most informative dimension ab out the

input ensemble Then in the high noise limit this redundant representa

tion of the most informationrich dimension will maximize the information

carried ab out the input ensemble

Thus given a correlationbased learning rule a completely lo calized rep

resentation can maximize information in the lownoise limit while a com

pletely delo calized representation can maximize information in the high

noise limit Intermediate levels of lo calization should be appropriate for

intermediate signaltonoise ratios this has recently b een demonstrated

quantitatively It seems likely that biology rather than designing an

antiHebbian learning rule has used its own correlationbased rules and

has made use of its natural tendency to form partially lo calized receptive

elds in order to ensure eciency of representation

SelfOrganizing Maps and Associative

Memories

The ab o ve ideas ab out eciency consider only the summed information

in the resp onses of the p ostsynaptic cells without regard for lo cation

or connectivity Alternative ideas ab out the computational signicance of

correlationbased rules fo cus on the spatial arrangement of p ostsynaptic

resp onse features and the connectivitybetween the p ostsynaptic cells

One set of such ideas stem from the study of the selforganizing feature

map SOFM of Kohonen and of related dimensionreducing map

pings As previously describ ed the SOFM corresp onds to a Hebbian

rule with a nonlinear lateral intracortical interaction suchthateachinput

xx Receptive Fields and Maps in the Visual Cortex

pattern leads to a single lo calized cluster of cortical activity The SOFM

and related algorithms lead to a mapping that matches the top ology and

geometry of the output space to that of the input space despite a p ossible

dimensional andor shap e mismatchbetween the two That is

nearbypoints in the output space corresp ond via the mapping to nearby

p oints in the input space and input patterns that o ccur more often develop

a larger representation than those that o ccur less often

A number of p ossible functions have b een assigned to such mappings

One is the minimization of wiring length assuming that cortical points

representing nearby input patterns need to b e connected to one another

Another is to represent the input data in a compressed form while

minimizing reconstruction error A sp ecic form of the latter idea

is as follows Supp ose that there is noise in the output layer that is distance

dep endent so that the probability of resp onse being centered at a given

output p oint falls o with its distance from the p oint that is correct for

that input Supp ose also that there is a metric on the input space and

the error in mistaking one input pattern for another is assigned as the

distance between the two patterns Then the SOFM can be interpreted

approximatelyasachieving the inputoutput mapping that minimizes the

average error in reconstructing the input pattern from the output resp onses

The ma jor problem in applying these ideas to biology is the diculty

in assigning biological meaning to the top ology and geometry of the non

retinotopic dimensions of the input space Given an ensemble of visual

input patterns on the retina for example how large is the corresp onding

o cular dominance or orientation dimension relativeto the retinotopic di

mensions Without a clear prescription for answering this question it is

dicult to make biological predictions from these ideas Nonetheless the

computational functions of selforganizing maps their close connection to

correlationbased mo dels and their ability to replicate many features of

cortical maps are intriguing

Another wellknown set of ideas concern the role of correlationbased

rules in establishing an asso ciative memory Supp ose one wishes to learn a

a a a a th

set of N inputoutput pairs u v where u and v are the a input

a a

and output vectors resp ectivelyLetv Mu for some synaptic matrix

a b

M If the input patterns are orthonormal u u then the input

ab

P

a a

T

output asso ciation is achieved by setting M v u eg

a

d

This relation will b e learned by a Hebbian rule M M N v u

ij ij i j

dt

a a

provided there is a teacher to clamp the output to v whenever u is

presented A fully connected network with activity states v will similarly

a

develop the activity states or memories v as stable attracting states

if the connection matrix between the cells is determined by the Hebbian

P

a a

T

prescription M v v eg Again to learn a sp ecic

a

set of memories a teacher is required to clamp the network into the

appropriate activity states during learning Given simple nonlinearities in

Kenneth D Miller xxi

neuronal activation the stored memories need not be orthogonal to one

another provided the memories are randomly chosen uncorrelated and

their numberissuciently small relativetothenumb er of cells eg

It is of biological interest to explore how asso ciative prop erties can develop

through correlationbased rules in the absence of a teacher as well as in

the presence of correlated input patterns for which see

OPEN QUESTIONS

This brief review can only p oint to a small sample of the rich literature on

this topic Among the many op en questions in the eld are How can bio

logically interpretable mo dels replicate the details of cortical maps Might

orientation selectivity arise from early oriented wave patterns of retinal ac

tivity or other mechanisms rather than through ONOFF comp eti

tion Might the initial development of orientation selectivity o ccur through

patterning of intracortical connections rather than through patterning of

LGN connections to cortex Howmightintracortical plasticity aect re

ceptive eld and map development Howmight input correlations aect

column size How will development b e altered by incorp oration of more

realistic cortical connectivity and more realistic nonlinear learning rules

For example might input correlations help determine the selforganization

of plastic intracortical connections or the size of nonlinearlydetermined

cortical activity clusters eachofwhichinturnwould shap e the pattern of

input synapses including column size Howcanwecharacterize the com

putational function of the correlationbased rules used biologically These

and other questions are likely to b e answered in the coming years

Acknowledgments

KD Miller is supp orted by grants from the National Eye Institute the

Searle Scholars Program and the Lucille P Markey Charitable Trust I

thank Ed Erwin Sergei Rebrik and Todd Troyer for helpful comments on

the manuscript



See for arguments that the early oriented waves of retinal activity are to o

large to drive the development of simple cells ie their wavelength is much wider

than the set of LGN inputs to a single simple cell but see for an argument

that the waves might nonetheless drivedevelopment of orientationselectivityby

determining the patterning of intracortical connections rather than of connections

