PHYSICAL REVIEW E 82, 021920 共2010兲
Theory for the alignment of cortical feature maps during development
Paul C. Bressloff
Mathematical Institute, University of Oxford, 24-29 St. Giles, Oxford OX1 3LB, United Kingdom
and Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA
Andrew M. Oster
Group for Neural Theory, INSERM 960, Département d’Études Cognitives,
École Normale Supérieure, 29 Rue d’Ulm, 75005, Paris, France
and Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA
共Received 20 June 2010; revised manuscript received 3 August 2010; published 23 August 2010兲
We present a developmental model of ocular dominance column formation that takes into account the
existence of an array of intrinsically specified cytochrome oxidase blobs. We assume that there is some
molecular substrate for the blobs early in development, which generates a spatially periodic modulation of
experience-dependent plasticity. We determine the effects of such a modulation on a competitive Hebbian
mechanism for the modification of the feedforward afferents from the left and right eyes. We show how
alternating left and right eye dominated columns can develop, in which the blobs are aligned with the centers
of the ocular dominance columns and receive a greater density of feedforward connections, thus becoming
defined extrinsically. More generally, our results suggest that the presence of periodically distributed anatomical markers early in development could provide a mechanism for the alignment of cortical feature maps.
DOI: 10.1103/PhysRevE.82.021920
PACS number共s兲: 87.19.lp
I. INTRODUCTION
One of the most striking manifestations of the columnar
organization of primary visual cortex 共V1兲 are the cytochrome oxidase 共CO兲 blobs found in both primates 关1,2兴 and
cats 关3兴. These regions, which on average are about 0.2 mm
in diameter and 0.5 mm apart, coincide with cells that are
more metabolically active and hence richer in their levels of
CO. The spatial distribution of CO blobs within cortex is
correlated with a number of periodically repeating feature
maps, in which local populations of neurons respond preferentially to stimuli with particular properties such as orientation, spatial frequency, and ocular dominance 共OD兲. For example, in the macaque monkey the blobs are found at evenly
spaced intervals along the center of OD columns 关2兴, and
neurons within the blobs tend to be less binocular and less
orientation selective 关4兴. The blobs are also linked with low
spatial frequency domains 关5兴 and appear to be coincident
with singularities in the orientation preference map 关6兴. Similar functional relationships have been shown in cat except
that they tend to be weaker 关3,7兴. That is, although blobs
avoid OD borders in cat visual cortex, they are not strictly
aligned along the center of the columns. The arrangement of
CO blobs is also reflected anatomically by the distribution of
intrinsic horizontal connections, which tend to link blobs to
blobs and interblobs to interblobs 关4,8,9兴, and by extrinsic
corticocortical connections linking blobs to specific compartments in prestriate cortex 共V2兲 and other extrastriate areas
关4,10兴. Taken together these observations suggest that the CO
blobs are sites of functionally and anatomically distinct
channels of visual processing.
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 are adultlike. Moreover, this
spatial relationship and the spacing of CO blobs do not appear to be influenced by visual experience 关11,12兴. In con1539-3755/2010/82共2兲/021920共10兲
trast, the cat’s visual cortex is quite immature at birth. For
example, supragranular layers of cat V1 differentiate postnatally and the blobs in these layers are normally first visible
around 2 weeks of age 共about 1 week after eye opening and
1 week before the critical period兲. This is approximately coincident with the earliest observation of OD columns in cats
关13兴. However, in contrast to the OD columns, altering visual
experience by monocular or binocular deprivation or dark
rearing does not affect the pattern of blobs, suggesting that
visual experience is not necessary for the initial expression
or early development of CO blobs in cat 关14兴. Thus, the lack
of influence of visual experience on CO blob development in
both cats and primates suggests that the blobs may reflect an
innate columnar organization within superficial layers of cortex 关12,14兴. An intrinsic cortical specification of the blobs,
which is distinct from the competition-driven development
of cortical circuitry, could be generated by a periodic array of
molecular markers. In addition to cytochrome oxidase, there
are a number of other molecular markers that are arranged in
a patchy fashion during early development. These include
N-methyl-D-asparatic acid 共NMDA兲 receptors 关15兴, which
play a key role in experience-dependent plasticity, serotonin
receptors, and synaptic zinc 关16兴. Interestingly, the distributions of these latter markers are sensitive to visual experience, and these latter markers are periodically distributed in a
manner precisely complementary to cytochrome oxidase,
thus tending to be located at OD column borders.
Activity-based developmental models of ocular dominance columns in visual cortex typically involve some
Hebbian-like competitive mechanism for the modification of
left 共L兲 and right 共R兲 eye afferents 关17–19兴. Intrinsic intracortical interactions consisting of short-range excitation and
longer-range inhibition mediate a pattern forming instability
with respect to the feedforward synaptic weights, leading to
the formation of alternating left and right eye dominated columns. The formation of the columns is sensitive to changes
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©2010 The American Physical Society
PHYSICAL REVIEW E 82, 021920 共2010兲
PAUL C. BRESSLOFF AND ANDREW M. OSTER
in the degree of correlation among the afferents serving the
two eyes. The intracortical interactions are usually taken to
be homogeneous, so there is no mechanism for aligning the
columns with an array of CO blobs. In this paper, we show
how such an alignment can occur when there is a spatially
periodic modulation of experience-dependent plasticity by an
array of anatomical markers, which are assumed to provide
an intrinsic substrate for the CO blobs early in development.
The basic alignment mechanism involves mode-locking between spatial frequency components of the distribution of
CO blobs and the emerging pattern of OD columns under the
assumption of commensurability between the lateral interactions and the spacing of the molecular marker. We also show
that the blobs develop a higher density of feedforward connections, so that they ultimately become defined extrinsically, as they are manifested in their staining in layer II/III of
cortex. Although not considered in this work, such a spatially
distributed marker might play a role in the development of
patchy long-range connections within cortex that have a
spacing inline with the CO blobs. Such an intrinsic patchiness of the long-range horizontal circuitry could then affect
the developing formation of feature maps in V1 关20–22兴.
From a more general perspective, our model is an example of a hybrid model that combines both activityindependent and activity-dependent processes. Most developmental models keep these processes separate, either by
positing them as alternative explanations for the same phenomenon or by assuming that they contribute to different
stages of development. Indeed, a common assumption is that
activity-independent effects are responsible for setting up a
coarse-grained version of a given feature map, which is then
further refined by activity. Another example of a hybrid
model occurs in the theory of topographic map formation,
that is, the mapping of spatial position from eye to brain
关23,24兴. The molecular cues in this type of model are assumed to set up some sort of chemical gradient that provides
a label for spatial position 关25兴. This should be contrasted
with our model of ocular dominance formation, in which
molecular cues are distributed in a periodic fashion. Such a
distribution could itself arise from a diffusion-driven instability 关26兴.
