Standard anatomical and visual space for the mouse retina: computational reconstruction and transformation of flattened retinae with the Retistruct package

Abstract

The concept of topographic mapping is central to the understanding of the visual system at many levels, from the developmental to the computational. It is important to be able to relate different coordinate systems, e.g. maps of the visual field and maps of the retina. Retinal maps are frequently based on flat-mount preparations. These use dissection and relaxing cuts to render the quasi-spherical retina into a 2D preparation. The variable nature of relaxing cuts and associated tears limits quantitative cross-animal comparisons. We present an algorithm, "Retistruct," that reconstructs retinal flat-mounts by mapping them into a standard, spherical retinal space. This is achieved by: stitching the marked-up cuts of the flat-mount outline; dividing the stitched outline into a mesh whose vertices then are mapped onto a curtailed sphere; and finally moving the vertices so as to minimise a physically-inspired deformation energy function. Our validation studies indicate that the algorithm can estimate the position of a point on the intact adult retina to within 8° of arc (3.6% of nasotemporal axis). The coordinates in reconstructed retinae can be transformed to visuotopic coordinates. Retistruct is used to investigate the organisation of the adult mouse visual system. We orient the retina relative to the nictitating membrane and compare this to eye muscle insertions. To align the retinotopic and visuotopic coordinate systems in the mouse, we utilised the geometry of binocular vision. In standard retinal space, the composite decussation line for the uncrossed retinal projection is located 64° away from the retinal pole. Projecting anatomically defined uncrossed retinal projections into visual space gives binocular congruence if the optical axis of the mouse eye is oriented at 64° azimuth and 22° elevation, in concordance with previous results. Moreover, using these coordinates, the dorsoventral boundary for S-opsin expressing cones closely matches the horizontal meridian.

Figure 1. Overview of the method. A,

The raw data: a retinal outline from an adult mouse (black), two types of data points (red and green circles) from paired injections into the superior colliculus and a landmark (blue line). B, Retinal outline with nasal pole (N) and cuts marked up. Each pair of dark cyan lines connects the vertices and apex of the four cuts. C, The outline after triangulation (shown by grey lines) and stitching, indicated by cyan lines between corresponding points on the cuts. Di, The initial projection of the triangulated and stitched outline onto a curtailed sphere. The strain of each edge is represented on a colour scale with blue indicating compression and red expansion. Cuts are shown in cyan. Dii, The strain plotted on the flat outline with lines of latitude and longitude superposed. Ei,ii, As Di,ii but after optimisation of the mapping. Fi, The data represented on a polar plot of the reconstructed retina. Mean locations of the different types of data points are indicated by diamonds. The nasal (N), dorsal (D), temporal (T) and ventral (V) poles are indicated. Cuts are shown in cyan. Fii, Data plotted on the flat representation, with lines of latitude and longitude superposed. All scale bars are 1 mm.

Figure 2. Examples of reconstructed retinae. A–D,

An example of a reconstruction of an adult retina with low deformation measure . A, Plot of length of edge on the sphere versus length of edge on the flat retina. Red indicates an edge that has expanded and blue a edge that has been compressed. B, The log strain indicated using the same colour scheme on the flat retina. C, The flat representation of lines of latitude and longitude with the optic disc (blue). D, The azimuthal equidistant (polar) representation showing the locations of the cuts and tears (cyan) and the location of the optic disc (blue). E–H, An example of a reconstruction of a P0 retina with high deformation energy . Meaning of E–H same as for corresponding panel in A–D. All scale bars are 1 mm.

Figure 3. Deformation of reconstructions and the effect of rim angle. A,

Histogram of the reconstruction error measure obtained from 288 successfully reconstructed retinae. B, Relationship between deformation measure and age. “A” indicates adult animals. C, Schematic diagram of eye, indicating the measurements and made on mouse eyes at different stages of development, and the rim angle derived from these measurements. Note that rim angle is measured from the optic pole (*). D, Rim colatitude that minimises reconstruction error versus the rim angle determined from eye measurements. Solid line shows equality and grey lines indicate ±10° and ±20° from equality. E, Minimum reconstruction error obtained by optimising rim angle versus reconstruction error obtained when using the rim angle from eye measurements. Solid line indicates equality.

Figure 4. Estimation of reconstruction error using optic disc location. A,

Inferred positions of optic discs from 72 adult reconstructed retinae plotted on the same polar projection. The colatitude and longitude of the Karcher mean is (3.7°, 95.4°). The standard deviation in the angular displacement from the mean is 7.4°. B, The same data plotted on a larger scale. C, The relationship between the deformation of the reconstruction and distance of the inferred optic disc from the population mean. There was a significant correlation between the two ().

