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The authors have declared that no competing interests exist.

Conceived and designed the experiments: DCS DL IDT. Performed the experiments: DL IDT. Analyzed the data: DCS DL IDT. Contributed reagents/materials/analysis tools: DCS. Wrote the paper: DCS DL DJW IDT. Wrote the software package: DCS.

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.

The retina projects directly and indirectly to a large number of areas in the central nervous system, such as the mammalian superior colliculus, lateral geniculate nucleus or visual cortex. Understanding the topographic mapping of these projections is a central feature of visual neuroscience

Historically, the anatomical organisation of the retina was frequently examined using serial sections, with the emphasis on example sections rather than reconstructions

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).

We describe a method to infer where points on a flat-mount retina would lie in a standard, intact retinal space. The standard retina is approximated as a partial sphere, with positions identified using spherical coordinates. Our results show that the method can estimate the location of a point on a flat-mounted retina to within 8° of arc of its original location, or 3.6% of the NT or DV axis. This has allowed us to define a standard retinal space for the adult and for the developing mouse eye. Establishing the orientation of retinal space for the mouse, whose retina contains no intrinsic markers, requires experimental intervention. In the Results, we focus on data from adult mice. We show that a mark based on the centre of the nictitating membrane is reliable and this mark can be related to the insertion of the rectus eye muscles. Furthermore, we transform standard retinal space into visuotopic space and use the geometry of binocular vision, together with anatomical tracing, to address questions about the projection of the optical axis of the mouse eye into visual space

Our Retistruct algorithm not only facilitates comparison of differential retinal distributions across animals but also allows analyses of distributions of labelled cells in spherical coordinates, obviating the distortions associated with 2D flat-mounts. Finally, transforming retinal coordinates into visual coordinates gives insights into the functional significance of retinal distributions.

In this section we give an overview of the reconstruction algorithm; a more detailed description is contained in the Supplemental Materials and Methods (

Age | |||

P0 | 1632±17 | 1308±17 | 127.13±1.92 |

P2 | 2146±13 | 1780±29 | 131.20±2.20 |

P4 | 2250±9 | 1857±20 | 130.58±1.40 |

P6 | 2450±41 | 1963±25 | 127.02±2.41 |

P8 | 2646±1 | 2088±34 | 125.31±1.81 |

P12 | 2786±15 | 2212±65 | 126.03±3.37 |

P16 | 2808±17 | 2043±17 | 117.10±0.96 |

P22 | 2958±35 | 2117±44 | 115.57±2.17 |

P64 | 3160±56 | 2161±30 | 111.56±1.89 |

The locations of mesh points in the flat-mount and their corresponding locations on the sphere define the relation between any point in the flat-mount and a standard spherical space. This relation is used to map the locations of data points and landmarks in the flat retina onto the standard retina. These can be visualised interactively on a 3D rendering of a sphere (see

In order to analyse data points on the standard retina, we used spherical statistics

To assess the amount of residual deformation at the end of the energy minimisation procedure, we plotted the length of each edge

An example of a reconstruction of an adult retina with low deformation measure

We used our algorithm on 297 flat-mounted retinae, 288 of which could be reconstructed successfully, 7 of which failed due to, as-yet unresolved, software problems and 2 of which were rejected because of unsatisfactory reconstructions (see below).

Thus the deformation measure

Histogram of the reconstruction error measure

It should be noted that retinae which have lost tissue due to poor dissection can be forced onto the spherical surface by the algorithm, albeit with high

To determine the rim angle of mouse eyes at varying stages of development we measured the distance

An alternative approach to setting the rim angle is to infer, for each individual retina, the rim angle that minimises the deformation. This was done by repeating the minimisation for rim angles at 1° intervals within a range

The deformation measure gives an indication of how easy it is to morph any particular flattened retina onto a partial sphere, but it does not indicate the error involved in the reconstruction, i.e. the difference between the inferred position on the spherical retina and the original position on the spherical retina. The ideal method for estimating the error would be to flatten a retina marked in known locations, and then compare the inferred with the known locations. However, this proved to be technically very difficult and so we tried another method of estimating the error that uses the inferred locations of the optic discs across a number of retinae. In mice, the optic disc is located “rather precisely in the geometric center of the retina”

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°.

We now describe the first application of the reconstruction algorithm. The relationship between the projections from the mouse retina to the ipsilateral and contralateral dLGN has been studied in retina flat-mounts following injection of retrograde tracer into the dLGN

The reconstruction method has enabled the quantification of the binocular projection from the retina to the dLGN across multiple animals. To label the projection, the dLGN in adult mice was injected either with Fluoro-Ruby or Fluoro-Emerald. Further, in some animals, the injections were bilateral (see

Schematic illustrating the retinal label resulting from bilateral injections of Fluoro-Ruby (magenta) and Fluoro-Emerald (cyan) dye into the dLGN.

Obtaining uniform and complete injections of tracer into the dLGN can be difficult and can result in variability in the pattern of label (e.g. the contralateral retinae in

Azimuthal equilateral projections in standard

The geometry of binocular vision implies that the ipsilateral decussation line should correspond to the vertical meridian in the adult mouse's visual field

In principle, the deviation of a ray by the eye can be estimated by means of a schematic model of the mouse eye

Using the above conventions to test whether the position of the ipsilateral decussation line corresponds to the vertical meridian in visuotopic space, we have transformed the retinotopic location of ipsilateral retinal ganglion cells, following bilateral injections, into head-centred visuotopic space

Given these observations and our group data for optic disc location (

To examine the orientation of the eye, the locations of the insertions of superior, lateral and inferior rectus into the globe of the eye were marked onto the retina (

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

The location of the optic axis at 64° azimuth and 22° elevation

S-opsin staining in dorsal, central and ventral retina. Images acquired at 20× magnification. Scale bar is 100 m.

In summary, with the location of the optic axis determined by the optimal match of the decussation line with the vertical meridian, the change in S-opsin staining nicely coincides with the horizontal meridian, at least for the central visual field, as predicted by Szél et al.

The reconstruction and transformation methods described here have been implemented in R

The program could be applied to flat-mount preparations of retinae from any vertebrate species at any age, provided the globe of the eye is approximately spherical, and it would be possible to add extra analysis routines. When examining retinae with mosaic labelling

It should be stressed that the current version is not intended for use on partial retinae. However, it might eventually be possible to locate incomplete retinae in standard retinal space, if there enough markers to orient the partial retina correctly. Similarly, it should be possible to locate sections (either histological or tomological) of known orientation and hemiretinal origin in standard space.

Complete source code for the Retistruct package (version 0.5.7). Includes demonstration data for

(ZIP)

Supplemental information including an extended Discussion, and Materials and Methods containing details of experiments and detailed description of the reconstruction algorithm and analysis of reconstructed data.

(PDF)

The authors would like to thank Andrew Lowe, Stephen Eglen, Johannes J. J. Hjorth, and Michael Herrmann for their very helpful comments throughout this work. We would also like to thank Nicholas Sarbicki for helping with the analysis of LGN projections and Henry Eynon-Lewis for assistance with dissections for muscle-insertion marking and S-opsin staining. We thank anonymous reviewers for alerting us to future possibilities of using Retistruct to map partial or sectioned retinae onto standard retinal space.

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