II. DEVELOPMENTAL MODEL
There are a variety of activity-dependent development
models that generate ocular dominance columns and that can
account for various visual deprivation experiments 关19兴.
Since the basic alignment mechanism we propose is independent of the details of any particular model, we will focus on
one of the simplest of such models, namely, the Swindale
model of ocular dominance column formation 关17兴. This
treats input layer IV of the cortex as a two-dimensional sheet
of neural tissue and considers competition between the synaptic densities of feedforward afferents from the left and
right eyes. Certain care has to be taken when extending such
a model to incorporate CO blobs since the latter are found in
superficial layers II/III of cortex rather than the input layer.
In our simplified model, we collapse these into a single hybrid layer, motivated by the fact that there are strong vertical
connections between layers early in development 关27兴.
Let nL共r , t兲 and nR共r , t兲 denote the densities of left and
right eye synaptic connections to a point r = 共x , y兲 on the
cortex at time t. The feedforward weights of the left and right
eyes are taken to evolve according to a slightly modified
version of the competitive Hebbian learning rule introduced
by Swindale 关17兴:
␶
⳵ ni
= ␮共− ni + M兲FN共ni兲
⳵t
+ FN共ni兲
兺
j=R,L
冕
wij共兩r − r⬘兩兲n j共r⬘,t兲dr⬘ ,
共1兲
where i = L / R. The logistic function FN共n兲 = n共N − n兲 ensures
that growth terminates at a maximum weight N and that the
weights remain positive, that is, 0 ⱕ n ⱕ N. The term
␮共−n + M兲 for ␮ ⬎ 0 represents a process that tends to stabilize the uniform binocular state nL = nR = M for some constant
M, which is identified as the initial state prior to the onset
of activity-driven synaptic modification. The transition to
the activity-driven phase could be modeled as a reduction
in the parameter ␮. The lateral interaction functions
wLL , wLR , wRL , wRR are taken to be difference of Gaussians of the form
w共r兲 = Ae−r
2/2␴2
E
− Be−r
2/2␴2
I
,
共2兲
where ␴E,I are space constants with ␴E ⬍ ␴I. Same-eye interactions 共RR and LL兲 have A ⬎ B ⬎ 0 and opposite-eye interactions 共RL and LR兲 have A ⬍ B ⬍ 0. The lateral interaction
functions encompass the correlation structure of the presynaptic and postsynaptic activities as well as the horizontal
interlaminar cortical connections 共see the review in 关19兴兲.
Equation 共1兲 reduces to the Swindale model if ␮ = 0 and the
maximal density N is spatially uniform.
We now incorporate the effects of an intrinsically defined
array of CO blobs by taking N to be spatially correlated with
the locations of the blobs. This is motivated by the recent
experimental data suggesting that the CO blobs are initially
expressed in terms of some patchy anatomical marker that
can modulate synaptic plasticity 关12,14,15兴. For example,
such markers could regulate NMDA subunit expression,
which is thought to be an important mechanism for metaplasticity 关28兴. As mentioned above, the upper bound for thalamic input is nonuniform in adult animals when considering
a compressed laminar model of V1. The CO blob marker is
directly coupled with additional thalamic input that innervates the supragranular layers in cortex 共layers II/III兲,
whereas the majority of direct thalamic input innervates
layer IV which stains uniformly for CO. In contrast to this,
early in development Murphy et al. observed weak patchy
CO staining in layer IV of normal young kittens 共2–2.5
weeks postnatally兲 关14兴. Since CO blobs are associated with
higher levels of metabolic activity and direct thalamic input,
these findings suggest that the thalamic innervation becomes
spatially periodic early in development, before the emergence of OD columns. Thus, we postulate that the marker for
the CO blobs locally enhances experience-dependent plastic-
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THEORY FOR THE ALIGNMENT OF CORTICAL FEATURE…
ity, which in the Swindale model can be interpreted as a local
increase in N, that is, we assume a periodic modulation of the
upper bound of the synaptic weights.
Letting u共r兲 denote the patchy distribution of the CO
marker, we set
N共r兲 = N̄ + ␬u共r兲,
共3兲
where N̄ is a base line level for the maximal density of feedforward afferents and ␬ is a positive constant that determines
the strength of the modulation. The distribution u共r兲 has
maxima at the CO blob centers, which are located at the sites
ᐉ of a two-dimensional lattice L. In order to elucidate the
basic principles underlying our theory of cortical
alignment, we will take L to be a regular planar lattice
ᐉ = 共m1ᐉ1 + m2ᐉ2兲d for integers m1 , m2 with ᐉ1 , ᐉ2 as the
generators of the lattice and d as the lattice spacing. We will
subsequently show in Sec. III that the proposed alignment
mechanism is robust with respect to disorder in the distribution of CO blobs 关12兴, as well as disorder in the OD column
pattern. If ␪ is the angle between the two basis vectors ᐉ1 and
ᐉ2, then there are three types of planar lattices distinguished
by the value of ␪: square lattice 共␪ = ␲ / 2兲, rhombic lattice
共0 ⬍ ␪ ⬍ ␲ / 2 , ␪ ⫽ ␲ / 3兲, and hexagonal lattice 共␪ = ␲ / 3兲.
The density u共r兲 is assumed to be a doubly periodic function
on the planar lattice L, that is, u共r + ᐉ兲 = u共r兲 for all lattice
vectors ᐉ.
Note that there are a number of other ways in which a
periodic modulation due to some chemical marker could be
introduced into the developmental model 共1兲. For example,
there could be an extra additive contribution to the level of
activity such that 兰wijn j → 兰wijn j + u. However, this presupposes that the CO blobs have already been established as
sites of enhanced activity, perhaps through an increase in the
density of feedforward afferents. On the other hand, we postulate that the chemical marker modifies some aspect of synaptic plasticity, which in the Swindale model we take to be
the maximum synaptic density N. In more detailed models of
the competitive process underlying synaptic modification
关29,30兴, this could be interpreted as a modulation in the local
level of neurotrophic factors; an analogous feature could also
be incorporated into the correlation-based model of Miller
et al. 关18兴 by modulating the weight constraints. Yet another
possibility would be to introduce a spatially periodic variation in the rate of synaptic development ␶−1. At the linear
level, all of these forms of modulation would provide a
mechanism for the spatial mode-locking of the OD columns
to an array of CO blobs 共see Sec. II A 2兲. However, in the
particular case of a periodic modulation in the total density
N, one finds that the developing pattern of OD columns naturally forms a higher density of feedforward afferents around
the CO blobs, which is consistent with what is found in adult
cortex. The other cases would require an additional mechanism to achieve this.
