**Abstract**:

**Abstract**
**In the human eye, all visual information must traverse the retinal ganglion cells. The most numerous subclass, the midget retinal ganglion cells, are believed to underlie spatial pattern vision. Thus the density of their receptive fields imposes a fundamental limit on the spatial resolution of human vision. This density varies across the retina, declining rapidly with distance from the fovea. Modeling spatial vision of extended or peripheral targets thus requires a quantitative description of midget cell density throughout the visual field. Through an analysis of published data on human retinal topography of cones and ganglion cells, as well as analysis of prior formulas, we have developed a new formula for midget retinal ganglion cell density as a function of position in the monocular or binocular visual field.**

- Along a given meridian, the cumulative distribution of RGC and RGCf must agree outside the displacement zone.
- In the fovea, it is likely that each cone connects (via bipolars) to exactly two mRGC (Kolb & Dekorver, 1991).
- Near the fovea midgets constitute most but not all of the ganglion cells. The ratio RGC/mRGC is given as about 1.12 by Drasdo et al. (2007).
- The hypothetical distribution of RGCf must be consistent with the measured distribution of RGC outside of the displacement zone.

*x*-coordinates refer to the temporal visual field of either eye, or the right binocular visual field.

*r*will indicate eccentricity in degree,

*d*will indicate density in deg

^{−2}, and

*s*will indicate spacing (of adjacent cells or receptive fields) in degrees. We use subscripts

*g*,

*m*, and

*c*to denote RGC, mRGC, and cones respectively, and

*gf*and

*mf*to denote RGC and mRGC receptive fields. A particular meridian will be indicated by an integer index

*k*.

^{2}in the four principal meridians at each of 34 eccentricities in mm (Curcio, 2006). Using the conversion formulas described in Appendix 6, we have converted the densities to cones/deg

^{2}as a function of eccentricity in degree as shown in Figure 1. Writing

*d*(

_{c}*r*,

*k*) for the cone density at eccentricity

*r*degree along meridian

*k*, we note that the foveal peak is

*d*(0, 1) =

_{c}*d*(0) = 14,804.6. This peak density is plotted at the upper left on this log-log plot.

_{c}^{2}in the four principal meridians at each of 35 eccentricities in millimeters. We have again converted these values to densities in RGC/deg

^{2}as a function of eccentricity in degree, and the results are plotted in Figure 2. We omit one point in the inferior meridian with a density below 1. The peak density of about 2,375 RGC/deg

^{2}occurs not at the foveal center but at an eccentricity of about 3.7°. This is because, as noted above, ganglion cell bodies within a displacement zone extending out as far as 17° are displaced centrifugally from their cone inputs. Thus the RGC densities within this zone cannot be used directly as an estimate of the densities of the RGCf.

*r*multiplied by 2

*πr*to account for the increasing area. The circular point on each curve marks the cumulative density at the approximate limit of the displacement zone (11 for temporal, 17 for the others). When we construct a candidate function for the density of RGCf, its cumulative value must approximately agree at these points. In other words, the total number of receptive fields must equal the total number of cell bodies within the displacement zone.

*d*(0) is the density at

_{gf}*r*= 0, and

*r*

_{2}is the eccentricity at which density is reduced by a factor of four (and spacing is doubled). By itself, this did not provide a good fit, especially at larger eccentricities. However we found that a simple modification, the addition of an exponential, yielded an acceptable fit. The new function is given by where

*a*is the weighting of the first term, and

_{k}*r*is the scale factor of the exponential. The meridian is indicated by the index

_{e,k}*k*. We have fit this expression separately for each meridian and optimized parameters relative to the error function described above. The results are shown in Figure 4. For each meridian, we show the average RGC densities reported by Curcio and Allen (1990), along with the fitted function. The vertical gray line in each figure shows the assumed limit of the displacement zone. Note that only data points outside the displacement zone are used in the fit. The estimated parameters, predicted cell counts, and fitting error are given in Table 1.

Meridian | k | a | r_{2} | r_{e} | Data count (× 1000) | Model count (× 1000) | Error | r_{z} |

Temporal | 1 | 0.9851 | 1.058 | 22.14 | 485.1 | 485.7 | 0.23 | 11 |

Superior | 2 | 0.9935 | 1.035 | 16.35 | 526.1 | 528.9 | 0.12 | 17 |

Nasal | 3 | 0.9729 | 1.084 | 7.633 | 660.9 | 661.1 | 0.01 | 17 |

Inferior | 4 | 0.996 | 0.9932 | 12.13 | 449.3 | 452.1 | 0.93 | 17 |

*h*(0) > 0° at 0° eccentricity. There is some disagreement about the value of

*h*(0). Drasdo stated that the first RGC are located 0.15 to 0.2 mm (0.53°–0.71°), whereas Curcio provides nonzero RGC densities at eccentricities as small as about 0.2° (see Figure 2). We have assumed a value of 0.5°, but a value of 0.3 gives very similar results.

