Morphogenesis of the Lateral Geniculate 
Nucleus: How Singularities Affect Global 
Structure 
Svilen Tzonev 
Beckman Institute 
University of Illinois 
Urbana, IL 61801 
svilen@ks.uiuc.edu 
Klaus Schulten 
Beckman Institute 
University of Illinois 
Urbana, IL 61801 
kschulte@ks.uiuc.edu 
Joseph G. Malpeli 
Psychology Department 
University of Illinois 
Champaign, IL 61820 
jmalp eli@uiuc.edu 
Abstract 
The macaque lateral geniculate nucleus (LGN) exhibits an intricate 
lamination pattern, which changes midway through the nucleus at a 
point coincident with small gaps due to the blind spot in the retina. 
We present a three-dimensional model of morphogenesis in which 
local cell interactions cause a wave of development of neuronal re- 
ceptive fields to propagate through the nucleus and establish two 
distinct lamination patterns. We examine the interactions between 
the wave and the localized singularities due to the gaps, and find 
that the gaps induce the change in lamination pattern. We explore 
critical factors which determine general LGN organization. 
1 INTRODUCTION 
Each side of the mammalian brain contains a structure called the lateral geniculate 
nucleus (LGN), which receives visual input from both eyes and sends projections to 
134 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli 
the primary visual cortex. In primates the LGN consists of several distinct layers 
of neurons separated by intervening layers of axons and dendrites. Each layer of 
neurons maps the opposite visual hemifield in a topographic fashion. The cells com- 
prising these layers differ in terms of their type (magnocellular and parvocellular), 
their input (from ipsilateral (same side) and contralateral (opposite side) eyes), and 
their receptive field organization (ON and OFF center polarity). Cells in one layer 
receive input from one eye only (Kaas et al., 1972), and in most parts of the nucleus 
have the same functional properties (Schiller 2z Malpeli, 1978). The maps are in 
register, i.e., representations of a point in the visual field are found in all layers, and 
lie in a narrow column roughly perpendicular to the layers (Figure 1). A prominent 
a projection 
column 
blind spot 
gaps 
Figure 1: A slice along the plane of symmetry of the macaque LGN. Layers are 
numbered ventral to dorsal. Posterior is to the left, where foveal (central) parts of 
the retinas are mapped; peripheral visual fields are mapped anteriorly (right). Cells 
in different layers have different morphology and functional properties: 6-P/C/ON; 
5-P/I/ON; 4-P/C/OFF; 3-P/I/OFF; 2-M/I/ON&OFF; 1-M/C/ON&OFF, where 
P is parvocellular, M is magnocellular, C is contralateral, I is ipsilateral, ON and 
OFF refer to polarities of the receptive-field centers. The gaps in layers 6, 4, and 1 
are images of the blind spot in the contralateral eye. Cells in columns perpendicular 
to the layers receive input from the same point in the visual field. 
feature in this laminar organization is the presence of cell-free gaps in some layers. 
These gaps are representations of the blind spot (the hole in the retina where the 
optic nerve exits) of the opposite retina. In the LGN of the rhesus macaque mon- 
key (Macaca mulatta) the pattern of laminar organization drastically changes at the 
position of the gaps -- foveal to the gaps there are six distinct layers, peripheral to 
the gaps there are four layers. The layers are extended two-dimensional structures 
whereas the gaps are essentially localized. However, the laminar transition occurs 
in a surface that extends far beyond the gaps, cutting completely across the main 
axis of the LGN (Malpeli 2z Baker., 1975). 
We propose a developmental model of LGN laminar morphogenesis. In particular, 
we investigate the role of the blind-spot gaps in the laminar pattern transition, and 
their extended influence over the global organization of the nucleus. In this model 
a wave of development caused by local cell interactions sweeps through the system 
(Figure 2). Strict enforcement of retinotopy maintains and propagates an initially 
localized foveal pattern. At the position of the gaps, the system is in a metastable 
Morphogenesis of the Lateral Geniculate Nucleus 13.5 
wave front 
maturing cells / immature cells 
_ I.-'  *,,.., - I horizontal 
,v ." . 
