92 Cowan and Friedman 
Development and Regeneration of Eye-Brain 
Maps: A Computational Model 
J.D. Cowan and A.E. Friedman 
Department of Mathematics, Committee on 
Neurobiology, and Brain Research Institute, 
The University of Chicago, 5734 S. Univ. Ave., 
Chicago, Illinois 60637 
ABSTRACT 
We outline a computational model of the development and regenera- 
tion of specific eye-brain circuits. The model comprises a self-organiz- 
ing map-forming network which uses local Hebb rules, constrained by 
molecular markers. Various simulations of the development of eye- 
brain maps in fish and frogs are described. 
1 INTRODUCTION 
The brain is a biological computer of immense complexity comprising highly specialized 
neurons and neural circuits. Such neurons are interconnected with high specificity in 
many regions of the brain, if not in all. There are also many observations which indicate 
that there is also considerable circuit plasticity. Both specificity and plasticity are found 
in the development and regeneration of eye-brain connections in vertebrates. Sperry 
(1944) first demonstrated specificity in the regeneration of eye-brain connections in frogs 
following optic nerve section and eye rotation; and Gaze and Sharma (1970) and Yoon 
(1972) found evidence for plasticity in the expanded and compressed maps which 
regenerate following eye and brain lesions in goldfish. There are now many experiments 
which indicate that the formation of connections involves both specificity and plasticity. 
Development and Regeneration of Eye-Brain Maps: A Computational Model 93 
1.1 EYE-BRAIN MAPS AND MODELS 
Fig. 1 shows the retinal map found in the optic lobe or tectum of fish and frog. The map 
is topological, i.e.; neighborhood relationships in the retina are preserved in the optic 
tectum. How does such a map develop? Initially there is considerable disorder in the 
1. rtias r. rtias 
1. oldtit r. 
Figure 1: The normal retino-tectal map in fish and frog. Temporal 
retina projects to (contralateral) rostral rectum; nasal retina to 
(contralateral) caudal tectum. 
pathway: retinal ganglion cells make contacts with many widely dispersed tectal neurons. 
However the mature pathway shows a high degree of topological order. How is such an 
organized map achieved? One answer was provided by Prestige & Willshaw (1975): 
retinal axons and tectal neurons are polarized by contact adhesion molecules distributed 
such that axons from one end of the retina are stickier than those from the other end, and 
neurons at one end of the tectum are (correspondingly) stickier than those at the other 
end. Of course this means that isolated retinal axons will all tend to stick to one end of 
the tectum. However if such axons compete with each other for tectal terminal sites (and 
if tectal sites compete for retinal axon terminals), less sticky axons will be displaced, and 
eventually a topological map will form. The Prestige-Willshaw theory explains many ob- 
servations indicating neural specificity. It does not provide for plasticity: the ability of 
retino-tectal systems to adapt to changed target conditions, and vice-versa. Willshaw and 
von der Malsburg (1976, 1977) provided a theory for the plasticity of map 
reorganization, by postulating the synaptic growth in development is Hebbian. Such a 
mechanism provides self-organizing properties in retino-tectal map formation and reor- 
ganization. Whitelaw & Cowan (1981) combined both sticky molecules and Hebbian sy- 
naptic growth to provide a theory which explains both the specificity and plasticity of 
map formation and reorganization in a reasonable fashion. 
There are many experiments, however, which indicate that such theories are too simple. 
Schmidt & Easter (1978) and Meyer (1982) have shown that retinal axons interact with 
94 Cowan and Friedman 
each other in a way which influences map formation. It is our view that there are 
(probably) at least two different types of sticky molecules in the system: those described 
above which mediate retino-tectal interactions, and an additional class which mediates 
axo-axonal interactions in a different way. In what follows we describe a model which 
incorporates such interactions. Some aspects of our model are similar to those introduced 
by Willshaw & yon der Malsburg (1979) and Fraser (1980). Our model can simulate 
almost all experiments in the literature, and provides a way to titrate the relative strenghts 
of intrinsic polarity markers mediating retino-tectal interactions, (postulated) positional 
markers mediating axo-axonal interactions, and stimulus-driven Hebbian synaptic 
changes. 
2 MODELS OF MAP FORMATION AND REGENERATION 
2.1. THE WHITELAW-COWAN MODEL 
Let sij be the strength or weight of the synapse made by the ith retinal axon with the jth 
rectal cell. Then the following differential equation expresses the changes in sij: 
ij = cij (ri' or) tj -  (Nr'l -i + Nt'l -j )(cij (ri' or) tj) (1) 
where N r is the number of retinal ganglion cells and N t the number of tectal neurons, cij 
is the "stickiness" of the ijth contact, r i denotes retinal activity and tj = :Eisijr i is the corre- 
sponding tectal activity, and ot is a constant measuring the rate of receptor alestabiliza- 
tion (see Whitelaw & Cowan (1981) for details). In addition both retinal and tectal ele- 
ments have fixed lateral inhibitory contacts. The dynamics described by eqn.1 is such 
that both Y_,isij and :Ejsij tend to constant values T and R respectively, where T is the total 
amount of tectal receptor material available per neuron, and R is the total amount of ax- 
onal material available per retinal ganglion cell: thus if sij increases anywhere in the net, 
other synapses made by the ith axon will decrease, as will other synapses on the jth tectal 
neuron. In the current terminology, this process is referred to as "winner-take-all". 
