Andrew M. Oster- Mathematical Models of Cortical Development

8/3/2019 Andrew M. Oster- Mathematical Models of Cortical Development

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MATHEMATICAL MODELS OF

CORTICAL DEVELOPMENT

by

Andrew M. Oster

A dissertation submitted to the faculty ofThe University of Utah

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Department of Mathematics

The University of Utah

December 2006


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Copyright c Andrew M. Oster 2006

All Rights Reserved


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THE UNIVERSITY OF UTAH GRADUATE SCHOOL

SUPERVISORY COMMITTEE APPROVAL

of a dissertation submitted by

Andrew M. Oster

This dissertation has been read by each member of the following supervisory committeeand by majority vote has been found to be satisfactory.

Chair: Paul C. Bressloff

Alessandra Angelucci

Aaron L. Fogelson

James P. Keener

Richard M. Normann


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THE UNIVERSITY OF UTAH GRADUATE SCHOOL

FINAL READING APPROVAL

To the Graduate Council of the University of Utah:

I have read the dissertation of Andrew M. Oster in its final formand have found that (1) its format, citations, and bibliographic style are consistent andacceptable; (2) its illustrative materials including figures, tables, and charts are in place;

and (3) the final manuscript is satisfactory to the Supervisory Committee and is readyfor submission to The Graduate School.

Date Paul C. BressloffChair, Supervisory Committee

Approved for the Major Department

Aaron J. BertramChair/Dean

Approved for the Graduate Council

David S. ChapmanDean of The Graduate School


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ABSTRACT

We extend classical activity-based developmental models of ocular dominance column

(ODC) formation in primary visual cortex (V1) to include cortical growth and cortexs

laminar structure. We show that with cortical growth the OD pattern exhibits a sequence

of pattern forming instabilities as the size of the cortex increases. Each instability results

in the insertion of an additional OD column such that over the course of development, the

mean width of an ODC is approximately preserved, consistent with recent experimental

observations of postnatal growth in cat. The other biologically motivated extension we

make is to consider a multilayer representation of V1 with thalamic and vertical connec-

tions taken to be modifiable by activity. By including the layer-specific thalamic input to

V1 along with the interlaminar projections to a correlationbased Hebbian learning rule,

our model allows for the joint development of OD columns and cytochrome oxidase (CO)

blobs in primate V1. The developed OD map in layer 4C is inherited by layer 2/3 via the

vertical projections. Competition between these projections and the direct thalamic input

to layer 2/3 then results in the formation of CO blobs superimposed upon the OD map.

The spacing of the OD columns is determined by the spatial profile of the intralaminar

connections within layer 4, while the spacing of CO blobs depends both on the width of

the ODCs inherited and the spatial distribution of intralaminar connections within the

superficial layer. These mathematical models of cortical development demonstrate that

simple models with key aspects of the biological system give us insight into the underlying

mechanisms for observed patterns in V1.


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To Dennis Stanton, my high school math teacher,

and to Thomas ONeil, my undergraduate advisor.


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CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

NOTATION AND SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

CHAPTERS

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. A SURVEY OF THE STRUCTURE AND DEVELOPMENT OFVISUAL CORTEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Structure of the primary visual cortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Retinotopic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Receptive fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Feature maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.4 Cortical circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Development of ocular dominance columns . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Refinement of columns during the critical period . . . . . . . . . . . . . . . . 172.2.2 Early development of OD columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.3 Development of CO blobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 The Hebbian synapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Excitatory synapses and NMDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Neurotrophins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3. MODELING CORTICALDEVELOPMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 LGN afferents terminating at a single neuron . . . . . . . . . . . . . . . . . . . . . . . 273.1.1 Swindale model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 Subtractive normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.3 Competition for neurotrophins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 ODC development on a cortical sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Subtractive normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.2 Inclusion of component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4. A DEVELOPMENTAL MODELOF OCULAR DOMINANCECOLUMN FORMATIONON A GROWINGCORTEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42


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4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Developmental model on a growing domain . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Linear stability analysis on a fixed domain . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.1 Stationary front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.2 Single stationary bump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.3 Periodic pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5 Correlationbased Hebbian model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5. LAMINAR NETWORK MODEL FOR THEJOINT DEVELOPMENT OF OCULARDOMINANCE COLUMNS ANDCYTOCHROME OXIDASEBLOBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Developmental model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2.1 Laminar architecture of V1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2.2 Reduced twolayer cortical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.3 Mathematical formulation of model . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3 Development of OD columnsand CO blobs in layer 2/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.1 O(1) analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.2 O() analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6. A THEORY FOR THE ALIGNMENT OF CORTICAL FEATURE

MAPS DURING DEVELOPMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2.1 Developmental model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.2 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2.2.1 Homogeneous case ( = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.2.2.2 Inhomogeneous case ( > 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2.3 Commensurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3.1 Pinning in a onedimensional network . . . . . . . . . . . . . . . . . . . . . . . . 1046.3.2 Pinning in a twodimensional network . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

vii


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NOTATION AND SYMBOLS

Uniformly Used

r, (x) cortical position (in 1d)w feedforward weights to layer 4W upperbound of synaptic weights into layer 4V postsynaptic activityI feedforward inputJ recurrent weight function strength of recurrent weight function

conversion factor for membrane potential to output firing rateL operator corresponding to the convolution with 1

J (r)G intracortical interaction function the domain (i.e., cortex)C correlation matrixS spatial correlations of inputH(r) effective intracortical interaction function, H(r) = G(r)S(r)p,q wavenumbers used as an eigenvalue and sometimes as the growth factor for spatial modes general subtractive normalization terma vector of 1s

f Fourier transform of f space constant of the weight functionA amplitude of weight function coefficient of inhibition noise term measure of L/R control over all V1c perturbation of solution used in linearization


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Cortical Growth cortical growth termHt interaction function that varies with timeH effective interaction function on original domain, depends upon growth growth rate, taken to be small ratio of initial to adult length of cortexu flow of cortexd 2 standard width of an ODC 2 growth functionL 1d length of cortex function used for stability test an auxiliary function effective space constant of the weight function

after mapping to original domain

Laminar model for V1 development

k koniocellular weightsm 1 weight of interlaminar projectionK upperbound of synaptic weights for koniocellular inputM upperbound of weight for vertical interlaminar projectionB correlation matrix between K and M inputC correlation matrix for koniocellular input subtr. normal term in layer 2/3 dynamicsL,R degree of L/R eye control at a cortical point with [0, 1]L,R convolution over layer 4 of the weight function of the L/R eye density

Ki mexican hat, O(), component of theith interaction function (Ki(r) = Ji(r)S(r))

CO blobs intrinsically defined via molecular markers

strength of binocular stabilizing term in modified SwindaleWb L/R density during initial binocular stated 2 distance between blobsv patchy distribution of the CO marker strength of modulation due to the patchy distribution

W baseline level for the maximal density of feed forward afferentsQ reciprocal lattice vector 2 disorder of CO marker lattice degree of pinning patch size of molecular marker

1,2 sometimes used as a summation index, used differently in other chapters

ix


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ACKNOWLEDGEMENTS

I would like to thank Paul Bressloff for his excellent direction that helped bring this

work to completion. Additionally, the many conversations with Alessandra Angelucci and

Jenny Lund were immensely helpful. I would also like to thank James Keener (Jim) for

the exciting classes he taught during my earlier years at the University of Utah. Lastly,

I would like to thank the University of Utah and the National Science Foundation (RTG

0354259) for their generous support during my studies. Cheers.


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CHAPTER 1

INTRODUCTION

When studying the large-scale functional and anatomical structure of cortex, two

distinct questions naturally arise: (I) how did the structure develop? and (II) what forms

of spontaneous and stimulus-driven neural dynamics are generated by such a cortical

structure? It turns out that in both cases the Turing mechanism for spontaneous pattern

formation plays an important role. Turing originally considered the problem of how

animal coat patterns develop, suggesting that chemical markers in the skin comprise a

system of diffusion-coupled chemical reactions among substances called morphogens [164].

