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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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2
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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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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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