The handshake between your eyes and brain feels seamless. It isn’t. The mapping is drastically compressed near the center of your gaze and your brain fills in gaps in real time, but it’s all describable – at least mathematically.
The relationship defining where something sits in your visual field and where it gets mapped in your visual cortex is logarithmic, and once you write down the equations you can predict, to the millimeter, where an apple lives in your brain. Five variables first. It’s important to note that your focal point is where your gaze is centered. So hopefully right now, it’s here -> *
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Eccentricity (ε) is the angle from your focal point (where you are looking right now) to any other point in your visual field. For example, if you are looking at the center of an apple, if the apple's edge is 2° offset from the center, the degree of eccentricity of the edge of the apple is 2°. Don't mix this up with ε₀, this is just a constant (typically 1°) to make the math work.
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Azimuth angle (α) is the angle offset from flat center line if you draw a circle around your focal point. So straight above your focal point would be 90° azimuth angle. If you imagine eccentricity as how long the clock hand is, then the azimuth angle is the number on the clock that the hand is pointing to.
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X is the lateral offset in millimeters from the center of your visual cortex. If we say that X1 is the left side of the object and X2 is the right side of the object, then the difference between them would be the physical amount of lateral space (in millimeters) that a given object takes up in your visual cortex.
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Y is the vertical offset in millimeters from the center of your visual cortex.
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Lambda (λ) is a constant variable depending on the species and size of the visual cortex.
- ~ 12mm for a macaque monkey
Now for the cool part. The relationship between where an object sits in your visual space (relative to your focal point) and where that object is mapped in your visual cortex, is logarithmic.
If you're interested, here are the equations
What this means practically: Imagine you are staring directly at an apple way off in the distance so that it is essentially just a red point to you. Now you bring the apple much closer so that the edge of the apple is 20° offset from the center. By this point you should have a pretty good view of the apple and its features. Since we know that the edge of the apple moves from about 0° to around 20° offset, we can know almost exactly the location and the size of the apple's representation in your visual cortex.
Math aside, what does it actually mean that this relationship is logarithmic? Why is it important that the brain works this way? The importance lies in the drastic difference between eccentricity going from 0°→1° and eccentricity going from 9° → 10°. Imagine 0° → 1° as the innermost ring of a target you are staring at, then 3° → 4° as the outermost ring of that target. What part of the target do you notice the most details about? Obviously the center, and it's not even close.
Now you move way closer to the target so that the center ring now takes up 0°→10° and the outer ring takes up 30° → 40°. You probably see where this is going; 0°→1° now only includes a tiny dot in the center of the center ring, so you start to notice the textures and scratches in the center. That tiny dot now takes up the same space in your cortex occupied by the entire center circle before – you hold more of its data, you notice more about it. Duh.
Now for the less intuitive implication of the logarithmic relationship. The target now takes up more space in your FOV, so it takes up more space in your cortex, this makes sense. The outer ring of the circle that previously covered 3° → 4°, now covers 30° → 40°, it takes up more space in your FOV, so it should take up more space in your cortex right…right? Nope, it takes up almost the exact same amount of space in your cortex, it just slides–laterally. If the ring was previously represented from 3mm → 4mm (and -3mm → -4mm since it's a ring), it may now be represented from 13mm → 14mm (and -13mm → -14mm since it's a ring). Same amount of representation, just moved sideways in the cortex. As objects move further into your peripheral (away from focal point), your brain logarithmically dedicates less cortex per area in your FOV. The target example provides a situation where the inner and outer edges of the ring scale by the same factor, so the cortical representation remains the exact same size, X just moves. Cool.
Important note: This 1:1 relationship is a mathematical simplification used to describe mapping when dealing with objects reasonably distanced from the focal point (Eccentricity (ε) is >1°) This is because ε/(ε+ε₀) approaches 1 when ε >> ε₀. The reality is that visuals changes are typically never perfect so the relationship is the logarithmic equation above
We've now established the relationship between object scale and location in FOV and lateral space in the visual cortex, X. But what about vertical space in the visual cortex, Y? Luckily this is a bit more intuitive, so I'll make it brief. Looking at the equation above, we see that there is a linear relationship between an object's Azimuth angle (α), and its corresponding representation in the visual cortex. In other words, if an object spans an entire 360° circle around your focal point, it spans the entire vertical height of your visual cortex. The rings of the target are an example of this. If an object only spans the top 180° above your focal point, then it will only span the top half of your visual cortex. Pretty straightforward.
Similar caveat as with X: In the full mapping, Y also has a minor relationship with eccentricity, but for mathematical purposes, the azimuth angle is what's doing the work.
Practically, the first relationship above allows for the brain to simulate resolution by mapping objects to the X axis, using a drastic and nonlinear curve. That being said, human vision is rotationally symmetrical – you see objects at the same resolution if they are 10° above, below, or to the side of your focal point. Because resolution doesn't change when travelling in a circle, every degree of angle can be represented by the exact same amount of brain, so we get Y.
P. S. What does it mean that an object "takes up space" in the visual cortex? To the extent of my knowledge, it works a bit like this. Imagine your visual cortex as a very dense area of neurons that "fire" at some rate, causing voltage fluctuations in their vicinity. If an apple randomly pops into your FOV, and your brain maps the apple onto X from 10mm to 11mm and Y from 0mm to -2mm, then the neurons in that area fire at a much higher rate. If the apple moves, the concentration of high firing rate neurons will move respectively. The thing is, those neurons only fire right when your FOV changes, like right when the apple appears. So what if you just stare perfectly at an apple with an unchanging background? The apple would disappear – actually though. Your brain is designed to notice changes in your environment, not constants. Try it. Did the apple disappear? Of course it didn't. Your eyes are always vibrating with microscopic, involuntary twitches called microsaccades.