He showed that in a two-component reaction-diffusion system, a state of uniform chemical

concentration can undergo a diffusion-driven instability leading to the formation of a

spatially inhomogeneous state. Ever since the pioneering work of Turing on morphogenesis

[164], there has been a great deal of interest in spontaneous pattern formation in physical

and biological systems [46, 130]. In the neural context, Wilson and Cowan [179] proposed

a nonlocal version of Turings diffusiondriven mechanism, based on competition between

short-range excitation and longer-range inhibition. Here interactions are mediated, not

by molecular diffusion, but by long-range axonal connections. Since then this neural

version of the Turing instability has been applied to a number of problems concerning the

dynamics and development of cortex. Examples in visual neuroscience include the Marr

Poggio model of stereopsis [120], developmental models of retinotopic, ocular dominance

and isoorientation maps [177, 155, 156, 158, 124], and cortical models of geometric visual

hallucinations [53, 24].

In this thesis we focus on the role of pattern formation in models of the activitydriven

development of primary visual cortex. We begin by briefly outlining how this fits in with

other stages of neural development. During embryogenesis, a thin sheet from the ectoderm

on the dorsal surface becomes specified as neural tissue. From this initial substrate, called

the neural plate, the nervous system develops. Through a combination of cell movement,

changes in cell shape, and differential cell adhesion, the lateral edges of the neural plate


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elevate and later fuse to form a hollow tube, called the neural tube. During the neural

tubes formation, a portion from the edges of the neural plate becomes pinched off to form

the neural crest. Attractive and repulsive cues guide these cells away from the forming

neural tube and act as the seeds to the peripheral nervous system. Other moleculesdiffusing from the middle layer of the embryo create molecular gradients that migrating

cells can use to navigate and modify gene expression. The caudal portion of the tube

is to become the spinal cord, whereas the rostral portion goes on to divide to form two

telencephalic vesicles and a diencephalon vesicle, upon which two evaginations develop

that form the optic vesicles that later become cups along the inner walls of which the

retinas form.

Retinal cells then form axons that travel through the optic tract guided by attractive

and repulsive molecular concentration gradients. In general, molecular gradients play akey role in the navigation of the axonal projections to their destination cortical areas.

The projections from the retinas travel to a portion of the thalamus and are then

forwarded on to the primary visual cortex (V1). The mapping from the retina to the

cortex, called the retinotopic map, preserves the the topography of the visual field. In

1947, Roger Sperry [153] demonstrated in the frog that molecular cues in the form of

chemical gradients play a role in the development of the retinotopic map, i.e., axon-target

recognition relied on chemical matching. However, the initial termination of the axonal

projections to V1 is wide and disperse yet has some general order owing to arrangementvia molecular gradients within V1. The activity within the system is involved in a process

that selectively prunes and refines connections so as to create effective connections. The

mapping of the visual field to the retinas onto the visual cortex is a prototypical example

of the ubiquitous cortical maps in cortex. For example, in the somatosensory system in

rodent, the twodimensional array of whiskers on the snout projects through midbrain

nuclei to form a somatotopic map in the somatosensory cortex [140], called the barrel

field because of its honeycomb-like structure that resembles an array of barrels. Similar

to the retinotopic map, the somatotopic map preserves the spatial arrangement of the

whisker array and encodes for the frequency and force of stimulation.

Not only are neurons in mature V1 selective to stimuli from specific regions in the

visual fields, they also respond preferentially to a variety of features associated with

visual stimuli, including orientation, ocular dominance, spatial frequency, and direction

selectivity. As one progresses tangentially across V1, the response properties vary in a


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nearly continuous fashion. Each feature has a corresponding selectivity map, that is,

across the cortex the selectivity or preference for a particular feature varies. Since there

are multiple feature maps across V1, these feature maps overlay one another in an intricate

manner, discussed further in Chapter 2. Furthermore, neurons through the cortical layershave similar response properties, so that visual cortex is thought to be organized in

a columnar fashion. However, this conjectured homogeneity is an oversimplification,

as highlighted in Chapter 5. Hubel and Wiesel [82, 84] conjectured that the feature

preferences of cortical neurons are generated by the convergence of thalamic afferents on to

input layer 4, and then passed on to other layers through vertical interlaminar projections.

Along these lines, the formation of feature preference maps could be understood in terms

of the development of feedforward connections from thalamus to layer 4 as many models

for cortical development assume (see the reviews of [158, 165]).The most studied cortical map is that of ocular dominance (OD), characterized by

a significant influence of, say, left-eye over right-eye activation determining a neurons

response properties, that is, the feedforward connections originating from left-eye thala-

mic regions are stronger than the connections from the right-eye thalamic regions. The

segregation of ocular streams may play a crucial role in stereopsis, i.e., depth perception.

OD maps appear not to be predetermined and depend critically upon the driving activity

during early development. As such, the OD maps for different species take on a variety

of patterns from a stripe-like pattern in macaque monkey and humans where the leftand right eye drives are believed to be approximately balanced to a patchy pattern in

cats where the contralateral eye more powerfully drives V1 during early development.

OD maps from animals that have been monocularly deprived at young age, either via

strabismus, suturing, or enucleation, are qualitatively and quantitatively different from

OD maps from normally raised animals. This dependence on activity in the system

motivates a class of models for the development of cortical maps, i.e., activitydriven

developmental models (see the reviews of [158, 165]). In this work, we extend the

existing activitydriven developmental models of Swindale and Miller [155, 124] to include

biological substrates that could affect development, specifically, cortical growth, the

laminar structure of cortex, and molecular markers.

We begin this work by giving a review of the visual system, principally for macaque

monkey and cat, and its plasticity in Chapter 2. In the following Chapter, we review

prevalent models for OD formation and highlight a mathematical sleightofhand used


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in all of these models that has important consequences for the underlying pattern forming

process. Although considerable cortical growth occurs after the initial formation of the

OD pattern, previous models have neglected the possible effects this growth has on the

OD map. Experimentalists [138] have shown that the OD pattern in cat V1 does notmerely expand with growth, since the width of an OD stripe in adult cortex is similar in

size to that of a kitten. Motivated by work by Crampin et al. [44] on the reaction-diffusion

of pigment cells, postulated to determine stripe formation during the growth of marine

angelfish, in Chapter 4 we consider the effects of cortical growth on OD map formation.

The resulting OD map undergoes a series of pattern forming instabilities as the cortex

grows, leading to the insertion of additional OD stripes.

In Chapter 5, we address the ubiquitous assumption of homogeneity within cortical

columns. For instance, staining for cytochrome oxidase (CO) in the superficial layers2/3 of macaque monkey V1 results in a periodic distribution of blob-like formations.

These stains have been given a variety of names: CO blobs, CO patches, and CO puffs.

Fascinatingly, there is a strong association (particularly in macaque) with the array of CO

blobs and the centers of the OD stripes, suggesting a strong relationship between these

structures. We consider a multilayer, activitydependent model for the joint development

of ocular dominance columns and cytochrome oxidase blobs in primate V1. We include

both thalamic and interlaminar vertical projections and take both to be modifiable by

activity. By including this more detailed biologically motivated framework, we are able toobtain CO blob distributions, defined by direct thalamic input to the superficial layers,

consistent with experimental data that are aligned with the underlying OD pattern.

Finally, in Chapter 6, we take a single hybrid layer representation of V1 and consider

an alternate approach that assumes the existence of a periodic distribution (lattice) of

molecular markers. The molecular markers will go on to intrinsically define the CO blob

positions and weakly affects the developing OD pattern so that it aligns to the same

underlying lattice. The virtue of the laminar approach in Chapter 5, instead of the

aligning of the CO blobs and OD pattern to a periodic distribution of molecular marker

as in Chapter 6, is that the former more closely mirrors detectable biological substrates.


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CHAPTER 2

A SURVEY OF THE STRUCTURE AND

DEVELOPMENT OF VISUAL CORTEX

2.1 Structure of the primary visual cortex

The primary visual cortex (V1) is the first cortical area to receive visual information

from the retina (see Figure 2.1(a)). The output from the retina is conveyed by ganglion

cells whose axons form the optic nerve. The optic nerve conducts the output spike trains

of the retinal ganglion cells to the lateral geniculate nucleus (LGN) of the thalamus, which

acts as a relay station between retina and primary visual cortex. Prior to arriving at the

LGN, some ganglion cell axons cross the midline at the optic chiasm. This allows the

left and right sides of the visual fields from both eyes to be represented on the right and

left sides of the brain, respectively. Note that signals from the left and right eyes are

segregated in the LGN and in input layers of V1, as seen in Figure 2.1(b). This means

that the corresponding LGN and cortical neurons are monocular, in the sense that they

only respond to stimuli presented to one of the eyes but not the other (ocular dominance).