*h*(0) and by the peak value, which we set to the maximum of the fitted values.

Meridian | α | β(deg) | γ | δ | μ(deg) |

Temporal | 1.8938 | 2.4598 | 0.91565 | 14.904 | −0.09386 |

Nasal | 2.4607 | 1.7463 | 0.77754 | 15.111 | −0.15933 |

*f*(

*r*) is the fraction of retinal ganglion cells that are midgets, as introduced in Equation 2. Two estimates of

*f*(

*r*) have been provided in the literature.

*f*(

*r*) was provided by Drasdo et al. (2007) as the formula where

*f*(0) = 1/1.12 = 0.8928 and

*r*= 41.03°. The formula is shown by the curve in Figure 8. This formula was derived from iterative fit of a more elaborate expression involving both psychophysical measures and anatomical measures (Drasdo et al., 2007). Drasdo and Dacey's measures agree generally that the fraction declines with eccentricity, and roughly agree in foveal and peripheral asymptotes.

_{m}*x*,

*y*} in the retina. To do this we make the assumption that within any one quadrant of the retina the iso-spacing contours are ellipses. This is consistent with the idea that spacing changes smoothly with the angle of a ray extending from the visual center. An example is shown in Figure 12. The eccentricities at the intersections of the ellipse with the two enclosing meridians are

*r*

_{x}and

*r*

_{y}.

*x*and

*y*, we can solve Equations 11 and 12 together to find numerical solutions for

*r*

_{x}and

*r*

_{y}, and then we can compute

*x*values to mean the temporal visual field. This corresponds to the right visual field for the right eye, and the left visual field for the left eye.

*s*

_{B}:

*r*= 0 is that these data were collected with Gabor targets that extended (at half height) well over 0.5°, so that performance may reflect the averaging spacing over that area. The precise relationship between mRGCf spacing and acuity is beyond the scope of this paper (Anderson & Thibos, 1999), here we only point to the general agreement in both the shape and absolute level of the calculations.

^{−2}(7,385 to 24,372 deg

^{−2}) (coefficient of variation ∼0.46). When two anomalous eyes are excluded, the lower bound only increases to 166,000 mm

^{−2}(12,483 deg

^{−2}). This variation largely disappeared at eccentricities beyond 1°, so that the total number of cones within a radius of about 3.6° (or over the entire retina) was nearly constant (coefficient of variation ∼0.1). However, more recent density estimates from in vivo measurement show a fairly consistent coefficient of variation (∼0.2) regardless of eccentricity (Song, Chui, Zhong, Elsner, & Burns, 2011). These latter authors have also shown an up to 25% decrement in density with age, primarily at eccentricities less than 1.6°. Curcio's data, and our formula, are consistent with the data for their younger group of observers.

*Visual Neuroscience*, 20 (2), 189– 209. [CrossRef] [PubMed]

*Journal of the Optical Society of America A*, 16 (10), 2321– 2333, http://josaa.osa.org/abstract.cfm?URI=josaa-16-10-2321. [CrossRef]

*Journal of Physiology*, 442, 47– 64, http://www.ncbi.nlm.nih.gov/pubmed/1798037. [CrossRef] [PubMed]

*Archiv für Ophthalmologie*, 3, 1– 37.

*Contrast sensitivity of the human eye and its effects on image quality*. Bellingham, WA: SPIE Optical Engineering Press.

*Visual optics and instrumentation*( pp. 1– 26). Boca Raton: CRC Press.

*Journal of Comparative Neurology*, 300 (1), 5– 25. [CrossRef] [PubMed]

*Journal of Comparative Neurology*, 292 (4), 497– 523, http://www.ncbi.nlm.nih.gov/pubmed/2324310. [CrossRef] [PubMed]

*Journal of Neuroscience*, 13 (12), 5334– 5355, http://www.ncbi.nlm.nih.gov/pubmed/8254378. [CrossRef] [PubMed]

*Proceedings of the National Academy of Sciences, USA*, 89 (20), 9666– 9670, http://www.ncbi.nlm.nih.gov/pubmed/1409680. [CrossRef]

*British Journal of Ophthalmology*, 58 (8), 709– 714, http://www.ncbi.nlm.nih.gov/pubmed/4433482. [CrossRef] [PubMed]

*Vision Research*, 47 (22), 2901– 2911, http://www.ncbi.nlm.nih.gov/pubmed/17320143. [CrossRef] [PubMed]

*Visual optics*(5th ed.). London: Hatton Press.

*Callithrix jacchus*.