Figure 5. Measurement of the ipsilateral projection. A,

Schematic illustrating the retinal label resulting from bilateral injections of Fluoro-Ruby (magenta) and Fluoro-Emerald (cyan) dye into the dLGN. B, Flat-mounted retina with label resulting from bilateral injections of Fluoro-Ruby (magenta) and Fluoro-Emerald (cyan) into left and right dLGN, respectively. C–D, Azimuthal equilateral projection of reconstructed retina in B. Isodensity contours for 5%, 25%, 50%, 75% & 95% are plotted using the kernel density estimates (KDEs) for fully sampled retinae (C) or kernel regression (KR) estimates for partially sampled retinae (D). Blue diamond is the peak density and magenta (C) or cyan (D) diamond is the Karcher mean. Yellow circle is the optic disc. E, Composite plot with ipsilateral label from unilateral injections (). Black dashed lines represent the median angle from the optic axis to the peripheral edges of the 5% isodensity contour. Coloured diamonds represent the Karcher means of the label in the individual retinae and large coloured circles are the optic discs for the individual retinae. White square and circle represent the average Karcher mean. The central dashed angle represents the median central edge of the 5% isodensity contour. F, Mean distances from either optic disc or optic axis of reconstructed retinae to the central edge of the 5% isodensity contour along a line passing through the Karcher mean of the label. G, The extent of the ipsilateral label and the distance beyond the horizontal and vertical axes. Grid spacing is 20°. In F–G, line represents the mean and error bars are standard error of the mean. Scale bar in C & D is 1 mm.

Figure 6. Alignment of the binocular zone in visuotopic coordinates. A,

Azimuthal equilateral projections in standard retinal space of left and right retinae with ipsilateral (upper) and contralateral (lower) label resulting from bilateral injections of Fluoro-Ruby (magenta) and Fluoro-Emerald (cyan) into left and right dLGN, respectively, of the same mouse. Plots were generated from stitched 10× epifluorescent images and cell locations detected using ImageJ. For this figure, we have abandoned the convention of always plotting nasal retina to the right. B, Schematic illustrating the approximate projection of retinal space onto visual space. When the orientation of the optic axis (grey line) is optimal, the ipsilateral crescent is projected entirely to the opposite visual field. Note that due to the refraction in the lens the visual field is estimated to be 180° for each eye. C–D, Orthographic projections in central visuotopic space of the two ipsilateral retinae in A with optic axis (*) at 64° azimuth; 22° elevation (C) and with optic disc at 60° azimuth; 35° elevation (D). E, Sinusoidal projection of contralateral retinae in B with the optic axis (*) at 64° azimuth; 22° elevation. Labels N, D, T, V indicate the projection of the corresponding pole of the retina. Grid spacing is 15°.

Figure 7. Measurement of muscle insertion angles. A,

Flat-mounted retina showing stitching and insertions for superior rectus (red), lateral rectus (green) and inferior rectus (blue). N indicates nasal cut. Plots on right represent the distortions introduced by reconstructing retina (see Figure 2 for explanation). B, Azimuthal equilateral projection of reconstructed retina in A. Dashed lines represent vectors connecting muscle insertion point to the optic disc. C, Muscle insertion points from 17 retinae. Solid black circles represent the optic discs for individual retinae. Dashed lines represent the line from each individual insertion point to its respective optic disc. Solid lines are from the Karcher mean insertion to the Karcher mean location of the optic disc. Grid Spacing is 15°. D, Plot of the angles of the angles of vectors connecting muscle insertions of Superior Rectus (SR), Lateral Rectus (LR) and Inferior Rectus (IR) to the individual optic discs. Bar represents the mean and error-bars are standard deviation.

Figure 8. Visuotopic axes with respect to S-opsin distribution. A,

S-opsin staining in dorsal, central and ventral retina. Images acquired at 20× magnification. Scale bar is 100 m. B–C, S-opsin distribution for right (A) and left (B) eyes plotted in orthographic projection centred on optic axis (*) at 22° elevation and 64° azimuth. Bottom left plot is flat-mounted retina. Bottom right plot is azimuthal equilateral plot. Plots were generated from stitched 10× epifluorescent images and cell locations detected using ImageJ . There are slight differences between the two eyes in the exact angle of density transition with respect to the horizontal meridian and in the density of staining around the optic disc. These will reflect experimental variance. Again our normal convention of showing nasal to the left has been relaxed. Scale bar is 1 mm. D, The average offset of the S-opsin density-transition from the horizontal meridian in central and peripheral visual field (). Error bars are SEM. E, S-opsin distribution for both eyes plotted in a sinusoidal projection with same optic axis (*) as in C. Yellow outline is edge of left retina; red outline is edge of right retina. Labels N, D, T, V indicate the projection of the corresponding pole of the retina. Grid spacing is 15°.