2.1.1 Retinotopic map

One of the striking features of the visual system is that the visual world is mapped

onto the cortical surface in a topographic manner. This means that neighboring points

in a visual image evoke activity in neighboring regions of visual cortex. Moreover, one

finds that the central region of the visual field has a larger representation in V1 than

the p eriphery, partly due to a nonuniform distribution of retinal ganglion cells. The

retinotopic map is defined as the coordinate transformation from points in the visualworld to locations on the cortical surface. In order to describe this map, we first need

to specify visual and cortical coordinate systems. Since objects located a fixed distance

from one eye lie on a sphere, we can introduce spherical coordinates with the north

pole of the sphere located at the fixation point, the image point that focuses onto the

fovea or center of the retina. In this system of coordinates, the latitude angle is called


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6

R

R

R

L

LL lateral

geniculatenucleusM

P {{

RL

KK

KK

KK

V1

(b)

V1

retina

opticchiasm

LGN

retina

LGN

(a)

Figure 2.1: Schematic of the wiring of the visual system. (a) Visual pathwaysfrom the retina through the lateral geniculate nucleus (LGN) of the thalamusto the primary visual cortex (V1). (b) A cartoon of the layers of LGN (inmacaque) and the LGN afferents to V1 terminating in a segregated fashion.We have labeled the magnocellular and parvocellular pathways and picturethe intercalated koniocellular pathway by magenta colored strips, discussedin section 5.2.1. The M, P, and K pathways in the macaque correspond toanalogous X, Y and W pathways in the cat visual system.

the eccentricity and the longitudinal angle measured from the horizontal meridian is

called the azimuth . In most experiments the image is on a flat screen such that, if

we ignore the curvature of the sphere, the pair (, ) approximately coincides with polar

coordinates on the screen. One can also represent p oints on the screen using Cartesiancoordinates (X, Y). In primary visual cortex the visual world is split in half with the

region 90o 90o represented on the left side of the brain, and the reflection of thisregion represented on the right side brain. Note that the eccentricity and Cartesian

coordinates (X, Y) are all based on measuring distance on the screen. However, it is

customary to divide these distances by the distance from the eye to the screen so that

they are specified in terms of angles. The structure of the retinotopic map in monkey is

shown in Figure 2.2, which was produced by imaging a radioactive tracer that was taken

up by active neurons while the monkey viewed a visual image consisting of concentriccircles and radial lines. The fovea is represented by the point F on the left hand side

of the cortex, and eccentricity increases to the right. Note that concentric circles are

approximately mapped to vertical lines and radial lines to horizontal lines.

Motivated by Figure 2.2, we assume that eccentricity is mapped onto the horizontal

coordinate x of the cortical sheet, and is mapped onto its y coordinate An approximate


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7

S

F H

3

2

1

1 cm

H

2

S

1

F

1

3

0 00

Figure 2.2: A deoxyglucose autoradiograph from the left side primary visualcortex of a macaque monkey brain. The radioactive trace displays the activitypattern evoked by the image shown to the left. Adapted from [160]

equation for the retinotopic map can then be obtained through specification of a quantity

known as the cortical magnification factor M(). This determines the distance across a

flattened sheet of cortex separating the activity evoked by two nearby image points. First

suppose that the two image points in question have eccentricities and + but thesame azimuthal coordinate . The corresponding distance on cortex is x = M() so

that

dx

d= M() (2.1)

Using experimental data such as shown in Figure 2.2 suggests that

M() =

0 + (2.2)

with 12 mm and 0 1o in macaque monkey. It follows that

x = ln(1 + /0) (2.3)

assuming x = 0 when = 0. Similarly, for two image points with the same eccentricity

but different azimuthal coordinates we find that


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8

dy

d=

180oM() (2.4)

and hence

y = a(0 + )1800

(2.5)

The minus sign appears because the visual field is inverted in cortex. For eccentricities

greater than 1o,

x ln(/0), y 180o

(2.6)

and the retinotopic map can be approximated by a complex logarithm [146]. That

is, introducing the complex representations Z = (/0)ei/180o

and z = x + iy thenz = log Z.

2.1.2 Receptive fields

Neurons in the retina, LGN and primary visual cortex respond to light stimuli in

restricted regions of the visual field called their classical receptive fields (RFs). Patterns

of illumination outside the RF of a given neuron cannot generate a response directly,

although they can significantly modulate responses to stimuli within the RF via long

range cortical interactions. The RF is divided into distinct ON and OFF regions. In an

ON (OFF) region illumination that is higher (lower) than the background light intensity

enhances firing. The spatial arrangement of these regions determines the selectivity of

the neuron to different stimuli.

The receptive fields of LGN cells are circular and are described as ON-center and

OFF-center. ON-center means that a cell responds to a light stimulus centered in the

receptive field with a darker region surrounding the contrasting disc of light in the center.

The higher the contrast between the center of the receptive field and its boundary, the

greater the response. An OFF-center receptive field has the roles of the light and dark

regions reversed; see Figure 2.3(a). Because of the circular shape of LGN receptive fields,

LGN cells are thus invariant to orientation. For example, consider an LGN cell with an

OFF-center receptive field. If a dark bar were placed across the center of the RF, the

neurons firing rate would increase. Keeping the bar centered in the RF yet changing its

orientation has no effect on the firing rate.


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V1 neurons, on the other hand, are generally orientation selective, i.e., a neuron will

only respond strongly to a bar in its receptive field if it is at a certain preferred orientation.

Orientation preference found in primary visual cortex is due to the fact that, unlike the

circular receptive fields for LGN cells, the receptive fields of V1 neurons are elongatedwith their principal axis at some orientation; see Figure 2.3(b). Thus, a neuron in V1 will

respond strongly to a bar at an orientation that matches the orientation of its receptive

field, whereas a bar at an oblique angle to the preferred orientation will elicit a reduced

response. Hubel and Wiesel [82, 84] made the observation that typical V1 receptive fields

could be constructed from the convergence of multiple feed forward inputs from LGN

cells with overlapping circular receptive fields onto a single V1 cell; see Figure 2.3(c). A

sinusoidal grating is a commonly used stimulus where both the orientation and spatial

frequency (spacing of the grating) can be varied. V1 receptive fields that are also selectivefor spatial frequency can be constructed from multiple LGN circular receptive fields.

2.1.3 Feature maps

In recent years much information has accumulated about the spatial distribution of

orientation selective cells in V1 [61]. Figure 2.4 shows a typical arrangement of such

cells, obtained via microelectrodes implanted in cat V1. The first panel shows how

orientation preferences rotate smoothly over the surface of V1, so that approximately

every 300m the same preference reappears, i.e., the distribution is periodic in the

orientation preference angle. The second panel shows the receptive fields of the cells,

-

--

--

---

-

--

--

+ +

++

Presynaptic cellsfrom the LGN

V1 simple cell

(a) (b) (c)

On-center Off-center

V1 RF

Figure 2.3: Typical receptive fields for LGN and V1 cells, (a) and (b)respectively. Note that LGN RFs are circularly symmetric, whereas V1 RFsare oriented and ovular. In (C), a sample construction of an elongated V1receptive field from LGN inputs [82].


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and how they change with V1 location. The third panel shows more clearly the rotation

of such fields with translation across V1. One also finds that cells with similar feature

preferences tend to arrange themselves in vertical columns so that to a first approximation

the layered structure of cortex can be ignored. For example, electrode track 1 in Figure2.4 is a vertical penetration of cortex that passes through a single column of cells with the

same orientation preference and ocular dominance. Thesituation regarding orientation

columns in macaque V1 is more complicated [115, 114]. For example, input layer 4 has

an additional sublaminar structure that reflects amongst other things the division of the

LGN afferents into parvocellular (P) and magnocellular (M) pathways (see Figures 2.1(b)

and 2.5). One finds that many cells in layer 4C are not orientation selective: orientation

preference emerges in a graded fashion as one moves to mid and upper layer 4 C. The M

1 2 3

y

x

2

y

x

1 3

Figure 2.4: Orientation tuned cells in layers of cat V1 which is shown incross-section. Note the constancy of orientation preference at each corticallocation [electrode tracks 1 and 3], and the rotation of orientation preferenceas cortical location changes [electrode track 2]. Redrawn from [61].


