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First published online 14 June 2006
doi: 10.1242/dev.02430

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Analysis of mouse EphA knockins and knockouts suggests that retinal axons programme target cells to form ordered retinotopic maps

Institute for Adaptive and Neural Computation, School of Informatics, University of Edinburgh, 5 Forrest Hill, Edinburgh EH1 2QL, Scotland, UK.

View larger version (10K): [in a new window]   Fig. 1. The projection of nasotemporal retina onto rostrocaudal superior colliculus in EphA3 knockin mice compared with normal. (A) Normal; (B) homozygote; (C) heterozygote. Black-filled triangles show projections from EphA3- RGCs; green-filled inverted triangles relate to EphA3+ RGCs. Redrawn from Brown et al. (Brown et al., 2000); original data kindly provided by Greg Lemke.

View larger version (23K): [in a new window]   Fig. 2. How the total densities of EphA receptor in the EphA3+ cells (green) and the EphA3- cells (black) vary along the nasotemporal axis of the retina, for the six experimental cases analysed.(A,B) Homozygous/heterozygous EphA3 knockin; (C,D) homozygous/heterozygous EphA3 knockin coupled with heterozygous EphA4 knockout; (E,F) homozygous/heterozygous EphA3 knockin coupled with homozygous EphA4 knockout. The two arrows at the side of each graph indicate the extent to which the range of variation in EphA density within the EphA3+ population overlaps with the range of variation within the EphA3- population. The six curves are calculated from the empirical result (Reber et al., 2004) that total EphA receptor density in a RGC at a distance x along the nasotemporal axis is 0.26e-2.3x + K 0 for the EphA3 cells and 0.26e-2.3x + K0 + K A3 for the EphA3 cells. The values of the constants K0 and KA3 in the six cases are: A, 1.05,1.86; B, 1.05,0.93; C, 0.54,1.77; D, 0.51,0.93; E, 0.0,1.80; F, 0.0,0.90.

View larger version (23K): [in a new window]   Fig. 3. Schematic of how synaptic connections and densities of ephrin in the tectum change continually according to the retinal induction model, illustrated for the EphB/ephrinB interaction. A one-dimensional retina innervates a one-dimensional tectum; not all connections are shown. (A) Changing the mapping. Each synaptic strength (red) is continually modified according to how closely the density of EphB (light blue) in the RGC resembles the density of ephrinB (dark blue) in the tectal cell. (B) Changing the tectal labels. An inductive signal is available at each tectal cell, made up from contributions from the individual axons innervating the cell. The magnitude of each contribution is in proportion to (1) the density of EphB in the parent RGC and (2) the strength of the synapse. The ephrinB density in each tectal cell is continually changed so as to reduce the difference between the current density of ephrinB (dark blue) and the density of induced EphB (light blue) at that cell. Note that the rules for the EphA/ephrinA system are more complicated: synaptic strengths are changed according to how close the product of the EphA density in the RGC with the ephrinA density in the tectal cell is to unity; similarly tectal ephrinA densities are changed so that the product of the induced EphA density and the actual tectal ephrinA density will tend towards unity. See Table 1 for a mathematical description of this process.

View larger version (25K): [in a new window]   Fig. 4. The mapping of the nasotemporal axis of mouse retina onto the rostrocaudal axis of the colliculus. (A-F) The six experimental cases investigated by Reber and co-workers (Reber et al., 2004). Green-filled inverted triangles are projection points from EphA3+ cells; black-filled triangles are points from EphA3- cells. Cubic fits to the data are shown. Data values kindly provided by Greg Lemke. (G-L) The plots obtained from computer simulations of the retinal induction model, assuming EphA distributions in the retina as given in Fig. 2; other parameter values are given in the legend to Table 1.

View larger version (20K): [in a new window]   Fig. 5. The density of EphA receptor in the axons innervating a particular collicular location, for the six cases considered. (A-F) Plots of the experimental data shown in Fig. 4A-F after converting retinal position into EphA receptor density using the empirical relationships given in Fig. 2. (G-L) Plots obtained from the simulation data shown in Fig. 4G-L in a similar way. The black lines show cubic fits to the data.

View larger version (55K): [in a new window]   Fig. 6. The development of the Xenopus retinotectal map according to the retinal induction model. Twenty cells are added to both retina and tectum at timesteps 0, 20000, 35000, 55000 and 80000. Each newly arriving axon makes its initial contacts at random within a 90° arc, centred at the retinotopically appropriate tectal position available to it. Top row. To the left are the colour-coded distributions of EphA and EphB over the retina. Different colour scales are used for EphA and EphB. To the right are shown the location of individual cells within the retina (circle) and within the tectum (square). The shading indicates the growth of the retina, from centre to periphery, and of the tectum, from rostrolateral to caudomedial. The required orientation of the map is that temporal (T), nasal (N), dorsal (D) and ventral (V) retina projects to rostral (R), caudal (C), lateral (L) and medial (M) tectum, respectively. Second row. The initial mapping. The two plots to the left show the distribution of the reciprocal of ephrinA density and the actual ephrinB density over the population of tectal cells existing at that time. Plotting the reciprocal of the ephrinA density eases the comparison between the distribution of EphA over the retina and that of ephrinA over the tectum. The two plots to the right show the initial map of tectum projected onto the retina through the connections made and the same information shown as a map of retina onto tectum. Each node of the lattice comprising the map of tectum onto retina marks the retinal location eliciting maximal response from a different tectal cell; each link connects the retinal positions associated with neighbouring tectal cells; similarly for the map of retina onto tectum. Third row. Similar plots at an intermediate stage in development. Bottom row. The final mapping. In the retinotectal maps, the projection from the oldest retinal cell, from central retina, is marked by a red disc and that of the most dorsotemporal retinal cell (which should project rostrolaterally) by a green disc. The distribution of EphA density over the nasotemporal retina is as given for wild-type retina in the legend to Fig. 2; EphB density along the dorsoventral axis is assumed to vary linearly.