␶
⳵ ni
= ␮共− ni + M兲FN共ni兲 + 2FN共ni兲
⳵t
w共兩r − r⬘兩兲n−共r⬘,t兲dr⬘ ,
共4兲
where n⫾ = 共nL ⫾ nR兲 / 2 with i = L / R. It follows that the only
nonzero homogeneous equilibrium is the binocular state
nLⴱ = nRⴱ = M. The linear stability of the uniform binocular state
can be determined by setting n j共r , t兲 = M + ⌬n j共r , t兲 and expanding Eq. 共4兲 to first order in ⌬n j. In terms of the sum and
difference densities ⌬n⫾ = 共⌬nL ⫾ ⌬nR兲 / 2 we find that
␶
␶
⳵ ⌬n+共r,t兲
= − ␮⌬n+共r,t兲H共r兲,
⳵t
冋
⳵ ⌬n−共r,t兲
= − ␮⌬n−共r,t兲 +
⳵t
冕
共5兲
w共兩r − r⬘兩兲⌬n−共r⬘,t兲dr⬘兴
⫻H共r兲,
共6兲
where
H共r兲 = M关N̄ + ␬u共r兲 − M兴.
共7兲
We will assume that N̄ ⬎ M so that H共r兲 is always positive. It
follows that if ␮ ⬎ 0 then ⌬n+共r , t兲 → 0 as t → ⬁, so that the
stability of the binocular state n j共r兲 = M will depend on the
asymptotic behavior of ⌬n−. Writing ⌬n−共r , t兲 = e␭t/␶a共r兲, we
obtain the linear eigenvalue equation
冋
␭a共r兲 = − ␮a共r兲 +
冕
册
w共兩r − r⬘兩兲a共r⬘兲dr⬘ H共r兲.
共8兲
The binocular state will be stable provided that all solutions
of Eq. 共8兲 satisfy 共Re兲␭ ⬍ 0. Note that if ␮ = 0 then the equilibrium is only marginally stable with respect to small perturbations ⌬n+ of the total density, and one has to impose an
additional constraint in order to specify the equilibrium
uniquely 关17兴.
1. Homogeneous case (␬ = 0)
Let us first ignore the modulatory effect of the CO blobs
by setting ␬ = 0 so that H共r兲 = M共N̄ − M兲. Since Eq. 共8兲 is
homogeneous when ␬ = 0, it has solutions of the form
a共r兲 = eik·r, where k denotes two-dimensional spatial frequency and the growth factor ␭ satisfies the dispersion relation
␭ = ␭共k兲 ⬅ 关− ␮ + Ŵ共k兲兴M共N̄ − M兲,
共9兲
with Ŵ共k兲 as the two-dimensional Fourier transform of
w共兩r兩兲, that is, Ŵ共k兲 = 兰e−ik·rw共兩r兩兲dr with k = 兩k兩. Since w satisfies Eq. 共2兲, it follows that Ŵ is a difference of Gaussians
with a positive maximum at a nonzero wave number kc:
2 2
/2
Ŵ共k兲 = 2␲共␴E2 Ae−␴Ek
A. Linear stability analysis
Following Swindale 共1980兲, we impose the left-right eye
symmetry constraints wLL = wRR = −wLR = −wRL = w. Equation
共1兲 reduces to the form
冕
2 2
− ␴I2Be−␴I k /2兲.
共10兲
If Ŵ共kc兲 ⬍ ␮ then the equilibrium is stable with respect to
local perturbations. On the other hand, if Ŵ共kc兲 ⬎ ␮ ⬎ 0 then
the equilibrium is unstable due to the growth of a finite band
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PAUL C. BRESSLOFF AND ANDREW M. OSTER
(a)
w(r)
(b)
h(k)
+=0
r
kc
k
+>0
FIG. 1. 共a兲 Difference of Gaussians lateral interaction function
w共r兲 showing short-range excitation and long-range inhibition. 共b兲
Dispersion curve ␭共k兲 = −␮ + Ŵ共k兲. The shaded regions indicate the
band of unstable eigenmodes around the critical wave number kc
where Ŵ共kc兲 = maxk兵Ŵ共k兲其. Black: unstable modes for ␮ ⬎ 0.
Black+ gray: unstable modes for ␮ = 0.
of Fourier modes lying in some interval containing kc
共see Fig. 1兲. Treating ␮ as a bifurcation parameter, we see
that there is a primary pattern forming instability at
␮ = ␮c ⬅ Ŵ共kc兲. Sufficiently close to the bifurcation point,
this instability leads to the formation of a spatially periodic
stripe pattern of the form n−共r兲 = A cos共k · r + ␾兲 with
兩k兩 ⬇ kc. This represents alternating left and right eye ocular
dominance columns of approximate width ␲ / kc, each of
which is a stripe running orthogonal to the direction of k.
Ignoring boundary effects, the direction arg共k兲 and spatial
phase ␾ of the stripe pattern are arbitrary when ␬ = 0, which
reflects the rotational and translational symmetries of the effective lateral interactions w. In the case ␮ = 0 considered by
Swindale 关17兴, the band of unstable modes extends to ⬁ such
that the dynamics is far from the primary bifurcation point
共see Fig. 1兲. The emerging pattern tends to have a more
disordered stripelike morphology, which is consistent with
the pattern of OD columns found in macaque.
2. Inhomogeneous case (␬ ⬎ 0)
Including the effects of the CO blobs by taking ␬ ⬎ 0
breaks continuous rotation and translation symmetries, so
that the spatial phase and direction of the pattern will no
longer be arbitrary. In fact, for certain critical wave vectors
the pattern locks to the lattice of blobs in an analogous fashion to the pinning of flow patterns observed in periodically
forced convective fluid systems 关31–33兴. A heuristic way to
understand this pinning is to note that the factor H共r兲 tends
to enhance the rate of pattern growth at the blob centers, that
is, at the lattice sites ᐉ 苸 L. Thus, there is a tendency for the
extrema of the resulting pattern to be located at the blobs,
and this effect will be strongest when the selected pattern is
commensurate with the lattice. The pinning mechanism essentially involves mode locking between spatial frequency
components of the emerging pattern of OD columns and the
distribution of CO blobs. The coupling between these frequency components can be analyzed at the linear level by
expanding Eq. 共8兲 as a Fourier series.
In order to elucidate further the pinning mechanism, we
consider the simpler case of a one-dimensional network with
a periodic distribution of CO blobs: u共x兲 = 兺qUqeiqx, where
q = 2␲m / d for integers m and d is the CO blob spacing. We
also impose the periodic boundary condition a共⌳兲 = a共0兲,
where ⌳ is the system size. In making this assumption, we
are assuming that we are sufficiently far away from both the
fovea and the boundary. The visual cortex is large compared
to the section of cortex that we are simulating; moreover, the
developed visual cortex has an intrinsic periodicity, suggesting that periodic boundary conditions are a natural choice.