from LGN to cortex The patterning of horizontal connections may take place

slightly later than the development of orientation selectivity but b oth o ccur

suciently early that their order remains unclear

xxii Receptive Fields and Maps in the Visual Cortex

References

K Albus and W Wolf Early p ostnatal development of neuronal

function in the kittens visual cortex A laminar analysis J Physiol

JJ Atick Could information theory provide an ecological theory

of sensory pro cessing In W Bialek editor Princeton Lectures on

Biophysics pages World Scientic Singap ore

HB Barlow Unsup ervised learning Neural Comp

BO Braastad and P Heggelund Developmentof spatial receptive

eld organization and orientation selectivity in kitten striate cortex

J Neurophysiol

EM Callaway and LC Katz Emergence and renement of clustered

horizontal connections in cat striate cortex J Neurosci

B Chapman and MP Stryker Developmentoforientation selectivity

in ferret visual cortex and eects of deprivation J Neurosci

M ConstantinePaton HT Cline and E Debski Patterned activity

synaptic convergence and the NMDA receptor in developing visual

pathways Ann Rev Neurosci

R Durbin and G Mitchison A dimension reduction framework for

understanding cortical maps Nature

E Erwin K Ob ermayer and K Schulten Mo dels of orientation and

o cular dominance columns in the visual cortex A critical comparison

Neural Comp

PFoldiak Adaptivenetwork for optimal linear feature extraction In

Proceedings IEEEINNS International Joint Conference on Neural

Networksvolume pages IEEE Press New York

Y Fregnac and M Imb ert Development of neuronal selectivityinthe

primary visual cortex of the cat Physiol Rev

GJ Go o dhill Top ography and o cular dominance A mo del exploring

p ositive correlations Biol Cybern

GJ Go o dhill and DJ Willshaw Application of the elastic net algo

rithm to the formation of o cular dominance strip es Network

Kenneth D Miller xxiii

CS Go o dman and CJ Shatz Developmental mechanisms that gen

erate precise patterns of neuronal connectivity Cel l Suppl

RW Guillery Bino cular comp etition in the control of geniculate cell

growth J Comp Neurol

RW Guillery and DJ Stelzner The dierential eects of unilateral

lid closure up on the mono cular and bino cular segments of the dorsal

lateral geniculate nucleus in the cat J Comp Neurol

JA Hertz AS Krogh and RG Palmer Introduction to the Theory

of Neural Computation AddisonWesley

JJ Hopeld Neural networks and physical systems with emergent

collective computational abilities Proc Natl Acad Sci USA

JJ Hopeld Neurons with graded resp onses have collectivecompu

tational prop erties like those of twostate neurons Proc Natl Acad

Sci USA

DH Hub el and TN Wiesel Receptiveelds of cells in striate cor

tex of very young visually inexp erienced kittens J Neurophysiol

M Idiart B Berk and LF Abb ott Reduced representation by neural

networks with restricted receptive elds Neural Comp

N Intrator Feature extraction using an unsup ervised neural network

Neural Computation

N Intrator and LN Co op er Ob jective function formulation of the

BCM theory of visual cortical plasticity Statistical connections sta

bility conditions Neural Networks

T Kohonen SelfOrganization and Associative Memory Springer

Verlag Berlin rd edition

S LeVay TN Wiesel and DH Hub el The development of ocu

lar dominance columns in normal and visually deprived monk eys J

Comp Neurol

Z Li and JJ Atick Ecient stereo co ding in the multiscale repre

sentation Network

xxiv Receptive Fields and Maps in the Visual Cortex

R Linsker From basic network principles to neural architecture

Emergence of spatialopp onent cells Proc Natl Acad Sci USA

R Linsker From basic network principles to neural architecture

Emergence of orientationselective cells Proc Natl Acad Sci USA

R Linsker From basic network principles to neural architecture

Emergence of orientation columns Proc Natl Acad Sci USA

R Linsker Selforganization in a p erceptual network Computer

March

R Linsker Lo cal synaptic learning rules suce to maximize mutual

information in a linear network Neural Comput

S Lowel Ocular dominance column development Strabismus changes

the spacing of adjacent columns in cat visual cortex J Neurosci