View larger version (64K): [in a new window]   Fig. 7. Simulations of the development of connections between a full-size retina and a full-size colliculus when the initial pattern of innervation is random and the desired polarity of the map is specified by initial weak gradients of ephrins over the colliculus. (A) The initial concentration C(x) of ephrinA at a distance x along the rostrocaudal axis of the colliculus is assumed to vary as C(x) = 0.6x + 0.5II where II is a random number between 0 and 1. Similarly, the initial concentration D(y) of ephrinB along the lateromedial axis is D(y) = 0.6y + 0.5II. (B) Control with random initial ephrin distributions.

View larger version (66K): [in a new window]   Fig. 8. The effects of introducing fixed counter-gradients of EphA over the rostrocaudal axis of the colliculus. (A) The simulation of Fig. 7A was repeated with a weaker initial concentration gradient over the rostrocaudal axis of 0.4x + 0.25II together with a weak counter-gradient of EphA. The contribution from the counter-gradient is assumed to have the same functional form as the EphA gradients in the retina. At a distance x this is 0.005 + 0.013e-2.3x + 0.005II. (B) As A but with a four times stronger counter-gradient resulting in a distorted map.

View larger version (24K): [in a new window]   Fig. 9. How the precision of the collicular map projected on the retina changes over time for the four cases described in Figs 7, 8. The receptive field size (blue line) is the mean diameter of the receptive fields of the collicular cells; the receptive field separation (dashed line) is the mean distance between receptive field centres from neighbouring collicular cells. Both measures are expressed as ratios to the mean spacing between neighbouring retinal cells. Time plotted on a logarithmic scale. (A) The development of the normal map shown in Fig. 7A. Both size and separation decrease with time until they are comparable with the average spacing between retinal cells. (B) The development of the disordered map shown in Fig. 7B. The receptive field size stays at a high level; initially all receptive field centres congregate in one small region of the retina before they increase to give a large field separation. (C) The development of an ordered map with weak counter-gradients (Fig. 8A). Precision measures evolve as in A. (D) The development of a distorted map under the influence of strong counter-gradients (Fig. 8B). The final field size is significantly greater than in C, with the map being more disordered.

View larger version (14K): [in a new window]   Fig. 10. Schematic of the mapping of nasotemporal retina to rostrocaudal colliculus. (A) Normal case. The bar chart along the horizontal axis shows the EphA receptor densities of cells positioned equidistantly along nasotemporal retina. According to an induction model, a monotonically rising profile of ephrinA is distributed across the rostrocaudal axis of the colliculus (also shown as a bar chart). The pattern of connections can be seen by finding, for each retinal cell, the collicular cells that have an ephrinA density that is the inverse of the retinal EphA density. Thus retinal cells with the largest amount of EphA connect to collicular cells with the smallest amount of ephrinA, and vice versa. (B) EphA3 homozygous knockin. There are two populations of retinal cells, each following a different EphA receptor density profile. Green: EphA3+ cells; black: EphA3- cells. According to an induction model, the distribution of ephrinA across the colliculus will also form a monotonically rising profile, with retinal cells with the largest amount of EphA projecting to collicular cells with the smallest amount of ephrinA, and vice versa. The variation of receptor density over the retina now involves both profiles, stretching from its largest value (in EphA3+ cells) to its smallest (in EphA3- cells). Because of the interleaved nature of the two overall EphA receptor density profiles in the retina, the retina will be projected twice over the colliculus, the colour coding indicating which contacts are made by the EphA3+ RGCs and which by the EphA3- RGCs.

View larger version (53K): [in a new window]   Fig. 11. Simulation of the development of the homozygous EphA3 knockin map in mouse colliculus compared to normal. The 100 retinal cells were assigned equiprobably to one of two categories, distinguished by the colours green (EphA3+) and black (EphA3-). The initial pattern of connectivity was random, and initially there were weak gradients of ephrinA and ephrinB over the colliculus. The maps shown are the projections from retina onto colliculus with the projections from the EphA3+ and the EphA3- cells shown separately. (A) Control. EphA densities in EphA3+ cells and the EphA3- cells are determined as for the wild-type case used in Fig. 2. A highly organised map (bottom left panel) develops from an initially disorganised state (middle left panel). (B) Homozygous EphA3 knockin (##/++). The EphA densities as shown in Fig. 2A were used. From the initially disorganised map, two separate ordered maps are formed (bottom row), with the green EphA3+ map more rostral than the black EphA3- map. The red rectangle in the top left hand figure of A,B encloses the set of retinal cells that were used to make the one-dimensional connectivity plots shown in Fig. 4G-L.