Further Studies of a Model for the 
Development and Regeneration 
of Eye-Brain Maps 
J.D. Cowan & 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 describe 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 
(genetically determined) molecular markers. Various simulations of 
the development and regeneration of eye-brain maps in fish and frogs 
are described, in particular successful simulations of experiments by 
Schmidt-Cicerone-Easter; Meyer; and Yoon. 
1 INTRODUCTION 
In a previous paper published in last years proceedings (Cowan & Friedman 1990) we 
outlined a new computational model for the development and regeneration of eye-brain 
maps. We indicated that such a model can simulate the results of a number of the more 
complicated surgical manipulations carried out on the visual pathways of goldfish and 
frogs. In this paper we describe in more detail some of these experiments, and our 
simulations of them. 
1.1 EYE-BRAIN MAPS 
We refer to figure 1 from the previous paper which shows the retinal map found in the 
optic lobe or rectum of a fish or frog. The map is topological, i.e.; neighborhood 
3 
4 Cowan and Friedman 
relationships in the retina are preserved in the optic tectum. As is well-known nearly 50 
years ago Sperry (1944) showed that such maps are quite precise and specific, in that 
maps (following optic nerve sectioning and eye rotation) regenerate in such a way that 
optic nerve fibers reconnect, more or less, to their previous rectal sites. Some 20 years 
ago 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 regeneration of 
connections involves both specificity and plasticity. 
1. 2. EXPANDED MAPS 
Such properties are seen in a series of more complicated experiments involving the 
expansion of a half-eye map to a whole rectum. These experiments were carried out by 
Schmidt, Cicerone and Easter (1978) on goldfish, in which following the expansion of 
retinal fibers from a half-eye over an entire (contralateral) rectum, and subsequent 
sectioning of the fibers, diverted retinal fibers from the other (intact) eye are found to 
expand over the rectum, as if they were also from a half-eye. This has been interpreted to 
imply that the rectum has no intrinsic positional markers to provide cues for incoming 
fibers, and that all its subsequent markers come from the retina (Chung & Cooke, 1978). 
However Schmidt et.al. also found that the diverted fibers also map normally. Figure 4 
of the previous paper shows the result. 
1. 3. COMPRESSED MAPS 
Compression is found in maps from entire eyes to ablated half recta (Gaze & Sharma, 
1970; Sharma & Gaze, 1971; Yoon, 1972). There has been considerable controversy 
concerning the results. Recently Meyer (1982) has shown that although 
electrophysiological techniques seem to provide evidence for smoothly expanded and 
compressed maps, autoradiographic techniques do not. Instead of a smooth map there 
are patches, and in many cases no real expansion or compression is seen in irradiated 
sections, at least not initially. An experiment by Yoon (1976) is relevant here. Yoon 
noticed that in the early stages of map formation under such conditions, the map is 
normal. Only after some considerable time does a compressed map form. However if 
the fibers are sectioned (cut) and allowed to regenerate a second time, compression is 
immediate. This result has been challenged (Cook, 1979), but it was subsequently 
confirmed by Schmidt (1983). 
1. 4. MISMATCHED MAPS 
In mismatch experiments, a half retina is confronted with an inappropriate half tectum. In 
Yoon's classic "mismatch" experiment (Yoon, 1972) fibers from a half-eye fragment are 
confronted with the "wrong" half-rectum: the resulting map is normally oriented, even 
though this involves displacement of retinal fibers from near the rectal positions they 
normally would occupy. 
Studies of a Model for the Development and Regeneration of Eye-Brain Maps 5 
About 12 years ago Meyer (1979) carried out another important mismatch experiment in 
which the left half of an eye and its attached retinal fibers were surgically removed, 
leaving an intact normal half-eye map. At the same time the right half the other eye and 
its attached fibers were removed, and the fibers from the remaining half eye were 
allowed to innervate the rectum with the left-half eye map. The result is shown in figure 
5 of our previous paper. Fibers from the right half-retina, labelled 1 through 5, would 
normally make contact with the corresponding tectal neurons. Instead they make contact 
with neurons 6 through 10, but in a reversed orientation. Meyer interprets this result to 
mean that optic nerve fibers show a tendency to aggregate with their nearest retinal 
neighbors. 
