Gradients for retinotectal mapping 
Geoffrey J. Goodhill 
Georgetown Institute for Cognitive and Computational Sciences 
Georgetown University Medical Center 
3970 Reservoir Road 
Washington DC 20007 
eof f@iccs. eoretown. edu 
Abstract 
The initial activity-independent formation of a topographic map 
in the retinotectal system has long been thought to rely on the 
matching of molecular cues expressed in gradients in the retina 
and the tectum. However, direct experimental evidence for the 
existence of such gradients has only emerged since 1995. The new 
data has provoked the discussion of a new set of models in the ex- 
perimental literature. Here, the capabilities of these models are an- 
alyzed, and the gradient shapes they predict in vivo are derived. 
1 Introduction 
During the early development of the visual system in for instance rats, fish and 
chickens, retinal axons grow across the surface of the optic tectum and establish 
connections so as to form an ordered map. Although later neural activity refines 
the map, it is not required to set up the initial topography (for reviews see Udin 
& Fawcett (1988); Goodhill (1992)). A long-standing idea is that the initial topog- 
raphy is formed by matching gradients of receptor expression in the retina with 
gradients of ligand expression in the tectum (Sperry, 1963). Particular versions of 
this idea have been formalized in theoretical models such as those of Prestige & 
Willshaw (1975), Willshaw &von der Malsburg (1979), Whitelaw & Cowan (1981), 
and Gierer (1983;1987). However, these models were developed in the absence 
of any direct experimental evidence for the existence of the necessary gradients. 
Since 1995, major breakthroughs have occurred in this regard in the experimental 
literature. These center around the Eph (Erythropoetin-producing hepatocellular) 
subfamily of receptor tyrosine kinases. Eph receptors and their ligands have been 
shown to be expressed in gradients in the developing retina and tectum respec- 
tively, and to play a role in guiding axons to appropriate positions. These exciting 
new developments have led experimentalists to discuss theoretical models differ- 
Gradients for Retinotectal Mapping 153 
ent from those previously proposed (e.g. Tessier-Lavigne (1995); Tessier-Lavigne 
& Goodman (1996); Nakamoto et al, (1996)). However, the mathematical conse- 
quences of these new models, for instance the precise gradient shapes they require, 
have not been analyzed. In this paper, it is shown that only certain combinations 
of gradients produce appropriate maps in these models, and that the validity of 
these models is therefore experimentally testable. 
2 Recent experimental data 
Receptor tyrosine kinases are a diverse class of membrane-spanning proteins. The 
Eph subfamily is the largest, with over a dozen members. Since 1990, many of the 
genes encoding Eph receptors and their ligands have been shown to be expressed 
in the developing brain (reviewed in Friedman & O'Leary, 1996). Ephrins, the 
ligands for Eph receptors, are all membrane anchored. This is unlike the majority 
of receptor tyrosine kinase ligands, which are usually soluble. The ephrins can be 
separated into two distinct groups A and B, based on the type of membrane anchor. 
These two groups bind to distinct sets of Eph receptors, which are thus also called 
A and B, though receptor-ligand interaction is promiscuous within each subgroup. 
Since many research groups discovered members of the Eph family independently, 
each member originally had several names. However a new standardized notation 
was recently introduced (Eph Nomenclature Committee, 1997), which is used in 
this paper 
With regard to the mapping from the nasal-temporal axis of the retina to the 
anterior-posterior axis of the tectum (figure 1), recent studies have shown the fol- 
lowing (see Friedman & O'Leary (1996) and Tessier-Lavigne & Goodman (1996) 
for reviews). 
 EphA3 is expressed in an increasing nasal to temporal gradient in the 
retina (Cheng et al, 1995). 
 EphA4 is expressed uniformly in the retina (Holash & Pasquale, 1995). 
 Ephrin-A2, a ligand of both EphA3 and EphA4, is expressed in an increas- 
ing rostral to caudal gradient in the tectum (Cheng et al, 1995). 
 Ephrin-A5, another ligand of EphA3 and EphA4, is also expressed in an 
increasing rostral to caudal gradient in the tectum, but at very low levels 
in the rostral half of the tectum (Drescher et al, 1995). 
All of these interactions are repulsive. With regard to mapping along the comple- 
mentary dimensions, EphB2 is expressed in a high ventral to low dorsal gradient 
in the retina, while its ligand ephrin-B1 is expressed in a high dorsal to low ventral 
gradient in the tectum (Braisted et al, 1997). Members of the Eph family are also 
beginning to be implicated in the formation of topographic projections between 
many other pairs of structures in the brain (Renping Zhou, personal communica- 
tion). For instance, EphA5 has been found in an increasing lateral to medial gradi- 
ent in the hippocampus, and ephrin-A2 in an increasing dorsal to ventral gradient 
in the septum, consistent with a role in establishing the topography of the map 
between hippocampus and septum (Gao et al, 1996). 
The current paper focusses just on the paradigm case of the nasal-temporal to 
anterior-posterior axis of the retinotectal mapping. Actual gradient shapes in this 
system have not yet been quantified. The analysis below will assume that certain 
gradients are linear, and derive the consequences for the other gradients. 