The one-dimensional version of Eq. 共8兲 can then be
analyzed by introducing the Fourier series expansion
a共x兲 = ⌳−1兺kAkeikx with k = 2␲m / ⌳ for integers m 共the periodic boundary conditions restrain the spectrum to be discrete兲. Substitution into Eq. 共8兲 yields the eigenvalue equation 共after rescaling ␭ and ␬兲
关␭ + ␮ − Ŵ共k兲兴Ak = ␬ 兺 Vq共k兲Ak−q ,
共11兲
q
where Ŵ共k兲 is now the one-dimensional Fourier transform
Ŵ共k兲 =
冕
⬁
e−ikxw共x兲dx = 冑2␲␴EAe−␴Ek
2 2
/2
− 冑2␲␴IBe−␴I k /2 ,
2 2
−⬁
共12兲
Vq共k兲 = 关− ␮ + Ŵ共k − q兲兴Uq .
共13兲
At this point, we note here that Eq. 共11兲 arises in previous
studies examining the role of long-range periodic intracortical connections on pinning cortical activity patterns to the
underlying cortical structure 关34兴. Equation 共11兲 implies that
the periodic modulation only couples together those coefficients Ak , Ak−q1 , Ak−q2 , . . . whose wave numbers differ by qm
= 2␲m / d. In other words, if we fix k then a共x兲 = eikxbk共x兲 with
bk共x兲 given by the d-periodic function bk共x兲 = 兺qAk−qe−iqx.
It is generally not possible to find exact solutions of the
eigenvalue equation 共11兲. However, one can proceed by
treating ␬ as a small parameter and carrying out a perturbation analysis along similar lines to a recent study of spontaneous cortical activity patterns 关21,34兴. Here, we will present
a perturbation argument for the occurrence of pinning. Let kc
be the critical wave number for a pattern forming instability
when ␬ = 0. That is, Ŵ共kc兲 ⬎ Ŵ共k兲 for all k ⬎ 0 , k ⫽ kc.
Suppose that kc is commensurate with the lattice spacing
d when ␬ = 0, that is, kc = m␲ / d for some integer m. It
follows that the pair of coefficients A⫾k associated with the
dominant wave number k = kc are coupled in Eq. 共11兲 since
−kc = kc − Q for Q = 2␲m / d. This implies that there is an approximate twofold degeneracy, and we must treat Eq. 共11兲
separately for the two cases k and k − Q. Thus, to first order
in ␬, Eq. 共11兲 reduces to a pair of equations for the coefficients Ak and Ak−Q:
冉
␭ − E共k兲
− ␬VQ共k兲
− ␬V−Q共k − Q兲 ␭ − E共k − Q兲
冊冉 冊
Ak
= 0,
Ak−Q
where E共k兲 = −␮ + Ŵ共k兲 + ␬V0共k兲. We will assume that
u共x兲 is an even function of x so that UQ = U−Q. Since
Ŵ共kc兲 = Ŵ共kc − Q兲 and Ŵ⬘共kc兲 = 0, it follows that for
k = m␲ / d + O共␬兲, the above matrix equation has solutions of
the form 共to first order in ␬兲
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THEORY FOR THE ALIGNMENT OF CORTICAL FEATURE…
even pattern
(a)
Q
k
CO blob
(b)
odd pattern
k-Q
Q2
(c)
Q2
Q1
B.Z.
B.Z.
Q1
FIG. 2. Even and odd eigenmodes localized around blob and
interblob regions.
␭⫾共k兲 = − ␮ + Ŵ共k兲 + ␬关V0共k兲 ⫾ VQ共k兲兴,
共14兲
with Ak−Q = ⫾ Ak. Thus, there is a splitting into the even and
odd eigenmodes,
a+共x兲 = A cos共n␲x/d兲,
a−共x兲 = A sin共n␲x/d兲,
共15兲
where A is an arbitrary amplitude 共at the linear level兲. The
even and odd eigenmodes for n = 1 are shown in Fig. 2. Note
that the even mode has extrema at sites corresponding to the
CO blobs, whereas the odd mode has extrema at sites corresponding to interblobs. Which of the two eigenmodes is selected by the pattern forming instability will depend on the
sign of VQ共kc兲 for Q = 2␲ / d. Assuming that UQ ⬎ 0 then the
even pattern will be selected leading to the formation of alternating left and right ocular dominance columns of width
␲ / d whose centers are located at the CO blob centers.
The major difference between the one- and twodimensional versions of Eq. 共8兲 is that in the latter case there
is an additional degeneracy arising from the isotropy of the
lateral interactions w. In principle the pattern forming instability can excite any eigenmode eik·r lying on the critical
circle 兩k兩 = kc for ␬ = 0. However, when ␬ ⬎ 0, continuous rotation symmetry is broken by the underlying lattice of CO
blobs, so that the OD stripes tend to favor particular directions that reflect the discrete symmetries of the lattice. The
modulation term u共r兲 can be expanded as a two-dimensional
Fourier series,
u共r兲 =
兺ˆ Uqeiq·r,
q苸L
q=
兺
j=1,2
2␲n j
ᐉ̂ j ,
d
共16兲
where ᐉ̂ j are the generators of the reciprocal lattice
L̂ = 兵q 兩 q · ᐉ = 0 for all ᐉ 苸 L其 with ᐉ̂i · ᐉ j = ␦i,j. Such a term
couples together eigenmodes with wave vectors k that differ
by a reciprocal lattice vector q. Extending the perturbation
analysis of the one-dimensional network, it can then be
shown that this leads to the following commensurability condition for the pinning of a two-dimensional OD stripe pattern
to the CO blobs: there exists a reciprocal lattice vector
Q 苸 L̂ such that 关21兴
兩k − Q兩 = 兩k兩 = kc .
共17兲
Geometrically this means that k must lie on the perpendicular bisector of the line joining the origin of the reciprocal
lattice L̂ to the lattice point Q 关see Fig. 3共a兲兴. In the theory of
FIG. 3. 共a兲 If k = 兩k − Q兩 then k must lie on the bisector of the
reciprocal lattice vector Q. 共b兲 and 共c兲 Construction of the first
Brillouin zone 共shaded region兲 in the reciprocal square and hexagonal lattices with Q j = 2␲ᐉ̂ j / d.
crystalline solids these perpendicular bisectors partition the
reciprocal lattice into domains known as Brillouin zones 关see
Figs. 3共b兲 and 3共c兲兴; commensurability then requires that k
lies on the boundary of a Brillouin zone 关35兴.