2 THE MODEL 
We introduced our model in last year's NIPS proceedings (Cowan & Friedman 1990). We 
here repeat some of the details. Let sij be the strength or weight of the synapse made by 
the ith retinal fiber with the jth rectal cell. Then the following system of differential 
equations expresses the changes in sij: 
ij = Xj + cij [J. lij + (r i - ot)tj] sij 
-  sij (T'li + R-Ij){X j + cij [Slij + (ri - o0tj] sij } (1) 
where i = 1, 2, .... , N r, the number of retinal ganglion cells and j = 1, 2, .... , N t, the 
number of tectal neurons, cij is the "stickiness" of the ijth contact, r i denotes retinal 
activity and tj = I;isijr i is the corresponding tectal activity, and o is a constant measuring 
the rate of receptor alestabilization (see Whitelaw & Cowan (1981) for details). In 
addition both retinal and tectal elements have fixed lateral inhibitory contacts. The 
dynamics described by eqn. 1 is such that both I;isij and I;jsij 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 axonal material available per retinal ganglion cell: 
thus if sij increases anywhere in the net, other synapses made by the ith fiber will 
decrease, as will other synapses on the jth tectal neuron. In the current terminology, this 
process is referred to as "winner-take-all". 
In addiiton Xj represents a general nonspecific growth of retinotectal contacts, presumed 
to be controlled and modulated by nerve growth factor (Campenot, 1982). Recent 
observations (Davies et. al., 1987) indicate that the first fibers to reach a given target 
neuron stimulate it to produce NGF, which in turn causes more fiber growth. We 
therefore set Xj = T-l_isijX where X is a constant. _isij is the instantaneous value of 
receptor material used to make contacts, and T is the total amount available, so Xj --> X 
as the jth neuron becomes innervated. The coefficient ij represents a postulated random 
depolarization which occurs at synapses due to the quantal release of neurotransmitter-- 
the analog of end-plate potentials (Walmsley et.al., 1987). Thus even if r i = 0, map 
formation can still occur. However the resulting maps are not as sharp as those formed in 
6 Cowan and Friedman 
the presence of retinal activity. Of course if ij = 0, as might be the case if o- 
bungarotoxin is administered, then ij = Xj(1- sij ) and sij --> 1, i.e.; all synapses of 
equal strength. 
It is the coefficients cij. which determine the nature of the solution to eqn. 1. These 
coefficients express the contact adhesion strengths of synapses. We suppose that such 
adhesions are generated by fixed distributions of molecules embedded in neural surface 
membranes. We postulate that the tips of retinal axons and the surfaces of tectal cells 
display at least two molecular species, labelled a and b, such that cij = .abaibj and the 
sum is over all possible combinations aa, ab etc. A number of possibilities exist in the 
choice of tab and of the spatial distribution of a and b. One possibility that is consistent 
with most of the assays which have been carded out (Trisler & Collins (1987), 
Bonhoffer and Huff (1980), Halfter, Claviez & Schwarz (1981), Boenhoffer & Huff 
(1985)) is aa = Ebb > 0 > tab = ba in which each species prefers itself and repels the 
other, the so-called homophilic case, with ai and bi as shown in figure 1. 
o o 
! i N r 
Figure 1: Postulated distribution of sticky molecules 
in the retina. A similar distribution is supposed to 
exist in the tectum. 
The mismatch and compound eye experiments indicate that map formation depends in 
part on a tendency for fibers tO stick to their retinal neighbors, in addition to their 
tendency to stick to tectal cell surfaces. We therefore append to cij the term 'k kj fik 
where -kj is a local average of Ski and its nearest rectal neighbors, where fik measures 
themutual stickiness of the ith and kth retinal fibers, and where -'k means -k  i- Fig. 2 
shows the postulated form of fik. {Again we suppose this stickiness is produced by the 
interaction of two molecular species etc.; specifically theneural contact adhesion 
molecules (nCAM) of the sort discovered by Edelman (1983)which seem to mediate the 
fiber-fiber adhesion observed in tissue cultures by Boenhoffer & Huff (1985), but we do 
not go into the details}. 
Studies of a Model for the Development and Regeneration of Eye-Brain Maps 7 
Figure 2: The fik surface. Retinal fibers are attracted 
only to themselves or to their immediate retinal 
neighbors. 
