Conformal geometry of the retinal nerve fiber layer

Abstract

The nerve fiber layer of the human retina is made up of the retinal segments of ganglion cell axons. Its geometry can be described mathematically as a fibration of a 2D domain: a partition of a certain region into smooth curves. Here, we present a simple family of curves that closely models the observed geometry of the nerve fiber layer. For each retina, the pattern depends on 2 parameters, A and B: A computer program determines A and B for a given retina and the theory matches the retina with a standard deviation of approximately 6-8 degrees . These particular curves turn out to be the curves that would be generated if the growing ganglion cell axon tip moved down a gradient toward a source of diffusible neuroattractant at the disk and away from a weaker macular diffusible repellant. Thus, this model provides morphological evidence that diffusible substances provide positional information to the embryonic ganglion cell axons in finding their way to the optic nerve head.

Fig. 1.

Definitions and technique. (A) A pen and ink drawing of the human retina by P.J.A. The dark disk on the left represents the optic disk and the rightmost disk is the perimacular area. The upper half has been overlaid by the matching curves of our mathematical model with parameters A = 0.379418 and B = 0.303835. (B) Graph of hypothetical potential function, F, with parameters A = 0.4 and B = 0.1. Nerve fibers are postulated to move in the direction of steepest decline downhill. The depression at coordinates (0, 0) represents the optic nerve sink and the peak (truncated) at (0, 1) is the macular source. Notice the ridge from the macular peak, creating the horizontal raphe watershed. (C) Retina 1 with graphic illustrating geometric definitions of the angles a and b. (D) Grid (spacing 0.2 in x–y coordinates) laid over photo of retina I. Red line segments indicate the NFL direction at points of lattice L.

Fig. 2.

Magnified view of Retina 6, illustrating NFL fibration. Overlay shows computed fibration using the parameters A = 0.444294 and B = 0.168305.

Fig. 3.

Retinal photos comparing measured NFL directions with theoretical curves. Lattice K of measured NFL directions is indicated by red line segments. The corresponding theoretical contour plots are calculated from the values for K in Table 1 for Retina 1 (A), Retina 2 (B), Retina 3 (C), Retina 4 (D).

Fig. 4.

Retinal photos with lattice K of and corresponding theoretical contour plots, calculated from K data for Retina5 (A), Retina6 (B), Retina7 (C), and Retina6 and lattice K (D), now showing theoretical overlay for streak potential using parameters in Table 2.

Fig. 5.

Magnified view of the raphe area of Retina7 with overlays comparing the streak fibration in the upper half with the dipole fibration in the lower half.