Reanalysis of EphA3 Knock-In Double Maps in Mouse Suggests That Stochasticity in Topographic Map Formation Acts at the Retina Rather than between Competing Mechanisms at the Colliculus

Examples from the database of the distribution of the variation of the azimuthal (nasotemporal) component of phase representing visual field position recorded along three straight line tracks drawn on the colliculus representing 1D sections at three standard elevations of −25° (orange), 0° (cyan), and +25° (brown) for WT, heterozygous (EphA3^ki/+) and homozygous (EphA3^ki/ki) knock-ins. A, B, WT. C–H, K, EphA3^ki/+. I, J, L, EphA3^ki/ki. Rows 1, 2 show the distributions of eligible (gray) and rejected (green) pixels on the colliculus (row 2) and corresponding visual field positions (row 1). Each colored line on the colliculus (row 2) is the straight line fit to the positions of all eligible pixels for which the corresponding visual field positions are at the given elevation. The colored points plotted on the visual field representation is the projection back into the visual field from all the pixels on the given straight line. These projections are not aligned perfectly with the appropriate elevations, suggesting that the nasotemporal and dorsoventral aspects of the map cannot be separated easily. Black dotted lines on the colliculus indicate the tracks used by Owens et al. (2015) in sampling the projection. Rows 3–5, Plots of the azimuthal component of phase against collicular position along the appropriate track. The orientation of the 1D sections vary but all are seen to be at ∼20° to the rostro-caudal axis. All three plots for the same dataset are drawn with the same horizontal scale, which varies from dataset to dataset. No data were recorded in the plot in K, row 3. Units of phase are degrees and those of position on the colliculus are mm.

2D maps from colliculus (bottom) to visual field (top) for WT and HETA datasets. Neighboring nodes on the colliculus and the corresponding nodes in the visual field map are connected by straight lines. Lines in the visual field which cross and the corresponding lines on the colliculus are colored red. To relate the 2D maps to the patterns of 1D profiles in Figure 2, in some figures three lines representing the projections at −25° (orange), 0° (cyan), and +25° (brown) are shown. A, B, Maps formed for the WT shown in Figure 2A. A, The complete projection with 162 nodes. B, The largest ordered submap, constructed by removing 4% of the nodes from the complete projection to eliminate all crossing lines giving a Map Quality of 96%. RC, ML polarities: all 90%. C, The largest ordered submap for the HETA shown in Figure 2C, with 215 nodes in the whole map and 183 in the largest ordered submap giving a Map Quality of 86%. RC, ML polarities: all 90%. Azimuthal and elevational magnifications: 77°/mm, 60°/mm. D–F, Maps formed for the WT shown in Figure 2B. D, The complete projection, with 165 nodes. E, F, The two largest ordered subpartmaps constructed on the two sets of nodes by partitioning the set of collicular nodes with a line drawn through the narrow strip of rejected pixels in rostral colliculus. E, Subpartmap on rostral colliculus. F, Subpartmap on the rest of the colliculus. RC, ML polarities are 94%. AM, EM: 115°/mm, 58°/mm. Scale bars: 20° (visual field); 200 μm (colliculus).

2D maps for two HETB datasets. A–C. Maps formed from the data shown in Figure 2E. A, Whole map with 148 nodes. Largest ordered submap has 124 nodes with Map Quality 84%. B, C, Subpartmaps formed after partitioning the colliculus through the green areas of rejected pixels into rostral (B) and caudal (C) areas; 3% more nodes were recruited in the two subpartmaps than in the largest ordered submap, giving a Map Quality of 87%. VFD, the area of visual field contained in both rostral and caudal projections: 3%. RC, ML polarities and AM, EM: 94%, 89%, 118°/mm, 89°/mm (B); 68%, 83%, 84°/mm, 75°/mm (C). D–F, Maps formed from the data shown in Figure 2F. D, Whole map with 186 nodes. The largest ordered submap has 169 nodes giving a Map Quality of 91%. E, F, Rostral and caudal subpartmaps, with 3% more nodes than in the largest ordered submap. VFD: 7%. RC, ML polarities and AM, EM: 96%, 92%, 101°/mm, 92°/mm (E); 68%, 92%, 86°/mm, 62°/mm (F). Conventions as in Figure 3.

2D maps for two HETC datasets. A–C, Maps formed from the data shown in Figure 2G. A, Whole map with 180 nodes in the complete map. Largest ordered submap has 135 nodes with Map Quality 75%. B, C, Subpartmaps, with Map Quality of 85%, 10% more than in the largest ordered submap. VFD: 29%. RC, ML polarities and AM, EM: 91%, 85%, 168°/mm, 84°/mm (B); 90%, 88%, 88°/mm, 73°/mm (C). The value of AM for B is almost twice that for C, primarily because of the high magnification in temporal field. D–F, Maps formed from the data shown in Figure 2H. D, Whole map with 207 nodes. Largest ordered submap has 188 nodes with Map Quality 91%. E, F, Subpartmaps, with Map Quality 98%, 7% more than in the largest ordered submap. VFD: 29%. All polarities between 90% and 93%. AM, EM: 160°/mm, 82°/mm (E); 86°/mm, 76°/mm (F). Conventions as in Figure 3.