B. Commensurability
So far we have assumed that the lateral interactions w, as
given by the Mexican hat function 共2兲, generate a stripe pattern of OD columns in which the width of each column is
approximately equal to a CO blob spacing d. Is there some
mechanism whereby such a matching of length scales could
be achieved? An idea proposed by Bressloff 关21兴 suggests
one possibility for such a commensurability to apply based
on the experimental observation 共in cats and ferrets兲 that
there exist crude patchy long-range horizontal connections
early in development 关36–38兴. In fact, three distinct phases
in the development of horizontal connections have been
identified: 共i兲 the initial appearance of long-range connections in the subplate and layer I, which could serve as a
scaffold for the patchy connections subsequently found in
other layers; 共ii兲 crude clustering of horizontal connections in
superficial layers II/III 共before eye opening in cat兲; this is not
affected by dark rearing, binocular deprivation, or blocking
of retinal activity; and 共iii兲 an activity-dependent refinement
of the patchy connections, in which the spacing of the
patches is approximately conserved. Combined with the observation that the patchy connections in the adult cortex tend
to link blobs in same eye OD columns 关8兴, it follows that the
spacing between horizontal patches transverse to the OD columns is approximately twice the blob spacing, and this relative spacing should also hold during early development when
the blobs are first detected. One intriguing possibility is that
intrinsic patchy chemical markers mediate both the initial
formation of CO blobs as well as the early development of
patchy horizontal connections, thus naturally leading to a
matching of length scales. The combination of the local
“Mexican hat” lateral circuitry with the modulatory effects of
the long-range patchy connections taken together would accentuate the effects of the lateral circuitry at the intrinsic
021920-5
PHYSICAL REVIEW E 82, 021920 共2010兲
PAUL C. BRESSLOFF AND ANDREW M. OSTER
(b)
8
(a)
space x (in units of d)
space x (in units of d)
degree of pinning
1
0
time t (in units of τ)
350
0.6
0.4
0.2
0
-0.2
0
0
(b)
0.8
8
0
time t (in units of τ)
0.5
0
-0.5
-1
0
0.1
0.2
κ
0.3
0.4
2
4
8 10 12 14 16
6
space x (in units of d)
350
FIG. 4. 共Color online兲 One-dimensional pinning of OD columns
to a periodic array of CO blobs. 共a兲 Space-time plot of the density
n−共x , t兲 obtained by numerically solving a one-dimensional
version of Eq. 共4兲 for N = 1 + 0.5␬关1 + cos共2␲x / d兲兴 with ␬ = 0.4,
␮ = 0.08⬇ ␮c, and a network of size ⌳ = 8d. The parameters of the
Mexican hat function 共2兲 are chosen to be A = 1.8, B = 1.0,
␴E = 0.29d, and ␴I = 0.72d, so that kc ⬇ ␲ / d. The initial binocular
state 共gray兲 evolves into alternating left and right eye ocular dominance columns 共colored white and black, respectively兲 whose centers align with the centers of the CO blobs 共indicated by red horizontal lines兲. 共b兲 Corresponding plot of n+共x , t兲 showing the
emergence of a periodic variation in the total density of feedforward
afferents 共white represents a high density of synaptic drive兲.
blob spacing, that is, the Fourier mode associated with the
blob spacing would be maximized.
III. RESULTS
As the amplitude of the selected pattern grows the dynamics will become dominated by nonlinearities arising from the
logistic function FN共n兲. To what extent does the pinning of
the OD pattern to the lattice of CO blobs persist when such
nonlinearities are included, and is the resulting pattern
stable? In order to investigate this issue, we numerically
solve Eq. 共1兲 for both one- and two-dimensional networks
using periodic boundary conditions.
A. Pinning in a one-dimensional network
Consider the one-dimensional version of Eq. 共1兲 with
N = 1 + ␬u共x兲 and CO blob density
u共x兲 = 0.5关1 + cos共2␲x/d兲兴.
density n-(x)
(a)
共18兲
We choose the parameters of the Mexican hat function 共2兲
such that kc ⬇ ␲ / d. Example space-time plots of the densities
n+共x , t兲 and n−共x , t兲 are shown in Fig. 4 for ␬ = 0.4. It can be
seen that the initial binocular state evolves into alternating
left and right eye ocular dominance columns whose centers
align with the CO blobs. A periodic variation in the total
density of feedforward afferents also emerges, which can be
understood as follows. If a point x lies within a left eye
ocular dominance column we expect nL共x , t兲 → N̄ + ␬u共x兲 and
nR共x , t兲 → 0 as t → ⬁, whereas we expect the opposite to occur if x lies within a right eye column. This suggests that
there will be a periodic variation in the total density of feedforward afferents such that n+共x兲 ⬇ 关N̄ + ␬u共x兲兴 / 2. Thus, the
pinning mechanism also provides a means for generating a
higher density of feedforward afferents to the CO blobs, so
FIG. 5. 共a兲 Plot of degree of pinning ␹ as a function of the
modulation strength ␬ for kc = ␲ / d and ␮ = 0.08 taken over 45 trials
with white noise about a binocular state. 共b兲 One-dimensional OD
pattern with a spatial phase defect. The steady-state density n−共x兲 is
plotted as a function of x for ⌳ = 16d. Other parameter values are as
in Fig. 4. It can be seen that pinning occurs except within a transition region 共shaded in gray兲 where the OD columns are gradually
shifted by an additional spatial phase equal to a single blob spacing.
The variation in the CO blob density u共x兲 is shown by the alternating light and dark dashed curve. The maxima of the density n−共x兲
shift from the dark to the light peaks of u共x兲 as x increases.
that they ultimately become defined extrinsically. This is
consistent with what is found in the mature cortex, where the
CO blobs in superficial layers are sites of enhanced metabolic activity resulting from a higher density of afferent inputs. 关For example, in macaque the CO blobs receive additional koniocellular inputs directly from the lateral geniculate
nucleus 共LGN兲.兴 When interpreting our numerical results,
however, it is important to remember that we have collapsed
superficial layers II/II and input layer IV into a single effective layer. Thus, the ocular dominance columns shown in
Fig. 4 are more consistent with input layer IV since each
column is predominately monocular. The variation in left or
right eye dominance is smoother in superficial layers, so that
there is a more gradual transition from monocular to binocular domains. On the other hand, the periodic variation in the
total synaptic density is more consistent with superficial layers where the CO blobs are located, since the total density of
synapses is approximately uniform in the input layer IV.
How robust is the pinning mechanism with respect to parameter variations in the strength ␬ of the periodic modulation or the mismatch ⌬k = dkc / ␲ − 1 between the pattern
wavelength and the blob spacing? In order to answer this
question we introduce the following index for the degree of
pinning:
␹=1−
4
Pd
冓
P
B
兩xOD
兺
p − xp 兩
p=1
冔
,
共19兲
denotes the center of the pth OD column,
where xOD
p
p = 1 , . . . , P, and xBp is the location of the nearest CO blob
center. The angular brackets 具 · 典 denote averaging over trials
with random initial conditions about the binocular state.