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PM

M2

P

M1

Figure 2.5: The parvocellular (P) and magnocellular (M) pathways ofmacaque V1. The P pathway innervates layer 4C, consisting of cells withsmaller receptive fields and slower responses. These send axons to layer 4A,which feeds into the CO blob regions of superficial layers 2/3. The M pathwayinnervates layer 4C, consisting of cells that have larger receptive fields andfaster responses. Upper 4C cells connect to layer 4B which itself connects tothe blob regions of layers 2/3. Midlayer 4C has a mixture of M and P neurons

and connects to the interblob regions of layers 2/3.

pathway is thought to contribute primarily to motion perception, whereas the P pathway

contributes primarily to form and color perception. However, there is some mixing of

the two pathways. Inhomogeneities in the laminar structure of cortex will be considered

further in Chapter 5.

A more complete picture of the twodimensional distribution of both orientation

preference and ocular dominance in layers 2/3 has been obtained using optical imaging

techniques [14, 18, 12]. The basic experimental procedure involves shining light directly

onto the surface of the cortex. The degree of light absorption within each patch of cortex

depends on the local level of activity. Thus, when an oriented image is presented across

a large part of the visual field, the regions of cortex that are particularly sensitive to

that stimulus will be differentiated. In the case of macaque V1, the topography revealed


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by these methods has a number of characteristic features [133], which are illustrated

in Figure 2.6: (i) Orientation preference changes continuously as a function of cortical

location, except at singularities or pinwheels. (ii) There exist linear zones, approximately

750 750 m2

in area, bounded by pinwheels, within which isoorientation regions formparallel slabs. (iii) Linear zones tend to cross the borders of ocular dominance stripes at

right angles; pinwheels tend to align with the centers of ocular dominance stripes.

Another important example of nonuniformity through the layers is the occurrence

of cytochrome oxidase (CO) blobs in superficial layers of primate and cat V1 [79, 76,

128]. These are regions of higher metabolic activity that receive a distinct class of direct

thalamic inputs [36, 71], with the density of CO staining being highly correlated with

the density of the thalamic afferents [107]. The spatial distribution of CO blobs within

cortex is also correlated with a number of stimulus feature preferences. For example, inold world monkeys such as macaques the blobs are found at evenly spaced intervals along

the center of OD columns [76], and neurons within the blobs tend to be less binocular

and less orientation selective [108]. The latter is probably due to their association with

orientation pinwheels; see Figure 2.6. The blobs are also linked with low spatial frequency

domains [159]. The arrangement of CO blobs is reflected anatomically by the distribution

of intrinsic horizontal connections (see section 2.3), which tend to link blobs-toblobs

and interblobstointerblobs [109, 183, 182], and by extrinsic corticocortical connections

Figure 2.6: Map of iso-orientation contours (yellow lines), ocular dominanceboundaries (dark gray lines) and CO blob regions (shaded areas) of macaqueV1. Adapted from [12].


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linking blobs to specific compartments in V2 and other extrastriate areas [109, 150].

Taken together these observations suggest that the CO blobs are sites of functionally and

anatomically distinct channels of visual processing. Some of the functional properties

of CO blobs, such as association with low spatial frequencies, persist in species otherthan old world monkeys. However, the spatial relationship b etween CO blobs and OD

columns is often less clear. For example, OD columns are less regular in cats compared

to macaques, and the periodicity of the CO blobs (around 1mm) appears to be too large

to provide sufficient coverage of CO blobs across all OD columns. Nevertheless, Murphy

et al. [128] found that CO blobs are more numerous near the centers of OD columns,

with nearby blobs tending to merge across OD borders; see Figure 2.7(a). The spatial

relationship between CO blobs and OD columns is weak or lacking in new world primates

such as the squirrel monkey [77] (pictured in Figure 2.7(b)), which also exhibit less strongOD segregation compared with their old world counterparts.

(a) (b)

1 mm

Figure 2.7: OD patterns and CO distributions in other mammalian systems.(a) Cat OD map with the centers of the CO blobs represented by gray circlessuperimposed over the OD pattern. The scale bar is 1mm. Adapted from[128]. In (b), OD segregation in squirrel monkey with superposition of theCO patches outlined. Adapted from [77].


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2.1.4 Cortical circuits

It turns out that the majority of synapses onto a cortical neuron arise from other

cortical cells rather than from feedforward LGN inputs. These include intralaminar con-

nections, interlaminar connections and feedback connections from higher cortical areas.There are at least two distinct classes of intralaminar connection. First, there is a local

circuit operating at submillimeter dimensions in which cells make connections with most

of their neighbors in a roughly isotropic fashion. It has been suggested that such circuitry

provides a substrate for the recurrent amplification and sharpening of the tuned response

of cells to local visual stimuli [151, 10]. The other circuit connects cells separated by

several millimeters of cortical tissue. The axons of these connections make terminal arbors

only every 0.7 mm or so along their tracks [139, 63], such that local populations of cells

are reciprocally connected in a patchy fashion to other cell populations. Optical imagingcombined with labeling techniques has generated considerable information concerning

the pattern of these connections in superficial layers of V1 [118, 183, 19], see Figure 2.8.

In particular, one finds that the patchy horizontal connections tend to link cells with

similar feature preferences. Stimulation of a neuron via lateral connections modulates

rather than initiates spiking activity [75, 161], suggesting that the long-range interactions

provide local cortical processes with contextual information about the global nature of

stimuli. As a consequence the horizontal connections have been invoked to explain a wide

variety of context-dependent visual processing phenomena [62, 58].One of the most conspicuous features of cortical circuits is their laminar organization.

Neurons within a layer send projections to only a subset of the other cortical layers with

a high degree of accuracy. Additionally, it is believed that each cortical layer provides its

primary output to just one other layer [32]. This assumption allows us to simplify the

circuitry into subunits (see Chapter 5). Additionally, the response properties of neurons

may vary through the layers due to a multitude of possible inputs to a particular cortical

circuit, which could consist of inputs due to interlaminar projections, projections from

other cortical areas, or from intralaminar connections. In Chapter 5, we provide a detailed

review of the properties of the layer specific thalamic drive to V1 as well as outline the

interlaminar circuits.

Finally, there are extensive feedforward and feedback connections linking V1 to ex-

trastriate areas such as V2, V3 and MT [143, 26]. As one proceeds to higher cortical areas

the receptive field size of neurons increases. Hence, feedback from these areas provides


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(a)

(d)(c)

(b)

Figure 2.8: Camera-lucida drawing of biocytin injection into layer 3 at blob(a) and interblob (b) regions adapted from [183]. The biocytin patches areoutlined in the drawings (a) and (b). Corresponding representations of (a)and (b) are given in (c) and (d), respectively. In (c,d), the mocha areas arethe injection sites at blob/interblob regions, respectively, and the gray areasare the biocytin patches. The CO blobs are outlined in dashed lines. Wecan see that interblob regions connect to other interblob regions (12 out of17) and in general that blobs connect to blobs (8 out of 12). The scale barsrepresent 200 m and the arrows point to blood vessels, which are used as

fiduciary landmarks.


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information to V1 neurons from a much larger region of visual space than expected

from the classical feedforward receptive fields. It is also important to note that the

feedforward and feedback connections are layer specific and reciprocal. Ongoing studies

of feedback connections from points in extrastriate areas back to area V1 [5, 6], show thatthe feedback connectional fields are also distributed in highly regular geometric patterns,

having a topographic spread of up to 13mm that is significantly larger than the spread of

intrinsic lateral connections. It is likely that the patchiness again signifies that feedback

correlates cells with similar feature preferences [148].