Meyer's mismatch experiment also indicate that existing fiber projections tend to exclude 
other fibers, especially inappropriate ones, from innervating occupied areas. One way to 
incorporate such geometric effects is to suppose that each fiber which establishes contact 
with a rectal neuron occludes rectal markers there by a factor proportional to its synaptic 
weight sij. Thus we subtract from the coefficient cij a fraction proportional to T' 1 .'kSkj ' 
With the introduction of occlusion effects and fiber-fiber interactions, it becomes ap- 
parent that debris in the form of degenerating fiber fragments adhering to rectal cells, 
following optic nerve sectioning, can also influence map formation. Incoming nerve 
fibers can stick to debris, and debris can occlude markers. There are in fact four possi- 
bilities: debris can occlude rectal markers, markers on other debris, or on incoming fibers; 
and incoming fibers can occlude markers on debris. All these possibilities can be in- 
cluded in the dependence of cij on sij, Ski etc. Note that such debris is supposed to 
decay, and eventually disappear. 
3 SIMULATIONS 
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. For purposes of illustration we consider the 
problem of connecting a line of N r retinal cells to a line of N t tectal 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 
fiber 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). 
8 Cowan and Friedman 
3.1 THE SCHMIDT ET. AL. EXPERIMENT 
Figure 3, for example shows a simulation of the retinal "induction" experiments of 
Schmidt et. al. This simulation generated both an expanded map and a nearly normal 
patch, interacting to form patches. These effects occur because some incoming retinal 
fibers stick to debris left over from the previous expanded map, and other fibers stick to 
non-occluded tectal markers. The fiber-fiber markers control the regeneration of the 
expanded map, whereas the retino-tectal markers control the formation of the nearly 
normal map. 
1 i lir 
1 
II t 
'%1 
Figure 3: Simulation of the Schmidt et.al. retinal in- 
duction experiment. A nearly normal map is interca- 
lated into an expanded map. 
, , arlnn,-,anr-rlnrla o   
 . . ,, Dn,-, an r-DnDDn D,', 
. . .  nClrn n a n n,-,nr-lr-lrn 
1200 $ooo 
r'] , 
  , e  n rqrqrqrqrq O []  ,n  ,   
..........   [] Cll-II-lCl 
16000 
Figure 4: Simulation of the Yoon second 
compression experiment (see text for details). 
Studies of a Model for the Development and Regeneration of Eye-Brain Maps 9 
3.2 THE YOON SECOND COMPRESSION EXPERIMENT 
Yoon's demonstration of immediate second compression can also be simulated. Figure 4 
shows details of the simulation. At an early stage just after the first cut, both a normal 
and a compressed map are forming. The normal map eventually disappears, leaving only 
a compressed map. After the second cut however, a compressed map forms immediately. 
Again it is the debris which carries fiber-fiber markers that control map formation. 
3.3 THE MEYER MISMATCH EXPERIMENT 
It is evident that fiber-fiber interactions are important in controlling map formation. The 
Meyer mismatch experiment shows this quite clearly. A simulation of this experiment 
also shows the effect. If fik, the mutual stickiness of neighboring fibers is not strong 
enough, retino-tectal markers dominate, and the mismatched map forms with normal 
polarity. However if fik is large enough, Meyer's result is found, the mismatched map 
forms with a reversed polarity. Figure 5 shows the details. 
1234-56789 987654.32! 1234.56789 :1t:)987654-32! 
Figure 5: Simulation of the Meyer mismatch 
experiment (see text for details). 
4 CONCLUSIONS 
The model we have outlined generates 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 et. al., can be produced by the effects of 
debris, as can Yoon's observations of immediate second compression. Affinity markers 
are seen to play a key role in such effects, as they do in the polarity reversal seen in 
Meyer's experiment. 
10 Cowan and Friedman 
In summary much of the complexity of the many regeneration experiments which have 
been carried out in the last fifty years can be understood in terms of the effects produced 
by contact adhesion molecules with differing affinities, acting to control an activity- 
dependent self-organizing mechanism. 
Acknowledgements 
We thank The University of Chicago Brain Research Foundation for partial support of 
this work. 
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