*Journal of Comparative Neurology*, 366 (1), 55– 75, http://www.ncbi.nlm.nih.gov/pubmed/8866846. [CrossRef] [PubMed]

*Vision Research*, 29 (9), 1095– 1101. [CrossRef] [PubMed]

*Journal of Comparative Neurology*, 303 (4), 617– 636, http://www.ncbi.nlm.nih.gov/pubmed/1707423. [CrossRef] [PubMed]

*Vision Research*, 26

*,*1751– 1761. [CrossRef] [PubMed]

*Vision Research*, 30 (7), 985– 991. [CrossRef] [PubMed]

*Acta Ophthalmologica, 6*(Suppl. VI), 1– 97.

*Vision Research*, 45 (17), 2331– 2338. [CrossRef] [PubMed]

*Vision Research*, 41 (10), 1313– 1319. [CrossRef] [PubMed]

*Nature Neuroscience*, 13 (2), 156– 157, http://dx.doi.org/10.1038/nn.2465. [CrossRef] [PubMed]

*Journal of Comparative Neurology*, 269 (4), 479– 505, http://www.ncbi.nlm.nih.gov/pubmed/3372725. [CrossRef] [PubMed]

*Vision Research*, 39 (18), 2987– 2998. [CrossRef] [PubMed]

*Graefes Archives for Clinical & Experimental Ophthalmology*, 237 (12), 1014– 1023.

*Investigative Ophthalmology & Visual Science*, 52 (10), 7376– 7384, http://www.iovs.org/content/52/10/7376. [Pubmed] [Article] [CrossRef] [PubMed]

*Journal of Vision*, 11( 5): 13, 1-82, http://www.journalofvision.org/content/11/5/13, doi:10.1167/11.5.13. [Pubmed] [Article]

*Journal of the Optical Society of America A*, 4 (8), 1524– 1529. [CrossRef]

*Vision Research*, 30 (11), 1897– 1911, http://www.ncbi.nlm.nih.gov/pubmed/2288097. [CrossRef] [PubMed]

*Mathematica*(Version 9.0). Champaign, IL: Author.

Symbol | Definition | Unit |

r | Eccentricity in deg relative to the visual axis | deg |

k | Meridian index | |

d(r, k) | Density of cells or receptive fields at eccentricity r along meridian k | deg^{−2} |

s(r, k) | Spacing between receptive fields at eccentricity r along meridian k | deg |

c, g, m, gf, mf | Subscripts to indicate cones, RGC, mRGC, RGCf, and mRGCf | |

h(r) | Displacement of RGC from RGCf at eccentricity r | deg |

f(r) | Fraction of RGC that are midget, as a function of eccentricity | dimensionless |

Δ_{k} | Offset between optic and visual axis along the specified meridian | mm |

r′ | Eccentricity in deg relative to the optic axis | deg |

r_{mm} | Eccentricity in mm relative to the visual axis | mm |

r′_{mm} | Eccentricity in mm relative to the optic axis | mm |

N | Nyquist frequency of hexagonal lattice | cycles/deg |

S | Point spacing of hexagonal lattice | deg |

R | Row spacing of hexagonal lattice | deg |

D | Point density of hexagonal lattice | deg^{−2} |

Item | Value | Unit |

Peak cone density | 14,804.6 | deg^{−2} |

Peak RGCf density | 33,162.3 | deg^{−2} |

Peak mRGCf density | 29,609.2 | deg^{−2} |

Minimum on-center mRGCf (or cone) spacing | 0.5299 | arcmin |

Peak on-center mRGCf (or cone) Nyquist | 65.37 | cycles/deg |

f(0), midget fraction at zero eccentricity | 1/1.12 = 0.8928 | |

r_{m}, scale factor for decline in midget fraction with eccentricity | 41.03 | deg |

*S*(deg) = the spacing between adjacent points,

*R*(deg) = the spacing between rows of points,

*D*(deg

^{−2}) = the density of points, and

*N*(c/deg) = the Nyquist frequency of the lattice (the highest frequency that can be supported by a particular row spacing). These formulas will allow us to convert among these metrics. Then

*r*′

_{mm}refers to distance in mm, while

*r*′ is the comparable measurement in degree. The prime marking indicates a measurement relative to the optic axis.

^{2}to deg

^{2}

^{2}to deg

^{2}. In their figure 5, Drasdo and Fowler (1974) show “variation of retinal area per solid degree with peripheral angle from the optic axis.” We have fit this with a polynomial where

*a*is the ratio of areas mm

^{2}/deg

^{2}. We have used this to convert cell densities in mm

^{−2}to deg

^{−2}. The function is illustrated in Figure A2.

*m*is the index of the meridian, and the corresponding offsets Δ from optic to visual axis in mm are given by