S Luttrell A Bayesian analysis of selforganizing maps Neural

Comp

DJC MacKay and KD Miller Analysis of Linskers applications of

Hebbian rules to linear networks Network

DJC MacKay and KD Miller Analysis of Linskers simulations of

Hebbian rules Neural Comput

L Maei and L GalliResta Correlation in the discharges of neigh

b oring retinal ganglion cells during prenatal life Proc Nat Acad

Sci USA

DN Mastronarde Correlated ring of retinal ganglion cells Trends

Neurosci

M Meister ROL W ong DA Baylor and CJ Shatz Synchronous

bursts of actionp otentials in ganglion cells of the developing mam

malian retina Science

KD Miller Correlationbased mo dels of neural development In MA

Gluck and DE Rumelhart editors Neuroscience and Connectionist

Theory pages Lawrence Erlbaum Ass Hillsdale NJ

KD Miller Derivation of linear Hebbian equations from a nonlin

ear Hebbian mo del of synaptic plasticity Neural Comput

Kenneth D Miller xxv

KD Miller A mo del for the development of simple cell receptive

elds and the ordered arrangement of orientation columns through

activitydep endent comp etition b etween ON and OFFcenter inputs

J Neurosci

KD Miller JB Keller and MP Stryker Ocular dominance column

development Analysis and simulation Science

KD Miller and DJC MacKay The role of constraints in Hebbian

learning Neural Comput

KD Miller and MP Stryker The development of o cular dominance

columns Mechanisms and mo dels In SJ Hanson and CR Olson

editors Connectionist Modeling and Brain Function The Developing

Interface pages MIT PressBradford Cambridge MA

M Miyashita and S Tanaka A mathematical mo del for the self

organization of orientation columns in visual cortex NeuroReport

K Ob ermayer GG Blasdel and K Schulten A statistical mechani

cal analysis of selforganization and pattern formation during the de

velopment of visual maps Phys Rev A

E Oja A simplied neuron mo del as a principal comp onent analyzer

J Math Biol

H Ritter T Martinetz and K Schulten Neural Computation and

SelfOrganizing Maps An Introduction AddisonWesley Reading

MA

J Rubner and K Schulten Development of feature detectors by self

organization Biol Cybern

TD Sanger An optimality principle for unsup ervised learning In

D Touretzky editor Advances in Neural Information Processing Sys

tems v olume pages Morgan Kaufmann San Mateo CA

CJ Shatz The developing brain Scientic Am

CJ Shatz and MP Stryker Ocular dominance in layer IV of the cats

visual cortex and the eects of mono cular deprivation J Physiol

J Sirosh and R Mikkulainen A unied neural network mo del for

the selforganization of top ographic receptive elds and lateral inter

actions Neural Comput to app ear

xxvi Receptive Fields and Maps in the Visual Cortex

MP Stryker and SL Strickland Physiological segregation of ocu

lar dominance columns dep ends on the pattern of aerent electrical

activity Inv Opthal Supp

NV Swindale A mo del for the co ordinated development of columnar

systems in primate striate cortex Biol Cyb

S Tanaka Theory of ocular dominance column formation Mathe

matical basis and computer simulation Biol Cybern

C von der Malsburg Selforganization of orientation selective cells in

the striate cortex Kybernetik

C von der Malsburg Network selforganization in the ontogene

sis of the mammalian visual system Internal Rep ort Ruhr

Univ ersitat Bo chum Institut fur Neuroinformatik Bo chum

Germany

C von der Malsburg and DJ Willshaw A mechanism for pro ducing

continuous neural mappings o cularity dominance strip es and ordered

retinotectal pro jections Exp Brain Res Supp

TN Wiesel and DH Hub el Comparison of the eects of unilat

eral and bilateral eye closure on cortical unit resp onses in kittens J

Neurophysiol

TN Wiesel and DH Hub el Ordered arrangement of orientation

columns in monkeys lacking visual exp erience J Comp Neurol

F Wolf HU Bauer and T Geisel Formation of eld discontinuities

and islands in visual cortical maps Biol Cyb

F Wolf K Pawelzik T Geisel DS Kim and T Bonho eer Op

timal smo othness of orientation preference maps In Computation in

Neurons and Neural Systems pages Boston Kluwer

RO Wong M Meister and CJ Shatz Transient p erio d of corre

lated bursting activity during development of the mammalian retina

Neuron