{"title": "Gradients for Retinotectal Mapping", "book": "Advances in Neural Information Processing Systems", "page_first": 152, "page_last": 158, "abstract": "", "full_text": "Gradients for retinotectal mapping \n\nGeoffrey J. Goodhill \n\nGeorgetown Institute for Cognitive and Computational Sciences \n\nGeorgetown University Medical Center \n\n3970 Reservoir Road \nWashington IX: 20007 \n\ngeoff@giccs.georgetown.edu \n\nAbstract \n\nThe initial activity-independent formation of a topographic map \nin the retinotectal system has long been thought to rely on the \nmatching of molecular cues expressed in gradients in the retina \nand the tectum. However, direct experimental evidence for the \nexistence of such gradients has only emerged since 1995. The new \ndata has provoked the discussion of a new set of models in the ex(cid:173)\nperimentalliterature. Here, the capabilities of these models are an(cid:173)\nalyzed, and the gradient shapes they predict in vivo are derived. \n\n1 \n\nIntroduction \n\nDuring the early development of the visual system in for instance rats, fish and \nchickens, retinal axons grow across the surface of the optic tectum and establish \nconnections so as to form an ordered map. Although later neural activity refines \nthe map, it is not required to set up the initial topography (for reviews see Udin \n& Fawcett (1988); Goodhill (1992\u00bb. A long-standing idea is that the initial topog(cid:173)\nraphy is formed by matching gradients of receptor expression in the retina with \ngradients of ligand expression in the tectum (Sperry, 1963). Particular versions of \nthis idea have been formalized in theoretical models such as those of Prestige & \nWillshaw (1975), Willshaw & von der Malsburg (1979), Whitelaw & Cowan (1981), \nand Gierer (1983;1987). However, these models were developed in the absence \nof any direct experimental evidence for the existence of the necessary gradients. \nSince 1995, major breakthroughs have occurred in this regard in the experimental \nliterature. These center around the Eph (Erythropoetin-producing hepatocellular) \nsubfamily of receptor tyrosine kinases. Eph receptors and their ligands have been \nshown to be expressed in gradients in the developing retina and tectum respec(cid:173)\ntively, and to playa role in guiding axons to appropriate positions. These exciting \nnew developments have led experimentalists to discuss theoretical models differ-\n\n\fGradients/or Retinotectal Mapping \n\n153 \n\nent from those previously proposed (e.g. Tessier-Lavigne (1995); Tessier-Lavigne \n& Goodman (1996); Nakamoto et aI, (1996)). However, the mathematical conse(cid:173)\nquences of these new models, for instance the precise gradient shapes they require, \nhave not been analyzed. In this paper, it is shown that only certain combinations \nof gradients produce appropriate maps in these models, and that the validity of \nthese models is therefore experimentally testable. \n\n2 Recent experimental data \n\nReceptor tyrosine kinases are a diverse class of membrane-spanning proteins. The \nEph subfamily is the largest, with over a dozen members. Since 1990, many of the \ngenes encoding Eph receptors and their ligands have been shown to be expressed \nin the developing brain (reviewed in Friedman & O'Leary, 1996). Ephrins, the \nligands for Eph receptors, are all membrane anchored. This is unlike the majority \nof receptor tyrosine kinase ligands, which are usually soluble. The ephrins can be \nseparated into two distinct groups A and B, based on the type of membrane anchor. \nThese two groups bind to distinct sets of Eph receptors, which are thus also called \nA and B, though receptor-ligand interaction is promiscuous within each subgroup. \nSince many research groups discovered members of the Eph family independently, \neach member originally had several names. However a new standardized notation \nwas recently introduced (Eph Nomenclature Committee, 1997), which is used in \nthis paper. \nWith regard to the mapping from the nasal-temporal axis of the retina to the \nanterior-posterior axis of the tectum (figure 1), recent studies have shown the fol(cid:173)\nlowing (see Friedman & O'Leary (1996) and Tessier-Lavigne & Goodman (1996) \nfor reviews). \n\n\u2022 EphA3 is expressed in an increasing nasal to temporal gradient in the \n\nretina (Cheng et aI, 