2.2 Development of ocular dominance columns

The existence of a set of overlapping cortical feature maps raises a number of inter-

esting questions. What are the anatomical substrates for cortical feature maps and howdo they develop, to what extent are these maps genetically predetermined, and what role

does neural activity play? Since our work is mainly concerned with the development

of ocular dominance columns (and the joint development of CO blobs), we focus our

discussion on this particular feature.

Recall that ocular dominance is characterized by one eye predominantly driving a

section of cortex. This, in turn, is intimately related to the segregation of the left and right

LGN afferents terminating in layer 4 of the primary visual cortex. Thus, understanding

the development of LGN afferent termination sites in V1 is key to the understanding of

how OD columns develop. Hubel and Wiesel [175], who first detected ocular dominance

columns using electro-physiological recordings and transneuronal tracers, theorized that

molecular cues or genetic markers predetermine the initial wiring of LGN afferents and,

consequently, the arrangement of ocular dominance columns in early development [83].

They further postulated that neural activity subsequently refines the ocular dominance

pattern during a critical period later in development. More recently, experimental ev-

idence has come to light that seems to suggest that activity-dependent plasticity may

still play a role in the initial development of OD columns, e.g., the existence of prenatal

retinal waves [117] and recurrent thalamocortical activity [47] could both be underlying

mechanisms for activitydependent development (reviewed in [28]).

The idea that neural activity fashions the OD map draws upon a seminal conjecture

of Donald Hebb regarding learning and synaptic plasticity [70]: When an axon of cell A

is near enough to excite cell B or repeatedly or persistently takes part in firing it, some


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growth process or metabolic change takes place in one or both cells such that As efficiency,

as one of the cells firing B, is increased. A more modern interpretation of this postulate

is that synaptic modification is driven by correlations in the firing activity of presynaptic

and postsynaptic neurons. As stated, Hebbs rule is unstable and would result in allthe synaptic weights saturating: thus some sort of normalization constraint is required in

order to maintain stability, i.e., a modified Hebbian rule is one in which the strengthening

of one synapse comes at a cost to others. A Hebbianlike competition b etween LGN

afferents is the backbone of the conjecture that activity-dependent plasticity plays the

main role in the development of V1, a view contrary to a genetically predetermined

structure of V1 as Hubel and Wiesel proposed. In this section, we present a brief history

of the discovery of ocular dominance columns and arguments that support and challenge

the above hypotheses regarding genetic versus activitybased mechanisms.

2.2.1 Refinement of columns during the critical period

In the late 1970s, Hubel and Wiesel performed a series of experiments examining

the cellular mechanism by which patterned visual stimulation affects the development of

visual perception [86, 87, 103, 104, 81] during a critical period in later development. They

examined both kittens and monkeys by making electro-physiological recordings of neural

responses in V1 elicited from stimuli presented to one or both eyes. They found that

neurons in V1 responded almost exclusively to input from a single eye, in other words,

most neurons are effectively monocularly driven. Nearby neurons in cortex were typically

driven by the same eye, yielding patterns of left and right eye driven regions of cortex,

ocular dominance patterns. In a key experiment [86], Hubel and Wiesel sutured shut

an eye of an infant monkey, and hence eliminated much of the retinal activity, referred

to as monocular deprivation (MD). After 6 months, the stitches were removed, and the

previously sutured eye had been rendered blind. In a normal animal the ocular dominance

stripes are of equal width, but in the monkey with one eye sutured the width of the stripes

had changed: the stripes driven by the sutured eye were dramatically thinner than the

remaining eye (see Figure 2.9). In binocular deprivation, on the other hand, the resulting

ocular dominance pattern closely resembled the normal case. This suggested some sort

of activitydriven competition between the left- and right-eye afferents for termination

sites in V1 during the critical period.


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(a) (b)

Figure 2.9: Activity or retinal drive during the critical period greatly altersthe final architecture of V1. In (a), the OD pattern in a normally rearedmonkey with OD stripes of approximately equal width. In (b), the OD patternfrom a monocularly deprived monkey that displays a stripe-like pattern, butthe stripes corresponding to the remaining open eye (white) are significantlywider, [86].

2.2.2 Early development of OD columns

LeVay et al. [103] used transneuronal tracers in multiple stages of development in

order to ascertain how OD patterns develop in cat. The technique of tracking transneu-

ronal transport of tritiated amino acids begins with the injection of a tracer into the

retina. The tracer then proceeds through the retinal ganglia to the LGN and continues

through the LGN afferents to terminate in layer 4C of V1. Experimenters use this

technique to visualize the ocular dominance columns. LeVay et al. found that beforethe onset of the critical period, the injected tracers stained continuous rather than

alternating bands of cortex, implying binocular input to V1 cells (see Figure 2.10).

This calls into question Hubel and Wiesels conjecture of an innate architecture early in

development. Sequential experiments in time unveiled alternating band patterns emerging

at the onset of the critical period and b ecoming more distinct as the animal aged. This

suggests that binocularity is the initial state of V1 and that, in time, activity-dependent,

competition-based plasticity rules develop the OD pattern, i.e., afferent arbors initially

overlap and are selectively pruned or strengthened during the critical period. However, itmay still be the case that there is an initial bias in the L/R thalamocortical connections

as depicted in Figure 2.11. In subsequent experiments, LeVay et al. [104] discovered that

leakage or spillover of the tracer occurs between the LGN layers of young animals (more

severely in younger animals), suggesting that perhaps spillover in the tracing process was

the cause of the continuous band, rather than the absence of a genetically predetermined


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2 weeks

3 weeks

5.5 weeks

13 weeks

1 mm

Figure 2.10: Audiographs of four stages in the development of the visualcortex in cat show the postnatal development of ocular dominance columns.Initially V1 appears binocularly driven and through the critical period theOD bands become distinct [103].

segregation of LGN afferents. So there may in fact exist an initial segregation of LGN

afferents, but due to complications in labeling it is not detectable. Nevertheless, LeVay

et al. concluded that in the cat the two sets of afferents were intermixed initially since

spillover would not have been sufficient to mask a columnar pattern had it been present

[104].

The time period for the initial formation of ocular dominance columns varies from

species to species. In macaque monkeys, connections from LGN to the primary visual

cortex begin to segregate into stripes prenatally. The finding of OD columns in prenatalmacaque monkeys supports the argument for genetically-driven development. However,

local correlations in the firing of retinal ganglion cells have been found to exist in dark-

reared mammals [121]. Furthermore, other studies found prenatal, spontaneously gener-

ated, correlated patterns of activity from the retina, or retinal waves, independently

generated from each eye [117]. These waves could, in theory, drive an activity-dependent,

competition-based developmental process for prenatal pruning of LGN afferents.

In cats, the formation of OD columns was originally thought to occur at the beginning

of their critical period, postnatal day 21 (P21) [103]. Stryker and Harris [154] performedpivotal experiments on cats where they made binocular injections of tetrodotoxin (TTX)

to block all forms of retinal activity from P14 to P45, in order to test the hypothesis

that the segregation of overlapping afferents requires activity to induce competition. At

P45, there was no evidence of OD columns [154]. The label in layer 4 of an animal

at P45 was continuous, prompting Stryker and Harris to conclude that blocking retinal


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Birth

2 wk

3 wk

6 wk

Figure 2.11: Assuming that molecular cues bias the afferents into overlapping,alternating bands of left and right control, throughout the critical periodactivity drives the further segregation of inputs to a final state of alternatingmonocularly driven bands of cortex. Timeline given as speculated in cat.Adapted from [91].


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activity prevented activity-driven competition that should have segregated the afferents.

Though the natural conclusion at the time, subsequent experiments by Crair et al. [42],

using more advanced optical and single-unit recording techniques, provided evidence

that OD columns are present at postnatal day 14, a week before the onset of the catscritical period. Thus an alternate conclusion to Stryker and Harris experiments is that

geniculocortical afferents were already segregated in P14 and that TTX desegregated the

existing OD columns.

More support for the claim that OD columns are present before P21 is derived from

comparisons to the ferrets developmental process. Ferret and cat development takes place

on a similar time course [89], with ferrets having a 21-day lag behind cat with respect to

birth dates. This makes ferrets good candidates for developmental studies because more

of the developmental process can be observed. Using transneuronal tracing, OD columnswere found in ferrets at P37 (2 days after critical period onset) [57, 141] and at P14 in

cat (a week before critical period onset) [47]. However using methods of direct injection,

ferret thalamus is found to have segregated LGN columns at P16, implying cats have

them 5 days before birth [47].