154 G. J. Goodhill 
RETINA 
TECTUM 
N 
T R 
c 
Figure 1: This shows the mapping that is normally set up from the retina to the 
rectum. Distance along the nasal-temporal axis of the retina is referred to as x and 
receptor concentration as R(x). Distance along the rostral-caudal axis of the tectum 
is referred to as/4 and ligand concentration as 
3 Mathematical models 
Let R be the concentration of a receptor expressed on a growth cone or axon, and 
L the concentration of a ligand present in the tectum. Refer to position along the 
nasal-temporal axis of the retina as x, and position along the rostral-caudal axis of 
the tectum as/4, so that R = R(x) and L = L(y) (see figure 1). Gierer (1983; 1987) 
discusses how topographic information could be signaled by interactions between 
ligands and receptors. A particular type of interaction, proposed by Nakamoto et 
al (1996), is that the concentration of a "topographic signal", the signal that tells 
axons where to stop, is related to the concentration of receptor and ligand by the 
law of mass action: 
c(x,y) = kR(x)(y) (1) 
where G(x, y) is the concentration of topographic signal produced within an axon 
originating from position x in the retina when it is at position y in the tecturn, 
and k is a constant. In the general case of multiple receptors and ligands, with 
promiscuous interactions between them, this equation becomes 
G(x,y) = E kijti(x)Lj(y) 
i,j 
(2) 
Whether each receptor-ligand interaction is attractive or repulsive is taken care of 
by the sign of the relevant kij. 
Two possibilities for how G(x, y) might produce a stop (or branch) signal in the 
growth cone (or axon) are that this occurs when (1) a "set point" is reached (dis- 
cussed in, for example, Tessier-Lavigne & Goodman (1996); Nakamoto et al (1996)) 
, i.e. G(x, y) = c where c is a constant, or (2) attraction (or repulsion) reaches a local 
maximum (or minimum), i.e. 0G(,y) = 0 (Gierer, 1983; 1987). For a smooth, uni- 
Oy 
Gradients for Retinotectal Mapping 155 
form mapping, one of these conditions must hold along a line//cx z. For simplicity 
assume the constant of proportionality is unity. 
3.1 Set point rule 
For one gradient in the retina and one gradient in the tectum (i.e. equation 1), this 
requires that the ligand gradient be inversely proportional to the receptor gradient: 
 
If R(z) is linear (c.f. the gradient of EphA3 in the retina), the ligand concentration 
is required to go to infinity at one end of the tectum (see figure 2). One way round 
this is to assume R(z) does not go to zero at x = 0: the experimental data is not 
precise enough to decide on this point. However, the addition of a second receptor 
gradient gives 
 
If R1 (x) is linear and R2 (x) is flat (c.f. the gradient of EphA4 in the retina), then 
/`(y) is no longer required to go to infinity (see figure 2). For two receptor and two 
ligand gradients many combinations of gradient shapes are possible. As a special 
case, consider R1 (x) linear, R2 (x) flat, and L1 (y) linear (c.f. the gradient of Elfl in 
the tectum). Then/,2 is required to have the shape 
L2(y) = ay + by 
dy+e 
where a, b, d, e are constants. This shape depends on the values of the constants, 
which depend on the relative strengths of binding between the different receptor 
and ligand combinations. An interesting case is where R1 binds only to L1 and R2 
binds only .to L, i.e. there is no promiscuity. In this case we have 
L(y) cx y 
(see figure 2). This function somewhat resembles the shape of the gradient that 
has been reported for ephrin-A5 in the tectum. However, this model requires one 
gradient to be attractive, whereas both are repulsive. 
3.2 Local optimum rule 
For one retinal and one tectal gradient we have the requirement 
O/`(y) =0 
Oy 
This is not generally true along the line y = x, therefore there is no map. The same 
problem arises with two receptor gradients, whatever their shapes. For two recep- 
tor and two ligand gradients many combinations of gradient shapes are possible. 
(Gierer (1983; 1987) investigated this case, but for a more complicated reaction law 
for generating the topographic signal than mass action.) For the special case intro- 
duced above, L (y) is required to have the shape 
L2(y) = ay + blog(dy + e) + f 
where a, b, d, e, and .f are constants as before. Considering the case of no promis- 
cuity, we again obtain 
L2(y) cx y2 
i.e. the same shape for L (y) as that specified by the set point rule. 
156 G. J. Goodhill 
A 
B 
c 
Figure 2: Three combinations of gradient shapes that are sufficient to produce a 
smooth mapping with the mass action rule. In the left column the horizontal axis 
is position in the retina while the vertical axis is the concentration of receptor. In 
the right column the horizontal axis is position in the tectum while the vertical axis 
is the concentration of ligand. Models A and B work with the set point but not the 
local optimum rule, while model C works with both rules. For models B and C, 
one gradient is negative and the other positive. 
Gradients for Retinotectal Mapping 157 
4 Discussion 
For both rules, there is a set of gradient shapes for the mass-action model that is 
consistent with the experimental data, except for the fact that they require one gra- 
dient in the tectum to be attractive. Both ephrin-A2 and ephrin-A5 have repulsive 
effects on their receptors expressed in the retina, which is clearly a problem for 
these models. The local optimum rule is more restrictive than the set point rule, 
since it requires at least two ligand gradients in the tectum. However, unlike the set 
point rule, it supplies directional information (in terms of an appropriate gradient 
for the topographic signal) when the axon is not at the optimal location. 
In conclusion, models based on the mass action assumption in conjunction with ei- 
ther a "set point" or "local optimum" rule can be true only if the relevant gradients 
satisfy the quantitative relationships described above. A different theoretical ap- 
proach, which analyzes gradients in terms of their ability to guide axons over the 
maximum possible distance, also makes predictions about gradient shapes in the 
retinotectal system (Goodhill & Baier, 1998). Advances in experimental technique 
should enable a more quantitative analysis of the gradients in situ to be performed 
shortly, allowing these predictions to be tested. In addition, analysis of particular 
Eph and ephrin knockout mice (for instance ephrin-A5 (Yates et al, 1997)) is now 
being performed, which should shed light on the role of these gradients in normal 
map development. 
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