, blind spot gap 
._ _:x - .'"L,..  n-ell') 
Figure 2: Top view of a single layer. As a wave of development sweeps through the 
LGN the foveal part matures first and the more peripheral parts develop later. The 
shape of the developmental wave front is shown schematically by lines of "equal 
development" 
state, and the perturbation in retinotopy caused by the gaps is sufficient to change 
the state of the system to its preferred four-layered pattern. We study the critical 
factors in this model, and make some predictions about LGN morphogenesis. 
2 MODEL OF LGN MORPHOGENESIS 
We will consider only the upper four (parvocellular) layers since the laminar tran- 
sition does not involve the other two layers. This transition results simply from a 
reordering of the four parvocellular strata (Figure 1). Foveal to the gaps, the strata 
form four morphologically distinct layers (6, 5, 4 and 3) because adjacent strata 
receive inputs from opposite eyes, which "repel" one another. Peripheral to the 
gaps, the reordering of strata reduces the number of parvocellular eye alternations 
to one, resulting in two parvocellular layers (6+4 and 5+3). 
2.1 GEOMETRY AND VARIABLES 
LGN cells ci are labeled by indices i - 1,2,... ,N. The cells have fixed, 
quazi-random and uniformly distributed locations ri  V C 3, where 
V = { (x,y,z)I0 < x < Sz,O < y < Sy, O < z < Sz}, and belong to one projection 
column Cab, a = 1,2,...,A and b = 1,2,...,B, (Figure 3). Functional 
properties of the neurons change in time (denoted by ), and are described 
by eye specificity and receptive-field polarity, ei(), and pi(), respectively: 
ei('r),pi(-) e [--1,1] C R, i-1,2,...,N, -=O,l,...,Tmaz. 
The values of eye specificity and polarity represent the prop options of synapses from 
competing types of retinal ganglion cells (there are four type of ganglion cells -- 
from different eyes and with ON or OFF polarity). i -- --1 (i -- 1) denotes that 
the i-th cell is receiving input solely from the opposite (same side) retina. Similarly, 
pi = -1 (pi = 1) denotes that the cell input is pure ON (OFF) center. Intermediate 
values of i and Pi imply that the cell does not have pure properties (it receives 
136 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli 
x 
Cab 
b e{1,..., B} 
v 
y a {1,..., A} 
Figure 3: Geometry of the model. LGN cells ci (i = 1, 2,.. N) have fixed random, 
and uniformly-distributed locations ri within a volume V ' , and belong to one 
projection column Cab. 
input from retinal ganglion cells of both eyes and with different polarities). Initial- 
ly, at r = 0, all LGN cells are characterized by el, Pi = O. This corresponds to 
two possibilities: no retinal ganglion cells synapse on any LGN cell, or proportions 
of synapses from different ganglion cells on all LGN neurons are equal, i.e., neu- 
rons possess completely undetermined functionality because of competing inputs of 
equal strength. As the neurons mature and acquire functional properties, their eye 
specificity and polarity reach their asymptotic values, -I-1. 
Even when cells are not completely mature, we will refer to them as being of four 
different types, depending on the signs of their functional properties. Following 
accepted anatomical notation, we will label them as 6, 5, 4, and 3. We denote 
eye specificity of cell types 6 and 4 as negative, and cell types 5 and 3 as positive. 
Polarity of cell types 6 and 5 is negative, while polarity of types 4 and 3 is positive. 