For purposes of illustration consider the problem of connecting a line of N r retinal 
ganglion cells to a line of N t rectal cells. The resulting maps can then be represented by 
two-dimensional matrices, in which the area of the square at the ijth intersection 
represents the weight of the synapse between the ith retinal axon and the jth tectal cell. 
The normal retino-tectal map is represented by large squares along the matrix diagonal., 
(see Whitelaw & Cowan (1981) for terminology and further details). It is fairly obvious 
that the only solutions to eqn. (1) lie along the matrix diagonal, or the anti-diagonal, as 
shown in fig. 2. These solutions correspond, respectively, to normal and inverted 
topological maps. It follows that if the affinity cij of the ith retinal ganglion cell for the 
jth tectal neuron is constant, a map will form consisting of normal and inverted local 
patches. To obtain a globally normal map iris necessary to bias the system. One way to 
do this is to suppose that cij = aia j, where a i and aj are respectively, the concentrations 
Development and Regeneration of Eye-Brain Maps: A Computational Model 95 
N t 
1 i N r 1 i N 
Figure 2: Diagonal and anti-diagonal solutions to eqn. 1. Such 
solutions correspond, respectively, to normal and inverted maps. 
of sticky molecules on the tips of retinal axons and on the surfaces of tectal neurons, and 
 is a constant. A good candidate for such a molecule is the recently discovered 
toponymic or TOP molecule found in chick retina and tectum (Trisler & Collins, 1987). 
If a i and aj are distributed in the graded fashion shown in fig. 3, then the system is 
biased in favor of the normally oriented map. 
2 
0 
1 i 1,1 r 
Figure 3: Postulated distribution of sticky molecules in the retina. 
similar distribution is supposed to exist in the tectum. 
A 
2.2 INADEQUACIES 
The Whitelaw-Cowan model simulates the normal development of monocular retinotec- 
tal maps, starting from either diffuse or scrambled initial maps, or from no map. In addi- 
tion it simulates the compressed, expanded, translocated, mismatched and rotated maps 
which have been described in a variety of surgical contexts. However it fails in the 
following respects: a. Although tetrodotoxin (TTX) blocks the refinement of refinotopic 
maps in salamanders, a coarse map can still develop in the absence of retinal activity 
Harris (1980). The model will not simulate this effect. b. Although the model simulates 
the formation of double maps in "classical" compound eyes {made from a half-left and a 
half right eye} (Gaze, Jacobson, & Szekely, 1963), it fails to account for the 
reprogramming observed in "new" compound eyes {made by cutting a slit down the 
middle of a tadpole eye} (Hunt & Jacobson, 1974), and fails to simulate the forming of a 
96 Cowan and Friedman 
normal retinotopic map to a compound tectum {made from two posterior halves} 
(Sharma, 1975). 
109 87 65 43 21 109 87 65 43 21 
1 2 34 56 78 910 1 2 34 56 78 910 
r'i' ttctm 
Figure 4: The normal and expanded maps which form after the prior 
expansion of axons from a contralateral half-eye. The two maps are 
actually superposed, but for ease of exposition are shown separately. 
hR reti.u rig reti.u 
12345 54321 
1 2 34 5 6 78 910 
rJ,g' tectv. 
Figure 5: Results of Meyer's experiment. Fibers from the right half- 
retina fail to contact their normal targets and instead make contact with 
available targets, but with reversed polarity. 
c. More significantly, it fails to account for the apparent retinal induction reported by 
Schmidt, Cicerone & Easter (1978) in which following the expansion of retinal axons 
from a goldfish half-eye over an entire (contralateral) tectum, and subsequent sectioning 
of the axons, diverted retinal axons from the other (intact) eye are found to expand over 
the tectum, as if they were also from a half-eye. This has been interpreted to imply that 
the tectum has no intrinsic markers, and that all its markers come from the retina (Chung 
& Cooke, 1978). However Schmidt et.al. also found that the diverted axons also map 
normally. Fig. 4 shows the result. d. There is also an important mismatch experiment 
Development and Regeneration of Eye-Brain Maps: A Computational Model 97 
carded out by Meyer (1979) which the model cannot simulate. In this experiment the 
left half of an eye and its attached retinal axons are surgically removed, leaving an intact 
normal half-eye map. At the same time the right half the other eye and its attached axons 
are removed, and the axons from the remaining half eye are allowed to innervate the 
tectum with the left-half eye map. The result is shown in fig. 5. e. Finally, there are 
now a variety of chemical assays of the nature of the affinities which retinal axons have 
for each other, and for tectal target sites. Thus Bonhoffer and Huff (1980) found that 
growing retinal axons stick preferentially to rostral tectum. This is consistent with the 
model. However, using a different assay Halfter, Claviez & Schwarz (1981) also found 
that tectal fragments tend to stick preferentially to that part of the retina which 
corresponds to caudal rectum, i.e.; to nasal retina. This appears to contradict the model, 
and the first assay. 