2.2.3 Development of CO blobs

In macaque both CO blobs and OD columns emerge prenatally, so that at birth the

pattern of OD columns and their spatial relationship with blobs is adult-like. CO blobs

in cat are normally first visible around 2 weeks of age [126], which is approximately

coincident with the earliest observation of OD columns in cat [43]. Thus it is possible

that CO blobs and OD columns develop at about the same time and thus interact with

each other. In contrast to OD columns, however, CO blobs are not significantly altered

by visual experience during the critical period. Modifying visual experience by either

dark-rearing, monocularly or binocularly depriving a kitten has little or no effect on the

cytochrome oxidase blob lattice [126]. In primates, further experiments by Murphy et

al. [127] found no significant difference in the spacing of CO blobs between normal and

strabismic macaque monkeys, monkeys that have one eye that cannot focus on an object

because of an imbalance of the eye muscles. The independence of the CO blob lattice to

visual experience could be interpreted in at least two ways:

1. The CO blob lattice is defined intrinsically via molecular markers and acts as a

roadmap to the developing visual cortex


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2. The CO blob lattice is not predefined but there is a mechanism which constricts

the blobs to the center of the OD regions. MD may change the width of the OD

regions but the centers of the OD regions remain unchanged.

We consider the first interpretation in Chapter 6 and the second in Chapter 5.

2.3 The Hebbian synapse

As we have already mentioned, the activity-driven development of OD columns in-

volves competition between left and right eye LGN afferents. A possible cellular substrate

for such competition is a set of synapses modified according to a form of Hebbs rule. Here

we briefly review some of the biophysical evidence for a Hebbian synapse and its role in

cortical development.

2.3.1 Excitatory synapses and NMDA

The basic stages of synaptic processing induced by the arrival of an action potential

at an axon terminal are shown in Figure 2.12. (See [29] for a more detailed description).

An action potential arriving at the terminal of a presynaptic axon causes voltage-gated

Ca2+ channels within an active zone to open. The influx of Ca2+ produces a high con-

centration of Ca2+ near the active zone [60, 11], which in turn causes vesicles containing

neurotransmitter to fuse with the presynaptic cell membrane and release their contents

into the synaptic cleft (a process known as exocytosis). The released neurotransmittermolecules then diffuse across the synaptic cleft and bind to specific receptors on the

postsynaptic membrane. These receptors cause ion channels to open, thereby changing

the membrane conductance and membrane potential of the postsynaptic cell.

The predominant fast, excitatory neurotransmitter of the vertebrate central nervous

system is the amino acid glutamate, whereas in the peripheral nervous system it is acetyl-

choline. Glutamate-sensitive receptors in the postsynaptic membrane can be subdivided

into two major types, namely, NMDA and AMPA [29]. At an AMPA receptor the

postsynaptic channels open very rapidly. The resulting increase in conductance peakswithin a few hundred microseconds, with an exponential decay of around 1 msec. In

contrast to an AMPA receptor, the NMDA receptor operates about 10 times slower

and the amplitude of the conductance change depends on the postsynaptic membrane

potential. If the postsynaptic potential is at rest and glutamate is bound to the NMDA

receptor then the channel opens but it is physically obstructed by Mg2+ ions (see Figure


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Action potential innerve terminalopens Ca channels

Ca entry causesvesicle fusion andtransmitter release

Transmitter

Receptor-channels open,Na enters the postsynapticcell and vesicles recycle

Presynaptic

nerveterminal

Receptor-channel

2+

2+

Ca2+

PostsynapticcellNa

+

+

Na+

Na+

Figure 2.12: Schematic of synaptic transmission. Adapted from [91].

2.13). As the membrane is depolarized, the Mg2+ ions move out and the channel becomes

permeable to Na+ and Ca2+ ions. The NMDA receptor is thus well placed to act as a

detector of correlations in the firing activity of presynaptic and postsynaptic neurons,

which forms the basic principle of a Hebbian synapse. Indeed, the rapid influx of calcium

ions due to the opening NMDA channels is thought to be the critical trigger for the onset

of long term potentiation (LTP) and long term depression (LTD), two major components

of bidirectional synaptic plasticity [16, 48, 15, 106, 119]. LTP is a p ersistent increase in

synaptic efficacy produced by high-frequency stimulation of presynaptic afferents or by the

pairing of low frequency presynaptic stimulation with robust postsynaptic depolarization.

LTD is a long-lasting decrease in synaptic strength induced by low-frequency stimulation

of presynaptic afferents.

The existence of LTP and LTD has been well documented in the visual cortex (e.g.,

see the review of Malenka and Bear [119]) and that the NMDA receptors play a key

role. One of the most striking demonstrations of the role of the NMDA receptors in the

development of the visual system occurs in the frog. We compare cases where the NMDA

receptors are blocked or overly activated to the control case. In normal development

of the frog retinotectal map, the L/R eye afferents pass through the optic chiasm and

terminate in the R/L hemispheres of the tectum, which is the analog of V1. However,

with the transplant of an additional right eye, the two right eye afferents traverse the


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Resting membranepotential

NMDAR AMPAR

Mg2+

Na+

Na+

Ca2+

Glu

Depolarized membranepotential

NMDAR AMPAR

Mg2+

Na+

Na+

Ca2+

Glu

Figure 2.13: Schematic of NMDA receptor.

optic track to the left hemisphere tectum. The afferents then compete for termination

sites in the tectum resulting in a stripe-like ocular dominance pattern, as pictured in

Figure 2.14(a). Constantine-Paton et al. (reviewed in [39]) studied the role of the NMDA-

type glutamate receptors by coadministration of the NMDA channel blocker MK801, i.e.,

effectively turning off the NMDA receptors. With the NMDA receptors blocked, the two

right-eye pathways fail to segregate, as seen in Figure 2.14(c). On the other hand, an

application of NMDA sharpens the segregation [39] as shown in Figure 2.14(d) compared

to the untreated tectum at the same magnification shown in Figure 2.14(b).

2.3.2 Neurotrophins

Neurotrophins are proteins that initiate essential biological functions required for

the maintenance and/or strengthening of neural connections. Included in the class of

neurotrophins are nerve growth factor (NGF), brain-derived neurotrophic factor (BDNF),

neurotrophin-3 (NT-3), and NT-4/5. For example, a sufficient amount of neurotrophic

factor is required for sustaining axonal arbors and determines the degree of the arboriza-

tions (e.g., [40]). Interestingly, neurotrophins also affect synaptic weights. Thus, it

has been postulated that the regulation of the neurotrophins and their receptors could

be a mechanism involved in plasticity. There is an array of examples from multiple


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(a) (b)

(c) (d)

Figure 2.14: Demonstration of the effects of NMDA. In (a) we show anuntreated tectum and the resulting segregated pattern, whereas in (c) thetectum is treated with MK801, an NMDA antagonist. We note that theretinotectal projections are disperse and without order. In (b) is a moremagnified view of the untreated tectum. In (d) we show the effects of anNMDA agonist and note the sharpness of the pattern when contrasted with(b). Scale bars in (a,c) are 200 m and 100 m in (b,d). Adapted from [39].


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cortical areas that demonstrate the substantial role of neurotrophins on plasticity. In

the visual cortex, BDNF has been shown to facilitate LTP [3, 88] and to attenuate LTD

in layer 2/3 synapses of young adult rats [2, 88, 95, 100], suggesting that one role of

BDNF during development is to modulate the properties of synaptic plasticity, enhancingsynaptic strengthening and reducing synaptic weakening processes which contribute to

the formation of specific synaptic connections [88].

Interestingly, neuroelectrical activity regulates the expression of neurotrophins and

their receptors, i.e., the expression of neurotrophins is associated with active synapses.

For instance, the regulation by BDNF of dendritic arborization requires both neuronal

activity and the Ca2+ influx due to Mg2+ NMDA receptors [122]. The interdependence

of the expression of neurotrophic factor and postsynaptic activity makes it a leading

candidate as a principal mechanism for activity-dependent synaptic plasticity [122].It has been suggested that neurotrophins could be thought of as a resource for which

postsynaptic connections compete (e.g., see [116] and the review [165] on modeling).