Cell functional properties are subject to the dynamics described in the following 
section. The process of LGN development starts from its foveal part, since in the 
retina it is the fovea that matures first. As more peripheral parts of the retina 
mature, their ganglion cells start to compete to establish permanent synapses on 
LGN cells. In this sorting process, each LGN cell gradually emerges with permanent 
synapses that connect only to several neighboring ganglions of the same type. A 
wave of gradual development of functionality sweeps through the nucleus. The 
driving force for this maturation process is described by localized cell interactions 
modulated by external influences. The particular pattern of the foveal lamination 
is shaped by external forces, and later serves as a starting point for a "propagation 
of sameness" of cell properties. Such a sameness propagation produces clustering of 
similar cells and formation of layers. It should be stressed that cells do not move, 
only their characteristics change. 
2.2 DYNAMICS 
The variables describing cell functional properties are subject to the following dy- 
namics 
i(r+l) = ei(r)+Aei(r)+rl, 
Pi (f q- 1) -- Pi (r) + Api (r) + r/p, i = 1,2,... ,N. (1) 
Morphogenesis of the Lateral Geniculate Nucleus 13 7 
In Eq. (1), there are two contributions to the change of the intermediate variables 
6i(r) and/Si(r). The first is deterministic, given by 
+ 
2 t 
(1-p(r)) t 
J3ab. (2) 
The second is a stochastic contribution corresponding to fluctuations in the growth 
of the synapses between retinal ganglion cells and LGN neurons. This noise in 
synaptic growth plays both a driving and a stabilizing role to be explained below. 
We explain the meaning of the variables in Eq. (2) only for the eye specificity variable 
el. The corresponding parameters for polarity Pi have similar interpretations. 
The parameter c (ri) is the rate of cell development. This rate is the same for eye 
specificity and polarity. It depends on the position ri of the cell in order to allow 
for spatially non-uniform development. The functional form of c (ri) is given in the 
Appendix. 
The term Ein (ri) = E7= 1 eif (Iri- rjl ) is effectively a cell force field. This field 
influences the development of nearby cells and promotes clustering of same type 
of cells. It depends on the maturity of the generating cells and on the distance 
between cells through the interaction function f(6). We chose for f(6) a Gaussian 
form, i.e., f (6) = exp (-82/a2), with characteristic interaction distance a. 
The external influences on cell development are incorporated in the term for the ex- 
ternal field Eet (ri). This external field plays two roles: it launches a particular lam- 
inar configuration of the system (in the foveal part of the LGN), and determines its 
peripheral pattern. It has, thus, two contributions E (ri) = Eef (ri) + EePt (ri). 
p 
The exact forms of ELt(ri) and Et(ri) are provided in the Appendix. 
The nonlinear term (1- e) in Eq. (2) ensures that 4-1 are the only stable fixed 
points of the dynamics. The neuronal properties gradually converge to either of 
these fixed points capturing the maturation process. This term also stabilizes the 
dynamic variables and prevents them from diverging. 
The last term/3,b (r) reflects the strict columnar organization of the maps. At each 
step of the development the proportion of all four types of LGN cells is calculated 
within a single column Ca, and/3,o(r) for different types t is adjusted such that all 
types are equally represented. Without this term, the cell organization degenerates 
to a non-laminar pattern (the system tries to minimize the surfaces between cell 
clusters of different type). The exact form of/3, (r) is given in the Appendix. 
At each stage of LGN development, cells receive input from retinal ganglion cells 
of particular types. This means that eye specificity and polarity of LGN cells 
are not independent variables. In fact, they are tightly coupled in the sense that 
le()l = Ip)l should hold for all cells at all times. This gives rise to coupled 
dynamics described by 
138 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli 
= l, I) 
= 
pi(q-1) = mi(-q-1) sgn(15i(-q-1)),i-- 1,2,...,N. (3) 
The blind spot gaps are modeled by not allowing cells in certain columns to acquire 
types of functionality for which retinal projections do not exist, e.g., from the blind 
spot of the opposite eye. Accordingly, ei is not allowed to become negative. Thus, 
some cells never reach a pure state ei,Pi - +1. It is assumed that in reality such cells 
die out. Of all quantities and parameters, only variables describing the neuronal 
receptive fields (el and Pi) are time-dependent. 