Thus, ␹ = 1 for perfect pinning, ␹ = −1 for perfect antipinning,
B
and ␹ = 0 for a random alignment 共in which 兩xOD
p − x p 兩 is uniformly distributed over the interval 关0 , d / 2兴兲. In Fig. 5共a兲 we
plot ␹ as a function of ␬ in the case ␮ ⬇ ␮c. It can be seen
that the degree of pinning tends to grow with the strength of
modulation ␬ for fixed kc = ␲ / d although, even for quite large
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THEORY FOR THE ALIGNMENT OF CORTICAL FEATURE…
(a)
0.2
0.8
2
0.6
0.4
0.2
0.8
0.6
0.2
0.4
κ
0.6
0.8
1
-0.4
-0.2
0
Δk
0.2
0.2
0.4
FIG. 6. 共a兲 Plot of degree of pinning ␹ as a function of the
modulation strength ␬ for kc = ␲ / d. 共b兲 Plot of degree of pinning ␹
as a function of the wavelength mismatch ⌬k = dkc / ␲ − 1 for
␬ = 1.0. Black dots show pinning with respect to OD centers. Weight
parameters are as in Fig. 4 except that ␮ = 0. Averages were performed over 40 trials.
values of ␬, perfect pinning does not occur. The latter is due
to the occurrence of phase defects in a small number of trials; the existence of phase defects is a well-known phenomenon in the theory of spatial pattern formation 关39兴. In our
particular case, a phase defect corresponds to a localized
region over which the centers of the ocular dominance columns are gradually shifted relative to the CO blob centers by
a net spatial phase equal to a single blob spacing. An example is shown in Fig. 5共b兲 for a larger system size where
the phase defect can be seen more clearly.
Now suppose that we take ␮ = 0 as in the Swindale model
关17兴. Since the system is operating far from the bifurcation
point when ␮ = 0, it is outside the perturbative regime considered in Sec. II A. In particular, we find that ␬ = O共1兲 in
order for good pinning to occur 关see Fig. 6共a兲兴. Another interesting feature also emerges for larger ␬ as illustrated in
Fig. 6共b兲. Here, we plot the degree of pinning ␹ as a function
of ⌬k for fixed ␬ = 1 共black dots兲. As one might expect from
the linear analysis of Sec. II B, pinning is maximal when
⌬k ⬇ 0 or kc ⬇ ␲ / d, that is, when the pattern wavelength is
commensurate with the CO blob spacing.
As we have already commented, the actual distribution of
CO blobs is not given by a perfect lattice. We find that the
pinning mechanism is robust to certain levels of irregularity
in the distribution of CO blobs. One way to introduce a degree of irregularity in the one-dimensional model is by taking the CO blob density to be of the form
0
-1
0.4
0
0
1
density n-(x)
0.4
degree of pinning χ
degree of pinning
0.6
0
(b)
1
1
0.8
degree of pinning
(a)
(b)
-2
0
0.1
0.2
0.3 0.4
disorder Γ
0.5
0.6
1
0.7
2
3
4
5
6
7
space x (in units of d)
8
FIG. 7. 共a兲 Plot of degree of pinning ␹ versus degree of
irregularity ⌫ in distribution of CO blobs as given by Eq. 共20兲.
Here, ␬ = 1.0, ␮ = 0, and kc = ␲ / d. 共b兲 One-dimensional OD pattern
共black curve兲 pinning to an irregular distribution 共⌫ = 0.7兲 of CO
blobs 共solid gray curve兲. The corresponding regular distribution of
spacing d is shown by the dashed gray curve. Same parameter values as 共a兲 with ⌫ = 0.7.
␬ this shift would be negligible. From an experimental perspective, the current finding in macaque is that the CO blobs
tend to lie in the centers of OD columns and also tend to
coincide with the most monocular cell populations. As far as
we are aware of, the possibility of a statistically significant
difference between these two aspects has not been investigated.
B. Pinning in a two-dimensional network
All of the above observations extend to the more realistic
case of two-dimensional networks. In Fig. 8 we show examples of two-dimensional OD patterns obtained by numerically solving Eq. 共4兲 for CO blobs on a square lattice. We
take N = 1 + ␬u共x , y兲 with
u共x,y兲 = 0.25关2 + cos共2␲x/d兲 + cos共2␲y/d兲兴,
共21兲
representing the two-dimensional distribution of CO blobs
and ␬ = 1.0, ␮ = 0. The parameters of the weight distribution
(a)
(b)
8
8
6
6
y
y
4
4
2
2
P
u共x兲 = 兺 exp关− 共x − x p兲2/2␥2兴,
共20兲
p=1
2
where ␥ determines the patch size 共assumed to be smaller
than d兲. The center x p of the pth blob is taken to be
x p = pd + ⌫␰ p with ␰ p as a random variable generated from a
uniform distribution over the interval 关−0.5, 0.5兴. Thus, ⌫ is a
measure of the degree of disorder. In Fig. 7共a兲 we plot the
degree of pinning ␹ as a function of ⌫. We see that pinning
persists for irregular lattices with the most monocular regions lining up with the blobs as illustrated in Fig. 7共b兲.
Again our model predicts a shift between the center of the
OD column and the most monocular region, depending on
the degree of disorder and the size of ␬; for smaller values of
4
x
6
8
2
4
x
6
8
FIG. 8. Two-dimensional pinning of OD columns to a
square lattice of CO blobs with N = 1 + ␬u共x , y兲 and
u共x , y兲 = 0.25关2 + cos共2␲x / d兲 + cos共2␲y / d兲兴. We take ␮ = 0 and
␬ = 1 and consider a radial weight distribution 共2兲 with A = 3.8,
B = 3.3, ␴e = 0.51d, and ␴i = 0.64d such that kc ⬇ ␲ / d, where d is the
blob spacing. The two examples correspond to different initial conditions obtained by introducing small random perturbations of the
uniform binocular state. For visualization purposes, the color scale
was chosen so that white corresponds to left eye dominance and
gray corresponds to right eye dominance, with the lattice shown in
black.
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PHYSICAL REVIEW E 82, 021920 共2010兲
PAUL C. BRESSLOFF AND ANDREW M. OSTER
1
(a)
(b)
8
8
8
6
6
6
y
y
y
4
4
2
2
4
2
0
2
0
2
4 x
6
8
1
FIG. 9. Two-dimensional pinning of OD columns to an irregular
lattice of CO blobs. Distribution of CO blobs is given by Eq. 共22兲
with ⌫ = 0.3.
w共r兲 given by Eq. 共2兲 are chosen so that kc ⬇ ␲ / d. The resulting OD stripes tend to be oriented in directions corresponding to symmetries of the lattice, and the centers of the
OD columns tend to coincide with CO blob centers.
Interestingly, pinning occurs even though the patterns are
quite irregular with a number of defects and branches. Pinning also persists when there is a certain level of disorder in
the two-dimensional distribution of the blobs. This is illustrated in Fig. 9 for the Gaussian CO blob distribution
u共x,y兲 = 兺 e关−共x − xp兲
2/2␥2兴 关−共y − y 兲2/2␥2兴
p
e
.