Specifically, neurotrophins play a role in activity-dependent synaptic modification in the

development of ocular dominance columns [116, 30, 80]. For example, the continuous

infusion of the neurotrophins NT 4/5 or BDNF to V1 prevents the formation of OD

columns in cat [30], presumably because the LGN axon branches fail to retract, yet the

application of a NT antagonist prevents OD column segregation by eliminating inputs to

both eyes [31]. In monocular deprivation experiments on rats, Maffei et al. showed thatwith additional NGF injected, the OD distribution fails to produce shifts towards open-eye

dominance[116], i.e., the OD distribution remains approximately the same, suggesting

that with an abundance of neurotrophic factor, the L/R neural afferents need not compete

for resource. These results support the notion that neurotrophins are a substrate for which

postsynaptic connections compete, yet may also, because of their regulation by acivity,

be involved in a type of Hebbian plasticity.


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CHAPTER 3

MODELING CORTICAL

DEVELOPMENT

3.1 LGN afferents terminating at a single neuron

Consider a single cortical neuron that receives two inputs, IL and IR, with correspond-

ing synaptic weights wL and wR. These could represent, for example, LGN afferents from

the left and right eyes. Suppose that the membrane potential of the cell evolves according

to the linear equation

vdV

dt= V + w I = V + wLIL + wRIR, (3.1)

where w = (wL, wR)T, I = (IL, IR)

T, and v is a membrane time constant. Assuming

that IL and IR are in units of1V , then wL and wR have the same units as the membrane

potential V. Since development takes place on a much slower time scale than the dynamics

of the feedforward input, we take V to be at steady-state. Thus,

V = wLIL + wRIR. (3.2)

Recall Donald Hebbs conjecture that if input from neuron A often contributes to the

firing of neuron B, then the synapse connecting neuron A to B should be strengthened

[70], i.e., if neuron j drives neuron B to fire then wj , the synaptic weight between j and

B should increase. These dynamics are described by

wd

dt wLwR

= V

ILIR

. (3.3)

where w determines the timescale of the weight dynamics with w V. Equation(3.3) is the mathematical representation of the basic Hebb rule. Note that activity of

the postsynaptic cell is specified by the membrane potential V, which is consistent with

the basic operation of the NMDA receptor (see section 2.3). Because development takes

place on a slower time scale then the feedforward dynamics, it is the long term statistics


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of the activity pattern that matter. Therefore, we consider an averaged version of the

Hebb rule.dw

dt=< V I >, (3.4)

where < . > denotes the ensemble average over the distribution of inputs.

From equation (3.2), we can replace V with w I and obtaindw

dt= Cw =

< IL IL > < IR IL >< IL IR > < IR IR >

w, (3.5)

where C is the input correlation matrix with matrix elements given by Ci,j =< Ij Ii >.

Within the context of ODC formation, C represents the correlation of the activities within

an eye and between the eyes. Ocular dominance occurs when the synaptic connections

originating from one eye are driven to zero while the connections from the alternate

eye grow or strengthen. As it stands, the simple Hebb rule does not generate such

competition. Moreover, it is inherently unstable. We now describe some of the most

common modifications of Hebbs rule that remove these limitations.

3.1.1 Swindale model

Swindale rectified the problem of unbounded synaptic weights by introducing a logistic

term to the right hand side of equation (3.5) [155], i.e.,

dw

dt= Cw

F(w) (3.6)

where F(w) = w(W w) ensures that the left and right synaptic densities remain bothnonnegative and bounded above by some maximum density W.

One expects sameeye correlations CRR and CLL to be positive and stronger than the

oppositeeye correlations CRL and CLR. Swindale took the eyes to be anticorrelated, i.e.,

CLR, CRL < 0 in order to induce competition b etween the L/R eyes. The reversal in sign

of opposite eye interactions is supposed to reflect negative statistical correlations between

left and right eye inputs. However, the existence of negative correlations is difficult to

justify from a neurobiological perspective. The problem of negative correlations can

be avoided by using a linear Hebbian model with subtractive normalization (see below)

instead of the Swindale model [124]. It turns out that both models exhibit very similar

behavior and can be analyzed in almost an identical fashion.

As a further simplification, suppose that the total synaptic weight at a point in cortex

is constant with wL + wR = W. This condition implies dwR/dt = dwL/dt, which is


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guaranteed if the eyes are symmetrically anticorrelated, CRR = CLR and CLL = CRL.With this correlation structure, the correlation matrix C has a single positive eigenvalue

with corresponding eigenvector, e = (1, 1)T, and growth in this direction corresponds

to the segregation of the L and R pathways. When intracortical interactions are included,this model yields an OD column pattern consistent with experimental observations, see

section 3.2 and [155].

3.1.2 Subtractive normalization

Another approach is to assume that the total synaptic weight that connects to each

point in cortex remains conserved. We can accomplish this by subtracting an equal

amount from each synaptic weight according to the subtractive normalization scheme.

dwdt = Cw (w)a (3.7)

where a = (1, 1)T and (w) enforces the conservation constraint. Equation (3.7) is

supplemented with additional constraints that ensure that the weights remain positive

and bounded

0 wL,R W. (3.8)

For illustrative purposes, suppose that the left and right eye inputs are symmetric.

That is, we take CLL = CRR = CS and CLR = CRL = CD so that C is a positive

symmetric matrix. Note that unlike the Swindale model, no assumption of anticorrelatedeyes is needed. We then set

(w) = [wL + wR] . (3.9)

Exploiting the fact that the input correlation matrix C has eigenvalues = CS CDwith corresponding eigenvectors e = (1, 1), it is straightforward to show that thevector equation (3.7) decomposes into the pair of decoupled equations

wdw+(r, t)

dt= (CS+ CD 2)w+(t) (3.10)

wdw

(r, t)

dt = (CS CD)w(t) (3.11)

with w = wL wR. Conservation of total synaptic density is thus achieved by setting = +/2 so that dw+(t)/dt = 0 for all t. For a more general choice of correlation matrix

C, the subtractive normalization term takes the form

(w) =1

a a (a Cw) . (3.12)


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Note that the subtractive normalization must be modified when there are more than

two populations vying for termination sites. We discuss this further in Chapter 5. This

phenomenological model could be interpreted as a simple model for the left and right eye

afferents competing for a common pool of resources, neurotrophins, which are requiredfor the maintenance and strengthening of synapses.

3.1.3 Competition for neurotrophins

Instead of introducing a mathematical abstraction to maintain total synaptic strength

(as in subtractive normalization), Ermentrout and Osan [55] consider a single pool of

resources that synaptic populations require in order to maintain or strengthen synaptic

connections (akin to a previous more detailed approach in [65]), which could, for example,

correspond to a p ool of neurotrophic factor at the postsynaptic location, see section 2.3.2.

One assumes mass action kinetics between the resource and the amount of synaptic

weight,

fK+

Kw (3.13)

where f is the available pool of substance needed for producing lasting synaptic connec-

tions and w is the strength of a synapse. Suppose that the total amount of neurotrophic

factor is conserved (both bound and unbound). So that

f + w = W (3.14)

where in this context W denotes a constant corresponding to the total amount of resource,

an upperbound on synaptic weight. Without any loss of generality, we can let W = 1.

Now consider a pair of synaptic connections each having its own pool of resources.