3 RESULTS 
We simulated the dynamics described by Eqs. (1, 2, 3), typically for 100,000 time 
steps. Depending on the rate of cell development, mature states were reached in 
about 10,000 steps. The maximum value of a was 0.0001. We used an interaction 
function with a = 1. 
First, we considered a two-dimensional LGN, V 
with S - 10 and Sz -- 6. There were ten projection columns (with equal size) along 
the  axis. An initial pattern was started in the foveal part by the external field. 
The size of the gaps g measured in terms of the interaction distance rr was crucial 
for pattern development. When the developmental wave reached the gaps, layer 6 
could "jump" its gap and continued to spread peripherally if the gap was sufficiently 
narrow (g/rr < 1.5). If its gap was not too narrow (g/rr > 0.5), layer 4 completely 
stopped (since cells in the gaps were not allowed to acquire negative eye specificity) 
and so layers 5 and 3 were able to merge. Cells of type 4 reappeared after the gaps 
(Figure 4, right side, shows behavior similar to the two-dimensional model) because 
of the required equal representation of all cell types in the projection columns, and 
because of noise in cell development. Energetically, the most favorable position 
of cell type 4 would be on top of type 6, which is inconsistent with experimental 
observations. Therefore, one must assume the existence of an external field in the 
peripheral part that will drive the system away from its otherwise preferred state. 
If the gaps were too large (g/rr > 1.5), cells of type 6 and 4 reappeared after the 
gaps in a more or less random vertical position and caused transitions of irregular 
nature. On the other hand, if the gaps were too narrow (g/rr < 0.5), both layers 
6 and 4 could continue to grow past their gaps, and no transition between laminar 
patterns occurred at all. When g/rr was close to the above limits, the pattern after 
the gaps differed from trial to trial. For the two-dimensional system, a realistic 
peripheral pattern always occurred for 0.7 < g/a < 1.2. 
We simulated a three-dimensional system with size S = 10, Su -- 10, and Sz = 6, 
and projection columns ordered in a 10 by 10 grid. The topology of the system 
is different in two and three dimensions: in two dimensions the gaps interrupt the 
layers completely and, thus, induce perturbations which cannot be by-passed. In 
three dimensions the gaps are just holes in a plane and generate localized pertur- 
bations: the layers can, in principle, grow around the gaps maintaining the initial 
laminar pattern. Nevertheless, in the three-dimensional case, an extended transi- 
Morphogenesis of the Lateral Geniculate Nucleus I $9 
5 
4 
3 
gap 
5 
4 
3 
Figure 4: Left: Mature state of the macaque LGN -- result of the three-dimensional 
model with 4,800 cells. Spheres with different shades represent cells with different 
properties. Gaps in strata 6 and 4 (this gap is not visible) are coded by the darkest 
color, and coincide with the transition surface between 4- and 2-1ayered patterns. 
Right: A cut of the three-dimensional structure along its plane of symmetry. A 
two-dimensional system exhibits similar organization. Compare with upper layers 
in Figure 1. Spatial segregation between layers is not modeled explicitly. 
tion was triggered by the gaps. The transition surface, vhich passed through the 
gaps and vas oriented roughly perpendicularly to the x axis, cut completely across 
the nucleus (Figure 4). 
Several factors were critical for the general behavior of the system. As in two dimen- 
sions, the size of the gaps must be within certain limits: typically 0.5 < g/a < 1.0. 
These limits depend on the curvature of the wavefront. The gaps must lie in a certain 
"inducing" interval along the x axis. If they were too close to the origin, the foveal 
pattern vas still more stable, so no transition could be induced there. However, a 
spontaneous transition might occur dovnstream. If the gaps vere too far from the 
origin, a spontaneous transition might occur before them. The occurrence and lo- 
cation of a spontaneous transition, (therefore, the limits of the "inducing" interval) 
depended on the external-field parameters. A realistic transition was observed only 
when the front of the developmental wave had sufficient curvature when it reached 
the gaps. Underlying anatomical reasons for a sufficiently curved front along the 
main axis could be the curvature of the nucleus, differences in layer thickness, or 
differences in ganglion-cell densities in the retinas. 