1995). \n\n\u2022 EphA4 is expressed uniformly in the retina (Holash & Pasquale, 1995). \n\n\u2022 Ephrin-A2, a ligand of both EphA3 and EphA4, is expressed in an increas(cid:173)\n\ning rostral to caudal gradient in the tectum (Cheng et aI, 1995). \n\n\u2022 Ephrin-A5, another ligand of EphA3 and EphA4, is also expressed in an \nincreasing rostral to caudal gradient in the tectum, but at very low levels \nin the rostral half of the tectum (Drescher et aI, 1995). \n\nAll of these interactions are repulsive. With regard to mapping along the comple(cid:173)\nmentary dimensions, EphB2 is expressed in a high ventral to low dorsal gradient \nin the retina, while its ligand ephrin-B1 is expressed in a high dorsal to low ventral \ngradient in the tectum (Braisted et aI, 1997). Members of the Eph family are also \nbeginning to be implicated in the formation of topographic projections between \nmany other pairs of structures in the brain (Renping Zhou, personal communica(cid:173)\ntion). For instance, EphA5 has been found in an increasing lateral to medial gradi(cid:173)\nent in the hippocampus, and ephrin-A2 in an increasing dorsal to ventral gradient \nin the septum, consistent with a role in establishing the topography of the map \nbetween hippocampus and septum (Gao et aI, 1996). \nThe current paper focusses just on the paradigm case of the nasal-temporal to \nanterior-posterior axis of the retinotectal mapping. Actual gradient shapes in this \nsystem have not yet been quantified. The analysis below will assume that certain \ngradients are linear, and derive the consequences for the other gradients. \n\n\fRETINA \n\nTECTUM \n\n154 \n\nN \n\nG. J. Goodhill \n\nc \n\nR(x) \n\nL(y) \n\n\"'-----.......;;;~x k:==::y \n\nFigure 1: This shows the mapping that is normally set up from the retina to the \ntectum. Distance along the nasal-temporal axis of the retina is referred to as x and \nreceptor concentration as R( x). Distance along the rostral-caudal axis of the tectum \nis referred to as y and ligand concentration as L(y). \n\n3 Mathematical models \n\nLet R be the concentration of a receptor expressed on a growth cone or axon, and \nL the concentration of a ligand present in the tectum. Refer to position along the \nnasal-temporal axis of the retina as x, and position along the rostral-caudal axis of \nthe tectum as y, so that R = R(x) and L = L(y) (see figure 1). Gierer (1983; 1987) \ndiscusses how topographic information could be signaled by interactions between \nligands and receptors. A particular type of interaction, proposed by Nakamoto et \nal (1996), is that the concentration of a \"topographic signal\", the signal that tells \naxons where to stop, is related to the concentration of receptor and ligand by the \nlaw of mass action: \n\nG(x, y) = kR(x)L(y) \n\n(1) \nwhere G(x, y) is the concentration of topographic signal produced within an axon \noriginating from position x in the retina when it is at position y in the tectum, \nand k is a constant. In the general case of multiple receptors and ligands, with \npromiscuous interactions between them, this equation becomes \n\nG(x, y) = L: kijRi(X)Lj(Y) \n\ni,j \n\n(2) \n\nWhether each receptor-ligand interaction is attractive or repulsive is taken care of \nby the sign of the relevant kij \u2022 \nTwo possibilities for how G(x, y) might produce a stop (or branch) signal in the \ngrowth cone (or axon) are that this occurs when (1) a \"set point\" is reached (dis(cid:173)\ncussed in, for example, Tessier-Lavigne & Goodman (1996); Nakamoto et al (1996\u00bb \n,i.e. G (x, y) = c where c is a constant, or (2) attraction (or repulsion) reaches a local \nmaximum (or minimum), i.e. &G~~,y) = 0 (Gierer, 1983; 1987). For a smooth, uni-\n\n\fGradients for Retinotectal Mapping \n\n155 \n\nform mapping, one of these conditions must hold along a line y ex: x. For simplicity \nassume the constant of proportionality is unity. \n\n3.1 Set point rule \n\nFor one gradient in the retina and one gradient in the tectum (i.e. equation 1), this \nrequires that the ligand gradient be inversely proportional to the receptor gradient: \n\nc \n\nL(x) = R(x) \n\nIf R(x) is linear (c.f. the gradient of EphA3 in the retina), the ligand concentration \nis required to go to infinity at one end of the tectum (see figure 2). One way round \nthis is to assume R(x) does not go to zero at x = 0: the experimental data is not \nprecise enough to decide on this point. However, the addition of a second receptor \ngradient gives \n\nL(x) = k1R1 (x) + k2R2(X) \n\nc \n\nIf R1 (x) is linear and R2(x) is flat (c.f. the gradient of EphA4 in the retina), then \nL (y) is no longer required to go to infinity (see figure 2). For two receptor and two \nligand gradients many combinations of gradient shapes are possible. As a special \ncase, consider R1 (x) linear, R2(x) flat, and L 1(y) linear (c.f. the gradient of Elfl in \nthe tectum). Then L2 is required to have the shape \nL ( ) = ay2 + by \ndy + e \n2 Y \n\nwhere a, b, d, e are constants. This shape depends on the values of the constants, \nwhich depend on the relative strengths of binding between the different receptor \nand ligand combinations. An interesting case is where R1 binds only to L1 and R2 \nbinds only .to L 2 , i.e. there is no promiscuity. In this case we have \n\nL2(y) ex: y2 \n\n(see figure 2). This function somewhat resembles the shape of the gradient that \nhas been reported for ephrin-AS in the tectum. However, this model requires one \ngradient to be attractive, whereas both are repulsive. \n\n3.2 Local optimum rule \n\nFor one retinal and one tectal gradient we have the requirement \n\nR(x) aL(y) = 0 \n\nay \n\nThis is not generally true along the line y = x, therefore there is no map. The same \nproblem arises with two receptor gradients, whatever their shapes. For two recep(cid:173)\ntor and two ligand gradients many combinations of gradient shapes are possible. \n(Gierer (1983; 1987) investigated this case, but for a more complicated reaction law \nfor generating the topographic signal than mass action.) For the special case intro(cid:173)\nduced above, L 2 (y) is required to have the shape \n\nL2(y) = ay + blog(dy + e) + f \n\nwhere a, b, d, e, and f are constants as before. Considering the case of no promis(cid:173)\ncuity, we again obtain \n\nL2(y) ex: y2 \n\ni.e. the same shape for L2 (y) as that specified by the set point rule. \n\n\f156 \n\nA \n\nB \n\nc \n\nG. 1. Goodhill \n\nL \n\nL \n\nFigure 2: Three combinations of gradient shapes that are sufficient to produce a \nsmooth mapping with the mass action rule. In the left column the horizontal axis \nis position in the retina while the vertical axis is the concentration of receptor. In \nthe right column the horizontal axis is position in the tectum while the vertical axis \nis the concentration of ligand. Models A and B work with the set point but not the \nlocal optimum rule, while model C works with both rules. For models B and C, \none gradient is negative and the other positive. \n\n\fGradients for Retinotectal Mapping \n\n4 Discussion \n\n157 \n\nFor both rules, there is a set of gradient shapes for the mass-action model that is \nconsistent with the experimental data, except for the fact that they require one gra(cid:173)\ndient in the tectum to be attractive. Both ephrin-A2 and ephrin-A5 have repulsive \neffects on their receptors expressed in the retina, which is clearly a problem for \nthese models. The local optimum rule is more restrictive than the set point rule, \nsince it requires at least two ligand gradients in the tectum. However, unlike the set \npoint rule, it supplies directional information (in terms of an appropriate gradient \nfor the topographic signal) when the axon is not at the optimal location. \nIn conclusion, models based on the mass action assumption in conjunction with ei(cid:173)\nther a \"set point\" or \"local optimum\" rule can be true only if the relevant gradients \nsatisfy the quantitative relationships described above. A different theoretical ap(cid:173)\nproach, which analyzes gradients in terms of their ability to guide axons over the \nmaximum possible distance, also makes predictions about gradient shapes in the \nretinotectal system (Goodhill & Baier, 1998). Advances in experimental technique \nshould enable a more quantitative analysis of the gradients in situ to be performed \nshortly, allowing these predictions to be tested. 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Retinal axon guidance defects in mice lacking ephrin-A5 (AL(cid:173)\nl/RAGS). Soc. Neurosci. Abstracts, 23, 324. \n\n\f", "award": [], "sourceid": 1423, "authors": [{"given_name": "Geoffrey", "family_name": "Goodhill", "institution": null}]}