# Canonical Coordinates for Retino-Cortical Magnification

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Modeling the Sensorium

**Figure 1.**Schematic representation of the optic pathways from each of the four quadrants of view for both eyes. Adapted from Wikimedia Commons, original illustration by Ratznium. LGN, lateral geniculate nucleus.

#### 2.2. Modeling Retino-Cortical Magnification

**Figure 2.**Retino-cortical magnification, ${V}^{\prime}(t,T)$ (

**left**), and its integral, $V(t,T)$ (

**right**), as a function of dimensionless eccentricity, t, illustrated for the case $T=95$ (dashed vertical line); recall Equations (12)–(15). The peak on the left occurs at ${t}_{+}=1$ and marks the border ${\rho}_{+}=a$ of the geometric foveola. The half maximum on the right is reached at ${t}_{\frac{1}{2}}\phantom{\rule{-0.166667em}{0ex}}\approx \phantom{\rule{-0.166667em}{0ex}}\sqrt{T}$, corresponding to the geometric equipartitioning radius (left vertical line), ${\rho}_{\frac{1}{2}}\phantom{\rule{-0.166667em}{0ex}}\approx \phantom{\rule{-0.166667em}{0ex}}\sqrt{a\phantom{\rule{0.166667em}{0ex}}R}$. With our choice of parameters (motivated in the text), the tiny geometric foveola has a relative processing capacity $v({t}_{+}=1,T=95)\approx 8\%$; recall Equation (13).

**Figure 3.**Retino-cortical mapping of macaque monkey. (

**Left**) Retina with spoke-wheel stimulus. (

**Right**) Stimulus image retinotopically mapped onto the posterior part of (hemifield) striate cortex, a.k.a. calcarine sulcus. Source: Tootell et al. [21].

#### 2.3. Canonical Coordinates

**Figure 4.**The canonical $(p,q)$-domain is the region between the graphs of ${q}_{\pm \frac{\pi}{2}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\pm \frac{\pi}{2}tanhp$ and the lines $p\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$ and $p\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\mathrm{arcsinh}\phantom{\rule{0.166667em}{0ex}}T$. On the

**left**, $(t,\varphi )$ are dimensionless radial and azimuthal coordinates. On the

**right**, the canonical $(p,q)$-coordinates are plotted as Cartesian coordinates, with p on the horizontal axis. Recall Equation (28), and compare with Figure 3.

## 3. Conclusion and Discussion

## Conflicts of Interest

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## Appendix: Biological Terms

**avascular zone**the central region of the retina without blood vessels, roughly coinciding with the fovea.

**eccentricity**the radial distance from the foveal center, usually measured in degrees.

**fovea (centralis)**the central part of the retina, accounting for sharp vision.

**foveola**the central part of the fovea, accounting for the highest visual acuity.

**lateral geniculate nucleus**the relay center for visual information in-between the retina and V1, cf. Figure 1.

**LGN**lateral geniculate nucleus.

**optic radiation**a.k.a. the geniculostriate pathway; the collection of axons connecting LGN and V1, cf. Figure 1.

**receptive field**the support region and sensitivity profile of a light-sensitive cell.

**retina**the light-sensitive inner surface of the eye.

**retinotopic mapping**mapping (i.c.retina to cortex) preserving neighborhood relations, cf. Figure 3 and Figure 4.

**striate cortex**same as the primary visual cortex.

**V1**same as the primary visual cortex.

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Florack, L.
Canonical Coordinates for Retino-Cortical Magnification. *Axioms* **2014**, *3*, 70-81.
https://doi.org/10.3390/axioms3010070

**AMA Style**

Florack L.
Canonical Coordinates for Retino-Cortical Magnification. *Axioms*. 2014; 3(1):70-81.
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**Chicago/Turabian Style**

Florack, Luc.
2014. "Canonical Coordinates for Retino-Cortical Magnification" *Axioms* 3, no. 1: 70-81.
https://doi.org/10.3390/axioms3010070