Then the weight dynamics are given by

dwidt

= K+(1 wi) Kwi, i = L,R. (3.15)

The positive influence of trophic factor on LTP has been suggested by experiments

in the rat hippocampal system in knockout mice [98]. Additionally, in slices of rat visual

cortex, Korte et al. showed that a type of neurotrophin, BDNF, plays a modulatory

role in synaptic plasticity [99]. In simple terms, these experiments suggest that the more

trophic factor available, the faster its uptake and subsequent increases in connection

strength. Motivated by the experiments of Korte et al., we take the forward binding rate

to be a nonnegative monotonically increasing sigmoidal function that depends upon a


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Hebbian term, that is high pre- and post-synaptic activity leads to a strengthening of

cortical connections. The steepness of the binding rate could be related to the uptake of

neurotrophic factor by the presynaptic afferent (see [65] for a detailed description). Yet

since there is only a finite amount of resource, we assume an active form of decay wherethe backward binding rate, i.e., decay of synaptic weights, is a function of the averaged

postsynaptic activity and similarly represent it as a nonnegative monotonically increasing

sigmoidal function. Then the dynamics can be written as

dwidt

= K+

j=L,R

Ci,jwj

(1 wi) K j=L,R

wj

wi, i = L,R, (3.16)so that the forward binding rate has an LTP-like term and the backward binding rate

has an LTD-like behavior. Note that we are considering L/R inputs that are statistically

equivalent, so that when one averages over the developmental period, the L/R eyes have

the same amount of presynaptic drive. Thus, the total postsynaptic activity due to

L/R drive is proportional to the feedforward synaptic weights. If the synaptic weights

competed for the same pool, then 1 wL,R would be replaced by 1 wL wR, thoughconsidering the first choice does not lead to a substantial difference in the models

behavior.

A binocular steady state, i.e., wL = wR = w, exists provided the function g(w),

g(w) = K+(+w)(1

w)

K(2w)w, (3.17)

where + = CS+ CD, has a root in the interval (0,1). Since K are both nonnegative

functions, g(0) = K+(0) > 0 and g(1) = K(2) < 0. Thus by the intermediate valuetheorem, there exists w (0, 1) such that g(w) = 0. In order to study the stability of thisfixed point, we linearize equation (3.16) about the binocular fixed point, w, to obtain

dcLdt

= AccL + BccRdcRdt

= BccL + AccR (3.18)

where

Ac = CSK+

(+w)(1 w) K+

(+w) K

(2w) K

(2w)wBc = CDK

+(+w)(1 w) K(2w)w, (3.19)

and K

= dKdw . The system yields two eigenvectors, e = (1, 1)T, corresponding tosymmetric and antisymmetric growth (note antisymmetric growth is synonymous with

competition) with the symmetric eigenvalue

s = Ac+ Bc = (CS+ CD)K+(+w)(1 w) K+(+w) K(2w) K(2w)w (3.20)


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and the antisymmetric eigenvalue

a = Ac Bc = (CS CD)K+(+w)(1 w) K+(+w) K(2w). (3.21)

In order for the L and R pathways to segregate, we require s < 0 and a > 0. SincedK

dw > 0, there are two ways that this could occur: either CD < 0, which corresponds

to negative correlations between the eyes and is not a realistic assumption, or K(2w)

becomes very large. The physical meaning of the latter case is tied to a requirement for

strong activity-dependent synaptic depression. That is, when the postsynaptic activity

is high (equivalent to the feedforward weights being strong), the decay term increases

sharply resulting in a decrease in the total synaptic weight.

We consider as an example, the case CS > 0, CD = 1 CS > 0. Take the reaction

rates to be of the form

K+(u) =1

1 + e(u0.5)and K(u) =

1

1 + e(u/20.5). (3.22)

In this example, the binocular fixed point is located at w = 1/2. The steepness of

the forward and backward reaction rates is determined by , which we will treat as a

bifurcation parameter. Large corresponds to strong LTP with strong active decay of the

synaptic weights. From a near 0 value, as increases the binocular state loses stability at

a subcritical pitchfork bifurcation (plotted in Figure 3.1(a)), meaning that for large the

binocular state quickly destabilizes and goes to either a left or right eye dominated state

dependent upon the initial conditions. Since the bifurcation is a subcritical pitchfork,

there exists a bistable region where both the binocular and L/R eye dominated states are

stable. In order to study the size of the basins of attraction for the fixed points, we plot

nullclines and some sample trajectories in phase space (pictured in Figure 3.1(b)). We

find that only a narrow set of initial conditions around the origin result in a binocular

state. So while the binocular state in this parameter regime is stable, any significant

perturbation should result in a L or R eye dominated state.

3.2 ODC development on a cortical sheet

Extending the analysis of a single postsynaptic neuron with multiple inputs to a system

of postsynaptic neurons modeled continuously is relatively straightforward. Suppose we

have a two-dimensional cortical sheet. To model the interactions between neighboring

cells, we introduce a recurrent weight function J that depends upon cortical distance; we


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wL

1.0

0.75

0.25

0.5

05 10 15 20 25 30

(a) (b)

0 1.20.80.4

1.2

0.8

0.4

0

wR

wL

w =0L.

w =0R.

Figure 3.1: Stability properties of synaptic weights in a system where thereis competition for a pool of neurotrophic factor. (a) Bifurcation diagramwith CS = 0.8 and CD = 0.2 that demonstrates that increasing results in

a subcritical pitchfork bifurcation at 13.33. On the unstable branches,it undergoes fold bifurcations at values of 10.665. Hence, there exists anarrow parameter regime that exhibits bistability. Note that solid lines rep-resent stable solutions, whereas the dashed line represents unstable solutions.(b) Phase diagram for solutions in wL, wR space in the bistable regime with = 12.4. Stable fixed points are circled, whereas the unstable fixed points aredenoted by triangles. Additionally, we plot a few sample tra jectories, shownin blue, about the origin. Both (a) and (b) were obtained using XPP.

denote cortical position by r. The recurrent weight function is meant to represent the

lateral cortico-cortical interactions. As outlined in section 2.1.4, the lateral circuitry has

at least two spatial components: the local, submillimeter connections and the nonlocal

patchy connections that link cortical points several millimeters apart. In this work, we

consider only the local connectivity and assume it is an innate feature of V1, so that J

represents solely the effects due to the submillimeter interactions, which are postulated

to amplify and sharpen tuned responses [151] and does not vary with time or cortical

location. Further, assume that cortico-cortical interactions are weak. Thus, we introduce

a small parameter to scale recurrent connection strengths.

Analogous to equation (3.1), the activity of postsynaptic cells changes with time as

vdV

dt(r, t) = V(r, t) + I(r) w(r) +

J(r r)V (r, t)dr, (3.23)

where V, I, and w now vary with space and is a factor that converts membrane potential

to output firing rate, assuming these neurons operate in a linear regime.


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Since development takes place on a much slower timescale than the dynamics of

cortical activity, we can take V to be given by its steadystate value. However, calculating

the steadystate explicitly requires inverting the nonlocal linear operator

LV(r) = V(r)

J(rr )V (r)dr. In the case of weak intracortical interactions, this inversion can becarried out by performing a perturbation expansion in (see appendix). The firstorder

approximation is thus

V(r) =

G(r r) wL(r)IL(r) + wR(r)IR(r) dr (3.24)

with G(r r) (r r) +

J (r r) and is the Dirac delta function. Note that weexplicitly include the -function component of the inverted operator L1. Usually thisterm is ignored and G is treated as a smooth weight function [124, 158]. However, in

order to preserve the invertibility of L, it is necessary to include the -component and torestrict the strength of intracortical interactions as determined by (see appendix). If is

too large there does not exist a stable steadystate solution and the linear approximation

of correlationbased learning breaks down. As we show below, the -component can have

a significant effect on the weight dynamics. In particular, it leads to a sensitivity to initial

conditions, namely, the initial balance of left and right eye afferents.

Substitute equation (3.24) into equation (3.4) and assume that the input correlations

are of the form

IL(r)IL(r) IL(r)IR(r)IR(r)IL(r) IR(r)IR(r) = S(r r)C, C = CS CDCD CS (3.25)to obtain the subsequent averaged Hebb rule

wdw

dt=

G(r r)S(r r)Cwdr, (3.26)

where S represents the spatial correlations of inputs and is taken to be a Gaussian and

w = (wL, wR)T. Thus, we can incorporate S into G and express the averaged Hebb rule

as

wdw

dt = H(r r)Cwdr, (3.27)where H(r) = G(r)S(r).

The averaged Hebbs rule in a spatially extended system has the same problems as

its single cell representation, i.e., a lack of competition and being unbounded. As such,

we provide the derivation for subtractive normalization on a cortical sheet, but note that

the underlying principles hold for the other normalization schemes.


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3.2.1 Subtractive normalization

Introducing a subtractive normalization term to (3.27) leads to t

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Andrew M. Oster- Mathematical Models of Cortical Development