Propagation of the developmental wave away from the gaps was quite stable. Before 
and after the gaps, the wave simply propagated the already established patterns. 
In a system without gaps, transitions of variable shape and location occurred vhen 
the peripheral contribution to the external fields vas sufficiently large; a weaker 
contribution allowed the foveal pattern to propagate through the entire nucleus. 
4 SUMMARY 
present a model that successfully captures the most important features of 
macaque LGN morphogenesis. It produces realistic laminar patterns and supports 
140 Svilen Tzonev, Klaus Schulten, Joseph G. Malpeli 
the hypothesis (Lee & Malpeli, 1994) that the blind spot gaps trigger the transition 
between patterns. It predicts that critical factors in LGN development are the size 
and location of the gaps, cell interaction distances, and shape of the front of the 
developmental wave. The model may be general enough to incorporate the LGN 
organizations of other primates. Small singularities, similar to the blind spot gaps, 
may have an extended influence on global organization of other biological systems. 
Acknowledgements 
This work has been supported by a Beckman Institute Research Assistantship, and 
by grants PHS 2P41 RR05969 and NIH EY02695. 
References 
J.H. Kaas, R.W. Guillery & J.M. Allman. (1972) Some principles of organization 
in the dorsal lateral geniculate nucleus, Brain Behar. Evol., 6: 253-299. 
D.Lee & J.G.Malpeli. (1994) Global Form and Singularity: Modeling the Blind 
Spot's Role in Lateral Geniculate Morphogenesis, Science, 263: 1292-1294. 
J.G. Malpeli & F.H. Baker. (1975) The representation of the visual field in the 
lateral geniculate nucleus of Macaca mulatta, J. Comp. Neurol., 161: 569-594. 
P.H. Schiller & J.G. Malpeli. (1978) Functional specificity of LGN of rhesus monkey, 
J. Neurophysiol., 41: 788-797. 
APPENDIX 
The form of a(x,y,z) (with ao = 0.0001)was chosen as 
a(x,y,z) = a0 (0.1 +exp(-(y- Su/2)2)). 
(4) 
Foveal external fields of the following form were used: 
E f t'x , z) -- 1010(z-d)-2O(z-2d)+2O(z-3d)-O(d-z)]exp(-x) 
ext \ , Y, -- 
Peft(x, y, z) -- 10 [20(z- 2d)- 1] exp(-x), (5) 
where d = Sz/4 is the layers' thickness and the "theta" function is defined as 
O(x) = 1, x > 0 and O(x) = 0, x < 0. Peripheral external fields (in fact they are 
present everywhere but determine the pattern in the peripheral part only) were 
chosen as 
 z) - 5120(z-2d)-l] 
Eet (x, y, 
Pf(x,y,z) = 5[O(z-d)-2O(z-2d)+2O(z-3d)-O(d-z)]. (6) 
/tab(r ) was calculated in the following ;vay: at any given time r, within the column 
Cab, we counted the number Nt(r) of cells, that could be classified as one of the 
four types t - 3,4,5,6. Cells with ei(r) or pi(7-) exactly zero were not counted. 
The total number of classified cells is then Nab (r) 6 
= Et:a Nb ('r). If there were no 
classified cells (Nab(r) = 0), then s(r) was set to one for all t. Otherwise the ratio 
of different types was calculated: t - 
na, - N,(r)/Na,(r). In this way we calculated 
/3t& (r) = 4 - 12 nab, t = 3, 4, 5, 6. (7) 
If/3tb(r) was negative it was replaced by zero. 