2D maps for two HOM datasets. A–C, Maps formed from the data shown in Figure 2I. A, Whole map with 148 nodes. Largest ordered submap has 123 nodes with Map Quality 83%. B, C, Subpartmaps, with Map Quality of 93%, 10% more than in the largest ordered submap. VFD: 17%. All polarities between 86% and 91%. AM, EM: 212°/mm, 85°/mm (B); 79°/mm, 85°/mm (C). D–F, Maps formed from the data shown in Figure 2J. D, Whole map with 177 nodes. Largest ordered submap has 131 nodes with Map Quality 74%. E, F, Subpartmaps, with Map Quality of 98%; 24% more than in the largest ordered submap. VFD: 63%. All polarities between 91% and 93%. AM, EM: 174°/mm, 117°/mm (E); 140°/mm, 85°/mm (F). Conventions as in Figure 3. Note that the homozygote map in A–C is based on the data presented by Owens et al. (2015) as the paradigm heterozygote with a double map (their Fig. 1N).

Single runs of the TK2011 model (Triplett et al., 2011). Connections developed between 10,000 retinal cells and 10,000 collicular cells, using the code developed by Hjorth et al. (2015). Maps were computed for different values of DR, the amount of EphA added to the Islet2 cells, and constructed using the Lattice Method (Willshaw et al., 2014). Largest ordered subpartmaps are shown. Retina is plotted as visual field to aid comparison with the experimental plots. For the mapping from colliculus to retina, colored black, the collicular lattice was divided into two by a line through the center of the green (rejected) pixels. The left-hand plot is the map from the rostral part of the colliculus; the right-hand plot that from the caudal part. For the mapping from retina to colliculus, colored blue, the left-hand plot is the projection from EphA3+ cells and the right-hand plot that for the EphA3− cells. Map orientation shown as in the experimental plots. A, DR = 0. Colliculus to retina map only. The collicular lattice was divided into equal-sized rostral and caudal halves. B–D, Projections in both directions are shown for DR = 0.22, 0.34, 0.56. For the projection from colliculus to retina the shaded region in the retina is represented in both rostral and caudal collicular areas. E, DR = 1.12. Colliculus to retina projection only. Each retinal cell was chosen to be EphA3+ with probability 0.5. Parameter values: α = 90, β = 135, a = 0.03, b = 0.11 (Triplett et al., 2011; Hjorth et al., 2015); γ = 0.00625, 5 × 10⁸ iterations (Gale, 2022). The EphA gradient is derived from Reber et al. (2004). Values of DR quoted are expressed as a fraction of the span of the values of EphA over the WT retina. On this scale, the values of DR for an EphA3^ki/+ and an EphA3^ki/ki map (Reber et al., 2004) are 0.37 and 0.74, respectively.

Data of the type used to generate Figure 7 from the TK2011 model was plotted to show a sequence of six 1D projections of the retina measured along a straight line located at dorsoventral position 0.5 and running nasotemporally onto the colliculus measured along the matching straight line running rostrocaudally, for increasing values of DR. Retina plotted as visual field. Each subfigure has five separate plots. Row 1, Distribution of EphA3− (purple) and EphA3+ (orange) cells over the retina. The horizontal line indicates the line along which the 1D profile was taken. Blue marks the site of a simulated injection of 1% of the retinal cells (row 1) and its projection(s) on the colliculus (row 5), which define(s) the mediolateral position of the matching line. Row 2, The distribution of EphA along the nasotemporal axis of the retina for both EphA3− cells (purple) and EphA3+ cells (orange) and of ephrinA cells (magenta) across the rostrocaudal axis of the colliculus. Nasal visual field corresponds to rostral colliculus and temporal field to caudal colliculus. Row 3, Nasotemporal component of the positions of all retinal cells with contacts from the cells lying along the rostrocaudal axis of the colliculus at mediolateral position 0.5, marked in row 5. Row 4, Rostrocaudal component of the positions of all collicular cells with contacts from the retinal cells lying along the nasotemporal axis of the retina at dorsoventral position 0.5, shown in row 5. The projections from the EphA3− cells (purple) and the EphA3+ cells (orange) can be distinguished. Row 5, Distribution of the average positions on the colliculus of the projections from each retinal cell, with the projections from the injected cells in blue. Initially a large number of contacts are made at random and then contacts are withdrawn selectively, those that lower the overall energy being maintained. The difference in energy between the contacts made by two retinal cells on the same collicular cell is the product of the amounts of EphA and ephrinA present. The contact with more EphA3 tends to be withdrawn as this decreases the energy by a larger amount. Note that, as in all Markov Chain Monte Carlo methods (Brooks et al., 2011), this is not a true minimization model which converges to a single configuration, but rather to a distribution of possible configurations (Gale, 2022). A, DR = 0.0. EphA3− and EphA3+ cells at the same retinal location have the same amount of EphA and so there is a single map (Fig. 7A). B, DR = 0.16. Some EphA3− cells from nasal retina (temporal visual field) have values of EphA lower than those in the EphA3+ population at the same collicular position (row 2, gray bar) and so survive at the expense of the contacts from the EphA3+ cells there. The contacts made more centrally by these same EphA3+ cells now compete against the EphA3− cells there, which have higher energy. The EphA3+ cells tend to survive, and the overall effect is that the EphA3+ projection from nasal retina moves more centrally. C, DR = 0.32 (Fig. 7B). D, DR = 0.48 (Fig. 7C). There is a growing population of EphA3− cells with unique values of EphA, increasing the extent of the single EphA3− projection caudally and leading to contacts from more and more EphA3+ cells being removed from this region of the colliculus. There is still a double projection far rostrally, where the activity mechanism favoring neighbor-neighbor connections dominates over the weaker chemoaffinity mechanism. E, DR = 0.64. F, DR = 0.78 (Fig. 7D). As DR is increased, the effect of chemoaffinity rostrally becomes stronger, diminishing the superposed projection. For sufficiently large DR, all the EphA3− cells have values of EphA lower than the EphA3+ cells (row 2) and a double projection results. EphA3+ cells project rostrally and EphA3− cells caudally.