共22兲
p
Here, the center 共x p , y p兲 of the pth blob is given by
x p = pd + ⌫␰ p, y p = pd + ⌫␩ p with ␰ p , ␩ p as random variables
generated from a uniform distribution over the interval
关−0.5, 0.5兴. As in the one-dimensional case, the CO blobs are
strongly correlated with the most monocular regions of the
OD column. Similar results also hold for regular or irregular
rectangular and hexagonal lattices.
Since the CO blobs are intrinsically defined in our model,
it follows that their spatial relationship with the ocular dominance columns is robust with respect to interventions that
mimic the effects of monocular deprivation 共MD兲 during the
critical period in macaque 关11兴. Following Swindale 关17,19兴,
monocular deprivation can be modeled as a temporary reduction in the strength of the central part of the lateral interaction function 共2兲 originating from one eye. In the case of a
monocularly deprived right eye, this would involve a reduction in ARR and ARL, shown in Fig. 10共a兲. Although the width
of the stripes associated with the deprived eye shrinks at the
expense of the nondeprived eye, the CO blobs still emerge as
the sites of highest-density afferents associated with the most
monocular parts of the OD columns. Finally, one can also
introduce a left-right eye asymmetry that tends to produce
OD patterns more consistent with cat, that is, circular islands
of OD columns from one eye embedded in a complementary
pattern of OD columns from the other eye 关19兴, shown in
Fig. 10共b兲. There is still a tendency for the blobs to coincide
with the centers of the OD columns but the patchy nature of
the OD columns leads to a weaker correspondence than occurs for the stripe patterns.
4
x
8
6
2
4
x
6
8
FIG. 10. Examples of pinning to the CO blob lattice in the cases
of MD and “catlike” conditions, shown in 共a兲 and 共b兲, respectively.
In both 共a兲 and 共b兲, the CO blob lattice and the primary weight
distribution are the same as in Fig. 8. In 共a兲, we simulate MD
conditions by reducing the excitation level of the deprived eye to
60% of its original value. The area that the nondeprived eye then
innervates is 67% of V1. In 共b兲, since the contralateral visual input
is dominant in early cat development, in order to simulate catlike
conditions, both excitation and inhibition levels of the ipsilateral
drive are reduced to 60% of their original values for the initial
phase of development 共here, ␬ is taken to be ␬ = 0.1兲. For visualization purposes, the color scale was chosen so that white corresponds
to left eye dominance and gray corresponds to right eye dominance,
with the lattice shown in black.
IV. DISCUSSION
In this paper we have presented a simple model of ocular
dominance 共OD兲 column formation that incorporates the effects of a spatially periodic modulation in synaptic plasticity.
We have suggested that such a modulation could arise from a
distribution of patchy chemical markers early in development, corresponding to an array of intrinsically defined cytochrome oxidase 共CO兲 blobs. Spatial coupling between the
emerging pattern of OD columns and the periodic markers
provides a mechanism for aligning the OD columns with the
CO blobs. Such a mechanism is robust with respect to a
certain degree of disorder in the distribution of the blobs as
well as irregularities 共defects and branches兲 in the OD pattern.
An obvious extension of our work is to investigate the
possible role of CO blobs in the alignment of other cortical
feature maps. For example, one could carry out an analogous
modification of Swindale’s model for the development of
orientation columns 关40兴. The latter model is formulated in
terms of a complex-valued function z共r兲, with arg z共r兲 specifying the orientation preference at r 共restricted to lie in the
interval 关0 , ␲兲兲 and 兩z共r兲兩 determining the associated orientation selectivity. The evolution equation for z takes the form
共for ␮ = 0兲
⳵z
=
⳵t
冕
w共r − r⬘兲z共r⬘兲dr⬘F共z兲,
共23兲
with F共z兲 = N − 兩z兩. The effects of a CO blob distribution could
be incorporated into the model by taking N to satisfy Eq. 共3兲.
One could then investigate to what extent the resulting orientation pinwheels align with the CO blobs as suggested by
optical imaging data 关6兴. Swindale’s model for the joint development of ocular dominance and orientation columns 关41兴
could be modified along similar lines. The possible role of
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PHYSICAL REVIEW E 82, 021920 共2010兲
CO blobs in the joint alignment of OD and orientation columns has also been briefly reported in a computational study
by Jones and Leyton-Brown 关42兴.
It is possible that the coalignment of multiple feature
maps could occur in the absence of a spatially periodic distribution of molecular markers. Indeed, it has previously
been demonstrated that OD and orientation columns can develop simultaneously, such that orientation pinwheels align
with the centers of the OD columns and iso-orientation contours cross OD column boundaries orthogonally 关41,43兴. The
inclusion of an additional feedforward label for the distribution of CO blobs 共perhaps by taking into account the koniocellular pathway that feeds directly to the CO blobs in
macaque兲 could result in the simultaneous alignment of the
OD columns, the orientation preference map, and the CO
blobs. In contrast to the mode-locking mechanism proposed
in this paper, such an alignment would arise spontaneously
through the competitive process that drives the development
of the multiple feature maps. There are, however, two potential difficulties with the latter approach. First, the coalignment of multiple feature maps tends to be sensitive to the
choice of parameter values 关43兴. Second, it does not take into
account the experimental finding that the spatial distribution
of the CO blobs appears early in development and is not
affected by activity 关12,14兴. We also note that an abstract
approach of using elastic neural nets can produce results in
line with cortical feature maps seen in V1 关44兴; however,
here we have attempted to develop a biologically inspired
mathematical model in line with the anatomy.
Finally, it is tempting to argue that a spatially periodic
distribution of molecular markers early in development orchestrates the coalignment of multiple feature maps in order
to achieve a modular architecture. This is consistent with
computational models that treat V1 as a system of coupled
hypercolumns 关45,46兴. Each hypercolumn processes local
features of an image by carrying out some form of local
Fourier decomposition, say; the results of these local computations are then correlated within V1 through long-range
horizontal connections and transmitted to extrastriate areas
for further processing. It would be interesting to determine
whether or not animals without such modular architectures
still have periodically distributed markers.
This paper was based on work supported in part by the
NSF 共Grant No. DMS-0209824兲 and by Award No. KUKC1-013-4 made by the King Abdullah University of Science
and Technology 共KAUST兲. P.C.B. was also partially supported by the Royal Society–Wolfson Foundation, and
A.M.O. was partially supported by the Neuropôle de Recherche Francilien 共NeRF兲 and an ANR MNP ‘Dopanic’ grant.
关1兴 J. C. Horton and D. H. Hubel, Nature 共London兲 292, 762
共1981兲.
关2兴 J. C. Horton, Philos. Trans. R. Soc. London, Ser. B 304, 199
共1984兲.
关3兴 K. M. Murphy, D. G. Jones, and R. C. Van Sluyters, J. Neurosci. 15共6兲, 4196 共1995兲 关http://www.jneurosci.org/cgi/
content/abstract/15/6/4196兴.