3 A NEW MODEL FOR MAP FORMATION 
The Whitelaw-Cowan model can be modified and extended to replicate much of the data 
described above. The first modification is to replace eqn. 1 by a more nonlinear equation. 
The reason for this is that the above equation has no threshold below which contacts can- 
not get established. In practice Whitelaw and I modified the equations to incorporate a 
small threshold effect. Another way is to make synaptic growth and decay exponential 
rather than linear. An equation expressing this can be easily formulated, which also in- 
corporates axo-axonal interactions, presumed to be produced by neural contact 
adhesion molecules (nCAM) of the sort discovered by Edelman (1983) which seem to 
mediate the axo-axonal adhesion observed in tissue cultures by Boenhoffer & Huff 
(1985). The resulting equations take the form: 
ij = Xj + cij [Slij + (r i - c)tj] sij 1 
- sij (T-1Ei + R' Ej ){Xj + cij [Slij + (r i - c)tj] sij ) 
(2) 
where Xj represents a general nonspecific growth of retinotectal contacts, presumed to 
be controlled and modulated by nerve growth factor (Campenot, 1982). The main 
difference between eqns. 1 and 2 however, lies in the coefficients cij. In eqn. 1, cij = 
aia j. In eqn. 2, cij expresses several different effects: (a). Instead of just one molecular 
species on the tips of retinal axons and on corresponding tectal cell surfaces, as in eqn. 1, 
two molecular species or two states of one species can be postulated to exist on these 
sites. In such a case the term aia j is replaced by _abaibj where a and b are the 
different species, and the sum is over all possible combinations aa, ab etc. A number of 
possibilities exist in the choice of ab- One possibility that is consistent with most of the 
biochemical assays described earlier is aa = bb < ab = ba in which each species 
prefers the other, the so-called heterophilic case. (b) The mismatch experiment cited 
earlier (Meyer, 1979) indicates that existing axon projections tend to exclude other axons, 
especially inappropriate ones, from innervating occupied areas. One way to incorporate 
such geometric effects is to suppose that each axon which establishes contact with a 
tectal neuron occludes tectal markers there by a factor proportional to its synaptic 
98 Cowan and Friedman 
weight sij. Thus we subtract from the coefficient cij a fraction proportional to T-1 .'kSkj 
where -'k means -k  i. (c) The mismatch experiment also indicates that map for- 
marion depends in part on a tendency for axons to stick to their retinal neighbors, in addi- 
tion to their tendency to stick to rectal cell surfaces. We therefore append to cij the term 
-'k kj fik where Skj is a local average of Skj and its nearest tectal neighbors, and where 
fik measures the mutual stickiness of the ith and kth retinal axons: non-zero only for 
nearest retinal neighbors. {Again we suppose this stickiness is produced by the 
interaction of two molecular species etc.; specifically the neuronal CAMs discovered by 
Edelman, but we do not go into the details}. (d) With the introduction of occlusion 
effects and axo-axonal interactions, it becomes apparent that debris in the form of 
degenerating axon fragments adhering to rectal cells, following optic nerve sectioning, 
can also influence map formation. Incoming nerve axons can stick to debris, and debris 
can occlude markers. There are in fact four possibilities: debris can occlude tectal 
markers, markers on other debris, or on incoming axons; and incoming axons can 
occlude markers on debris. All these possibilities can be included in the dependence of 
cij on sij, Skj etc. 
The model which results from all these modifications and extensions is much more com- 
plex in its mathematical structure than any of the previous models. However computer 
simulation studies show it to be capable of correctly reproducing the observed details of 
almost all the experiments cited above. Fig. 6, for example shows a simulation of the 
retinal "induction" experiments of Schmidt et.al. 
1 i 
Figure 6: Simulation of the Schmidt et.al. retinal induction 
experiment. A nearly normal map is intercalated into an expanded map. 
This simulation generated both a patchy expanded and a patchy nearly normal map. 
These effects occur because some incoming retinal axons stick to debris left over from 
Development and Regeneration of Eye-Brain Maps: A Computational Model 99 
the previous expanded map, and other axons stick to non-occluded tectal markers. The 
axo-axonal positional markers control the formation of the expanded map, whereas the 
retino-tectal polarity markers control the formation of the nearly normal map. 
4 CONCLUSIONS 
The model we have outlined combines Hebbian plasticity with intrinsic, genetic eye- 
brain and axo-axonic markers, to generate correctly oriented retinotopic maps. It permits 
the simulation of a large number of experiments, and provides a consistent explanation of 
almost all of them. In particular it shows how the apparent induction of central markers 
by peripheral effects, as seen in the Schmidt-Cicerone-Easter experiment (Schmidt et.al. 
1978), can be produced by the effects of debris; and the polarity reversal seen in Meyer's 
experiment (Meyer 1979), can be produced by axo-axonal interactions. 
Acknowledgements 
We thank the System Development Foundation, Palo Alto, California, and The 
University of Chicago Brain Research Foundation for partial support of this work. 
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