关4兴 M. S. Livingstone and D. H. Hubel, J. Neurosci. 4共11兲, 2830
共1984兲
关http://www.jneurosci.org/cgi/content/abstract/4/11/
2830兴.
关5兴 R. B. Tootell, M. S. Silverman, S. L. Hamilton, E. Switkes,
and R. L. De Valois, J. Neurosci. 8共5兲, 1610 共1988兲 关http://
www.jneurosci.org/cgi/content/abstract/8/5/1610兴.
关6兴 K. Obermayer and G. G. Blasdel, J. Neurosci. 13共10兲, 4114
共1993兲 关http://www.jneurosci.org/cgi/content/abstract/13/10/
4114兴.
关7兴 D. Shoham, M. Hubener, S. Schulze, A. Grinvald, and T. Bonhoeffer, Nature 共London兲 385, 529 共1997兲.
关8兴 T. Yoshioka, G. G. Blasdel, J. B. Levitt, and J. S. Lund, Cereb.
Cortex 6, 297 共1996兲.
关9兴 N. H. Yabuta and E. M. Callaway, Visual Neurosci. 15, 1007
共1998兲.
关10兴 L. C. Sincich and J. C. Horton, Science 295, 1734 共2002兲.
关11兴 J. C. Horton and D. R. Hocking, J. Neurosci. 17共10兲, 3684
共1997兲 关http://www.jneurosci.org/cgi/content/abstract/17/10/
3684兴.
关12兴 K. M. Murphy, D. G. Jones, S. B. Fenstemeaker, V. D. Pegado,
L. Kiorpes, and J. A. Movshon, Cereb. Cortex 8, 237 共1998兲.
关13兴 M. C. Crair, J. C. Horton, and A. Antonini, J. Comp. Neurol.
430, 235 共2001兲.
关14兴 K. M. Murphy, K. R. Duffy, D. G. Jones, and D. E. Mitchell,
Cereb. Cortex 11, 122 共2001兲.
关15兴 C. Trepel, D. R. Duffy, V. D. Pegado, and K. M. Murphy,
J. Neurosci. 18共9兲, 3404 共1998兲 关http://www.jneurosci.org/cgi/
content/abstract/18/9/3404兴.
关16兴 R. H. Dyck and M. S. Cynader, Proc. Natl. Acad. Sci. U.S.A.
90, 9066 共1993兲.
关17兴 N. V. Swindale, Proc. R. Soc. London, Ser. B 208, 243 共1980兲.
关18兴 K. D. Miller, J. B. Keller, and M. P. Stryker, Science 245, 605
共1989兲.
关19兴 N. V. Swindale, Network 7, 161 共1996兲.
关20兴 H. Z. Shouval, D. H. Goldberg, J. P. Jones, M. Beckerman, and
L. N. Cooper, J. Neurosci. 20共3兲, 1119 共2000兲 关http://
www.jneurosci.org/cgi/content/abstract/20/3/1119兴.
关21兴 P. C. Bressloff, Physica D 185, 131 共2003兲.
关22兴 J. J. Wright, D. M. Alexander, and P. D. Bourke, Vision Res.
46, 2703 共2006兲.
关23兴 V. A. Whitelaw and J. D. Cowan, J. Neurosci. 1共12兲, 1369
共1981兲
关http://www.jneurosci.org/cgi/content/abstract/1/12/
1369兴.
关24兴 C. Weber, H. Ritter, J. D. Cowan, and K. Obermayer, Philos.
Trans. R. Soc. London, Ser. B 352, 1603 共1997兲.
关25兴 D. Willshaw and D. Price, Modeling Neural Development
共MIT Press, Cambridge, MA, 2003兲, pp. 213–244.
关26兴 A. M. Turing, Philos. Trans. R. Soc. London, Ser. B 237, 37
共1952兲.
关27兴 E. M. Callaway, J. Neurosci. 18共4兲, 1505 共1998兲 关http://
www.jneurosci.org/cgi/content/abstract/18/4/1505兴.
ACKNOWLEDGMENTS
021920-9
PHYSICAL REVIEW E 82, 021920 共2010兲
PAUL C. BRESSLOFF AND ANDREW M. OSTER
关28兴 K. Yashiro and B. D. Philpot, Neuropharmacology 55, 1081
共2008兲.
关29兴 A. E. Harris, G. B. Ermentrout, and A. L. Small, J. Comput.
Neurosci. 8, 227 共2000兲.
关30兴 A. van Ooyen and R. R. Ribchester, Modeling Neural Development 共MIT Press, Cambridge, MA, 2003兲, pp. 183–212.
关31兴 M. Lowe and J. P. Gollub, Phys. Rev. A 31, 3893 共1985兲.
关32兴 P. Coullet, Phys. Rev. Lett. 56, 724 共1986兲.
关33兴 D. Walgraef, Spatio-Temporal Pattern Formation 共SpringerVerlag, Berlin, 1997兲.
关34兴 P. C. Bressloff, Phys. Rev. Lett. 89, 088101 共2002兲.
关35兴 N. W. Ashcroft and N. D. Mermin, Solid State Physics 共Saunders College Publishing, Philadelphia, PA, 1976兲.
关36兴 E. M. Callaway and L. C. Katz, J. Neurosci. 10共4兲, 1134
共1990兲
关http://www.jneurosci.org/cgi/content/abstract/10/4/
1134兴.
关37兴 E. S. Ruthazer and M. P. Stryker, J. Neurosci. 16共22兲, 7253
关38兴
关39兴
关40兴
关41兴
关42兴
关43兴
关44兴
关45兴
关46兴
021920-10
共1996兲 关http://www.jneurosci.org/cgi/content/abstract/16/22/
7253兴.
C. Chiu and M. Weliky, Neuron 35, 1123 共2002兲.
P. Bak, Rep. Prog. Phys. 45, 587 共1982兲.
N. V. Swindale, Proc. R. Soc. London, Ser. B 215, 211 共1982兲.
N. V. Swindale, Biol. Cybern. 66, 217 共1992兲.
D. G. Jones and K. Leyton-Brown, Soc. Neurosci. Abstr. 24,
813 共1998兲.
E. Erwin and K. D. Miller, J. Neurosci. 18共23兲, 9870 共1998兲
关http://www.jneurosci.org/cgi/content/abstract/18/23/9870兴.
M. Á. Carreira-Perpiňán, R. J. Lister, and G. J. Goodhill,
Cereb. Cortex 15, 1222 共2005兲.
S. LeVay and S. B. Nelson, The Neural Basis of Visual Function 共CRC Press, Boca Raton, FL, 1991兲, pp. 266–315.
J. S. Lund, A. Angelucci, and P. C. Bressloff, Cereb. Cortex
13共1兲, 15 共2003兲.